Research ArticleMultiple Model-Based Synchronization Approaches for TimeDelayed Slaving Data in a Space Launch Vehicle Tracking System

Haryong Song and Yongtae Choi

Flight Safety Technology Team Korea Aerospace Research Institute 169-84 Gwahangno Yuseong-guDaejeon 305-806 Republic of Korea

Correspondence should be addressed to Haryong Song hrsongkarirekr

Received 19 May 2016 Revised 13 July 2016 Accepted 19 July 2016

Academic Editor Quanmin Zhu

Copyright copy 2016 H Song and Y ChoiThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Due to the inherent characteristics of the flight mission of a space launch vehicle (SLV) which is required to fly over very largedistances and have very high fault tolerances in general SLV tracking systems (TSs) comprise multiple heterogeneous sensors suchas radars GPS INS and electrooptical targeting systems installed over widespread areas To track an SLV without interruption andto hand over the measurement coverage between TSs properly the mission control system (MCS) transfers slaving data to eachTS through mission networks When serious network delays occur however the slaving data from the MCS can lead to the failureof the TS To address this problem in this paper we propose multiple model-based synchronization (MMS) approaches whichtake advantage of the multiple motionmodels of an SLV Cubic spline extrapolation prediction through an 120572-120573-120574 filter and a singlemodel Kalman filter are presented as benchmark approachesWe demonstrate the synchronization accuracy and effectiveness of theproposed MMS approaches using the Monte Carlo simulation with the nominal trajectory data of Korea Space Launch Vehicle-I

1 Introduction

The range safety system (RSS) [1] for the flight mission of aspace launch vehicle (SLV) consists ofmultiple heterogeneoustracking systems (TSs) with mission control systems (MCSs)Critical launch mission details such as time-space-positioninformation (TSPI) launch mission status data that is quicklook message (QLM) and flight safety information canbe acquired from the RSS Tracking an extensive missiontrajectory of an SLV requires widespread multiple TSs so thatthe RSS covers the entire trajectory of the SLV flight missionGenerally multiple TSs are spread out over different sites andthey automatically hand over observation coverage accordingto the flight of the SLV In this circumstance one of the mostimportant roles of the RSS is to distribute slaving data to eachTS for continuous tracking of the SLV If a critical networkdelay results in time delayed slaving data to be sent to theTSs the MCS will not receive accurate SLV tracking dataThis problem can lead to significant difficulties for the SLVmission progress and analysis Therefore the motivation of

this research is to enhance slaving data accuracy by compen-sating for possible network time delays The basic solution tothis problem is simple (linear or nonlinear) extrapolation ofthe filtered data In this case extrapolation methods simplypropagate the TSPI data without regard to the system dynam-ics of the SLV On the other hand a Kalman filter (KF) and itsprediction capability [2 3] can reflect the system dynamics ofthe SLVwhich results in better synchronization performanceHowever since a KF only utilizes a single dynamic model ingeneral the tracking performance of a KF for a maneuveringtarget is inferior tomultiplemodel estimators [4] In additiondue to stage separation the flight phase of an SLV is separatedinto two parts the propelled flight phase (PFP) and the coast-ing flight phase (CFP) Hence the dynamic model of an SLVcan be described using multiple models so that the multipleflight phases are properly accounted for To adaptively selectone of the multiple dynamic models according to the flightphase of the SLV multiple model estimators (MME) suchas an interacting multiple model (IMM) [5] and a multiplemodel adaptive estimator (MMAE) [6] could significantly

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 9319282 12 pageshttpdxdoiorg10115520169319282

2 Mathematical Problems in Engineering

improve SLV trackingwhen anetwork delay results in delayedslaving data transmission

In the past several decades considerable research hasbeen undertaken in the field of launch vehicle tracking basedonmultiple dynamicmodels researchers have shown interestin various applications such as the tracking of reentry vehi-cles short-range projectiles and sounding rockets [7ndash13] Areentry vehicle tracking problem known as highly nonlineardynamics was conducted using a modified IMM with adifferent algorithm cycle compared with an ordinary IMMwith multiple modes of diverse ballistic coefficients [7]Short-range ballistic munitions or projectiles with multiplemodels such as spin-stabilized and fin-stabilizedmodels wereimplemented using an IMM [8 9] Both research alternativescan be applied in the impact point prediction of projectilesTactical ballistic missile tracking was carried out using anIMM estimator with three modes [10] The first mode wasa constant axial force model for the boosting and reentryphases The second mode was a ballistic acceleration modelthat incorporated the gravitational Coriolis and centripetalforces for the exoatmospheric phase The final mode wasa standard autocorrelated acceleration Singerrsquos model formalfunction motions of missiles such as reentry tumbling Amultiple IMM algorithm with an unbiased mixing approachfor multiple modes of thrusting or for ballistic projectiles waspresented [11 12] A sounding rocket with multiple modes ofpropelled flight or free fall flight was tracked using a multiplemodel adaptive estimation approach [14]

In this paper multiple model-based synchronization(MMS) approaches are proposed to synchronize the timedelayed slaving data of the RSSThe proposed approaches canbe expressed via two distinct multiple models a nonlinearmodel and a linear model The nonlinear model considerscomprehensive factors such as thrust gravity drag coeffi-cient Mach number and air density [10ndash13] Although thenonlinear model precisely describes the motion of the SLVit requires complex information concerning the SLV specifi-cations to be collected in advance In contrast in the case ofthe linear dynamic model a simple constant velocity (CV) orconstant acceleration (CA) model with multiple hypotheseswhich takes advantage of Singerrsquos model [16 17] can beutilized [14] To describe the motion of the SLV the motionmodes of both models are separated into two parts PFP andCFP We propose a slaving data synchronization approachfor the RSS based on MME so that the MCS can adaptivelyfind an appropriate dynamic model at an arbitrary timeindex where time delay has occurred The performance ofslaving data synchronization is compared to various bench-mark methods such as cubic spline extrapolation predictionthrough an120572-120573-120574 filter and a singlemodel KF to demonstratethe effectiveness of the proposed algorithm

The remainder of the paper is organized as followsSection 2 presents a statement of the problem for delayed slav-ing data in RSS In Section 3 conventional synchronizationapproaches are illustrated In Section 4 two proposedMME-based synchronization approaches are derived Section 5presents simulation settings and results the comparisonbetween different types of synchronization approaches is

depicted as an aspect of RMS error of the state vector Finallyin Section 6 the conclusions of this paper are presented

2 Problem Statement for Delayed SlavingData in RSS

The transmission of slaving data from the MCS to multipleTSs facilitates seamless tracking of the SLV in a sparselylocatedmultiple TS environment If a data transmission delayproblem occurs it can cause an SLV tracking failure Asdepicted in Figure 1 the antennas of TSs are pointing at theSLV by controlling their attitude according to slaving datafrom the MCS In this situation a slight time delay in slavingdata can give rise to large differences between the antennabeam and the SLV due to the fast movement of the SLV Tosolve this problem we propose MMS approaches Hence thegoal of this paper is to find a synchronized state

sync119896+119904

at time119896+119904 (where s is a known delay) based on delayed slaving data

delay119896

at time k such that

sync119896+119904

= 119891119901(119904

delay119896

) 119883 = [119909 119910 119911]

119879 (1)

where 119891119901 is a linear or nonlinear propagation function and

119883 = [119909 119910 119911]

119879 is a slaving state vector that is composed ofx-axis y-axis and z-axis positions but is not limited to theposition components

3 Conventional Synchronization Approachesfor Slaving Data

31 Synchronization Using Cubic Spline Extrapolation [18 19]Let us assume a synchronized slaving state vector to be anunknown function of the delayed slaving state vector whosevalues are known only until time 119896 We then define a cubicspline extrapolation function119891

ep119899

(119896+119904) where 119899 = 119909 119910 119911 at aspecific synchronization time 119896 + 119904 that is extrapolated basedon the known function 119891119899(119896) that has a real value with119873+1

points where 1198960 le 119896 le 119896119873We approximate 119891

ep119899

(119896 + 119904) as a three-order polynomialbased on the interval [119896119894 119896119894+1] where 119894 = 0 119873 minus 1 Then119891ep119899

(119896 + 119904) can be defined as follows

119891ep119899

(119896 + 119904)

= 119891119899 (119896) 119896 isin [119896119894 119896119894+1] 119894 = 0 119873 minus 1

119891119899 (119896) = 1198861198991198941198963+ 119887119899119894119896

2+ 119888119899119894119896 + 119889119899119894

(2)

where 119886119899119894 119887119899119894 119888119899119894 and 119889119899119894 are unknown coefficients To findthe unknown coefficients wemake the assumption that119891119899(119896)

should be continuous in [1198960 119896119873] Therefore the followingequation containing unknown coefficients 119886119899119894 119887119899119894 119888119899119894 and119889119899119894 is obtained

119891119899119894 (119896119894) = 119891119899119894+1 (119896119894)

1198911015840

119899119894(119896119894) = 119891

1015840

119899119894+1(119896119894)

Mathematical Problems in Engineering 3

On-board GPS

Trajectory

Delay compensation(synchronization)All antennas are pointing exactly at the SLV

No delay compensationAntennas are pointing to

a ghost target (tracking failure)

Tracking radar number 1(station number 1)

Tracking radar number 2(station number 2)

Mission control system

Telemetry

TSPITSPI

QLM

Slaving data

Slaving data

Slaving data

Range system

Figure 1 Illustration of problem statement for delayed slaving data in RSS

11989110158401015840

119899119894(119896119894) = 119891

10158401015840

119899119894+1(119896119894)

119899 = 119909 119910 119911 119894 = 1 119873 minus 1

(3)

On combining (2) and (3) the cubic polynomials 119891ep119899

(119896 + 119904)

are reconstructed by solving the linear equations obtainedOnce we find the coefficients 119886119899119894 119887119899119894 119888119899119894 and 119889119899119894 where119894 = 0 119873 minus 1 we can evaluate 119891

ep119899

(119896 + 119904) where 119904 is anarbitrary lead-time for future points in [1199090 119909119873+119904]

32 Synchronization Using an 120572-120573-120574 Filter When the stateestimation covariance for a time invariant system convergesunder suitable conditions to a steady-state value explicitexpressions of the steady-state covariance and filter gaincan be obtained The resulting steady-state filters for noisykinematic models are known as 120572-120573 and 120572-120573-120574 filters [20] Inthis paper a combined 120572-120573-120574 filter using both an expandingmemory polynomial filter (EMF) and a fading memorypolynomial filter (FMF) was used [21 22] At first the EMF isrepresented as follows

119901119864

119899119896= 119901

119864

119899119896minus1+ Δ119905V119864

119899119896minus1+

Δ1199052

2

119886119864

119899119896minus1

+

3 (31198962+ 3119896 + 2)

(119896 + 3) (119896 + 2) (119896 + 1)

120576119864

119899119896

V119864119899119896

= V119864119899119896minus1

+ Δ119886V119864119899119896minus1

+

1

Δ119905

18 (2119896 + 1)

(119896 + 3) (119896 + 2) (119896 + 1)

120576119864

119899119896

119886119864

119899119896= 119886

119864

119899119896minus1+

1

Δ1199052

60

(119896 + 3) (119896 + 2) (119896 + 1)

120576119864

119899119896

120576119864

119899119896= 119910119899119896 minus 119901

119864

119899119896minus1minus Δ119905V119864

119899119896minus1minus

Δ1199052

2

119886119864

119899119896minus1

(4)

where Δ119905 is the sampling time and 119901119864

119899119896 V119864

119899119896 and 119886

119864

119899119896are

the position velocity and acceleration estimates of the EMFrespectively In addition the variance reduction factor (VRF)[22] of the EMF that is VRF119864 can be represented as

VRF119864=

91198962+ 27119896 + 24

119896 (119896 + 1) (119896 minus 1)

(5)

On the other hand the FMF and its VRF (VRF119865) can bewritten as follows

119901119865

119899119896= 119901

119865

119899119896minus1+ Δ119905V119865

119899119896minus1+

Δ1199052

2

119886119865

119899119896minus1+ (1 minus 120582

3) 120576

119865

119899119896

V119865119899119896

= V119865119899119896minus1

+ Δ119886V119865119899119896minus1

+

3

2Δ119905

(1 minus 120582)2(1 + 120582) 120576

119865

119899119896

119886119865

119899119896= 119886

119865

119899119896minus1+

1

Δ1199052(1 minus 120582)

3120576119865

119899119896

120576119865

119899119896= 119910119899119896 minus 119901

119865

119899119896minus1minus Δ119905V119865

119899119896minus1minus

Δ1199052

2

119886119865

119899119896minus1

VRF119865=

1 minus 120582

(1 + 120582)5(19 + 24120582 + 16120582

2+ 6120582

3+ 120582

4)

(6)

where 0 lt 120582 lt 1 Both filters are conducted in parallel butonly one of them is selected as a final estimate by comparing

4 Mathematical Problems in Engineering

the VRFs During the early tracking phase the EMF isselected as the final estimate however at a certain timewhereVRF119865 is larger than VRF119864 FMF is selected as the finalestimate

120572120573120574

119899119896= 119891119899

120572120573120574

119899119896minus1+ 119870[119910119896 minus 119867

120572120573120574

120572120573120574

119899119896minus1]

[

[

[

119901119899119896

V119899119896119886119899119896

]

]

]

=

[

[

[

[

[

1 Δ119905

Δ1199052

2

0 1 Δ119905

0 0 1

]

]

]

]

]

[

[

[

119901119899119896minus1

V119899119896minus1119886119899119896minus1

]

]

]

+

[

[

[

[

[

[

120572

120573

Δ119905

120574

Δ1199052

]

]

]

]

]

]

[

[

[

119910119896 minus 119867120572120573120574 [

[

[

119901119899119896minus1

V119899119896minus1119886119899119896minus1

]

]

]

]

]

]

where 119883120572120573120574

119899119896= [119901119899119896 V119899119896 119886119899119896]

119879 119867

120572120573120574= [1 0 0]

VRF119864ge VRF119865

997888rarr

119901119899119896 = 119901119864

119899119896

V119899119896 = V119864119899119896

119886119899119896 = 119886119864

119899119896

VRF119865gt VRF119864

997888rarr

119901119899119896 = 119901119865

119899119896

V119899119896 = V119865119899119896

119886119899119896 = 119886119865

119899119896

(7)

Finally synchronization using an 120572-120573-120574 filter is completed bylinear propagation using the system matrix 119891119899 such that

120572120573120574

119899119896+119904= (

119904

prod

119871=1

119891119899) sdot

120572120573120574

119899119896 (8)

33 Kalman Predictor The motion of the SLV is simplydepicted as a discretized Wiener process acceleration model[20]

119883119896+1 = 119865119883119896 + 119908119896 (9)

where the state vector 119883119896 isin R9 consists of theposition velocity and acceleration components along 119909-axis 119910-axis and 119911-axis respectively that is 119883119896 =

[119909119901119896 119909V119896 119909119886119896 119910119901119896 119910V119896 119910119886119896 119911119901119896 119911V119896 119911119886119896]119879 The system

matrix and covariance matrix of the system noise 119908119896 arerepresented as (10) and (11) respectively

119891119899=119909119910119911 =

[

[

[

[

[

1 Δ119905

Δ1199052

2

0 1 Δ119905

0 0 1

]

]

]

]

]

119865 =[

[

[

119891119909 03 0303 119891119910 0303 03 119891119911

]

]

]

(10)

119864 [119908119896119908119879

119897] = 119876120575119896minus119897

119902119871=119909119910119911 =

[

[

[

[

[

[

[

[

[

Δ1199055

20

Δ1199054

8

Δ1199052

6

Δ1199054

8

Δ1199053

3

Δ1199052

2

Δ1199053

6

Δ1199052

2

Δ119905

]

]

]

]

]

]

]

]

]

119876 =[

[

[

119902119909 03 0303 119902119910 0303 03 119902119911

]

]

]

(11)

A radar measurement for the SLV gives the spherical coordi-nate observations such that

119885119896 = ℎ (119883119896) + V119904119896

119885119896 isin R3 V119904119896 isin R

3

119885119896 =[

[

[

119903119896

120593119896

120579119896

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

radic1199092

119901119896+ 119910

2

119901119896+ 119911

2

119901119896

tanminus1(

119910119901119896

119909119901119896

)

tanminus1(

119910119901119896

radic1199092

119901119896+ 119910

2

119901119896

)

]

]

]

]

]

]

]

]

]

]

]

+[

[

[

V119903V120593V120579

]

]

]

(12)

where V119904119896 sim N(0 119877119904119896) and 119877119904119896 = diag(1205902

119903 120590

2

120593 120590

2

120579) Using a 3D

debiased converted measurement [23] we can transform theoriginal nonlinear equations (12) into their linear form as

z119896 = 119867119883119896 + V119896 V119896 isin R3

z119896 =[

[

[

119909119896

119910119896

119911119896

]

]

]

=[

[

[

1 01times5 01times301times3 1 01times501times5 1 01times2

]

]

]

119883119896 +[

[

[

V119909119896V119910119896V119911119896

]

]

]

(13)

Here V119888119896 is the converted measurement noise expressed interms of Cartesian coordinates that is V119896 sim N(0 119877119896)

119877119896 =

[

[

[

[

119877119909119909119896 119877119909119910119896 119877119909119911119896

119877119910119909119896 119877119910119910119896 119877119910119911119896

119877119911119909119896 119877119911119910119896 119877119911119911119896

]

]

]

]

(14)

Mathematical Problems in Engineering 5

Prediction or fixed-lead prediction inmean squaremeans thesynchronization of the slaving data is the estimation of thestate at a future time 119896+119904 where 119904 gt 0 beyond the observationinterval that is based on data up to an earlier time [20 24]

119896+119904|119896 = 119864 (119883119896+119904|119896 | 119885119896) 119885119896 = z0 z119896 (15)

The optimal predictor or synchronized state 119896+119904|119896 ≜

KP119896+119904|119896

and its error covariance 119875119896+119904|119896 ≜ 119875KP119896+119904|119896

are given by theKalman predictor equations [2 24]

KP119896+119904|119896

= 119865119896+119904minus1

KP119896+119904minus1|119896

= sdot sdot sdot = Φ119896+119904119896

KP119896|119896

KP119896|119896

≜

KF119896|119896

119875KP119896+119904|119896

= 119865119896+119904minus1119875KP119896+119904minus1|119896

119865119879

119896+119904minus1+ 119876119896+119904minus1

= Φ119896+119904119896119875KP119896|119896

Φ119879

119896+119904119896

+

119904minus1

sum

119895=0

Φ119896+119904119896+119895+1119876119896+119895Φ119879

119896+119904119896+119895+1 119875

KP119896|119896

≜ 119875KF119896|119896

Φ119896+119904119896 = 119865119896+119904minus1119865119896+119904minus2 sdot sdot sdot 119865119896

Φ119896119896 = 119868119899

119896 gt 1

(16)

where

KF119896|119896

and 119875KF119896|119896

are the KF estimate and covariancerespectively

4 Proposed MME-Based SynchronizationApproaches

41 SynchronizationUsing an IMMwith Singerrsquos LinearModelSinger [16] described 2D manned maneuvering targets inrange-bearing coordinates This model can be adapted to theSLV kinematics in 3D Cartesian coordinates with multipleflight phases

[

[

[

119899

V119899119899

]

]

]

= 119865[

[

[

119901119899

V119899119886119899

]

]

]

+ 119866119908119899

119865 =[

[

[

0 1 0

0 0 1

0 0 minus120572119899

]

]

]

119866 =[

[

[

0

0

0

]

]

]

(17)

where119908119899 isin R3times1 is a white noise process along the Cartesianaxis 119899 = 119909 119910 119911 The parameter 120573 = 1120572119899 is the maneuvercorrelation time constant and 120590

2

119886119899is the acceleration variance

describing maneuver intensity In a steady state

1205902

119908119899

= 21205721205902

119886119899 (18)

To describe SLV kinematics the model must cope with cru-cial nonzero mean acceleration maneuvers during the pro-pelled phase In addition after each stagersquos engine burns outa multiple model approach is applied to the coasting flightthat is one model describes PFP whereas the other depictsthe CFP Empirically tuned independent probability densityfunctions (PDFs) represented by TUM describe the accel-erations of the SLV in the local coordinate frame Figure 2shows the means and variances of the acceleration processesof the SLV in this paper The discrete-time model with statetransition matrix Ψ119899(120572119899 Δ119905) is as follows

[

[

[

119901119899119896+1

V119899119896+1119886119899119896+1

]

]

]

= Ψ119899 (120572119899 Δ119905)[

[

[

119901119899119896

V119899119896119886119899119896

]

]

]

+ 119908119899119896

Ψ119899 (120572119899 Δ119905) = 119890119865Δ119905

[

[

[

[

[

[

[

1 Δ119905 (120572119899Δ119905 minus 1 + 119890minus120572119899Δ119905

)

0 1

(1 minus 119890minus120572119899Δ119905

)

120572119899

0 0 119890minus120572119899Δ119905

]

]

]

]

]

]

]

119908119899119896 = int

(119896+1)Δ119905

119896Δ119905

119890119865[(119896+1)Δ119905minus119903]

119866119908119899 119889119903

(19)

In the case of the PFP 119908119899119896 is a nonzero mean white noisesequence caused by the nonzero mean acceleration 120583119886119899 seenin Figure 2 A nonzeromeanwhite noise sequence for the PFPshould be considered in the target kinematics when imple-menting the state propagation stage in theKFThus the deter-ministic input 119906119899119896 caused by 119908119899119896 along 119909-axis 119910-axis and119911-axis is derived as follows

119906119909119896 = 119864 [119908119909119896] = 119864int

(119896+1)Δ119905

119896Δ119905

119890119865[(119896+1)Δ119905minus119903]

119866119908119909119889119903

asymp 119864

int

(119896+1)Δ119905

119896Δ119905

[

[

[

[

119890minus120572119909[(119896+1)Δ119905minus119903]

0

0

]

]

]

]

119908119909 119889119903

119906119909119896 =

120583119886119909

120572119899 [1minus119890minus120572119909Δ119905

00

]

(20)

119910-axis and 119911-axis can be derived in the samemanner as shownin (20) The maneuver excitation covariance [10] whichrepresents the uncertainty of the SLV kinematics model is

119876119899119896 (120572119899 Δ119905)

= 119864 (119908119899119896 minus 119864 [119908119899119896]) (119908119899119896 minus 119864 [119908119899119896])119879

= 21205721198991205902

119886119899

[

[

[

11990211 11990212 11990213

11990212 11990222 11990223

11990213 11990223 11990233

]

]

]

(21)

The specific components of 11990211 11990233 are illustrated in [25]In addition the measurement matrix 119867119871 for Singerrsquos modelcan be depicted as

6 Mathematical Problems in Engineering

01

minus20 4minus6 0

0028

120583 =

1205902 =

p(ax)

minus6ms2

4246

a (m2s4)

m2s4

(a)

01

minus5 810

005

120583 = 1ms2

1205902 = 106

p(ay)

a (m2s4)

m2s4

(b)

01

minus40 4minus7 0

0008

a (m2s4)

120583 = minus7ms2

1205902 = 12674 m2s4

p(az)

(c)

01

005 005

minus4 40

008

120583 = 0ms2

1205902 = 183

p(ax) p(ay) p(az)

a (m2s4)

m2s4

(d)

Figure 2 Nonzero mean acceleration PDF in the PFP model (a) along 119909-axis (b) along 119910-axis and (c) along 119911-axis Zero mean accelerationPDF in the CFP Model (d) along 119909-axis 119910-axis and 119911-axis

119911119871119896 = 119867119871x119871119896 + ]119896 119867119871 =[

[

[

1 0 0

0 03times2 1 03times2 0 03times20 0 1

]

]

]

x119871119896 = [119909119896 119896 119896 119910119896 119896

119896

119911119896 119896 119896]119879 (22)

where ]119896 is measurement noise (as shown in (14)) with errorcovariance

411 IMM with Singerrsquos Linear Model From (19)ndash(22) inSection 41 we can rewrite the Markov jump linear systemswhere the 119894th model of the finite multiple model set M =

119898(1)

119898(119872)

obeys the following equations

x119871119896+1 = Ψ(119894)

119896x119871119896 + 119908

(119894)

119896

119911119871119896 = 119867119871x119871119896 + ]119896(23)

where

cov (119908(119894)

119896) = 119876

(119894)

119896

cov (]119896) = 119877119896

Ψ(119894)

119896= diag (Ψ119909119896 (120572

(119894)

119909 Δ119905) Ψ119910119896 (120572

(119894)

119910 Δ119905)

Ψ119911119896 (120572(119894)

119911 Δ119905))

119876(119894)

119896= diag (119876119909119896 (120572

(119894)

119909 Δ119905) 119876119910119896 (120572

(119894)

119910 Δ119905)

119876119911119896 (120572(119894)

119911 Δ119905))

(24)

Here 1205902

119886119899for the SLV can be represented by TUM as in

Figure 2 The PDFs of Figures 2(a)ndash2(d) are experimentallysampled from the nominal acceleration profile of the SLVThesuperscript (i) denotes quantities pertinent to model 119898(119894) inM and the jumps of the system mode are assumed to havetransition probabilities

Pr 119898(119895)

119896+1| 119898

(119894)

119896 ≜ 120587119894119895

(25)

where 119898(119894)

119896denotes the event in which model 119898

(119894) matchesthe system mode in effect at time 119896 In our application 119872 =

2 and 119898(1) and 119898

(2) are the propelled and coasting modesrespectively

Finally complete recursion of the IMM with modematched KF for the SLV tracking is summarized as follows

(i) Model-conditioned reinitialization (for 119894 = 1 2

119872)

Mathematical Problems in Engineering 7

(a) predicted mode probability 120583(119894)

119871119896|119896minus1≜ Pr119898(119894)

119896|

119911119871119896minus1 = sum119895120587119895119894120583

(119895)

119871119896minus1

(b) mixingweight 120583119895|119894

119871119896minus1≜ Pr119898(119895)

119896minus1| 119898

(119894)

119896 119911119871119896minus1 =

sum119895120587119895119894120583

(119895)

119871119896minus1120583

(119894)

119871119896minus1

(c) mixing estimate and covariance

x(119894)119871119896minus1|119896minus1

≜ 119864 [x119871119896minus1|119896minus1 | 119898(119894)

119896 119911119871119896minus1]

= sum

119895

x(119895)119871119896minus1|119896minus1

120583119895|119894

119871119896minus1

119875

(119894)

119871119896minus1|119896minus1= sum

119895

[119875(119894)

119871119896minus1|119896minus1+ (x(119894)

119871119896minus1|119896minus1minus x(119895)

119871119896minus1|119896minus1)

sdot (x(119894)119871119896minus1|119896minus1

minus x(119895)119871119896minus1|119896minus1

)

119879

] 120583119895|119894

119871119896minus1

(26)

(ii) Model-conditioned filtering (for 119894 = 1 2 119872)

(a) predicted estimate and covariance

x(119894)119871119896|119896minus1

= Ψ(119894)

119896minus1x(119894)119871119896minus1|119896minus1

+ u(119894)

119896minus1

where u(1)

119896minus1= [119906

119879

119909119896minus1119906119879

119910119896minus1119906119879

119911119896minus1]

119879

u(2)

119896minus1= 03times1

119875(119894)

119871119896|119896minus1= Ψ

(119894)

119896minus1119875

(119894)

119871119896minus1|119896minus1Ψ

(119894)119879

119896minus1+ 119876

(119894)

119896minus1

(27)

(b) measurement residual (119894)

119871119896= 119911119871119896 minus119867119871x

(119894)

119871119896|119896minus1minus

]119896(c) residual covariance 119878(119894)

119871119896= 119867119871119875

(119894)

119871119896|119896minus1119867

119879

119871minus 119877119896

(d) filter gain 119870(119894)

119871119896= 119875

(119894)

119871119896|119896minus1119867

119879

119871(119878

(119894)

119871119896)minus1

(e) update of state and covariance x(119894)119871119896|119896

= x(119894)119871119896|119896minus1

+

119870(119894)

119871119896(119894)

119871119896 119875

(119894)

119871119896|119896= 119875

(119894)

119871119896|119896minus1minus 119870

(119894)

119871119896119878(119894)

119871119896119870

(119894)119879

119871119896

(iii) Mode probability update (for 119894 = 1 2 119872)

(a) mode likelihood Λ(119894)

119871119896≜ 119901[119911

(119894)

119871119896| 119898

(119894)

119896

119885119871119896minus1]assume

= N(119911(119894)

119871119896 0 119878

(119894)

119871119896)

(b) mode probability 120583(119894)

119871119896= 120583

(119894)

119871119896|119896minus1Λ

(119894)

119871119896

(sum119895120583(119895)

119871119896|119896minus1Λ

(119895)

119871119896)

(iv) Combination (for 119894 = 1 2 119872)

x119871119896|119896 = sum

119894

x(119894)119871119896|119896

120583(119894)

119871119896

(119894)

119871119896|119896

= sum

119894

[119875(119894)

119871119896|119896+ (x119871119896|119896 minus x(119894)

119871119896|119896) (x119871119896|119896 minus x(119894)

119871119896|119896)

119879

] 120583(119894)

119871119896

(28)

Singerrsquos MMS of time delayed slaving data is completed bypropagating the combined estimate based on the currentmodersquos dynamic model such that

xsync119871119896+119904|119896

= Φsync119896+119904119896

x119871119896|119896

Φsync119896+119904119896

= 119865sync119896+119904minus1

119865sync119896+119904minus2

sdot sdot sdot 119865sync119896

Φsync119896119896

= 119868119899

119896 gt 1

(29)

where 119865sync119896+119904minus1

119865sync119896+119904minus2

119865sync119896

are systemmatrices dependingon a flight phase mode at current time 119896

42 Synchronization Using an IMM with a Nonlinear BallisticModel For a nonlinear ballistic model the state vector forthe propelled mode is denoted as

x119905 = [119909119905 119910119905 119911119905 119905 119905

119905 120585119905 120591119905]119879 (30)

where 120585119905 is the drag coefficient and 120591119905 is the thrust Generallythe drag coefficient varies significantlywith theMachnumberregime subsonic transonic and supersonic Therefore wetake advantage of the dynamic model considering a Machnumber-dependent multiplier [11 12] such that

[

[

[

]

]

]

=

120591

119881

[

[

[

]

]

]

thrust term

+ 120585120585119898119863[

[

[

]

]

]

drag term

+ 119892[

[

[

0

0

1

]

]

]

gravity term

+ 1

120585 = 2

= 3

(31)

The first term on the right side of (31) represents the thrust(m2

s) of the SLV in 119909 119910 and 119911 directions Two distinctmultiplemodes of a nonlinear SLVmodel can be separated bythe existence of thrust In other words the existence of thrustsignifies propelled flight mode whereas zero thrust signifiescoasting flight mode 119881 is the magnitude of the velocity V =

[ ]

119879 that is the SLV speed (ms) The second part of(31) is the drag term which is related to velocity and altitudethat is 119863 = minus120588(119911)1198812 where 120588(119911) = 1205880119890

minus119888119911 is the air density(kgm3) at an altitude 119911 (m) and 119888 is the air density constant(mminus1) [9] 120585 is the drag coefficient and 120585119898 is theMachnumber-dependent drag coefficient multiplier which is approximatedby the cubic spline curve shown in Figure 3 In this paperfor the drag characteristics of the SLV which are applied inthe subsequent simulation section drag coefficients of theSaturn119881 launch vehicle [15] are usedThe third part of (31) isa gravity term Gravity 119892 is the standard acceleration due togravity at sea level which is assumed to be constant through-out the trajectory with a value of 9812 ms2 1 2 and 3

are assumed to be continuous time zeromeanwhite Gaussiannoises The drag coefficient and thrust acceleration are rep-resented as Wiener processes with a slow variation [11 12]

8 Mathematical Problems in Engineering

0 2 4 6 8 1004

06

08

1

12

14

16

18

2

Mach number

Nor

mal

ized

dra

g co

effici

ent

Selected pointsLinear interpolationCubic spline interpolation

Figure 3 Normalized drag coefficient [15]

We can modify the dynamic equations (30) and (31) as acompact form such that

x119905 = 119891 [x119905] + 119905 (32)

where

119891 [x119905] =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119905

119905

119905

120591

119905

119881119905

+ 120585119905119863119905119905

120591

119905

119881119905

+ 120585119905119863119905119905

120591

119905

119881119905

+ 120585119905119863119905119905 minus 119892

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

119905 = [1119905 2119905 3119905]119879

(33)

The state vector equation (33) is discretized by a second-order Taylor expansion [26] Then (33) can be written as

a discretized continuous time systemwithwhite process noisesuch that

x119896+1 = x119896 + 119891 [x119896] Δ119905 + 119860119896119891 [x119896]Δ119905

2

2

+ 120596119896(34)

where 119860119896 is the Jacobian of (33) evaluated at x119896 [26] and120596119896 is the discretized continuous time process noise for thesampling interval Δ119905 The corresponding covariance matrixof the discretized process noise is

119876 =[

[

[

1198761119902V 06times1 06times101times6 Δ119905119902120585 0

01times6 0 Δ119905119902120591

]

]

]

1198761 =

[

[

[

[

Δ1199053

3

1198683

Δ1199052

2

1198683

Δ1199052

2

1198683 Δ1199051198683

]

]

]

]

(35)

where 1198683 is the 3 times 3 identity matrix and the continuous timeprocess noise intensities 119902V 119902120585 and 119902120591 are the correspondingpower spectral densities

The measurement matrix 119867NL for the nonlinear ballisticmultiple model can be depicted as

119911NL119896 = 119867NLxNL119896 + ]119896

119867NL = [1198683 0] xNL119896 = [119909119896 119910119896 119911119896 119896 119896

119896 120585119896 120591119896]119879

(36)

where ]119896 is measurement noise with error covariance119877119896120575119896minus119895 = 119864[]119896]

119879

119895] that is (14)

An IMMalgorithm for nonlinear dynamics with differentsizes of the mode state vector is summarized as follows [25]

(v) Model-conditioned reinitialization (for 119894 = 1 2

119872)

(a) predicted mode probability 120583(119894)

NL119896|119896minus1 ≜

Pr119898(119894)

119896| 119911NL119896minus1 = sum

119895120587119895119894120583

(119895)

NL119896minus1(b) mixing weight

120583119895|119894

NL119896minus1 ≜ Pr 119898(119895)

119896minus1| 119898

(119894)

119896 119911NL119896minus1 =

sum119895120587119895119894120583

(119895)

NL119896minus1

120583(119894)

NL119896minus1

(37)

(c) unbiased mixing estimate and covariance

x(119894)NL119896minus1|119896minus1 ≜ 119864 [xNL119896minus1|119896minus1 | 119898(119894)

119896 119911NL119896minus1] = sum

119895

(119895)

NL119896minus1|119896minus1120583119895|119894

NL119896minus1

where (1)

NL119896minus1|119896minus1 ≜ x(1)NL119896minus1|119896minus1 (2)

NL119896minus1|119896minus1 ≜ [x(2)119879

NL119896minus1|119896minus1 120591119896minus1]119879

119875

(119894)

NL119896minus1|119896minus1 = sum

119895

[119875(119894)

NL119896minus1|119896minus1 + (x(119894)NL119896minus1|119896minus1 minus (119895)

NL119896minus1|119896minus1) (x(119894)NL119896minus1|119896minus1 minus (119895)

NL119896minus1|119896minus1)119879

] 120583119895|119894

NL119896minus1

(38)

Mathematical Problems in Engineering 9

(vi) Model-conditioned filtering (for 119894 = 1 2 119872)

(a) predicted estimate and covariance

x(119894)NL119896|119896minus1 = x(119894)NL119896minus1|119896minus1 + 119891 [x(119894)NL119896minus1|119896minus1] Δ119905

+

(119894)

NL119896minus1119891 [x(119894)NL119896minus1|119896minus1]Δ119905

2

2

(119894)

119896=

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x=x(119894)119896

119875(119894)

NL119896|119896minus1 = Ω(119894)

119896minus1119875

(119894)

NL119896minus1|119896minus1Ω(119894)119879

119896minus1+ 119876119896minus1

Ω(119894)

119896minus1= 119868 +

(119894)

NL119896minus1

(39)

(b) measurement residual (119894)

NL119896 = 119911NL119896 minus

119867NLx(119894)

NL119896|119896minus1 minus ]119896

(c) residual covariance 119878(119894)NL119896 = 119867NL119875(119894)

NL119896|119896minus1119867119879

NL minus

119877119896(d) filter gain 119870(119894)

NL119896 = 119875(119894)

NL119896|119896minus1119867119879

NL(119878(119894)

NL119896)minus1

(e) update of state and covariance

x(119894)NL119896|119896 = x(119894)NL119896|119896minus1 + 119870(119894)

NL119896(119894)

NL119896

119875(119894)

NL119896|119896 = 119875(119894)

NL119896|119896minus1 minus 119870(119894)

NL119896119878(119894)

NL119896119870(119894)119879

NL119896

(40)

(vii) Mode probability update (for 119894 = 1 2 119872)

(a) mode likelihood

Λ(119894)

NL119896 ≜ 119901 [119911(119894)

NL119896 | 119898(119894)

119896 119885NL119896minus1]

assume= N (119911

(119894)

NL119896 0 119878(119894)

NL119896)

(41)

(b) mode probability 120583(119894)

NL119896 = 120583(119894)

NL119896|119896minus1Λ(119894)

NL119896

(sum119895120583(119895)

NL119896|119896minus1Λ(119895)

NL119896)

(viii) Combination (for 119894 = 1 2 119872)

xNL119896|119896 = sum

119894

x(119894)NL119896|119896120583(119894)

NL119896

(119894)

NL119896|119896 = sum

119894

[119875(119894)

NL119896|119896

+ (xNL119896|119896 minus x(119894)NL119896|119896) (xNL119896|119896 minus x(119894)NL119896|119896)119879

] 120583(119894)

NL119896

(42)

The nonlinearMMS of time delayed slaving data is completedby propagating the combined estimate based on the currentmodersquos estimated vector such that

xNL119896+119904|119896 = xNL119896|119896 + 119891 [xNL119896|119896] sdot 119904 + NL119896119891 [xNL119896|119896]

sdot

1199042

2

(43)

where 119904 is a lead-time for synchronization (119904 = 119899 sdot Δ119905) If theSLV is in the PFP (120583(1)

NL119896 gt 120583(2)

NL119896) the current state estimateincludes the thrust term that is the state vector xNL119896 =

[119909119896 119910119896 119911119896 119896 119896

119896 120585119896 120591119896]119879 If the SLV is not in the PFP

the state vector does not include the thrust term (120591119896 = 0)

5 Simulation Results

To demonstrate the performance of the proposed synchro-nization approaches for delayed slaving data we simulatedthe SLV tracking problem based on the nominal flight tra-jectory of the Korea Space Launch Vehicle-I (KSLV-I) In thesimulation the radar measurement noise intensities of (12)are selected as 120590119903 = 15m 120590120593 = 001 deg and 120590120579 = 001 degThe nominal flight sequence of KSLV-I is as follows First thepayload fairing separates during the first stage flight at 2154 sAfter the first stage engine shutdown at 2287 s the upperstage separates from the first stage and enters the CFP Thesecond stage continues in the CFP until the kick motor igni-tion at 395 s and the vehicle then enters the PFP At the endof kickmotor combustion (4527 s) the upper stage enters thetarget orbit in CFP as in the previous separation Finally thesatellite is inserted into the target orbit after it separates fromthe upper stage during CFP at 540 s Since delayed slavingdata from the MCS to the TSs can occur in both the PFP andCFP synchronization simulations are conducted at arbitrarypoints of the flight phases Figure 4 shows the synchroniza-tion error of the delayed slaving data with respect to thebenchmark and the proposed approaches for delays of 119904 =

01 02 1 s Figures 4(a)ndash4(c) illustrate synchronizationerrors where delayed slaving data occurred at 155 s which isthe first stage of PFP and Figures 4(d)ndash4(f) show synchro-nization errors where delayed slaving data occurred at 280 swhich is the first stage of CFP As shown in Figures 4(a)ndash4(c)the extrapolation and120572-120573-120574filter-based approach cannot findthe proper synchronized slaving data The synchronizationerrors of these approaches exponentially increase accordingto increasing delay 119904 in particular the error of the extrapo-lation approaches an out-of-figure bound (2500m) On theother hand the Kalman prediction and the proposedapproaches have stable synchronization errors even if thedelay time 119904 is larger Furthermore we can observe thatthe errors in the synchronization approach of the Kalmanprediction are relatively larger than the proposed approachesespecially in the PFP As explained in the flight sequence ofKSLV-I in general the motion of the SLV is described bymultiple dynamics (in our case PFP and CFP) rather thansingle dynamics However the dynamic model of the KF andprediction capability used in this paper is the constant accel-eration (CA) model This means that the filter is optimizedwith respect to CAmotion which is actually not occurring inour application This is why the performance of the Kalmanprediction-based synchronization is worse than the proposedapproaches Nonetheless we can see that the 119910-axis synchro-nization errors shown in Figures 4(b) and 4(e) are smallThis result is observed because the 119910-axis motion of KSLV-I in a local coordinate is relatively smaller than the otheraxis motion and the CA model approximately expresses this

10 Mathematical Problems in Engineering

01 02 03 04 05 06 07 08 09 10

500

1000

1500

2000

2500

Delay (sec)

x-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

Cubic spline extrapolation

Kalman filter

(a)

(b)

(c)

(d)

(e)

(f)

Linear MMNonlinear MM120572-120573-120574 filter 120572-120573-120574 filter

120572-120573-120574 filter120572-120573-120574 filter

120572-120573-120574 filter 120572-120573-120574 filter

01 02 03 04 05 06 07 08 09 1Delay (sec)

0

500

1000

1500

2000

2500

y-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

z-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

x-p

ositi

on ab

solu

te er

ror i

n CF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500y

-pos

ition

abso

lute

erro

r in

CFP

(m)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

z-p

ositi

on ab

solu

te er

ror i

n CF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

Figure 4 Synchronization errors at PFP and CFP

Mathematical Problems in Engineering 11

small motion A single KF prediction model cannot exactlydescribe the motion of both PFP and CFP whereas theproposed multiple model-based approaches work well Asshown in Figure 4 regardless of the delay 119904 the proposedmultiplemodel-based approaches find synchronized positionvectors In the comparison of synchronization performancebetween linear IMM and nonlinear IMM synchronizationthe nonlinear IMM-based synchronization approach showsthe best performance except for 119910-axis where CA motion isdominant In addition the difference between the proposedmultiple model-based approaches is very small as shown inFigure 4 but the complexities of the algorithms for real-timeapplications are quite dissimilar Hence the operator mayadaptively select one of the proposed approaches accordingto onersquos environment

6 Conclusions

In this paper we investigated the time synchronizationapproaches of delayed slaving data in the RSS for SLV track-ingOne of themost important roles of the RSS is to distributeslaving data to each TS for continuous tracking of the SLV Ifthere is a critical network delay resulting in time delayed slav-ing data being sent to each TS theMCS will not receive accu-rate SLV tracking data This problem can give rise to signifi-cant difficulties for the SLVmission progress and analysis Toovercome this problemwe proposedMMSapproacheswhichtake advantage of the multiple motion models of an SLVThelinear IMM-based synchronization approach was developedusing Singerrsquos model with ternary uniform mixtures and thenonlinear IMM-based synchronization approachwas derivedfrom a nonlinear ballistic model with a drag coefficient Forverification of the proposed algorithms SLV tracking simu-lations using KSLV-I and the radar measurement data gen-erated from nominal trajectory were conducted To demon-strate the superiority of time synchronization performancein these simulations we compared the proposed algorithmwith benchmark approaches for absolute error between thenominal trajectory data and the synchronized slaving datathe simulation results demonstrated that the proposed MMSapproaches performed competitively

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] R Varaprasad and V Seshagiri Rao ldquoRange safety real-timesystem for satellite launch vehicle missions-testing methodolo-giesrdquo Defence Science Journal vol 56 no 5 pp 693ndash700 2006

[2] H R Song M G Jeon T S Choi and V Shin ldquoTwo fusionpredictors for discrete-time linear systems with different typesof observationsrdquo International Journal of Control Automationand Systems vol 7 no 4 pp 651ndash658 2009

[3] H Song V Shin and M Jeon ldquoMobile node localization usingfusion prediction-based interacting multiple model in cricketsensor networkrdquo IEEE Transactions on Industrial Electronicsvol 59 no 11 pp 4349ndash4359 2012

[4] T Kirubarajan and Y Bar-Shalom ldquoKalman filter versus IMMestimator when do we need the latterrdquo IEEE Transactions onAerospace and Electronic Systems vol 39 no 4 pp 1452ndash14572003

[5] E Mazor A Averbuch Y Bar-Shalom and J Dayan ldquoInteract-ing multiple model methods in target tracking a surveyrdquo IEEETransactions on Aerospace and Electronic Systems vol 34 no 1pp 103ndash123 1998

[6] P D Hanlon and P S Maybeck ldquoMultiple-model adaptiveestimation using a residual correlation Kalman filter bankrdquoIEEE Transactions on Aerospace and Electronic Systems vol 36no 2 pp 393ndash406 2000

[7] S-J Shin ldquoRe-entry vehicle tracking with a new multiplemodel estimation applicable to highly non-linear dynamicsrdquoIET Radar Sonar amp Navigation vol 9 no 5 pp 581ndash588 2015

[8] S Conover J Kerce G Brown L Ehrman and D HardimanldquoImpact point prediction of small ballistic munitions with aninteracting multiple model estimatorrdquo in Acquisition TrackingPointing and Laser Systems Technologies XXI vol 6569 ofProceedings of SPIE Orlando Fla USA 2007

[9] V C Ravindra Y Bar-Shalom and P Willett ldquoProjectileidentification and impact point predictionrdquo IEEE Transactionson Aerospace and Electronic Systems vol 46 no 4 pp 2004ndash2021 2010

[10] W J Farrell III ldquoInteracting multiple model filter for tacticalballistic missile trackingrdquo IEEE Transactions on Aerospace andElectronic Systems vol 44 no 2 pp 418ndash426 2008

[11] T Yuan Y Bar-Shalom PWillett E Mozeson S Pollak and DHardiman ldquoA multiple IMM estimation approach with unbi-ased mixing for thrusting projectilesrdquo IEEE Transactions onAerospace and Electronic Systems vol 48 no 4 pp 3250ndash32672012

[12] T Yuan Y Bar-Shalom P Willett and D Hardiman ldquoImpactpoint prediction for thrusting projectiles in the presence ofwindrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 50 no 1 pp 102ndash119 2014

[13] X Li and V Jilkov ldquoSurvey of maneuvering target trackingpart II motion models of ballistic and space targetsrdquo IEEETransactions on Aerospace and Electronic Systems vol 46 no1 pp 96ndash119 2010

[14] J Cesar Bolzani de Campos Ferreira and JWaldmann ldquoCovari-ance intersection-based sensor fusion for sounding rockettracking and impact area predictionrdquo Control Engineering Prac-tice vol 15 no 4 pp 389ndash409 2007

[15] httpwwwbraeunigusapollosaturnVhtm[16] RA Singer ldquoEstimating optimal tracking filter performance for

manned maneuvering targetsrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 6 no 4 pp 473ndash483 1970

[17] R A Singer and KW Behnke ldquoReal-time tracking filter evalu-ation and selection for tactical applicationsrdquo IEEE Transactionson Aerospace and Electronic Systems vol AES-7 no 1 pp 100ndash110 1971

[18] W Press B Flannery S Teukolsky and W Vetterling Numer-ical Recipes in FORTRAN The Art of Scientific ComputingCambridge University New York NY USA 2nd edition 1992

[19] S Dyer and J Dyer ldquoCubic-spline interpolation part Irdquo IEEEInstrumentation amp Measurement Magazine pp 44ndash46 2001

[20] Y Bar-Shalom X Li and T Kirubarajan Estimation with Appli-cations to Tracking and Navigation John Wiley amp Sons NewYork NY USA 2001

12 Mathematical Problems in Engineering

[21] N Morrison Introduction to Sequential Smoothing and Predic-tion McGraw-Hill 1969

[22] E Brookner Tracking and Kalman Filtering Made Easy JohnWiley amp Sons 1998

[23] P Suchomski ldquoExplicit expressions for debiased statistics of 3Dconverted measurementsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 35 no 1 pp 368ndash370 1999

[24] H R Song I Y Song and V Shin ldquoMultisensory predictionfusion of nonlinear functions of the state vector in discrete-timesystemsrdquo International Journal of Distributed Sensor Networksvol 2015 Article ID 249857 13 pages 2015

[25] H Song andYHan ldquoComparison of space launch vehicle track-ing using different types of multiple modelsrdquo in Proceedingsof the International Conference on Information Fusion pp 1ndash8Heidelberg Germany July 2016

[26] R KMehra ldquoA comparison of several nonlinear filters for reen-try vehicle trackingrdquo IEEE Transactions on Automatic Controlvol 16 no 4 pp 307ndash319 1971

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Function Spaces

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

improve SLV trackingwhen anetwork delay results in delayedslaving data transmission

In the past several decades considerable research hasbeen undertaken in the field of launch vehicle tracking basedonmultiple dynamicmodels researchers have shown interestin various applications such as the tracking of reentry vehi-cles short-range projectiles and sounding rockets [7ndash13] Areentry vehicle tracking problem known as highly nonlineardynamics was conducted using a modified IMM with adifferent algorithm cycle compared with an ordinary IMMwith multiple modes of diverse ballistic coefficients [7]Short-range ballistic munitions or projectiles with multiplemodels such as spin-stabilized and fin-stabilizedmodels wereimplemented using an IMM [8 9] Both research alternativescan be applied in the impact point prediction of projectilesTactical ballistic missile tracking was carried out using anIMM estimator with three modes [10] The first mode wasa constant axial force model for the boosting and reentryphases The second mode was a ballistic acceleration modelthat incorporated the gravitational Coriolis and centripetalforces for the exoatmospheric phase The final mode wasa standard autocorrelated acceleration Singerrsquos model formalfunction motions of missiles such as reentry tumbling Amultiple IMM algorithm with an unbiased mixing approachfor multiple modes of thrusting or for ballistic projectiles waspresented [11 12] A sounding rocket with multiple modes ofpropelled flight or free fall flight was tracked using a multiplemodel adaptive estimation approach [14]

In this paper multiple model-based synchronization(MMS) approaches are proposed to synchronize the timedelayed slaving data of the RSSThe proposed approaches canbe expressed via two distinct multiple models a nonlinearmodel and a linear model The nonlinear model considerscomprehensive factors such as thrust gravity drag coeffi-cient Mach number and air density [10ndash13] Although thenonlinear model precisely describes the motion of the SLVit requires complex information concerning the SLV specifi-cations to be collected in advance In contrast in the case ofthe linear dynamic model a simple constant velocity (CV) orconstant acceleration (CA) model with multiple hypotheseswhich takes advantage of Singerrsquos model [16 17] can beutilized [14] To describe the motion of the SLV the motionmodes of both models are separated into two parts PFP andCFP We propose a slaving data synchronization approachfor the RSS based on MME so that the MCS can adaptivelyfind an appropriate dynamic model at an arbitrary timeindex where time delay has occurred The performance ofslaving data synchronization is compared to various bench-mark methods such as cubic spline extrapolation predictionthrough an120572-120573-120574 filter and a singlemodel KF to demonstratethe effectiveness of the proposed algorithm

The remainder of the paper is organized as followsSection 2 presents a statement of the problem for delayed slav-ing data in RSS In Section 3 conventional synchronizationapproaches are illustrated In Section 4 two proposedMME-based synchronization approaches are derived Section 5presents simulation settings and results the comparisonbetween different types of synchronization approaches is

depicted as an aspect of RMS error of the state vector Finallyin Section 6 the conclusions of this paper are presented

2 Problem Statement for Delayed SlavingData in RSS

The transmission of slaving data from the MCS to multipleTSs facilitates seamless tracking of the SLV in a sparselylocatedmultiple TS environment If a data transmission delayproblem occurs it can cause an SLV tracking failure Asdepicted in Figure 1 the antennas of TSs are pointing at theSLV by controlling their attitude according to slaving datafrom the MCS In this situation a slight time delay in slavingdata can give rise to large differences between the antennabeam and the SLV due to the fast movement of the SLV Tosolve this problem we propose MMS approaches Hence thegoal of this paper is to find a synchronized state

sync119896+119904

at time119896+119904 (where s is a known delay) based on delayed slaving data

delay119896

at time k such that

sync119896+119904

= 119891119901(119904

delay119896

) 119883 = [119909 119910 119911]

119879 (1)

where 119891119901 is a linear or nonlinear propagation function and

119883 = [119909 119910 119911]

119879 is a slaving state vector that is composed ofx-axis y-axis and z-axis positions but is not limited to theposition components

3 Conventional Synchronization Approachesfor Slaving Data

31 Synchronization Using Cubic Spline Extrapolation [18 19]Let us assume a synchronized slaving state vector to be anunknown function of the delayed slaving state vector whosevalues are known only until time 119896 We then define a cubicspline extrapolation function119891

ep119899

(119896+119904) where 119899 = 119909 119910 119911 at aspecific synchronization time 119896 + 119904 that is extrapolated basedon the known function 119891119899(119896) that has a real value with119873+1

points where 1198960 le 119896 le 119896119873We approximate 119891

ep119899

(119896 + 119904) as a three-order polynomialbased on the interval [119896119894 119896119894+1] where 119894 = 0 119873 minus 1 Then119891ep119899

(119896 + 119904) can be defined as follows

119891ep119899

(119896 + 119904)

= 119891119899 (119896) 119896 isin [119896119894 119896119894+1] 119894 = 0 119873 minus 1

119891119899 (119896) = 1198861198991198941198963+ 119887119899119894119896

2+ 119888119899119894119896 + 119889119899119894

(2)

where 119886119899119894 119887119899119894 119888119899119894 and 119889119899119894 are unknown coefficients To findthe unknown coefficients wemake the assumption that119891119899(119896)

should be continuous in [1198960 119896119873] Therefore the followingequation containing unknown coefficients 119886119899119894 119887119899119894 119888119899119894 and119889119899119894 is obtained

119891119899119894 (119896119894) = 119891119899119894+1 (119896119894)

1198911015840

119899119894(119896119894) = 119891

1015840

119899119894+1(119896119894)

Mathematical Problems in Engineering 3

On-board GPS

Trajectory

Delay compensation(synchronization)All antennas are pointing exactly at the SLV

No delay compensationAntennas are pointing to

a ghost target (tracking failure)

Tracking radar number 1(station number 1)

Tracking radar number 2(station number 2)

Mission control system

Telemetry

TSPITSPI

QLM

Slaving data

Slaving data

Slaving data

Range system

Figure 1 Illustration of problem statement for delayed slaving data in RSS

11989110158401015840

119899119894(119896119894) = 119891

10158401015840

119899119894+1(119896119894)

119899 = 119909 119910 119911 119894 = 1 119873 minus 1

(3)

On combining (2) and (3) the cubic polynomials 119891ep119899

(119896 + 119904)

are reconstructed by solving the linear equations obtainedOnce we find the coefficients 119886119899119894 119887119899119894 119888119899119894 and 119889119899119894 where119894 = 0 119873 minus 1 we can evaluate 119891

ep119899

(119896 + 119904) where 119904 is anarbitrary lead-time for future points in [1199090 119909119873+119904]

32 Synchronization Using an 120572-120573-120574 Filter When the stateestimation covariance for a time invariant system convergesunder suitable conditions to a steady-state value explicitexpressions of the steady-state covariance and filter gaincan be obtained The resulting steady-state filters for noisykinematic models are known as 120572-120573 and 120572-120573-120574 filters [20] Inthis paper a combined 120572-120573-120574 filter using both an expandingmemory polynomial filter (EMF) and a fading memorypolynomial filter (FMF) was used [21 22] At first the EMF isrepresented as follows

119901119864

119899119896= 119901

119864

119899119896minus1+ Δ119905V119864

119899119896minus1+

Δ1199052

2

119886119864

119899119896minus1

+

3 (31198962+ 3119896 + 2)

(119896 + 3) (119896 + 2) (119896 + 1)

120576119864

119899119896

V119864119899119896

= V119864119899119896minus1

+ Δ119886V119864119899119896minus1

+

1

Δ119905

18 (2119896 + 1)

(119896 + 3) (119896 + 2) (119896 + 1)

120576119864

119899119896

119886119864

119899119896= 119886

119864

119899119896minus1+

1

Δ1199052

60

(119896 + 3) (119896 + 2) (119896 + 1)

120576119864

119899119896

120576119864

119899119896= 119910119899119896 minus 119901

119864

119899119896minus1minus Δ119905V119864

119899119896minus1minus

Δ1199052

2

119886119864

119899119896minus1

(4)

where Δ119905 is the sampling time and 119901119864

119899119896 V119864

119899119896 and 119886

119864

119899119896are

the position velocity and acceleration estimates of the EMFrespectively In addition the variance reduction factor (VRF)[22] of the EMF that is VRF119864 can be represented as

VRF119864=

91198962+ 27119896 + 24

119896 (119896 + 1) (119896 minus 1)

(5)

On the other hand the FMF and its VRF (VRF119865) can bewritten as follows

119901119865

119899119896= 119901

119865

119899119896minus1+ Δ119905V119865

119899119896minus1+

Δ1199052

2

119886119865

119899119896minus1+ (1 minus 120582

3) 120576

119865

119899119896

V119865119899119896

= V119865119899119896minus1

+ Δ119886V119865119899119896minus1

+

3

2Δ119905

(1 minus 120582)2(1 + 120582) 120576

119865

119899119896

119886119865

119899119896= 119886

119865

119899119896minus1+

1

Δ1199052(1 minus 120582)

3120576119865

119899119896

120576119865

119899119896= 119910119899119896 minus 119901

119865

119899119896minus1minus Δ119905V119865

119899119896minus1minus

Δ1199052

2

119886119865

119899119896minus1

VRF119865=

1 minus 120582

(1 + 120582)5(19 + 24120582 + 16120582

2+ 6120582

3+ 120582

4)

(6)

where 0 lt 120582 lt 1 Both filters are conducted in parallel butonly one of them is selected as a final estimate by comparing

4 Mathematical Problems in Engineering

the VRFs During the early tracking phase the EMF isselected as the final estimate however at a certain timewhereVRF119865 is larger than VRF119864 FMF is selected as the finalestimate

120572120573120574

119899119896= 119891119899

120572120573120574

119899119896minus1+ 119870[119910119896 minus 119867

120572120573120574

120572120573120574

119899119896minus1]

[

[

[

119901119899119896

V119899119896119886119899119896

]

]

]

=

[

[

[

[

[

1 Δ119905

Δ1199052

2

0 1 Δ119905

0 0 1

]

]

]

]

]

[

[

[

119901119899119896minus1

V119899119896minus1119886119899119896minus1

]

]

]

+

[

[

[

[

[

[

120572

120573

Δ119905

120574

Δ1199052

]

]

]

]

]

]

[

[

[

119910119896 minus 119867120572120573120574 [

[

[

119901119899119896minus1

V119899119896minus1119886119899119896minus1

]

]

]

]

]

]

where 119883120572120573120574

119899119896= [119901119899119896 V119899119896 119886119899119896]

119879 119867

120572120573120574= [1 0 0]

VRF119864ge VRF119865

997888rarr

119901119899119896 = 119901119864

119899119896

V119899119896 = V119864119899119896

119886119899119896 = 119886119864

119899119896

VRF119865gt VRF119864

997888rarr

119901119899119896 = 119901119865

119899119896

V119899119896 = V119865119899119896

119886119899119896 = 119886119865

119899119896

(7)

Finally synchronization using an 120572-120573-120574 filter is completed bylinear propagation using the system matrix 119891119899 such that

120572120573120574

119899119896+119904= (

119904

prod

119871=1

119891119899) sdot

120572120573120574

119899119896 (8)

33 Kalman Predictor The motion of the SLV is simplydepicted as a discretized Wiener process acceleration model[20]

119883119896+1 = 119865119883119896 + 119908119896 (9)

where the state vector 119883119896 isin R9 consists of theposition velocity and acceleration components along 119909-axis 119910-axis and 119911-axis respectively that is 119883119896 =

[119909119901119896 119909V119896 119909119886119896 119910119901119896 119910V119896 119910119886119896 119911119901119896 119911V119896 119911119886119896]119879 The system

matrix and covariance matrix of the system noise 119908119896 arerepresented as (10) and (11) respectively

119891119899=119909119910119911 =

[

[

[

[

[

1 Δ119905

Δ1199052

2

0 1 Δ119905

0 0 1

]

]

]

]

]

119865 =[

[

[

119891119909 03 0303 119891119910 0303 03 119891119911

]

]

]

(10)

119864 [119908119896119908119879

119897] = 119876120575119896minus119897

119902119871=119909119910119911 =

[

[

[

[

[

[

[

[

[

Δ1199055

20

Δ1199054

8

Δ1199052

6

Δ1199054

8

Δ1199053

3

Δ1199052

2

Δ1199053

6

Δ1199052

2

Δ119905

]

]

]

]

]

]

]

]

]

119876 =[

[

[

119902119909 03 0303 119902119910 0303 03 119902119911

]

]

]

(11)

A radar measurement for the SLV gives the spherical coordi-nate observations such that

119885119896 = ℎ (119883119896) + V119904119896

119885119896 isin R3 V119904119896 isin R

3

119885119896 =[

[

[

119903119896

120593119896

120579119896

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

radic1199092

119901119896+ 119910

2

119901119896+ 119911

2

119901119896

tanminus1(

119910119901119896

119909119901119896

)

tanminus1(

119910119901119896

radic1199092

119901119896+ 119910

2

119901119896

)

]

]

]

]

]

]

]

]

]

]

]

+[

[

[

V119903V120593V120579

]

]

]

(12)

where V119904119896 sim N(0 119877119904119896) and 119877119904119896 = diag(1205902

119903 120590

2

120593 120590

2

120579) Using a 3D

debiased converted measurement [23] we can transform theoriginal nonlinear equations (12) into their linear form as

z119896 = 119867119883119896 + V119896 V119896 isin R3

z119896 =[

[

[

119909119896

119910119896

119911119896

]

]

]

=[

[

[

1 01times5 01times301times3 1 01times501times5 1 01times2

]

]

]

119883119896 +[

[

[

V119909119896V119910119896V119911119896

]

]

]

(13)

Here V119888119896 is the converted measurement noise expressed interms of Cartesian coordinates that is V119896 sim N(0 119877119896)

119877119896 =

[

[

[

[

119877119909119909119896 119877119909119910119896 119877119909119911119896

119877119910119909119896 119877119910119910119896 119877119910119911119896

119877119911119909119896 119877119911119910119896 119877119911119911119896

]

]

]

]

(14)

Mathematical Problems in Engineering 5

Prediction or fixed-lead prediction inmean squaremeans thesynchronization of the slaving data is the estimation of thestate at a future time 119896+119904 where 119904 gt 0 beyond the observationinterval that is based on data up to an earlier time [20 24]

119896+119904|119896 = 119864 (119883119896+119904|119896 | 119885119896) 119885119896 = z0 z119896 (15)

The optimal predictor or synchronized state 119896+119904|119896 ≜

KP119896+119904|119896

and its error covariance 119875119896+119904|119896 ≜ 119875KP119896+119904|119896

are given by theKalman predictor equations [2 24]

KP119896+119904|119896

= 119865119896+119904minus1

KP119896+119904minus1|119896

= sdot sdot sdot = Φ119896+119904119896

KP119896|119896

KP119896|119896

≜

KF119896|119896

119875KP119896+119904|119896

= 119865119896+119904minus1119875KP119896+119904minus1|119896

119865119879

119896+119904minus1+ 119876119896+119904minus1

= Φ119896+119904119896119875KP119896|119896

Φ119879

119896+119904119896

+

119904minus1

sum

119895=0

Φ119896+119904119896+119895+1119876119896+119895Φ119879

119896+119904119896+119895+1 119875

KP119896|119896

≜ 119875KF119896|119896

Φ119896+119904119896 = 119865119896+119904minus1119865119896+119904minus2 sdot sdot sdot 119865119896

Φ119896119896 = 119868119899

119896 gt 1

(16)

where

KF119896|119896

and 119875KF119896|119896

are the KF estimate and covariancerespectively

4 Proposed MME-Based SynchronizationApproaches

41 SynchronizationUsing an IMMwith Singerrsquos LinearModelSinger [16] described 2D manned maneuvering targets inrange-bearing coordinates This model can be adapted to theSLV kinematics in 3D Cartesian coordinates with multipleflight phases

[

[

[

119899

V119899119899

]

]

]

= 119865[

[

[

119901119899

V119899119886119899

]

]

]

+ 119866119908119899

119865 =[

[

[

0 1 0

0 0 1

0 0 minus120572119899

]

]

]

119866 =[

[

[

0

0

0

]

]

]

(17)

where119908119899 isin R3times1 is a white noise process along the Cartesianaxis 119899 = 119909 119910 119911 The parameter 120573 = 1120572119899 is the maneuvercorrelation time constant and 120590

2

119886119899is the acceleration variance

describing maneuver intensity In a steady state

1205902

119908119899

= 21205721205902

119886119899 (18)

To describe SLV kinematics the model must cope with cru-cial nonzero mean acceleration maneuvers during the pro-pelled phase In addition after each stagersquos engine burns outa multiple model approach is applied to the coasting flightthat is one model describes PFP whereas the other depictsthe CFP Empirically tuned independent probability densityfunctions (PDFs) represented by TUM describe the accel-erations of the SLV in the local coordinate frame Figure 2shows the means and variances of the acceleration processesof the SLV in this paper The discrete-time model with statetransition matrix Ψ119899(120572119899 Δ119905) is as follows

[

[

[

119901119899119896+1

V119899119896+1119886119899119896+1

]

]

]

= Ψ119899 (120572119899 Δ119905)[

[

[

119901119899119896

V119899119896119886119899119896

]

]

]

+ 119908119899119896

Ψ119899 (120572119899 Δ119905) = 119890119865Δ119905

[

[

[

[

[

[

[

1 Δ119905 (120572119899Δ119905 minus 1 + 119890minus120572119899Δ119905

)

0 1

(1 minus 119890minus120572119899Δ119905

)

120572119899

0 0 119890minus120572119899Δ119905

]

]

]

]

]

]

]

119908119899119896 = int

(119896+1)Δ119905

119896Δ119905

119890119865[(119896+1)Δ119905minus119903]

119866119908119899 119889119903

(19)

In the case of the PFP 119908119899119896 is a nonzero mean white noisesequence caused by the nonzero mean acceleration 120583119886119899 seenin Figure 2 A nonzeromeanwhite noise sequence for the PFPshould be considered in the target kinematics when imple-menting the state propagation stage in theKFThus the deter-ministic input 119906119899119896 caused by 119908119899119896 along 119909-axis 119910-axis and119911-axis is derived as follows

119906119909119896 = 119864 [119908119909119896] = 119864int

(119896+1)Δ119905

119896Δ119905

119890119865[(119896+1)Δ119905minus119903]

119866119908119909119889119903

asymp 119864

int

(119896+1)Δ119905

119896Δ119905

[

[

[

[

119890minus120572119909[(119896+1)Δ119905minus119903]

0

0

]

]

]

]

119908119909 119889119903

119906119909119896 =

120583119886119909

120572119899 [1minus119890minus120572119909Δ119905

00

]

(20)

119910-axis and 119911-axis can be derived in the samemanner as shownin (20) The maneuver excitation covariance [10] whichrepresents the uncertainty of the SLV kinematics model is

119876119899119896 (120572119899 Δ119905)

= 119864 (119908119899119896 minus 119864 [119908119899119896]) (119908119899119896 minus 119864 [119908119899119896])119879

= 21205721198991205902

119886119899

[

[

[

11990211 11990212 11990213

11990212 11990222 11990223

11990213 11990223 11990233

]

]

]

(21)

The specific components of 11990211 11990233 are illustrated in [25]In addition the measurement matrix 119867119871 for Singerrsquos modelcan be depicted as

6 Mathematical Problems in Engineering

01

minus20 4minus6 0

0028

120583 =

1205902 =

p(ax)

minus6ms2

4246

a (m2s4)

m2s4

(a)

01

minus5 810

005

120583 = 1ms2

1205902 = 106

p(ay)

a (m2s4)

m2s4

(b)

01

minus40 4minus7 0

0008

a (m2s4)

120583 = minus7ms2

1205902 = 12674 m2s4

p(az)

(c)

01

005 005

minus4 40

008

120583 = 0ms2

1205902 = 183

p(ax) p(ay) p(az)

a (m2s4)

m2s4

(d)

Figure 2 Nonzero mean acceleration PDF in the PFP model (a) along 119909-axis (b) along 119910-axis and (c) along 119911-axis Zero mean accelerationPDF in the CFP Model (d) along 119909-axis 119910-axis and 119911-axis

119911119871119896 = 119867119871x119871119896 + ]119896 119867119871 =[

[

[

1 0 0

0 03times2 1 03times2 0 03times20 0 1

]

]

]

x119871119896 = [119909119896 119896 119896 119910119896 119896

119896

119911119896 119896 119896]119879 (22)

where ]119896 is measurement noise (as shown in (14)) with errorcovariance

411 IMM with Singerrsquos Linear Model From (19)ndash(22) inSection 41 we can rewrite the Markov jump linear systemswhere the 119894th model of the finite multiple model set M =

119898(1)

119898(119872)

obeys the following equations

x119871119896+1 = Ψ(119894)

119896x119871119896 + 119908

(119894)

119896

119911119871119896 = 119867119871x119871119896 + ]119896(23)

where

cov (119908(119894)

119896) = 119876

(119894)

119896

cov (]119896) = 119877119896

Ψ(119894)

119896= diag (Ψ119909119896 (120572

(119894)

119909 Δ119905) Ψ119910119896 (120572

(119894)

119910 Δ119905)

Ψ119911119896 (120572(119894)

119911 Δ119905))

119876(119894)

119896= diag (119876119909119896 (120572

(119894)

119909 Δ119905) 119876119910119896 (120572

(119894)

119910 Δ119905)

119876119911119896 (120572(119894)

119911 Δ119905))

(24)

Here 1205902

119886119899for the SLV can be represented by TUM as in

Figure 2 The PDFs of Figures 2(a)ndash2(d) are experimentallysampled from the nominal acceleration profile of the SLVThesuperscript (i) denotes quantities pertinent to model 119898(119894) inM and the jumps of the system mode are assumed to havetransition probabilities

Pr 119898(119895)

119896+1| 119898

(119894)

119896 ≜ 120587119894119895

(25)

where 119898(119894)

119896denotes the event in which model 119898

(119894) matchesthe system mode in effect at time 119896 In our application 119872 =

2 and 119898(1) and 119898

(2) are the propelled and coasting modesrespectively

Finally complete recursion of the IMM with modematched KF for the SLV tracking is summarized as follows

(i) Model-conditioned reinitialization (for 119894 = 1 2

119872)

Mathematical Problems in Engineering 7

(a) predicted mode probability 120583(119894)

119871119896|119896minus1≜ Pr119898(119894)

119896|

119911119871119896minus1 = sum119895120587119895119894120583

(119895)

119871119896minus1

(b) mixingweight 120583119895|119894

119871119896minus1≜ Pr119898(119895)

119896minus1| 119898

(119894)

119896 119911119871119896minus1 =

sum119895120587119895119894120583

(119895)

119871119896minus1120583

(119894)

119871119896minus1

(c) mixing estimate and covariance

x(119894)119871119896minus1|119896minus1

≜ 119864 [x119871119896minus1|119896minus1 | 119898(119894)

119896 119911119871119896minus1]

= sum

119895

x(119895)119871119896minus1|119896minus1

120583119895|119894

119871119896minus1

119875

(119894)

119871119896minus1|119896minus1= sum

119895

[119875(119894)

119871119896minus1|119896minus1+ (x(119894)

119871119896minus1|119896minus1minus x(119895)

119871119896minus1|119896minus1)

sdot (x(119894)119871119896minus1|119896minus1

minus x(119895)119871119896minus1|119896minus1

)

119879

] 120583119895|119894

119871119896minus1

(26)

(ii) Model-conditioned filtering (for 119894 = 1 2 119872)

(a) predicted estimate and covariance

x(119894)119871119896|119896minus1

= Ψ(119894)

119896minus1x(119894)119871119896minus1|119896minus1

+ u(119894)

119896minus1

where u(1)

119896minus1= [119906

119879

119909119896minus1119906119879

119910119896minus1119906119879

119911119896minus1]

119879

u(2)

119896minus1= 03times1

119875(119894)

119871119896|119896minus1= Ψ

(119894)

119896minus1119875

(119894)

119871119896minus1|119896minus1Ψ

(119894)119879

119896minus1+ 119876

(119894)

119896minus1

(27)

(b) measurement residual (119894)

119871119896= 119911119871119896 minus119867119871x

(119894)

119871119896|119896minus1minus

]119896(c) residual covariance 119878(119894)

119871119896= 119867119871119875

(119894)

119871119896|119896minus1119867

119879

119871minus 119877119896

(d) filter gain 119870(119894)

119871119896= 119875

(119894)

119871119896|119896minus1119867

119879

119871(119878

(119894)

119871119896)minus1

(e) update of state and covariance x(119894)119871119896|119896

= x(119894)119871119896|119896minus1

+

119870(119894)

119871119896(119894)

119871119896 119875

(119894)

119871119896|119896= 119875

(119894)

119871119896|119896minus1minus 119870

(119894)

119871119896119878(119894)

119871119896119870

(119894)119879

119871119896

(iii) Mode probability update (for 119894 = 1 2 119872)

(a) mode likelihood Λ(119894)

119871119896≜ 119901[119911

(119894)

119871119896| 119898

(119894)

119896

119885119871119896minus1]assume

= N(119911(119894)

119871119896 0 119878

(119894)

119871119896)

(b) mode probability 120583(119894)

119871119896= 120583

(119894)

119871119896|119896minus1Λ

(119894)

119871119896

(sum119895120583(119895)

119871119896|119896minus1Λ

(119895)

119871119896)

(iv) Combination (for 119894 = 1 2 119872)

x119871119896|119896 = sum

119894

x(119894)119871119896|119896

120583(119894)

119871119896

(119894)

119871119896|119896

= sum

119894

[119875(119894)

119871119896|119896+ (x119871119896|119896 minus x(119894)

119871119896|119896) (x119871119896|119896 minus x(119894)

119871119896|119896)

119879

] 120583(119894)

119871119896

(28)

Singerrsquos MMS of time delayed slaving data is completed bypropagating the combined estimate based on the currentmodersquos dynamic model such that

xsync119871119896+119904|119896

= Φsync119896+119904119896

x119871119896|119896

Φsync119896+119904119896

= 119865sync119896+119904minus1

119865sync119896+119904minus2

sdot sdot sdot 119865sync119896

Φsync119896119896

= 119868119899

119896 gt 1

(29)

where 119865sync119896+119904minus1

119865sync119896+119904minus2

119865sync119896

are systemmatrices dependingon a flight phase mode at current time 119896

42 Synchronization Using an IMM with a Nonlinear BallisticModel For a nonlinear ballistic model the state vector forthe propelled mode is denoted as

x119905 = [119909119905 119910119905 119911119905 119905 119905

119905 120585119905 120591119905]119879 (30)

where 120585119905 is the drag coefficient and 120591119905 is the thrust Generallythe drag coefficient varies significantlywith theMachnumberregime subsonic transonic and supersonic Therefore wetake advantage of the dynamic model considering a Machnumber-dependent multiplier [11 12] such that

[

[

[

]

]

]

=

120591

119881

[

[

[

]

]

]

thrust term

+ 120585120585119898119863[

[

[

]

]

]

drag term

+ 119892[

[

[

0

0

1

]

]

]

gravity term

+ 1

120585 = 2

= 3

(31)

The first term on the right side of (31) represents the thrust(m2

s) of the SLV in 119909 119910 and 119911 directions Two distinctmultiplemodes of a nonlinear SLVmodel can be separated bythe existence of thrust In other words the existence of thrustsignifies propelled flight mode whereas zero thrust signifiescoasting flight mode 119881 is the magnitude of the velocity V =

[ ]

119879 that is the SLV speed (ms) The second part of(31) is the drag term which is related to velocity and altitudethat is 119863 = minus120588(119911)1198812 where 120588(119911) = 1205880119890

minus119888119911 is the air density(kgm3) at an altitude 119911 (m) and 119888 is the air density constant(mminus1) [9] 120585 is the drag coefficient and 120585119898 is theMachnumber-dependent drag coefficient multiplier which is approximatedby the cubic spline curve shown in Figure 3 In this paperfor the drag characteristics of the SLV which are applied inthe subsequent simulation section drag coefficients of theSaturn119881 launch vehicle [15] are usedThe third part of (31) isa gravity term Gravity 119892 is the standard acceleration due togravity at sea level which is assumed to be constant through-out the trajectory with a value of 9812 ms2 1 2 and 3

are assumed to be continuous time zeromeanwhite Gaussiannoises The drag coefficient and thrust acceleration are rep-resented as Wiener processes with a slow variation [11 12]

8 Mathematical Problems in Engineering

0 2 4 6 8 1004

06

08

1

12

14

16

18

2

Mach number

Nor

mal

ized

dra

g co

effici

ent

Selected pointsLinear interpolationCubic spline interpolation

Figure 3 Normalized drag coefficient [15]

We can modify the dynamic equations (30) and (31) as acompact form such that

x119905 = 119891 [x119905] + 119905 (32)

where

119891 [x119905] =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119905

119905

119905

120591

119905

119881119905

+ 120585119905119863119905119905

120591

119905

119881119905

+ 120585119905119863119905119905

120591

119905

119881119905

+ 120585119905119863119905119905 minus 119892

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

119905 = [1119905 2119905 3119905]119879

(33)

The state vector equation (33) is discretized by a second-order Taylor expansion [26] Then (33) can be written as

a discretized continuous time systemwithwhite process noisesuch that

x119896+1 = x119896 + 119891 [x119896] Δ119905 + 119860119896119891 [x119896]Δ119905

2

2

+ 120596119896(34)

where 119860119896 is the Jacobian of (33) evaluated at x119896 [26] and120596119896 is the discretized continuous time process noise for thesampling interval Δ119905 The corresponding covariance matrixof the discretized process noise is

119876 =[

[

[

1198761119902V 06times1 06times101times6 Δ119905119902120585 0

01times6 0 Δ119905119902120591

]

]

]

1198761 =

[

[

[

[

Δ1199053

3

1198683

Δ1199052

2

1198683

Δ1199052

2

1198683 Δ1199051198683

]

]

]

]

(35)

where 1198683 is the 3 times 3 identity matrix and the continuous timeprocess noise intensities 119902V 119902120585 and 119902120591 are the correspondingpower spectral densities

The measurement matrix 119867NL for the nonlinear ballisticmultiple model can be depicted as

119911NL119896 = 119867NLxNL119896 + ]119896

119867NL = [1198683 0] xNL119896 = [119909119896 119910119896 119911119896 119896 119896

119896 120585119896 120591119896]119879

(36)

where ]119896 is measurement noise with error covariance119877119896120575119896minus119895 = 119864[]119896]

119879

119895] that is (14)

An IMMalgorithm for nonlinear dynamics with differentsizes of the mode state vector is summarized as follows [25]

(v) Model-conditioned reinitialization (for 119894 = 1 2

119872)

(a) predicted mode probability 120583(119894)

NL119896|119896minus1 ≜

Pr119898(119894)

119896| 119911NL119896minus1 = sum

119895120587119895119894120583

(119895)

NL119896minus1(b) mixing weight

120583119895|119894

NL119896minus1 ≜ Pr 119898(119895)

119896minus1| 119898

(119894)

119896 119911NL119896minus1 =

sum119895120587119895119894120583

(119895)

NL119896minus1

120583(119894)

NL119896minus1

(37)

(c) unbiased mixing estimate and covariance

x(119894)NL119896minus1|119896minus1 ≜ 119864 [xNL119896minus1|119896minus1 | 119898(119894)

119896 119911NL119896minus1] = sum

119895

(119895)

NL119896minus1|119896minus1120583119895|119894

NL119896minus1

where (1)

NL119896minus1|119896minus1 ≜ x(1)NL119896minus1|119896minus1 (2)

NL119896minus1|119896minus1 ≜ [x(2)119879

NL119896minus1|119896minus1 120591119896minus1]119879

119875

(119894)

NL119896minus1|119896minus1 = sum

119895

[119875(119894)

NL119896minus1|119896minus1 + (x(119894)NL119896minus1|119896minus1 minus (119895)

NL119896minus1|119896minus1) (x(119894)NL119896minus1|119896minus1 minus (119895)

NL119896minus1|119896minus1)119879

] 120583119895|119894

NL119896minus1

(38)

Mathematical Problems in Engineering 9

(vi) Model-conditioned filtering (for 119894 = 1 2 119872)

(a) predicted estimate and covariance

x(119894)NL119896|119896minus1 = x(119894)NL119896minus1|119896minus1 + 119891 [x(119894)NL119896minus1|119896minus1] Δ119905

+

(119894)

NL119896minus1119891 [x(119894)NL119896minus1|119896minus1]Δ119905

2

2

(119894)

119896=

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x=x(119894)119896

119875(119894)

NL119896|119896minus1 = Ω(119894)

119896minus1119875

(119894)

NL119896minus1|119896minus1Ω(119894)119879

119896minus1+ 119876119896minus1

Ω(119894)

119896minus1= 119868 +

(119894)

NL119896minus1

(39)

(b) measurement residual (119894)

NL119896 = 119911NL119896 minus

119867NLx(119894)

NL119896|119896minus1 minus ]119896

(c) residual covariance 119878(119894)NL119896 = 119867NL119875(119894)

NL119896|119896minus1119867119879

NL minus

119877119896(d) filter gain 119870(119894)

NL119896 = 119875(119894)

NL119896|119896minus1119867119879

NL(119878(119894)

NL119896)minus1

(e) update of state and covariance

x(119894)NL119896|119896 = x(119894)NL119896|119896minus1 + 119870(119894)

NL119896(119894)

NL119896

119875(119894)

NL119896|119896 = 119875(119894)

NL119896|119896minus1 minus 119870(119894)

NL119896119878(119894)

NL119896119870(119894)119879

NL119896

(40)

(vii) Mode probability update (for 119894 = 1 2 119872)

(a) mode likelihood

Λ(119894)

NL119896 ≜ 119901 [119911(119894)

NL119896 | 119898(119894)

119896 119885NL119896minus1]

assume= N (119911

(119894)

NL119896 0 119878(119894)

NL119896)

(41)

(b) mode probability 120583(119894)

NL119896 = 120583(119894)

NL119896|119896minus1Λ(119894)

NL119896

(sum119895120583(119895)

NL119896|119896minus1Λ(119895)

NL119896)

(viii) Combination (for 119894 = 1 2 119872)

xNL119896|119896 = sum

119894

x(119894)NL119896|119896120583(119894)

NL119896

(119894)

NL119896|119896 = sum

119894

[119875(119894)

NL119896|119896

+ (xNL119896|119896 minus x(119894)NL119896|119896) (xNL119896|119896 minus x(119894)NL119896|119896)119879

] 120583(119894)

NL119896

(42)

The nonlinearMMS of time delayed slaving data is completedby propagating the combined estimate based on the currentmodersquos estimated vector such that

xNL119896+119904|119896 = xNL119896|119896 + 119891 [xNL119896|119896] sdot 119904 + NL119896119891 [xNL119896|119896]

sdot

1199042

2

(43)

where 119904 is a lead-time for synchronization (119904 = 119899 sdot Δ119905) If theSLV is in the PFP (120583(1)

NL119896 gt 120583(2)

NL119896) the current state estimateincludes the thrust term that is the state vector xNL119896 =

[119909119896 119910119896 119911119896 119896 119896

119896 120585119896 120591119896]119879 If the SLV is not in the PFP

the state vector does not include the thrust term (120591119896 = 0)

5 Simulation Results

To demonstrate the performance of the proposed synchro-nization approaches for delayed slaving data we simulatedthe SLV tracking problem based on the nominal flight tra-jectory of the Korea Space Launch Vehicle-I (KSLV-I) In thesimulation the radar measurement noise intensities of (12)are selected as 120590119903 = 15m 120590120593 = 001 deg and 120590120579 = 001 degThe nominal flight sequence of KSLV-I is as follows First thepayload fairing separates during the first stage flight at 2154 sAfter the first stage engine shutdown at 2287 s the upperstage separates from the first stage and enters the CFP Thesecond stage continues in the CFP until the kick motor igni-tion at 395 s and the vehicle then enters the PFP At the endof kickmotor combustion (4527 s) the upper stage enters thetarget orbit in CFP as in the previous separation Finally thesatellite is inserted into the target orbit after it separates fromthe upper stage during CFP at 540 s Since delayed slavingdata from the MCS to the TSs can occur in both the PFP andCFP synchronization simulations are conducted at arbitrarypoints of the flight phases Figure 4 shows the synchroniza-tion error of the delayed slaving data with respect to thebenchmark and the proposed approaches for delays of 119904 =

01 02 1 s Figures 4(a)ndash4(c) illustrate synchronizationerrors where delayed slaving data occurred at 155 s which isthe first stage of PFP and Figures 4(d)ndash4(f) show synchro-nization errors where delayed slaving data occurred at 280 swhich is the first stage of CFP As shown in Figures 4(a)ndash4(c)the extrapolation and120572-120573-120574filter-based approach cannot findthe proper synchronized slaving data The synchronizationerrors of these approaches exponentially increase accordingto increasing delay 119904 in particular the error of the extrapo-lation approaches an out-of-figure bound (2500m) On theother hand the Kalman prediction and the proposedapproaches have stable synchronization errors even if thedelay time 119904 is larger Furthermore we can observe thatthe errors in the synchronization approach of the Kalmanprediction are relatively larger than the proposed approachesespecially in the PFP As explained in the flight sequence ofKSLV-I in general the motion of the SLV is described bymultiple dynamics (in our case PFP and CFP) rather thansingle dynamics However the dynamic model of the KF andprediction capability used in this paper is the constant accel-eration (CA) model This means that the filter is optimizedwith respect to CAmotion which is actually not occurring inour application This is why the performance of the Kalmanprediction-based synchronization is worse than the proposedapproaches Nonetheless we can see that the 119910-axis synchro-nization errors shown in Figures 4(b) and 4(e) are smallThis result is observed because the 119910-axis motion of KSLV-I in a local coordinate is relatively smaller than the otheraxis motion and the CA model approximately expresses this

10 Mathematical Problems in Engineering

01 02 03 04 05 06 07 08 09 10

500

1000

1500

2000

2500

Delay (sec)

x-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

Cubic spline extrapolation

Kalman filter

(a)

(b)

(c)

(d)

(e)

(f)

Linear MMNonlinear MM120572-120573-120574 filter 120572-120573-120574 filter

120572-120573-120574 filter120572-120573-120574 filter

120572-120573-120574 filter 120572-120573-120574 filter

01 02 03 04 05 06 07 08 09 1Delay (sec)

0

500

1000

1500

2000

2500

y-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

z-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

x-p

ositi

on ab

solu

te er

ror i

n CF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500y

-pos

ition

abso

lute

erro

r in

CFP

(m)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

z-p

ositi

on ab

solu

te er

ror i

n CF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

Figure 4 Synchronization errors at PFP and CFP

Mathematical Problems in Engineering 11

small motion A single KF prediction model cannot exactlydescribe the motion of both PFP and CFP whereas theproposed multiple model-based approaches work well Asshown in Figure 4 regardless of the delay 119904 the proposedmultiplemodel-based approaches find synchronized positionvectors In the comparison of synchronization performancebetween linear IMM and nonlinear IMM synchronizationthe nonlinear IMM-based synchronization approach showsthe best performance except for 119910-axis where CA motion isdominant In addition the difference between the proposedmultiple model-based approaches is very small as shown inFigure 4 but the complexities of the algorithms for real-timeapplications are quite dissimilar Hence the operator mayadaptively select one of the proposed approaches accordingto onersquos environment

6 Conclusions

In this paper we investigated the time synchronizationapproaches of delayed slaving data in the RSS for SLV track-ingOne of themost important roles of the RSS is to distributeslaving data to each TS for continuous tracking of the SLV Ifthere is a critical network delay resulting in time delayed slav-ing data being sent to each TS theMCS will not receive accu-rate SLV tracking data This problem can give rise to signifi-cant difficulties for the SLVmission progress and analysis Toovercome this problemwe proposedMMSapproacheswhichtake advantage of the multiple motion models of an SLVThelinear IMM-based synchronization approach was developedusing Singerrsquos model with ternary uniform mixtures and thenonlinear IMM-based synchronization approachwas derivedfrom a nonlinear ballistic model with a drag coefficient Forverification of the proposed algorithms SLV tracking simu-lations using KSLV-I and the radar measurement data gen-erated from nominal trajectory were conducted To demon-strate the superiority of time synchronization performancein these simulations we compared the proposed algorithmwith benchmark approaches for absolute error between thenominal trajectory data and the synchronized slaving datathe simulation results demonstrated that the proposed MMSapproaches performed competitively

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] R Varaprasad and V Seshagiri Rao ldquoRange safety real-timesystem for satellite launch vehicle missions-testing methodolo-giesrdquo Defence Science Journal vol 56 no 5 pp 693ndash700 2006

[2] H R Song M G Jeon T S Choi and V Shin ldquoTwo fusionpredictors for discrete-time linear systems with different typesof observationsrdquo International Journal of Control Automationand Systems vol 7 no 4 pp 651ndash658 2009

[3] H Song V Shin and M Jeon ldquoMobile node localization usingfusion prediction-based interacting multiple model in cricketsensor networkrdquo IEEE Transactions on Industrial Electronicsvol 59 no 11 pp 4349ndash4359 2012

[4] T Kirubarajan and Y Bar-Shalom ldquoKalman filter versus IMMestimator when do we need the latterrdquo IEEE Transactions onAerospace and Electronic Systems vol 39 no 4 pp 1452ndash14572003

[5] E Mazor A Averbuch Y Bar-Shalom and J Dayan ldquoInteract-ing multiple model methods in target tracking a surveyrdquo IEEETransactions on Aerospace and Electronic Systems vol 34 no 1pp 103ndash123 1998

[6] P D Hanlon and P S Maybeck ldquoMultiple-model adaptiveestimation using a residual correlation Kalman filter bankrdquoIEEE Transactions on Aerospace and Electronic Systems vol 36no 2 pp 393ndash406 2000

[7] S-J Shin ldquoRe-entry vehicle tracking with a new multiplemodel estimation applicable to highly non-linear dynamicsrdquoIET Radar Sonar amp Navigation vol 9 no 5 pp 581ndash588 2015

[8] S Conover J Kerce G Brown L Ehrman and D HardimanldquoImpact point prediction of small ballistic munitions with aninteracting multiple model estimatorrdquo in Acquisition TrackingPointing and Laser Systems Technologies XXI vol 6569 ofProceedings of SPIE Orlando Fla USA 2007

[9] V C Ravindra Y Bar-Shalom and P Willett ldquoProjectileidentification and impact point predictionrdquo IEEE Transactionson Aerospace and Electronic Systems vol 46 no 4 pp 2004ndash2021 2010

[10] W J Farrell III ldquoInteracting multiple model filter for tacticalballistic missile trackingrdquo IEEE Transactions on Aerospace andElectronic Systems vol 44 no 2 pp 418ndash426 2008

[11] T Yuan Y Bar-Shalom PWillett E Mozeson S Pollak and DHardiman ldquoA multiple IMM estimation approach with unbi-ased mixing for thrusting projectilesrdquo IEEE Transactions onAerospace and Electronic Systems vol 48 no 4 pp 3250ndash32672012

[12] T Yuan Y Bar-Shalom P Willett and D Hardiman ldquoImpactpoint prediction for thrusting projectiles in the presence ofwindrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 50 no 1 pp 102ndash119 2014

[13] X Li and V Jilkov ldquoSurvey of maneuvering target trackingpart II motion models of ballistic and space targetsrdquo IEEETransactions on Aerospace and Electronic Systems vol 46 no1 pp 96ndash119 2010

[14] J Cesar Bolzani de Campos Ferreira and JWaldmann ldquoCovari-ance intersection-based sensor fusion for sounding rockettracking and impact area predictionrdquo Control Engineering Prac-tice vol 15 no 4 pp 389ndash409 2007

[15] httpwwwbraeunigusapollosaturnVhtm[16] RA Singer ldquoEstimating optimal tracking filter performance for

manned maneuvering targetsrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 6 no 4 pp 473ndash483 1970

[17] R A Singer and KW Behnke ldquoReal-time tracking filter evalu-ation and selection for tactical applicationsrdquo IEEE Transactionson Aerospace and Electronic Systems vol AES-7 no 1 pp 100ndash110 1971

[18] W Press B Flannery S Teukolsky and W Vetterling Numer-ical Recipes in FORTRAN The Art of Scientific ComputingCambridge University New York NY USA 2nd edition 1992

[19] S Dyer and J Dyer ldquoCubic-spline interpolation part Irdquo IEEEInstrumentation amp Measurement Magazine pp 44ndash46 2001

[20] Y Bar-Shalom X Li and T Kirubarajan Estimation with Appli-cations to Tracking and Navigation John Wiley amp Sons NewYork NY USA 2001

12 Mathematical Problems in Engineering

[21] N Morrison Introduction to Sequential Smoothing and Predic-tion McGraw-Hill 1969

[22] E Brookner Tracking and Kalman Filtering Made Easy JohnWiley amp Sons 1998

[23] P Suchomski ldquoExplicit expressions for debiased statistics of 3Dconverted measurementsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 35 no 1 pp 368ndash370 1999

[24] H R Song I Y Song and V Shin ldquoMultisensory predictionfusion of nonlinear functions of the state vector in discrete-timesystemsrdquo International Journal of Distributed Sensor Networksvol 2015 Article ID 249857 13 pages 2015

[25] H Song andYHan ldquoComparison of space launch vehicle track-ing using different types of multiple modelsrdquo in Proceedingsof the International Conference on Information Fusion pp 1ndash8Heidelberg Germany July 2016

[26] R KMehra ldquoA comparison of several nonlinear filters for reen-try vehicle trackingrdquo IEEE Transactions on Automatic Controlvol 16 no 4 pp 307ndash319 1971

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

On-board GPS

Trajectory

Delay compensation(synchronization)All antennas are pointing exactly at the SLV

No delay compensationAntennas are pointing to

a ghost target (tracking failure)

Tracking radar number 1(station number 1)

Tracking radar number 2(station number 2)

Mission control system

Telemetry

TSPITSPI

QLM

Slaving data

Slaving data

Slaving data

Range system

Figure 1 Illustration of problem statement for delayed slaving data in RSS

11989110158401015840

119899119894(119896119894) = 119891

10158401015840

119899119894+1(119896119894)

119899 = 119909 119910 119911 119894 = 1 119873 minus 1

(3)

On combining (2) and (3) the cubic polynomials 119891ep119899

(119896 + 119904)

are reconstructed by solving the linear equations obtainedOnce we find the coefficients 119886119899119894 119887119899119894 119888119899119894 and 119889119899119894 where119894 = 0 119873 minus 1 we can evaluate 119891

ep119899

(119896 + 119904) where 119904 is anarbitrary lead-time for future points in [1199090 119909119873+119904]

32 Synchronization Using an 120572-120573-120574 Filter When the stateestimation covariance for a time invariant system convergesunder suitable conditions to a steady-state value explicitexpressions of the steady-state covariance and filter gaincan be obtained The resulting steady-state filters for noisykinematic models are known as 120572-120573 and 120572-120573-120574 filters [20] Inthis paper a combined 120572-120573-120574 filter using both an expandingmemory polynomial filter (EMF) and a fading memorypolynomial filter (FMF) was used [21 22] At first the EMF isrepresented as follows

119901119864

119899119896= 119901

119864

119899119896minus1+ Δ119905V119864

119899119896minus1+

Δ1199052

2

119886119864

119899119896minus1

+

3 (31198962+ 3119896 + 2)

(119896 + 3) (119896 + 2) (119896 + 1)

120576119864

119899119896

V119864119899119896

= V119864119899119896minus1

+ Δ119886V119864119899119896minus1

+

1

Δ119905

18 (2119896 + 1)

(119896 + 3) (119896 + 2) (119896 + 1)

120576119864

119899119896

119886119864

119899119896= 119886

119864

119899119896minus1+

1

Δ1199052

60

(119896 + 3) (119896 + 2) (119896 + 1)

120576119864

119899119896

120576119864

119899119896= 119910119899119896 minus 119901

119864

119899119896minus1minus Δ119905V119864

119899119896minus1minus

Δ1199052

2

119886119864

119899119896minus1

(4)

where Δ119905 is the sampling time and 119901119864

119899119896 V119864

119899119896 and 119886

119864

119899119896are

the position velocity and acceleration estimates of the EMFrespectively In addition the variance reduction factor (VRF)[22] of the EMF that is VRF119864 can be represented as

VRF119864=

91198962+ 27119896 + 24

119896 (119896 + 1) (119896 minus 1)

(5)

On the other hand the FMF and its VRF (VRF119865) can bewritten as follows

119901119865

119899119896= 119901

119865

119899119896minus1+ Δ119905V119865

119899119896minus1+

Δ1199052

2

119886119865

119899119896minus1+ (1 minus 120582

3) 120576

119865

119899119896

V119865119899119896

= V119865119899119896minus1

+ Δ119886V119865119899119896minus1

+

3

2Δ119905

(1 minus 120582)2(1 + 120582) 120576

119865

119899119896

119886119865

119899119896= 119886

119865

119899119896minus1+

1

Δ1199052(1 minus 120582)

3120576119865

119899119896

120576119865

119899119896= 119910119899119896 minus 119901

119865

119899119896minus1minus Δ119905V119865

119899119896minus1minus

Δ1199052

2

119886119865

119899119896minus1

VRF119865=

1 minus 120582

(1 + 120582)5(19 + 24120582 + 16120582

2+ 6120582

3+ 120582

4)

(6)

where 0 lt 120582 lt 1 Both filters are conducted in parallel butonly one of them is selected as a final estimate by comparing

4 Mathematical Problems in Engineering

the VRFs During the early tracking phase the EMF isselected as the final estimate however at a certain timewhereVRF119865 is larger than VRF119864 FMF is selected as the finalestimate

120572120573120574

119899119896= 119891119899

120572120573120574

119899119896minus1+ 119870[119910119896 minus 119867

120572120573120574

120572120573120574

119899119896minus1]

[

[

[

119901119899119896

V119899119896119886119899119896

]

]

]

=

[

[

[

[

[

1 Δ119905

Δ1199052

2

0 1 Δ119905

0 0 1

]

]

]

]

]

[

[

[

119901119899119896minus1

V119899119896minus1119886119899119896minus1

]

]

]

+

[

[

[

[

[

[

120572

120573

Δ119905

120574

Δ1199052

]

]

]

]

]

]

[

[

[

119910119896 minus 119867120572120573120574 [

[

[

119901119899119896minus1

V119899119896minus1119886119899119896minus1

]

]

]

]

]

]

where 119883120572120573120574

119899119896= [119901119899119896 V119899119896 119886119899119896]

119879 119867

120572120573120574= [1 0 0]

VRF119864ge VRF119865

997888rarr

119901119899119896 = 119901119864

119899119896

V119899119896 = V119864119899119896

119886119899119896 = 119886119864

119899119896

VRF119865gt VRF119864

997888rarr

119901119899119896 = 119901119865

119899119896

V119899119896 = V119865119899119896

119886119899119896 = 119886119865

119899119896

(7)

Finally synchronization using an 120572-120573-120574 filter is completed bylinear propagation using the system matrix 119891119899 such that

120572120573120574

119899119896+119904= (

119904

prod

119871=1

119891119899) sdot

120572120573120574

119899119896 (8)

33 Kalman Predictor The motion of the SLV is simplydepicted as a discretized Wiener process acceleration model[20]

119883119896+1 = 119865119883119896 + 119908119896 (9)

where the state vector 119883119896 isin R9 consists of theposition velocity and acceleration components along 119909-axis 119910-axis and 119911-axis respectively that is 119883119896 =

[119909119901119896 119909V119896 119909119886119896 119910119901119896 119910V119896 119910119886119896 119911119901119896 119911V119896 119911119886119896]119879 The system

matrix and covariance matrix of the system noise 119908119896 arerepresented as (10) and (11) respectively

119891119899=119909119910119911 =

[

[

[

[

[

1 Δ119905

Δ1199052

2

0 1 Δ119905

0 0 1

]

]

]

]

]

119865 =[

[

[

119891119909 03 0303 119891119910 0303 03 119891119911

]

]

]

(10)

119864 [119908119896119908119879

119897] = 119876120575119896minus119897

119902119871=119909119910119911 =

[

[

[

[

[

[

[

[

[

Δ1199055

20

Δ1199054

8

Δ1199052

6

Δ1199054

8

Δ1199053

3

Δ1199052

2

Δ1199053

6

Δ1199052

2

Δ119905

]

]

]

]

]

]

]

]

]

119876 =[

[

[

119902119909 03 0303 119902119910 0303 03 119902119911

]

]

]

(11)

A radar measurement for the SLV gives the spherical coordi-nate observations such that

119885119896 = ℎ (119883119896) + V119904119896

119885119896 isin R3 V119904119896 isin R

3

119885119896 =[

[

[

119903119896

120593119896

120579119896

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

radic1199092

119901119896+ 119910

2

119901119896+ 119911

2

119901119896

tanminus1(

119910119901119896

119909119901119896

)

tanminus1(

119910119901119896

radic1199092

119901119896+ 119910

2

119901119896

)

]

]

]

]

]

]

]

]

]

]

]

+[

[

[

V119903V120593V120579

]

]

]

(12)

where V119904119896 sim N(0 119877119904119896) and 119877119904119896 = diag(1205902

119903 120590

2

120593 120590

2

120579) Using a 3D

debiased converted measurement [23] we can transform theoriginal nonlinear equations (12) into their linear form as

z119896 = 119867119883119896 + V119896 V119896 isin R3

z119896 =[

[

[

119909119896

119910119896

119911119896

]

]

]

=[

[

[

1 01times5 01times301times3 1 01times501times5 1 01times2

]

]

]

119883119896 +[

[

[

V119909119896V119910119896V119911119896

]

]

]

(13)

Here V119888119896 is the converted measurement noise expressed interms of Cartesian coordinates that is V119896 sim N(0 119877119896)

119877119896 =

[

[

[

[

119877119909119909119896 119877119909119910119896 119877119909119911119896

119877119910119909119896 119877119910119910119896 119877119910119911119896

119877119911119909119896 119877119911119910119896 119877119911119911119896

]

]

]

]

(14)

Mathematical Problems in Engineering 5

Prediction or fixed-lead prediction inmean squaremeans thesynchronization of the slaving data is the estimation of thestate at a future time 119896+119904 where 119904 gt 0 beyond the observationinterval that is based on data up to an earlier time [20 24]

119896+119904|119896 = 119864 (119883119896+119904|119896 | 119885119896) 119885119896 = z0 z119896 (15)

The optimal predictor or synchronized state 119896+119904|119896 ≜

KP119896+119904|119896

and its error covariance 119875119896+119904|119896 ≜ 119875KP119896+119904|119896

are given by theKalman predictor equations [2 24]

KP119896+119904|119896

= 119865119896+119904minus1

KP119896+119904minus1|119896

= sdot sdot sdot = Φ119896+119904119896

KP119896|119896

KP119896|119896

≜

KF119896|119896

119875KP119896+119904|119896

= 119865119896+119904minus1119875KP119896+119904minus1|119896

119865119879

119896+119904minus1+ 119876119896+119904minus1

= Φ119896+119904119896119875KP119896|119896

Φ119879

119896+119904119896

+

119904minus1

sum

119895=0

Φ119896+119904119896+119895+1119876119896+119895Φ119879

119896+119904119896+119895+1 119875

KP119896|119896

≜ 119875KF119896|119896

Φ119896+119904119896 = 119865119896+119904minus1119865119896+119904minus2 sdot sdot sdot 119865119896

Φ119896119896 = 119868119899

119896 gt 1

(16)

where

KF119896|119896

and 119875KF119896|119896

are the KF estimate and covariancerespectively

4 Proposed MME-Based SynchronizationApproaches

41 SynchronizationUsing an IMMwith Singerrsquos LinearModelSinger [16] described 2D manned maneuvering targets inrange-bearing coordinates This model can be adapted to theSLV kinematics in 3D Cartesian coordinates with multipleflight phases

[

[

[

119899

V119899119899

]

]

]

= 119865[

[

[

119901119899

V119899119886119899

]

]

]

+ 119866119908119899

119865 =[

[

[

0 1 0

0 0 1

0 0 minus120572119899

]

]

]

119866 =[

[

[

0

0

0

]

]

]

(17)

where119908119899 isin R3times1 is a white noise process along the Cartesianaxis 119899 = 119909 119910 119911 The parameter 120573 = 1120572119899 is the maneuvercorrelation time constant and 120590

2

119886119899is the acceleration variance

describing maneuver intensity In a steady state

1205902

119908119899

= 21205721205902

119886119899 (18)

To describe SLV kinematics the model must cope with cru-cial nonzero mean acceleration maneuvers during the pro-pelled phase In addition after each stagersquos engine burns outa multiple model approach is applied to the coasting flightthat is one model describes PFP whereas the other depictsthe CFP Empirically tuned independent probability densityfunctions (PDFs) represented by TUM describe the accel-erations of the SLV in the local coordinate frame Figure 2shows the means and variances of the acceleration processesof the SLV in this paper The discrete-time model with statetransition matrix Ψ119899(120572119899 Δ119905) is as follows

[

[

[

119901119899119896+1

V119899119896+1119886119899119896+1

]

]

]

= Ψ119899 (120572119899 Δ119905)[

[

[

119901119899119896

V119899119896119886119899119896

]

]

]

+ 119908119899119896

Ψ119899 (120572119899 Δ119905) = 119890119865Δ119905

[

[

[

[

[

[

[

1 Δ119905 (120572119899Δ119905 minus 1 + 119890minus120572119899Δ119905

)

0 1

(1 minus 119890minus120572119899Δ119905

)

120572119899

0 0 119890minus120572119899Δ119905

]

]

]

]

]

]

]

119908119899119896 = int

(119896+1)Δ119905

119896Δ119905

119890119865[(119896+1)Δ119905minus119903]

119866119908119899 119889119903

(19)

In the case of the PFP 119908119899119896 is a nonzero mean white noisesequence caused by the nonzero mean acceleration 120583119886119899 seenin Figure 2 A nonzeromeanwhite noise sequence for the PFPshould be considered in the target kinematics when imple-menting the state propagation stage in theKFThus the deter-ministic input 119906119899119896 caused by 119908119899119896 along 119909-axis 119910-axis and119911-axis is derived as follows

119906119909119896 = 119864 [119908119909119896] = 119864int

(119896+1)Δ119905

119896Δ119905

119890119865[(119896+1)Δ119905minus119903]

119866119908119909119889119903

asymp 119864

int

(119896+1)Δ119905

119896Δ119905

[

[

[

[

119890minus120572119909[(119896+1)Δ119905minus119903]

0

0

]

]

]

]

119908119909 119889119903

119906119909119896 =

120583119886119909

120572119899 [1minus119890minus120572119909Δ119905

00

]

(20)

119910-axis and 119911-axis can be derived in the samemanner as shownin (20) The maneuver excitation covariance [10] whichrepresents the uncertainty of the SLV kinematics model is

119876119899119896 (120572119899 Δ119905)

= 119864 (119908119899119896 minus 119864 [119908119899119896]) (119908119899119896 minus 119864 [119908119899119896])119879

= 21205721198991205902

119886119899

[

[

[

11990211 11990212 11990213

11990212 11990222 11990223

11990213 11990223 11990233

]

]

]

(21)

The specific components of 11990211 11990233 are illustrated in [25]In addition the measurement matrix 119867119871 for Singerrsquos modelcan be depicted as

6 Mathematical Problems in Engineering

01

minus20 4minus6 0

0028

120583 =

1205902 =

p(ax)

minus6ms2

4246

a (m2s4)

m2s4

(a)

01

minus5 810

005

120583 = 1ms2

1205902 = 106

p(ay)

a (m2s4)

m2s4

(b)

01

minus40 4minus7 0

0008

a (m2s4)

120583 = minus7ms2

1205902 = 12674 m2s4

p(az)

(c)

01

005 005

minus4 40

008

120583 = 0ms2

1205902 = 183

p(ax) p(ay) p(az)

a (m2s4)

m2s4

(d)

Figure 2 Nonzero mean acceleration PDF in the PFP model (a) along 119909-axis (b) along 119910-axis and (c) along 119911-axis Zero mean accelerationPDF in the CFP Model (d) along 119909-axis 119910-axis and 119911-axis

119911119871119896 = 119867119871x119871119896 + ]119896 119867119871 =[

[

[

1 0 0

0 03times2 1 03times2 0 03times20 0 1

]

]

]

x119871119896 = [119909119896 119896 119896 119910119896 119896

119896

119911119896 119896 119896]119879 (22)

where ]119896 is measurement noise (as shown in (14)) with errorcovariance

411 IMM with Singerrsquos Linear Model From (19)ndash(22) inSection 41 we can rewrite the Markov jump linear systemswhere the 119894th model of the finite multiple model set M =

119898(1)

119898(119872)

obeys the following equations

x119871119896+1 = Ψ(119894)

119896x119871119896 + 119908

(119894)

119896

119911119871119896 = 119867119871x119871119896 + ]119896(23)

where

cov (119908(119894)

119896) = 119876

(119894)

119896

cov (]119896) = 119877119896

Ψ(119894)

119896= diag (Ψ119909119896 (120572

(119894)

119909 Δ119905) Ψ119910119896 (120572

(119894)

119910 Δ119905)

Ψ119911119896 (120572(119894)

119911 Δ119905))

119876(119894)

119896= diag (119876119909119896 (120572

(119894)

119909 Δ119905) 119876119910119896 (120572

(119894)

119910 Δ119905)

119876119911119896 (120572(119894)

119911 Δ119905))

(24)

Here 1205902

119886119899for the SLV can be represented by TUM as in

Figure 2 The PDFs of Figures 2(a)ndash2(d) are experimentallysampled from the nominal acceleration profile of the SLVThesuperscript (i) denotes quantities pertinent to model 119898(119894) inM and the jumps of the system mode are assumed to havetransition probabilities

Pr 119898(119895)

119896+1| 119898

(119894)

119896 ≜ 120587119894119895

(25)

where 119898(119894)

119896denotes the event in which model 119898

(119894) matchesthe system mode in effect at time 119896 In our application 119872 =

2 and 119898(1) and 119898

(2) are the propelled and coasting modesrespectively

Finally complete recursion of the IMM with modematched KF for the SLV tracking is summarized as follows

(i) Model-conditioned reinitialization (for 119894 = 1 2

119872)

Mathematical Problems in Engineering 7

(a) predicted mode probability 120583(119894)

119871119896|119896minus1≜ Pr119898(119894)

119896|

119911119871119896minus1 = sum119895120587119895119894120583

(119895)

119871119896minus1

(b) mixingweight 120583119895|119894

119871119896minus1≜ Pr119898(119895)

119896minus1| 119898

(119894)

119896 119911119871119896minus1 =

sum119895120587119895119894120583

(119895)

119871119896minus1120583

(119894)

119871119896minus1

(c) mixing estimate and covariance

x(119894)119871119896minus1|119896minus1

≜ 119864 [x119871119896minus1|119896minus1 | 119898(119894)

119896 119911119871119896minus1]

= sum

119895

x(119895)119871119896minus1|119896minus1

120583119895|119894

119871119896minus1

119875

(119894)

119871119896minus1|119896minus1= sum

119895

[119875(119894)

119871119896minus1|119896minus1+ (x(119894)

119871119896minus1|119896minus1minus x(119895)

119871119896minus1|119896minus1)

sdot (x(119894)119871119896minus1|119896minus1

minus x(119895)119871119896minus1|119896minus1

)

119879

] 120583119895|119894

119871119896minus1

(26)

(ii) Model-conditioned filtering (for 119894 = 1 2 119872)

(a) predicted estimate and covariance

x(119894)119871119896|119896minus1

= Ψ(119894)

119896minus1x(119894)119871119896minus1|119896minus1

+ u(119894)

119896minus1

where u(1)

119896minus1= [119906

119879

119909119896minus1119906119879

119910119896minus1119906119879

119911119896minus1]

119879

u(2)

119896minus1= 03times1

119875(119894)

119871119896|119896minus1= Ψ

(119894)

119896minus1119875

(119894)

119871119896minus1|119896minus1Ψ

(119894)119879

119896minus1+ 119876

(119894)

119896minus1

(27)

(b) measurement residual (119894)

119871119896= 119911119871119896 minus119867119871x

(119894)

119871119896|119896minus1minus

]119896(c) residual covariance 119878(119894)

119871119896= 119867119871119875

(119894)

119871119896|119896minus1119867

119879

119871minus 119877119896

(d) filter gain 119870(119894)

119871119896= 119875

(119894)

119871119896|119896minus1119867

119879

119871(119878

(119894)

119871119896)minus1

(e) update of state and covariance x(119894)119871119896|119896

= x(119894)119871119896|119896minus1

+

119870(119894)

119871119896(119894)

119871119896 119875

(119894)

119871119896|119896= 119875

(119894)

119871119896|119896minus1minus 119870

(119894)

119871119896119878(119894)

119871119896119870

(119894)119879

119871119896

(iii) Mode probability update (for 119894 = 1 2 119872)

(a) mode likelihood Λ(119894)

119871119896≜ 119901[119911

(119894)

119871119896| 119898

(119894)

119896

119885119871119896minus1]assume

= N(119911(119894)

119871119896 0 119878

(119894)

119871119896)

(b) mode probability 120583(119894)

119871119896= 120583

(119894)

119871119896|119896minus1Λ

(119894)

119871119896

(sum119895120583(119895)

119871119896|119896minus1Λ

(119895)

119871119896)

(iv) Combination (for 119894 = 1 2 119872)

x119871119896|119896 = sum

119894

x(119894)119871119896|119896

120583(119894)

119871119896

(119894)

119871119896|119896

= sum

119894

[119875(119894)

119871119896|119896+ (x119871119896|119896 minus x(119894)

119871119896|119896) (x119871119896|119896 minus x(119894)

119871119896|119896)

119879

] 120583(119894)

119871119896

(28)

Singerrsquos MMS of time delayed slaving data is completed bypropagating the combined estimate based on the currentmodersquos dynamic model such that

xsync119871119896+119904|119896

= Φsync119896+119904119896

x119871119896|119896

Φsync119896+119904119896

= 119865sync119896+119904minus1

119865sync119896+119904minus2

sdot sdot sdot 119865sync119896

Φsync119896119896

= 119868119899

119896 gt 1

(29)

where 119865sync119896+119904minus1

119865sync119896+119904minus2

119865sync119896

are systemmatrices dependingon a flight phase mode at current time 119896

42 Synchronization Using an IMM with a Nonlinear BallisticModel For a nonlinear ballistic model the state vector forthe propelled mode is denoted as

x119905 = [119909119905 119910119905 119911119905 119905 119905

119905 120585119905 120591119905]119879 (30)

where 120585119905 is the drag coefficient and 120591119905 is the thrust Generallythe drag coefficient varies significantlywith theMachnumberregime subsonic transonic and supersonic Therefore wetake advantage of the dynamic model considering a Machnumber-dependent multiplier [11 12] such that

[

[

[

]

]

]

=

120591

119881

[

[

[

]

]

]

thrust term

+ 120585120585119898119863[

[

[

]

]

]

drag term

+ 119892[

[

[

0

0

1

]

]

]

gravity term

+ 1

120585 = 2

= 3

(31)

The first term on the right side of (31) represents the thrust(m2

s) of the SLV in 119909 119910 and 119911 directions Two distinctmultiplemodes of a nonlinear SLVmodel can be separated bythe existence of thrust In other words the existence of thrustsignifies propelled flight mode whereas zero thrust signifiescoasting flight mode 119881 is the magnitude of the velocity V =

[ ]

119879 that is the SLV speed (ms) The second part of(31) is the drag term which is related to velocity and altitudethat is 119863 = minus120588(119911)1198812 where 120588(119911) = 1205880119890

minus119888119911 is the air density(kgm3) at an altitude 119911 (m) and 119888 is the air density constant(mminus1) [9] 120585 is the drag coefficient and 120585119898 is theMachnumber-dependent drag coefficient multiplier which is approximatedby the cubic spline curve shown in Figure 3 In this paperfor the drag characteristics of the SLV which are applied inthe subsequent simulation section drag coefficients of theSaturn119881 launch vehicle [15] are usedThe third part of (31) isa gravity term Gravity 119892 is the standard acceleration due togravity at sea level which is assumed to be constant through-out the trajectory with a value of 9812 ms2 1 2 and 3

are assumed to be continuous time zeromeanwhite Gaussiannoises The drag coefficient and thrust acceleration are rep-resented as Wiener processes with a slow variation [11 12]

8 Mathematical Problems in Engineering

0 2 4 6 8 1004

06

08

1

12

14

16

18

2

Mach number

Nor

mal

ized

dra

g co

effici

ent

Selected pointsLinear interpolationCubic spline interpolation

Figure 3 Normalized drag coefficient [15]

We can modify the dynamic equations (30) and (31) as acompact form such that

x119905 = 119891 [x119905] + 119905 (32)

where

119891 [x119905] =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119905

119905

119905

120591

119905

119881119905

+ 120585119905119863119905119905

120591

119905

119881119905

+ 120585119905119863119905119905

120591

119905

119881119905

+ 120585119905119863119905119905 minus 119892

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

119905 = [1119905 2119905 3119905]119879

(33)

The state vector equation (33) is discretized by a second-order Taylor expansion [26] Then (33) can be written as

a discretized continuous time systemwithwhite process noisesuch that

x119896+1 = x119896 + 119891 [x119896] Δ119905 + 119860119896119891 [x119896]Δ119905

2

2

+ 120596119896(34)

where 119860119896 is the Jacobian of (33) evaluated at x119896 [26] and120596119896 is the discretized continuous time process noise for thesampling interval Δ119905 The corresponding covariance matrixof the discretized process noise is

119876 =[

[

[

1198761119902V 06times1 06times101times6 Δ119905119902120585 0

01times6 0 Δ119905119902120591

]

]

]

1198761 =

[

[

[

[

Δ1199053

3

1198683

Δ1199052

2

1198683

Δ1199052

2

1198683 Δ1199051198683

]

]

]

]

(35)

where 1198683 is the 3 times 3 identity matrix and the continuous timeprocess noise intensities 119902V 119902120585 and 119902120591 are the correspondingpower spectral densities

The measurement matrix 119867NL for the nonlinear ballisticmultiple model can be depicted as

119911NL119896 = 119867NLxNL119896 + ]119896

119867NL = [1198683 0] xNL119896 = [119909119896 119910119896 119911119896 119896 119896

119896 120585119896 120591119896]119879

(36)

where ]119896 is measurement noise with error covariance119877119896120575119896minus119895 = 119864[]119896]

119879

119895] that is (14)

An IMMalgorithm for nonlinear dynamics with differentsizes of the mode state vector is summarized as follows [25]

(v) Model-conditioned reinitialization (for 119894 = 1 2

119872)

(a) predicted mode probability 120583(119894)

NL119896|119896minus1 ≜

Pr119898(119894)

119896| 119911NL119896minus1 = sum

119895120587119895119894120583

(119895)

NL119896minus1(b) mixing weight

120583119895|119894

NL119896minus1 ≜ Pr 119898(119895)

119896minus1| 119898

(119894)

119896 119911NL119896minus1 =

sum119895120587119895119894120583

(119895)

NL119896minus1

120583(119894)

NL119896minus1

(37)

(c) unbiased mixing estimate and covariance

x(119894)NL119896minus1|119896minus1 ≜ 119864 [xNL119896minus1|119896minus1 | 119898(119894)

119896 119911NL119896minus1] = sum

119895

(119895)

NL119896minus1|119896minus1120583119895|119894

NL119896minus1

where (1)

NL119896minus1|119896minus1 ≜ x(1)NL119896minus1|119896minus1 (2)

NL119896minus1|119896minus1 ≜ [x(2)119879

NL119896minus1|119896minus1 120591119896minus1]119879

119875

(119894)

NL119896minus1|119896minus1 = sum

119895

[119875(119894)

NL119896minus1|119896minus1 + (x(119894)NL119896minus1|119896minus1 minus (119895)

NL119896minus1|119896minus1) (x(119894)NL119896minus1|119896minus1 minus (119895)

NL119896minus1|119896minus1)119879

] 120583119895|119894

NL119896minus1

(38)

Mathematical Problems in Engineering 9

(vi) Model-conditioned filtering (for 119894 = 1 2 119872)

(a) predicted estimate and covariance

x(119894)NL119896|119896minus1 = x(119894)NL119896minus1|119896minus1 + 119891 [x(119894)NL119896minus1|119896minus1] Δ119905

+

(119894)

NL119896minus1119891 [x(119894)NL119896minus1|119896minus1]Δ119905

2

2

(119894)

119896=

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x=x(119894)119896

119875(119894)

NL119896|119896minus1 = Ω(119894)

119896minus1119875

(119894)

NL119896minus1|119896minus1Ω(119894)119879

119896minus1+ 119876119896minus1

Ω(119894)

119896minus1= 119868 +

(119894)

NL119896minus1

(39)

(b) measurement residual (119894)

NL119896 = 119911NL119896 minus

119867NLx(119894)

NL119896|119896minus1 minus ]119896

(c) residual covariance 119878(119894)NL119896 = 119867NL119875(119894)

NL119896|119896minus1119867119879

NL minus

119877119896(d) filter gain 119870(119894)

NL119896 = 119875(119894)

NL119896|119896minus1119867119879

NL(119878(119894)

NL119896)minus1

(e) update of state and covariance

x(119894)NL119896|119896 = x(119894)NL119896|119896minus1 + 119870(119894)

NL119896(119894)

NL119896

119875(119894)

NL119896|119896 = 119875(119894)

NL119896|119896minus1 minus 119870(119894)

NL119896119878(119894)

NL119896119870(119894)119879

NL119896

(40)

(vii) Mode probability update (for 119894 = 1 2 119872)

(a) mode likelihood

Λ(119894)

NL119896 ≜ 119901 [119911(119894)

NL119896 | 119898(119894)

119896 119885NL119896minus1]

assume= N (119911

(119894)

NL119896 0 119878(119894)

NL119896)

(41)

(b) mode probability 120583(119894)

NL119896 = 120583(119894)

NL119896|119896minus1Λ(119894)

NL119896

(sum119895120583(119895)

NL119896|119896minus1Λ(119895)

NL119896)

(viii) Combination (for 119894 = 1 2 119872)

xNL119896|119896 = sum

119894

x(119894)NL119896|119896120583(119894)

NL119896

(119894)

NL119896|119896 = sum

119894

[119875(119894)

NL119896|119896

+ (xNL119896|119896 minus x(119894)NL119896|119896) (xNL119896|119896 minus x(119894)NL119896|119896)119879

] 120583(119894)

NL119896

(42)

The nonlinearMMS of time delayed slaving data is completedby propagating the combined estimate based on the currentmodersquos estimated vector such that

xNL119896+119904|119896 = xNL119896|119896 + 119891 [xNL119896|119896] sdot 119904 + NL119896119891 [xNL119896|119896]

sdot

1199042

2

(43)

where 119904 is a lead-time for synchronization (119904 = 119899 sdot Δ119905) If theSLV is in the PFP (120583(1)

NL119896 gt 120583(2)

NL119896) the current state estimateincludes the thrust term that is the state vector xNL119896 =

[119909119896 119910119896 119911119896 119896 119896

119896 120585119896 120591119896]119879 If the SLV is not in the PFP

the state vector does not include the thrust term (120591119896 = 0)

5 Simulation Results

To demonstrate the performance of the proposed synchro-nization approaches for delayed slaving data we simulatedthe SLV tracking problem based on the nominal flight tra-jectory of the Korea Space Launch Vehicle-I (KSLV-I) In thesimulation the radar measurement noise intensities of (12)are selected as 120590119903 = 15m 120590120593 = 001 deg and 120590120579 = 001 degThe nominal flight sequence of KSLV-I is as follows First thepayload fairing separates during the first stage flight at 2154 sAfter the first stage engine shutdown at 2287 s the upperstage separates from the first stage and enters the CFP Thesecond stage continues in the CFP until the kick motor igni-tion at 395 s and the vehicle then enters the PFP At the endof kickmotor combustion (4527 s) the upper stage enters thetarget orbit in CFP as in the previous separation Finally thesatellite is inserted into the target orbit after it separates fromthe upper stage during CFP at 540 s Since delayed slavingdata from the MCS to the TSs can occur in both the PFP andCFP synchronization simulations are conducted at arbitrarypoints of the flight phases Figure 4 shows the synchroniza-tion error of the delayed slaving data with respect to thebenchmark and the proposed approaches for delays of 119904 =

01 02 1 s Figures 4(a)ndash4(c) illustrate synchronizationerrors where delayed slaving data occurred at 155 s which isthe first stage of PFP and Figures 4(d)ndash4(f) show synchro-nization errors where delayed slaving data occurred at 280 swhich is the first stage of CFP As shown in Figures 4(a)ndash4(c)the extrapolation and120572-120573-120574filter-based approach cannot findthe proper synchronized slaving data The synchronizationerrors of these approaches exponentially increase accordingto increasing delay 119904 in particular the error of the extrapo-lation approaches an out-of-figure bound (2500m) On theother hand the Kalman prediction and the proposedapproaches have stable synchronization errors even if thedelay time 119904 is larger Furthermore we can observe thatthe errors in the synchronization approach of the Kalmanprediction are relatively larger than the proposed approachesespecially in the PFP As explained in the flight sequence ofKSLV-I in general the motion of the SLV is described bymultiple dynamics (in our case PFP and CFP) rather thansingle dynamics However the dynamic model of the KF andprediction capability used in this paper is the constant accel-eration (CA) model This means that the filter is optimizedwith respect to CAmotion which is actually not occurring inour application This is why the performance of the Kalmanprediction-based synchronization is worse than the proposedapproaches Nonetheless we can see that the 119910-axis synchro-nization errors shown in Figures 4(b) and 4(e) are smallThis result is observed because the 119910-axis motion of KSLV-I in a local coordinate is relatively smaller than the otheraxis motion and the CA model approximately expresses this

10 Mathematical Problems in Engineering

01 02 03 04 05 06 07 08 09 10

500

1000

1500

2000

2500

Delay (sec)

x-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

Cubic spline extrapolation

Kalman filter

(a)

(b)

(c)

(d)

(e)

(f)

Linear MMNonlinear MM120572-120573-120574 filter 120572-120573-120574 filter

120572-120573-120574 filter120572-120573-120574 filter

120572-120573-120574 filter 120572-120573-120574 filter

01 02 03 04 05 06 07 08 09 1Delay (sec)

0

500

1000

1500

2000

2500

y-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

z-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

x-p

ositi

on ab

solu

te er

ror i

n CF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500y

-pos

ition

abso

lute

erro

r in

CFP

(m)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

z-p

ositi

on ab

solu

te er

ror i

n CF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

Figure 4 Synchronization errors at PFP and CFP

Mathematical Problems in Engineering 11

small motion A single KF prediction model cannot exactlydescribe the motion of both PFP and CFP whereas theproposed multiple model-based approaches work well Asshown in Figure 4 regardless of the delay 119904 the proposedmultiplemodel-based approaches find synchronized positionvectors In the comparison of synchronization performancebetween linear IMM and nonlinear IMM synchronizationthe nonlinear IMM-based synchronization approach showsthe best performance except for 119910-axis where CA motion isdominant In addition the difference between the proposedmultiple model-based approaches is very small as shown inFigure 4 but the complexities of the algorithms for real-timeapplications are quite dissimilar Hence the operator mayadaptively select one of the proposed approaches accordingto onersquos environment

6 Conclusions

In this paper we investigated the time synchronizationapproaches of delayed slaving data in the RSS for SLV track-ingOne of themost important roles of the RSS is to distributeslaving data to each TS for continuous tracking of the SLV Ifthere is a critical network delay resulting in time delayed slav-ing data being sent to each TS theMCS will not receive accu-rate SLV tracking data This problem can give rise to signifi-cant difficulties for the SLVmission progress and analysis Toovercome this problemwe proposedMMSapproacheswhichtake advantage of the multiple motion models of an SLVThelinear IMM-based synchronization approach was developedusing Singerrsquos model with ternary uniform mixtures and thenonlinear IMM-based synchronization approachwas derivedfrom a nonlinear ballistic model with a drag coefficient Forverification of the proposed algorithms SLV tracking simu-lations using KSLV-I and the radar measurement data gen-erated from nominal trajectory were conducted To demon-strate the superiority of time synchronization performancein these simulations we compared the proposed algorithmwith benchmark approaches for absolute error between thenominal trajectory data and the synchronized slaving datathe simulation results demonstrated that the proposed MMSapproaches performed competitively

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] R Varaprasad and V Seshagiri Rao ldquoRange safety real-timesystem for satellite launch vehicle missions-testing methodolo-giesrdquo Defence Science Journal vol 56 no 5 pp 693ndash700 2006

[2] H R Song M G Jeon T S Choi and V Shin ldquoTwo fusionpredictors for discrete-time linear systems with different typesof observationsrdquo International Journal of Control Automationand Systems vol 7 no 4 pp 651ndash658 2009

[3] H Song V Shin and M Jeon ldquoMobile node localization usingfusion prediction-based interacting multiple model in cricketsensor networkrdquo IEEE Transactions on Industrial Electronicsvol 59 no 11 pp 4349ndash4359 2012

[4] T Kirubarajan and Y Bar-Shalom ldquoKalman filter versus IMMestimator when do we need the latterrdquo IEEE Transactions onAerospace and Electronic Systems vol 39 no 4 pp 1452ndash14572003

[5] E Mazor A Averbuch Y Bar-Shalom and J Dayan ldquoInteract-ing multiple model methods in target tracking a surveyrdquo IEEETransactions on Aerospace and Electronic Systems vol 34 no 1pp 103ndash123 1998

[6] P D Hanlon and P S Maybeck ldquoMultiple-model adaptiveestimation using a residual correlation Kalman filter bankrdquoIEEE Transactions on Aerospace and Electronic Systems vol 36no 2 pp 393ndash406 2000

[7] S-J Shin ldquoRe-entry vehicle tracking with a new multiplemodel estimation applicable to highly non-linear dynamicsrdquoIET Radar Sonar amp Navigation vol 9 no 5 pp 581ndash588 2015

[8] S Conover J Kerce G Brown L Ehrman and D HardimanldquoImpact point prediction of small ballistic munitions with aninteracting multiple model estimatorrdquo in Acquisition TrackingPointing and Laser Systems Technologies XXI vol 6569 ofProceedings of SPIE Orlando Fla USA 2007

[9] V C Ravindra Y Bar-Shalom and P Willett ldquoProjectileidentification and impact point predictionrdquo IEEE Transactionson Aerospace and Electronic Systems vol 46 no 4 pp 2004ndash2021 2010

[10] W J Farrell III ldquoInteracting multiple model filter for tacticalballistic missile trackingrdquo IEEE Transactions on Aerospace andElectronic Systems vol 44 no 2 pp 418ndash426 2008

[11] T Yuan Y Bar-Shalom PWillett E Mozeson S Pollak and DHardiman ldquoA multiple IMM estimation approach with unbi-ased mixing for thrusting projectilesrdquo IEEE Transactions onAerospace and Electronic Systems vol 48 no 4 pp 3250ndash32672012

[12] T Yuan Y Bar-Shalom P Willett and D Hardiman ldquoImpactpoint prediction for thrusting projectiles in the presence ofwindrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 50 no 1 pp 102ndash119 2014

[13] X Li and V Jilkov ldquoSurvey of maneuvering target trackingpart II motion models of ballistic and space targetsrdquo IEEETransactions on Aerospace and Electronic Systems vol 46 no1 pp 96ndash119 2010

[14] J Cesar Bolzani de Campos Ferreira and JWaldmann ldquoCovari-ance intersection-based sensor fusion for sounding rockettracking and impact area predictionrdquo Control Engineering Prac-tice vol 15 no 4 pp 389ndash409 2007

[15] httpwwwbraeunigusapollosaturnVhtm[16] RA Singer ldquoEstimating optimal tracking filter performance for

manned maneuvering targetsrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 6 no 4 pp 473ndash483 1970

[17] R A Singer and KW Behnke ldquoReal-time tracking filter evalu-ation and selection for tactical applicationsrdquo IEEE Transactionson Aerospace and Electronic Systems vol AES-7 no 1 pp 100ndash110 1971

[18] W Press B Flannery S Teukolsky and W Vetterling Numer-ical Recipes in FORTRAN The Art of Scientific ComputingCambridge University New York NY USA 2nd edition 1992

[19] S Dyer and J Dyer ldquoCubic-spline interpolation part Irdquo IEEEInstrumentation amp Measurement Magazine pp 44ndash46 2001

[20] Y Bar-Shalom X Li and T Kirubarajan Estimation with Appli-cations to Tracking and Navigation John Wiley amp Sons NewYork NY USA 2001

12 Mathematical Problems in Engineering

[21] N Morrison Introduction to Sequential Smoothing and Predic-tion McGraw-Hill 1969

[22] E Brookner Tracking and Kalman Filtering Made Easy JohnWiley amp Sons 1998

[23] P Suchomski ldquoExplicit expressions for debiased statistics of 3Dconverted measurementsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 35 no 1 pp 368ndash370 1999

[24] H R Song I Y Song and V Shin ldquoMultisensory predictionfusion of nonlinear functions of the state vector in discrete-timesystemsrdquo International Journal of Distributed Sensor Networksvol 2015 Article ID 249857 13 pages 2015

[25] H Song andYHan ldquoComparison of space launch vehicle track-ing using different types of multiple modelsrdquo in Proceedingsof the International Conference on Information Fusion pp 1ndash8Heidelberg Germany July 2016

[26] R KMehra ldquoA comparison of several nonlinear filters for reen-try vehicle trackingrdquo IEEE Transactions on Automatic Controlvol 16 no 4 pp 307ndash319 1971

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Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

the VRFs During the early tracking phase the EMF isselected as the final estimate however at a certain timewhereVRF119865 is larger than VRF119864 FMF is selected as the finalestimate

120572120573120574

119899119896= 119891119899

120572120573120574

119899119896minus1+ 119870[119910119896 minus 119867

120572120573120574

120572120573120574

119899119896minus1]

[

[

[

119901119899119896

V119899119896119886119899119896

]

]

]

=

[

[

[

[

[

1 Δ119905

Δ1199052

2

0 1 Δ119905

0 0 1

]

]

]

]

]

[

[

[

119901119899119896minus1

V119899119896minus1119886119899119896minus1

]

]

]

+

[

[

[

[

[

[

120572

120573

Δ119905

120574

Δ1199052

]

]

]

]

]

]

[

[

[

119910119896 minus 119867120572120573120574 [

[

[

119901119899119896minus1

V119899119896minus1119886119899119896minus1

]

]

]

]

]

]

where 119883120572120573120574

119899119896= [119901119899119896 V119899119896 119886119899119896]

119879 119867

120572120573120574= [1 0 0]

VRF119864ge VRF119865

997888rarr

119901119899119896 = 119901119864

119899119896

V119899119896 = V119864119899119896

119886119899119896 = 119886119864

119899119896

VRF119865gt VRF119864

997888rarr

119901119899119896 = 119901119865

119899119896

V119899119896 = V119865119899119896

119886119899119896 = 119886119865

119899119896

(7)

Finally synchronization using an 120572-120573-120574 filter is completed bylinear propagation using the system matrix 119891119899 such that

120572120573120574

119899119896+119904= (

119904

prod

119871=1

119891119899) sdot

120572120573120574

119899119896 (8)

33 Kalman Predictor The motion of the SLV is simplydepicted as a discretized Wiener process acceleration model[20]

119883119896+1 = 119865119883119896 + 119908119896 (9)

where the state vector 119883119896 isin R9 consists of theposition velocity and acceleration components along 119909-axis 119910-axis and 119911-axis respectively that is 119883119896 =

[119909119901119896 119909V119896 119909119886119896 119910119901119896 119910V119896 119910119886119896 119911119901119896 119911V119896 119911119886119896]119879 The system

matrix and covariance matrix of the system noise 119908119896 arerepresented as (10) and (11) respectively

119891119899=119909119910119911 =

[

[

[

[

[

1 Δ119905

Δ1199052

2

0 1 Δ119905

0 0 1

]

]

]

]

]

119865 =[

[

[

119891119909 03 0303 119891119910 0303 03 119891119911

]

]

]

(10)

119864 [119908119896119908119879

119897] = 119876120575119896minus119897

119902119871=119909119910119911 =

[

[

[

[

[

[

[

[

[

Δ1199055

20

Δ1199054

8

Δ1199052

6

Δ1199054

8

Δ1199053

3

Δ1199052

2

Δ1199053

6

Δ1199052

2

Δ119905

]

]

]

]

]

]

]

]

]

119876 =[

[

[

119902119909 03 0303 119902119910 0303 03 119902119911

]

]

]

(11)

A radar measurement for the SLV gives the spherical coordi-nate observations such that

119885119896 = ℎ (119883119896) + V119904119896

119885119896 isin R3 V119904119896 isin R

3

119885119896 =[

[

[

119903119896

120593119896

120579119896

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

radic1199092

119901119896+ 119910

2

119901119896+ 119911

2

119901119896

tanminus1(

119910119901119896

119909119901119896

)

tanminus1(

119910119901119896

radic1199092

119901119896+ 119910

2

119901119896

)

]

]

]

]

]

]

]

]

]

]

]

+[

[

[

V119903V120593V120579

]

]

]

(12)

where V119904119896 sim N(0 119877119904119896) and 119877119904119896 = diag(1205902

119903 120590

2

120593 120590

2

120579) Using a 3D

debiased converted measurement [23] we can transform theoriginal nonlinear equations (12) into their linear form as

z119896 = 119867119883119896 + V119896 V119896 isin R3

z119896 =[

[

[

119909119896

119910119896

119911119896

]

]

]

=[

[

[

1 01times5 01times301times3 1 01times501times5 1 01times2

]

]

]

119883119896 +[

[

[

V119909119896V119910119896V119911119896

]

]

]

(13)

Here V119888119896 is the converted measurement noise expressed interms of Cartesian coordinates that is V119896 sim N(0 119877119896)

119877119896 =

[

[

[

[

119877119909119909119896 119877119909119910119896 119877119909119911119896

119877119910119909119896 119877119910119910119896 119877119910119911119896

119877119911119909119896 119877119911119910119896 119877119911119911119896

]

]

]

]

(14)

Mathematical Problems in Engineering 5

Prediction or fixed-lead prediction inmean squaremeans thesynchronization of the slaving data is the estimation of thestate at a future time 119896+119904 where 119904 gt 0 beyond the observationinterval that is based on data up to an earlier time [20 24]

119896+119904|119896 = 119864 (119883119896+119904|119896 | 119885119896) 119885119896 = z0 z119896 (15)

The optimal predictor or synchronized state 119896+119904|119896 ≜

KP119896+119904|119896

and its error covariance 119875119896+119904|119896 ≜ 119875KP119896+119904|119896

are given by theKalman predictor equations [2 24]

KP119896+119904|119896

= 119865119896+119904minus1

KP119896+119904minus1|119896

= sdot sdot sdot = Φ119896+119904119896

KP119896|119896

KP119896|119896

≜

KF119896|119896

119875KP119896+119904|119896

= 119865119896+119904minus1119875KP119896+119904minus1|119896

119865119879

119896+119904minus1+ 119876119896+119904minus1

= Φ119896+119904119896119875KP119896|119896

Φ119879

119896+119904119896

+

119904minus1

sum

119895=0

Φ119896+119904119896+119895+1119876119896+119895Φ119879

119896+119904119896+119895+1 119875

KP119896|119896

≜ 119875KF119896|119896

Φ119896+119904119896 = 119865119896+119904minus1119865119896+119904minus2 sdot sdot sdot 119865119896

Φ119896119896 = 119868119899

119896 gt 1

(16)

where

KF119896|119896

and 119875KF119896|119896

are the KF estimate and covariancerespectively

4 Proposed MME-Based SynchronizationApproaches

41 SynchronizationUsing an IMMwith Singerrsquos LinearModelSinger [16] described 2D manned maneuvering targets inrange-bearing coordinates This model can be adapted to theSLV kinematics in 3D Cartesian coordinates with multipleflight phases

[

[

[

119899

V119899119899

]

]

]

= 119865[

[

[

119901119899

V119899119886119899

]

]

]

+ 119866119908119899

119865 =[

[

[

0 1 0

0 0 1

0 0 minus120572119899

]

]

]

119866 =[

[

[

0

0

0

]

]

]

(17)

where119908119899 isin R3times1 is a white noise process along the Cartesianaxis 119899 = 119909 119910 119911 The parameter 120573 = 1120572119899 is the maneuvercorrelation time constant and 120590

2

119886119899is the acceleration variance

describing maneuver intensity In a steady state

1205902

119908119899

= 21205721205902

119886119899 (18)

To describe SLV kinematics the model must cope with cru-cial nonzero mean acceleration maneuvers during the pro-pelled phase In addition after each stagersquos engine burns outa multiple model approach is applied to the coasting flightthat is one model describes PFP whereas the other depictsthe CFP Empirically tuned independent probability densityfunctions (PDFs) represented by TUM describe the accel-erations of the SLV in the local coordinate frame Figure 2shows the means and variances of the acceleration processesof the SLV in this paper The discrete-time model with statetransition matrix Ψ119899(120572119899 Δ119905) is as follows

[

[

[

119901119899119896+1

V119899119896+1119886119899119896+1

]

]

]

= Ψ119899 (120572119899 Δ119905)[

[

[

119901119899119896

V119899119896119886119899119896

]

]

]

+ 119908119899119896

Ψ119899 (120572119899 Δ119905) = 119890119865Δ119905

[

[

[

[

[

[

[

1 Δ119905 (120572119899Δ119905 minus 1 + 119890minus120572119899Δ119905

)

0 1

(1 minus 119890minus120572119899Δ119905

)

120572119899

0 0 119890minus120572119899Δ119905

]

]

]

]

]

]

]

119908119899119896 = int

(119896+1)Δ119905

119896Δ119905

119890119865[(119896+1)Δ119905minus119903]

119866119908119899 119889119903

(19)

In the case of the PFP 119908119899119896 is a nonzero mean white noisesequence caused by the nonzero mean acceleration 120583119886119899 seenin Figure 2 A nonzeromeanwhite noise sequence for the PFPshould be considered in the target kinematics when imple-menting the state propagation stage in theKFThus the deter-ministic input 119906119899119896 caused by 119908119899119896 along 119909-axis 119910-axis and119911-axis is derived as follows

119906119909119896 = 119864 [119908119909119896] = 119864int

(119896+1)Δ119905

119896Δ119905

119890119865[(119896+1)Δ119905minus119903]

119866119908119909119889119903

asymp 119864

int

(119896+1)Δ119905

119896Δ119905

[

[

[

[

119890minus120572119909[(119896+1)Δ119905minus119903]

0

0

]

]

]

]

119908119909 119889119903

119906119909119896 =

120583119886119909

120572119899 [1minus119890minus120572119909Δ119905

00

]

(20)

119910-axis and 119911-axis can be derived in the samemanner as shownin (20) The maneuver excitation covariance [10] whichrepresents the uncertainty of the SLV kinematics model is

119876119899119896 (120572119899 Δ119905)

= 119864 (119908119899119896 minus 119864 [119908119899119896]) (119908119899119896 minus 119864 [119908119899119896])119879

= 21205721198991205902

119886119899

[

[

[

11990211 11990212 11990213

11990212 11990222 11990223

11990213 11990223 11990233

]

]

]

(21)

The specific components of 11990211 11990233 are illustrated in [25]In addition the measurement matrix 119867119871 for Singerrsquos modelcan be depicted as

6 Mathematical Problems in Engineering

01

minus20 4minus6 0

0028

120583 =

1205902 =

p(ax)

minus6ms2

4246

a (m2s4)

m2s4

(a)

01

minus5 810

005

120583 = 1ms2

1205902 = 106

p(ay)

a (m2s4)

m2s4

(b)

01

minus40 4minus7 0

0008

a (m2s4)

120583 = minus7ms2

1205902 = 12674 m2s4

p(az)

(c)

01

005 005

minus4 40

008

120583 = 0ms2

1205902 = 183

p(ax) p(ay) p(az)

a (m2s4)

m2s4

(d)

Figure 2 Nonzero mean acceleration PDF in the PFP model (a) along 119909-axis (b) along 119910-axis and (c) along 119911-axis Zero mean accelerationPDF in the CFP Model (d) along 119909-axis 119910-axis and 119911-axis

119911119871119896 = 119867119871x119871119896 + ]119896 119867119871 =[

[

[

1 0 0

0 03times2 1 03times2 0 03times20 0 1

]

]

]

x119871119896 = [119909119896 119896 119896 119910119896 119896

119896

119911119896 119896 119896]119879 (22)

where ]119896 is measurement noise (as shown in (14)) with errorcovariance

411 IMM with Singerrsquos Linear Model From (19)ndash(22) inSection 41 we can rewrite the Markov jump linear systemswhere the 119894th model of the finite multiple model set M =

119898(1)

119898(119872)

obeys the following equations

x119871119896+1 = Ψ(119894)

119896x119871119896 + 119908

(119894)

119896

119911119871119896 = 119867119871x119871119896 + ]119896(23)

where

cov (119908(119894)

119896) = 119876

(119894)

119896

cov (]119896) = 119877119896

Ψ(119894)

119896= diag (Ψ119909119896 (120572

(119894)

119909 Δ119905) Ψ119910119896 (120572

(119894)

119910 Δ119905)

Ψ119911119896 (120572(119894)

119911 Δ119905))

119876(119894)

119896= diag (119876119909119896 (120572

(119894)

119909 Δ119905) 119876119910119896 (120572

(119894)

119910 Δ119905)

119876119911119896 (120572(119894)

119911 Δ119905))

(24)

Here 1205902

119886119899for the SLV can be represented by TUM as in

Figure 2 The PDFs of Figures 2(a)ndash2(d) are experimentallysampled from the nominal acceleration profile of the SLVThesuperscript (i) denotes quantities pertinent to model 119898(119894) inM and the jumps of the system mode are assumed to havetransition probabilities

Pr 119898(119895)

119896+1| 119898

(119894)

119896 ≜ 120587119894119895

(25)

where 119898(119894)

119896denotes the event in which model 119898

(119894) matchesthe system mode in effect at time 119896 In our application 119872 =

2 and 119898(1) and 119898

(2) are the propelled and coasting modesrespectively

Finally complete recursion of the IMM with modematched KF for the SLV tracking is summarized as follows

(i) Model-conditioned reinitialization (for 119894 = 1 2

119872)

Mathematical Problems in Engineering 7

(a) predicted mode probability 120583(119894)

119871119896|119896minus1≜ Pr119898(119894)

119896|

119911119871119896minus1 = sum119895120587119895119894120583

(119895)

119871119896minus1

(b) mixingweight 120583119895|119894

119871119896minus1≜ Pr119898(119895)

119896minus1| 119898

(119894)

119896 119911119871119896minus1 =

sum119895120587119895119894120583

(119895)

119871119896minus1120583

(119894)

119871119896minus1

(c) mixing estimate and covariance

x(119894)119871119896minus1|119896minus1

≜ 119864 [x119871119896minus1|119896minus1 | 119898(119894)

119896 119911119871119896minus1]

= sum

119895

x(119895)119871119896minus1|119896minus1

120583119895|119894

119871119896minus1

119875

(119894)

119871119896minus1|119896minus1= sum

119895

[119875(119894)

119871119896minus1|119896minus1+ (x(119894)

119871119896minus1|119896minus1minus x(119895)

119871119896minus1|119896minus1)

sdot (x(119894)119871119896minus1|119896minus1

minus x(119895)119871119896minus1|119896minus1

)

119879

] 120583119895|119894

119871119896minus1

(26)

(ii) Model-conditioned filtering (for 119894 = 1 2 119872)

(a) predicted estimate and covariance

x(119894)119871119896|119896minus1

= Ψ(119894)

119896minus1x(119894)119871119896minus1|119896minus1

+ u(119894)

119896minus1

where u(1)

119896minus1= [119906

119879

119909119896minus1119906119879

119910119896minus1119906119879

119911119896minus1]

119879

u(2)

119896minus1= 03times1

119875(119894)

119871119896|119896minus1= Ψ

(119894)

119896minus1119875

(119894)

119871119896minus1|119896minus1Ψ

(119894)119879

119896minus1+ 119876

(119894)

119896minus1

(27)

(b) measurement residual (119894)

119871119896= 119911119871119896 minus119867119871x

(119894)

119871119896|119896minus1minus

]119896(c) residual covariance 119878(119894)

119871119896= 119867119871119875

(119894)

119871119896|119896minus1119867

119879

119871minus 119877119896

(d) filter gain 119870(119894)

119871119896= 119875

(119894)

119871119896|119896minus1119867

119879

119871(119878

(119894)

119871119896)minus1

(e) update of state and covariance x(119894)119871119896|119896

= x(119894)119871119896|119896minus1

+

119870(119894)

119871119896(119894)

119871119896 119875

(119894)

119871119896|119896= 119875

(119894)

119871119896|119896minus1minus 119870

(119894)

119871119896119878(119894)

119871119896119870

(119894)119879

119871119896

(iii) Mode probability update (for 119894 = 1 2 119872)

(a) mode likelihood Λ(119894)

119871119896≜ 119901[119911

(119894)

119871119896| 119898

(119894)

119896

119885119871119896minus1]assume

= N(119911(119894)

119871119896 0 119878

(119894)

119871119896)

(b) mode probability 120583(119894)

119871119896= 120583

(119894)

119871119896|119896minus1Λ

(119894)

119871119896

(sum119895120583(119895)

119871119896|119896minus1Λ

(119895)

119871119896)

(iv) Combination (for 119894 = 1 2 119872)

x119871119896|119896 = sum

119894

x(119894)119871119896|119896

120583(119894)

119871119896

(119894)

119871119896|119896

= sum

119894

[119875(119894)

119871119896|119896+ (x119871119896|119896 minus x(119894)

119871119896|119896) (x119871119896|119896 minus x(119894)

119871119896|119896)

119879

] 120583(119894)

119871119896

(28)

Singerrsquos MMS of time delayed slaving data is completed bypropagating the combined estimate based on the currentmodersquos dynamic model such that

xsync119871119896+119904|119896

= Φsync119896+119904119896

x119871119896|119896

Φsync119896+119904119896

= 119865sync119896+119904minus1

119865sync119896+119904minus2

sdot sdot sdot 119865sync119896

Φsync119896119896

= 119868119899

119896 gt 1

(29)

where 119865sync119896+119904minus1

119865sync119896+119904minus2

119865sync119896

are systemmatrices dependingon a flight phase mode at current time 119896

42 Synchronization Using an IMM with a Nonlinear BallisticModel For a nonlinear ballistic model the state vector forthe propelled mode is denoted as

x119905 = [119909119905 119910119905 119911119905 119905 119905

119905 120585119905 120591119905]119879 (30)

where 120585119905 is the drag coefficient and 120591119905 is the thrust Generallythe drag coefficient varies significantlywith theMachnumberregime subsonic transonic and supersonic Therefore wetake advantage of the dynamic model considering a Machnumber-dependent multiplier [11 12] such that

[

[

[

]

]

]

=

120591

119881

[

[

[

]

]

]

thrust term

+ 120585120585119898119863[

[

[

]

]

]

drag term

+ 119892[

[

[

0

0

1

]

]

]

gravity term

+ 1

120585 = 2

= 3

(31)

The first term on the right side of (31) represents the thrust(m2

s) of the SLV in 119909 119910 and 119911 directions Two distinctmultiplemodes of a nonlinear SLVmodel can be separated bythe existence of thrust In other words the existence of thrustsignifies propelled flight mode whereas zero thrust signifiescoasting flight mode 119881 is the magnitude of the velocity V =

[ ]

119879 that is the SLV speed (ms) The second part of(31) is the drag term which is related to velocity and altitudethat is 119863 = minus120588(119911)1198812 where 120588(119911) = 1205880119890

minus119888119911 is the air density(kgm3) at an altitude 119911 (m) and 119888 is the air density constant(mminus1) [9] 120585 is the drag coefficient and 120585119898 is theMachnumber-dependent drag coefficient multiplier which is approximatedby the cubic spline curve shown in Figure 3 In this paperfor the drag characteristics of the SLV which are applied inthe subsequent simulation section drag coefficients of theSaturn119881 launch vehicle [15] are usedThe third part of (31) isa gravity term Gravity 119892 is the standard acceleration due togravity at sea level which is assumed to be constant through-out the trajectory with a value of 9812 ms2 1 2 and 3

are assumed to be continuous time zeromeanwhite Gaussiannoises The drag coefficient and thrust acceleration are rep-resented as Wiener processes with a slow variation [11 12]

8 Mathematical Problems in Engineering

0 2 4 6 8 1004

06

08

1

12

14

16

18

2

Mach number

Nor

mal

ized

dra

g co

effici

ent

Selected pointsLinear interpolationCubic spline interpolation

Figure 3 Normalized drag coefficient [15]

We can modify the dynamic equations (30) and (31) as acompact form such that

x119905 = 119891 [x119905] + 119905 (32)

where

119891 [x119905] =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119905

119905

119905

120591

119905

119881119905

+ 120585119905119863119905119905

120591

119905

119881119905

+ 120585119905119863119905119905

120591

119905

119881119905

+ 120585119905119863119905119905 minus 119892

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

119905 = [1119905 2119905 3119905]119879

(33)

The state vector equation (33) is discretized by a second-order Taylor expansion [26] Then (33) can be written as

a discretized continuous time systemwithwhite process noisesuch that

x119896+1 = x119896 + 119891 [x119896] Δ119905 + 119860119896119891 [x119896]Δ119905

2

2

+ 120596119896(34)

where 119860119896 is the Jacobian of (33) evaluated at x119896 [26] and120596119896 is the discretized continuous time process noise for thesampling interval Δ119905 The corresponding covariance matrixof the discretized process noise is

119876 =[

[

[

1198761119902V 06times1 06times101times6 Δ119905119902120585 0

01times6 0 Δ119905119902120591

]

]

]

1198761 =

[

[

[

[

Δ1199053

3

1198683

Δ1199052

2

1198683

Δ1199052

2

1198683 Δ1199051198683

]

]

]

]

(35)

where 1198683 is the 3 times 3 identity matrix and the continuous timeprocess noise intensities 119902V 119902120585 and 119902120591 are the correspondingpower spectral densities

The measurement matrix 119867NL for the nonlinear ballisticmultiple model can be depicted as

119911NL119896 = 119867NLxNL119896 + ]119896

119867NL = [1198683 0] xNL119896 = [119909119896 119910119896 119911119896 119896 119896

119896 120585119896 120591119896]119879

(36)

where ]119896 is measurement noise with error covariance119877119896120575119896minus119895 = 119864[]119896]

119879

119895] that is (14)

An IMMalgorithm for nonlinear dynamics with differentsizes of the mode state vector is summarized as follows [25]

(v) Model-conditioned reinitialization (for 119894 = 1 2

119872)

(a) predicted mode probability 120583(119894)

NL119896|119896minus1 ≜

Pr119898(119894)

119896| 119911NL119896minus1 = sum

119895120587119895119894120583

(119895)

NL119896minus1(b) mixing weight

120583119895|119894

NL119896minus1 ≜ Pr 119898(119895)

119896minus1| 119898

(119894)

119896 119911NL119896minus1 =

sum119895120587119895119894120583

(119895)

NL119896minus1

120583(119894)

NL119896minus1

(37)

(c) unbiased mixing estimate and covariance

x(119894)NL119896minus1|119896minus1 ≜ 119864 [xNL119896minus1|119896minus1 | 119898(119894)

119896 119911NL119896minus1] = sum

119895

(119895)

NL119896minus1|119896minus1120583119895|119894

NL119896minus1

where (1)

NL119896minus1|119896minus1 ≜ x(1)NL119896minus1|119896minus1 (2)

NL119896minus1|119896minus1 ≜ [x(2)119879

NL119896minus1|119896minus1 120591119896minus1]119879

119875

(119894)

NL119896minus1|119896minus1 = sum

119895

[119875(119894)

NL119896minus1|119896minus1 + (x(119894)NL119896minus1|119896minus1 minus (119895)

NL119896minus1|119896minus1) (x(119894)NL119896minus1|119896minus1 minus (119895)

NL119896minus1|119896minus1)119879

] 120583119895|119894

NL119896minus1

(38)

Mathematical Problems in Engineering 9

(vi) Model-conditioned filtering (for 119894 = 1 2 119872)

(a) predicted estimate and covariance

x(119894)NL119896|119896minus1 = x(119894)NL119896minus1|119896minus1 + 119891 [x(119894)NL119896minus1|119896minus1] Δ119905

+

(119894)

NL119896minus1119891 [x(119894)NL119896minus1|119896minus1]Δ119905

2

2

(119894)

119896=

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x=x(119894)119896

119875(119894)

NL119896|119896minus1 = Ω(119894)

119896minus1119875

(119894)

NL119896minus1|119896minus1Ω(119894)119879

119896minus1+ 119876119896minus1

Ω(119894)

119896minus1= 119868 +

(119894)

NL119896minus1

(39)

(b) measurement residual (119894)

NL119896 = 119911NL119896 minus

119867NLx(119894)

NL119896|119896minus1 minus ]119896

(c) residual covariance 119878(119894)NL119896 = 119867NL119875(119894)

NL119896|119896minus1119867119879

NL minus

119877119896(d) filter gain 119870(119894)

NL119896 = 119875(119894)

NL119896|119896minus1119867119879

NL(119878(119894)

NL119896)minus1

(e) update of state and covariance

x(119894)NL119896|119896 = x(119894)NL119896|119896minus1 + 119870(119894)

NL119896(119894)

NL119896

119875(119894)

NL119896|119896 = 119875(119894)

NL119896|119896minus1 minus 119870(119894)

NL119896119878(119894)

NL119896119870(119894)119879

NL119896

(40)

(vii) Mode probability update (for 119894 = 1 2 119872)

(a) mode likelihood

Λ(119894)

NL119896 ≜ 119901 [119911(119894)

NL119896 | 119898(119894)

119896 119885NL119896minus1]

assume= N (119911

(119894)

NL119896 0 119878(119894)

NL119896)

(41)

(b) mode probability 120583(119894)

NL119896 = 120583(119894)

NL119896|119896minus1Λ(119894)

NL119896

(sum119895120583(119895)

NL119896|119896minus1Λ(119895)

NL119896)

(viii) Combination (for 119894 = 1 2 119872)

xNL119896|119896 = sum

119894

x(119894)NL119896|119896120583(119894)

NL119896

(119894)

NL119896|119896 = sum

119894

[119875(119894)

NL119896|119896

+ (xNL119896|119896 minus x(119894)NL119896|119896) (xNL119896|119896 minus x(119894)NL119896|119896)119879

] 120583(119894)

NL119896

(42)

The nonlinearMMS of time delayed slaving data is completedby propagating the combined estimate based on the currentmodersquos estimated vector such that

xNL119896+119904|119896 = xNL119896|119896 + 119891 [xNL119896|119896] sdot 119904 + NL119896119891 [xNL119896|119896]

sdot

1199042

2

(43)

where 119904 is a lead-time for synchronization (119904 = 119899 sdot Δ119905) If theSLV is in the PFP (120583(1)

NL119896 gt 120583(2)

NL119896) the current state estimateincludes the thrust term that is the state vector xNL119896 =

[119909119896 119910119896 119911119896 119896 119896

119896 120585119896 120591119896]119879 If the SLV is not in the PFP

the state vector does not include the thrust term (120591119896 = 0)

5 Simulation Results

To demonstrate the performance of the proposed synchro-nization approaches for delayed slaving data we simulatedthe SLV tracking problem based on the nominal flight tra-jectory of the Korea Space Launch Vehicle-I (KSLV-I) In thesimulation the radar measurement noise intensities of (12)are selected as 120590119903 = 15m 120590120593 = 001 deg and 120590120579 = 001 degThe nominal flight sequence of KSLV-I is as follows First thepayload fairing separates during the first stage flight at 2154 sAfter the first stage engine shutdown at 2287 s the upperstage separates from the first stage and enters the CFP Thesecond stage continues in the CFP until the kick motor igni-tion at 395 s and the vehicle then enters the PFP At the endof kickmotor combustion (4527 s) the upper stage enters thetarget orbit in CFP as in the previous separation Finally thesatellite is inserted into the target orbit after it separates fromthe upper stage during CFP at 540 s Since delayed slavingdata from the MCS to the TSs can occur in both the PFP andCFP synchronization simulations are conducted at arbitrarypoints of the flight phases Figure 4 shows the synchroniza-tion error of the delayed slaving data with respect to thebenchmark and the proposed approaches for delays of 119904 =

01 02 1 s Figures 4(a)ndash4(c) illustrate synchronizationerrors where delayed slaving data occurred at 155 s which isthe first stage of PFP and Figures 4(d)ndash4(f) show synchro-nization errors where delayed slaving data occurred at 280 swhich is the first stage of CFP As shown in Figures 4(a)ndash4(c)the extrapolation and120572-120573-120574filter-based approach cannot findthe proper synchronized slaving data The synchronizationerrors of these approaches exponentially increase accordingto increasing delay 119904 in particular the error of the extrapo-lation approaches an out-of-figure bound (2500m) On theother hand the Kalman prediction and the proposedapproaches have stable synchronization errors even if thedelay time 119904 is larger Furthermore we can observe thatthe errors in the synchronization approach of the Kalmanprediction are relatively larger than the proposed approachesespecially in the PFP As explained in the flight sequence ofKSLV-I in general the motion of the SLV is described bymultiple dynamics (in our case PFP and CFP) rather thansingle dynamics However the dynamic model of the KF andprediction capability used in this paper is the constant accel-eration (CA) model This means that the filter is optimizedwith respect to CAmotion which is actually not occurring inour application This is why the performance of the Kalmanprediction-based synchronization is worse than the proposedapproaches Nonetheless we can see that the 119910-axis synchro-nization errors shown in Figures 4(b) and 4(e) are smallThis result is observed because the 119910-axis motion of KSLV-I in a local coordinate is relatively smaller than the otheraxis motion and the CA model approximately expresses this

10 Mathematical Problems in Engineering

01 02 03 04 05 06 07 08 09 10

500

1000

1500

2000

2500

Delay (sec)

x-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

Cubic spline extrapolation

Kalman filter

(a)

(b)

(c)

(d)

(e)

(f)

Linear MMNonlinear MM120572-120573-120574 filter 120572-120573-120574 filter

120572-120573-120574 filter120572-120573-120574 filter

120572-120573-120574 filter 120572-120573-120574 filter

01 02 03 04 05 06 07 08 09 1Delay (sec)

0

500

1000

1500

2000

2500

y-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

z-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

x-p

ositi

on ab

solu

te er

ror i

n CF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500y

-pos

ition

abso

lute

erro

r in

CFP

(m)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

z-p

ositi

on ab

solu

te er

ror i

n CF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

Figure 4 Synchronization errors at PFP and CFP

Mathematical Problems in Engineering 11

small motion A single KF prediction model cannot exactlydescribe the motion of both PFP and CFP whereas theproposed multiple model-based approaches work well Asshown in Figure 4 regardless of the delay 119904 the proposedmultiplemodel-based approaches find synchronized positionvectors In the comparison of synchronization performancebetween linear IMM and nonlinear IMM synchronizationthe nonlinear IMM-based synchronization approach showsthe best performance except for 119910-axis where CA motion isdominant In addition the difference between the proposedmultiple model-based approaches is very small as shown inFigure 4 but the complexities of the algorithms for real-timeapplications are quite dissimilar Hence the operator mayadaptively select one of the proposed approaches accordingto onersquos environment

6 Conclusions

In this paper we investigated the time synchronizationapproaches of delayed slaving data in the RSS for SLV track-ingOne of themost important roles of the RSS is to distributeslaving data to each TS for continuous tracking of the SLV Ifthere is a critical network delay resulting in time delayed slav-ing data being sent to each TS theMCS will not receive accu-rate SLV tracking data This problem can give rise to signifi-cant difficulties for the SLVmission progress and analysis Toovercome this problemwe proposedMMSapproacheswhichtake advantage of the multiple motion models of an SLVThelinear IMM-based synchronization approach was developedusing Singerrsquos model with ternary uniform mixtures and thenonlinear IMM-based synchronization approachwas derivedfrom a nonlinear ballistic model with a drag coefficient Forverification of the proposed algorithms SLV tracking simu-lations using KSLV-I and the radar measurement data gen-erated from nominal trajectory were conducted To demon-strate the superiority of time synchronization performancein these simulations we compared the proposed algorithmwith benchmark approaches for absolute error between thenominal trajectory data and the synchronized slaving datathe simulation results demonstrated that the proposed MMSapproaches performed competitively

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] R Varaprasad and V Seshagiri Rao ldquoRange safety real-timesystem for satellite launch vehicle missions-testing methodolo-giesrdquo Defence Science Journal vol 56 no 5 pp 693ndash700 2006

[2] H R Song M G Jeon T S Choi and V Shin ldquoTwo fusionpredictors for discrete-time linear systems with different typesof observationsrdquo International Journal of Control Automationand Systems vol 7 no 4 pp 651ndash658 2009

[3] H Song V Shin and M Jeon ldquoMobile node localization usingfusion prediction-based interacting multiple model in cricketsensor networkrdquo IEEE Transactions on Industrial Electronicsvol 59 no 11 pp 4349ndash4359 2012

[4] T Kirubarajan and Y Bar-Shalom ldquoKalman filter versus IMMestimator when do we need the latterrdquo IEEE Transactions onAerospace and Electronic Systems vol 39 no 4 pp 1452ndash14572003

[5] E Mazor A Averbuch Y Bar-Shalom and J Dayan ldquoInteract-ing multiple model methods in target tracking a surveyrdquo IEEETransactions on Aerospace and Electronic Systems vol 34 no 1pp 103ndash123 1998

[6] P D Hanlon and P S Maybeck ldquoMultiple-model adaptiveestimation using a residual correlation Kalman filter bankrdquoIEEE Transactions on Aerospace and Electronic Systems vol 36no 2 pp 393ndash406 2000

[7] S-J Shin ldquoRe-entry vehicle tracking with a new multiplemodel estimation applicable to highly non-linear dynamicsrdquoIET Radar Sonar amp Navigation vol 9 no 5 pp 581ndash588 2015

[8] S Conover J Kerce G Brown L Ehrman and D HardimanldquoImpact point prediction of small ballistic munitions with aninteracting multiple model estimatorrdquo in Acquisition TrackingPointing and Laser Systems Technologies XXI vol 6569 ofProceedings of SPIE Orlando Fla USA 2007

[9] V C Ravindra Y Bar-Shalom and P Willett ldquoProjectileidentification and impact point predictionrdquo IEEE Transactionson Aerospace and Electronic Systems vol 46 no 4 pp 2004ndash2021 2010

[10] W J Farrell III ldquoInteracting multiple model filter for tacticalballistic missile trackingrdquo IEEE Transactions on Aerospace andElectronic Systems vol 44 no 2 pp 418ndash426 2008

[11] T Yuan Y Bar-Shalom PWillett E Mozeson S Pollak and DHardiman ldquoA multiple IMM estimation approach with unbi-ased mixing for thrusting projectilesrdquo IEEE Transactions onAerospace and Electronic Systems vol 48 no 4 pp 3250ndash32672012

[12] T Yuan Y Bar-Shalom P Willett and D Hardiman ldquoImpactpoint prediction for thrusting projectiles in the presence ofwindrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 50 no 1 pp 102ndash119 2014

[13] X Li and V Jilkov ldquoSurvey of maneuvering target trackingpart II motion models of ballistic and space targetsrdquo IEEETransactions on Aerospace and Electronic Systems vol 46 no1 pp 96ndash119 2010

[14] J Cesar Bolzani de Campos Ferreira and JWaldmann ldquoCovari-ance intersection-based sensor fusion for sounding rockettracking and impact area predictionrdquo Control Engineering Prac-tice vol 15 no 4 pp 389ndash409 2007

[15] httpwwwbraeunigusapollosaturnVhtm[16] RA Singer ldquoEstimating optimal tracking filter performance for

manned maneuvering targetsrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 6 no 4 pp 473ndash483 1970

[17] R A Singer and KW Behnke ldquoReal-time tracking filter evalu-ation and selection for tactical applicationsrdquo IEEE Transactionson Aerospace and Electronic Systems vol AES-7 no 1 pp 100ndash110 1971

[18] W Press B Flannery S Teukolsky and W Vetterling Numer-ical Recipes in FORTRAN The Art of Scientific ComputingCambridge University New York NY USA 2nd edition 1992

[19] S Dyer and J Dyer ldquoCubic-spline interpolation part Irdquo IEEEInstrumentation amp Measurement Magazine pp 44ndash46 2001

[20] Y Bar-Shalom X Li and T Kirubarajan Estimation with Appli-cations to Tracking and Navigation John Wiley amp Sons NewYork NY USA 2001

12 Mathematical Problems in Engineering

[21] N Morrison Introduction to Sequential Smoothing and Predic-tion McGraw-Hill 1969

[22] E Brookner Tracking and Kalman Filtering Made Easy JohnWiley amp Sons 1998

[23] P Suchomski ldquoExplicit expressions for debiased statistics of 3Dconverted measurementsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 35 no 1 pp 368ndash370 1999

[24] H R Song I Y Song and V Shin ldquoMultisensory predictionfusion of nonlinear functions of the state vector in discrete-timesystemsrdquo International Journal of Distributed Sensor Networksvol 2015 Article ID 249857 13 pages 2015

[25] H Song andYHan ldquoComparison of space launch vehicle track-ing using different types of multiple modelsrdquo in Proceedingsof the International Conference on Information Fusion pp 1ndash8Heidelberg Germany July 2016

[26] R KMehra ldquoA comparison of several nonlinear filters for reen-try vehicle trackingrdquo IEEE Transactions on Automatic Controlvol 16 no 4 pp 307ndash319 1971

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Prediction or fixed-lead prediction inmean squaremeans thesynchronization of the slaving data is the estimation of thestate at a future time 119896+119904 where 119904 gt 0 beyond the observationinterval that is based on data up to an earlier time [20 24]

119896+119904|119896 = 119864 (119883119896+119904|119896 | 119885119896) 119885119896 = z0 z119896 (15)

The optimal predictor or synchronized state 119896+119904|119896 ≜

KP119896+119904|119896

and its error covariance 119875119896+119904|119896 ≜ 119875KP119896+119904|119896

are given by theKalman predictor equations [2 24]

KP119896+119904|119896

= 119865119896+119904minus1

KP119896+119904minus1|119896

= sdot sdot sdot = Φ119896+119904119896

KP119896|119896

KP119896|119896

≜

KF119896|119896

119875KP119896+119904|119896

= 119865119896+119904minus1119875KP119896+119904minus1|119896

119865119879

119896+119904minus1+ 119876119896+119904minus1

= Φ119896+119904119896119875KP119896|119896

Φ119879

119896+119904119896

+

119904minus1

sum

119895=0

Φ119896+119904119896+119895+1119876119896+119895Φ119879

119896+119904119896+119895+1 119875

KP119896|119896

≜ 119875KF119896|119896

Φ119896+119904119896 = 119865119896+119904minus1119865119896+119904minus2 sdot sdot sdot 119865119896

Φ119896119896 = 119868119899

119896 gt 1

(16)

where

KF119896|119896

and 119875KF119896|119896

are the KF estimate and covariancerespectively

4 Proposed MME-Based SynchronizationApproaches

41 SynchronizationUsing an IMMwith Singerrsquos LinearModelSinger [16] described 2D manned maneuvering targets inrange-bearing coordinates This model can be adapted to theSLV kinematics in 3D Cartesian coordinates with multipleflight phases

[

[

[

119899

V119899119899

]

]

]

= 119865[

[

[

119901119899

V119899119886119899

]

]

]

+ 119866119908119899

119865 =[

[

[

0 1 0

0 0 1

0 0 minus120572119899

]

]

]

119866 =[

[

[

0

0

0

]

]

]

(17)

where119908119899 isin R3times1 is a white noise process along the Cartesianaxis 119899 = 119909 119910 119911 The parameter 120573 = 1120572119899 is the maneuvercorrelation time constant and 120590

2

119886119899is the acceleration variance

describing maneuver intensity In a steady state

1205902

119908119899

= 21205721205902

119886119899 (18)

To describe SLV kinematics the model must cope with cru-cial nonzero mean acceleration maneuvers during the pro-pelled phase In addition after each stagersquos engine burns outa multiple model approach is applied to the coasting flightthat is one model describes PFP whereas the other depictsthe CFP Empirically tuned independent probability densityfunctions (PDFs) represented by TUM describe the accel-erations of the SLV in the local coordinate frame Figure 2shows the means and variances of the acceleration processesof the SLV in this paper The discrete-time model with statetransition matrix Ψ119899(120572119899 Δ119905) is as follows

[

[

[

119901119899119896+1

V119899119896+1119886119899119896+1

]

]

]

= Ψ119899 (120572119899 Δ119905)[

[

[

119901119899119896

V119899119896119886119899119896

]

]

]

+ 119908119899119896

Ψ119899 (120572119899 Δ119905) = 119890119865Δ119905

[

[

[

[

[

[

[

1 Δ119905 (120572119899Δ119905 minus 1 + 119890minus120572119899Δ119905

)

0 1

(1 minus 119890minus120572119899Δ119905

)

120572119899

0 0 119890minus120572119899Δ119905

]

]

]

]

]

]

]

119908119899119896 = int

(119896+1)Δ119905

119896Δ119905

119890119865[(119896+1)Δ119905minus119903]

119866119908119899 119889119903

(19)

In the case of the PFP 119908119899119896 is a nonzero mean white noisesequence caused by the nonzero mean acceleration 120583119886119899 seenin Figure 2 A nonzeromeanwhite noise sequence for the PFPshould be considered in the target kinematics when imple-menting the state propagation stage in theKFThus the deter-ministic input 119906119899119896 caused by 119908119899119896 along 119909-axis 119910-axis and119911-axis is derived as follows

119906119909119896 = 119864 [119908119909119896] = 119864int

(119896+1)Δ119905

119896Δ119905

119890119865[(119896+1)Δ119905minus119903]

119866119908119909119889119903

asymp 119864

int

(119896+1)Δ119905

119896Δ119905

[

[

[

[

119890minus120572119909[(119896+1)Δ119905minus119903]

0

0

]

]

]

]

119908119909 119889119903

119906119909119896 =

120583119886119909

120572119899 [1minus119890minus120572119909Δ119905

00

]

(20)

119910-axis and 119911-axis can be derived in the samemanner as shownin (20) The maneuver excitation covariance [10] whichrepresents the uncertainty of the SLV kinematics model is

119876119899119896 (120572119899 Δ119905)

= 119864 (119908119899119896 minus 119864 [119908119899119896]) (119908119899119896 minus 119864 [119908119899119896])119879

= 21205721198991205902

119886119899

[

[

[

11990211 11990212 11990213

11990212 11990222 11990223

11990213 11990223 11990233

]

]

]

(21)

The specific components of 11990211 11990233 are illustrated in [25]In addition the measurement matrix 119867119871 for Singerrsquos modelcan be depicted as

6 Mathematical Problems in Engineering

01

minus20 4minus6 0

0028

120583 =

1205902 =

p(ax)

minus6ms2

4246

a (m2s4)

m2s4

(a)

01

minus5 810

005

120583 = 1ms2

1205902 = 106

p(ay)

a (m2s4)

m2s4

(b)

01

minus40 4minus7 0

0008

a (m2s4)

120583 = minus7ms2

1205902 = 12674 m2s4

p(az)

(c)

01

005 005

minus4 40

008

120583 = 0ms2

1205902 = 183

p(ax) p(ay) p(az)

a (m2s4)

m2s4

(d)

Figure 2 Nonzero mean acceleration PDF in the PFP model (a) along 119909-axis (b) along 119910-axis and (c) along 119911-axis Zero mean accelerationPDF in the CFP Model (d) along 119909-axis 119910-axis and 119911-axis

119911119871119896 = 119867119871x119871119896 + ]119896 119867119871 =[

[

[

1 0 0

0 03times2 1 03times2 0 03times20 0 1

]

]

]

x119871119896 = [119909119896 119896 119896 119910119896 119896

119896

119911119896 119896 119896]119879 (22)

where ]119896 is measurement noise (as shown in (14)) with errorcovariance

411 IMM with Singerrsquos Linear Model From (19)ndash(22) inSection 41 we can rewrite the Markov jump linear systemswhere the 119894th model of the finite multiple model set M =

119898(1)

119898(119872)

obeys the following equations

x119871119896+1 = Ψ(119894)

119896x119871119896 + 119908

(119894)

119896

119911119871119896 = 119867119871x119871119896 + ]119896(23)

where

cov (119908(119894)

119896) = 119876

(119894)

119896

cov (]119896) = 119877119896

Ψ(119894)

119896= diag (Ψ119909119896 (120572

(119894)

119909 Δ119905) Ψ119910119896 (120572

(119894)

119910 Δ119905)

Ψ119911119896 (120572(119894)

119911 Δ119905))

119876(119894)

119896= diag (119876119909119896 (120572

(119894)

119909 Δ119905) 119876119910119896 (120572

(119894)

119910 Δ119905)

119876119911119896 (120572(119894)

119911 Δ119905))

(24)

Here 1205902

119886119899for the SLV can be represented by TUM as in

Figure 2 The PDFs of Figures 2(a)ndash2(d) are experimentallysampled from the nominal acceleration profile of the SLVThesuperscript (i) denotes quantities pertinent to model 119898(119894) inM and the jumps of the system mode are assumed to havetransition probabilities

Pr 119898(119895)

119896+1| 119898

(119894)

119896 ≜ 120587119894119895

(25)

where 119898(119894)

119896denotes the event in which model 119898

(119894) matchesthe system mode in effect at time 119896 In our application 119872 =

2 and 119898(1) and 119898

(2) are the propelled and coasting modesrespectively

Finally complete recursion of the IMM with modematched KF for the SLV tracking is summarized as follows

(i) Model-conditioned reinitialization (for 119894 = 1 2

119872)

Mathematical Problems in Engineering 7

(a) predicted mode probability 120583(119894)

119871119896|119896minus1≜ Pr119898(119894)

119896|

119911119871119896minus1 = sum119895120587119895119894120583

(119895)

119871119896minus1

(b) mixingweight 120583119895|119894

119871119896minus1≜ Pr119898(119895)

119896minus1| 119898

(119894)

119896 119911119871119896minus1 =

sum119895120587119895119894120583

(119895)

119871119896minus1120583

(119894)

119871119896minus1

(c) mixing estimate and covariance

x(119894)119871119896minus1|119896minus1

≜ 119864 [x119871119896minus1|119896minus1 | 119898(119894)

119896 119911119871119896minus1]

= sum

119895

x(119895)119871119896minus1|119896minus1

120583119895|119894

119871119896minus1

119875

(119894)

119871119896minus1|119896minus1= sum

119895

[119875(119894)

119871119896minus1|119896minus1+ (x(119894)

119871119896minus1|119896minus1minus x(119895)

119871119896minus1|119896minus1)

sdot (x(119894)119871119896minus1|119896minus1

minus x(119895)119871119896minus1|119896minus1

)

119879

] 120583119895|119894

119871119896minus1

(26)

(ii) Model-conditioned filtering (for 119894 = 1 2 119872)

(a) predicted estimate and covariance

x(119894)119871119896|119896minus1

= Ψ(119894)

119896minus1x(119894)119871119896minus1|119896minus1

+ u(119894)

119896minus1

where u(1)

119896minus1= [119906

119879

119909119896minus1119906119879

119910119896minus1119906119879

119911119896minus1]

119879

u(2)

119896minus1= 03times1

119875(119894)

119871119896|119896minus1= Ψ

(119894)

119896minus1119875

(119894)

119871119896minus1|119896minus1Ψ

(119894)119879

119896minus1+ 119876

(119894)

119896minus1

(27)

(b) measurement residual (119894)

119871119896= 119911119871119896 minus119867119871x

(119894)

119871119896|119896minus1minus

]119896(c) residual covariance 119878(119894)

119871119896= 119867119871119875

(119894)

119871119896|119896minus1119867

119879

119871minus 119877119896

(d) filter gain 119870(119894)

119871119896= 119875

(119894)

119871119896|119896minus1119867

119879

119871(119878

(119894)

119871119896)minus1

(e) update of state and covariance x(119894)119871119896|119896

= x(119894)119871119896|119896minus1

+

119870(119894)

119871119896(119894)

119871119896 119875

(119894)

119871119896|119896= 119875

(119894)

119871119896|119896minus1minus 119870

(119894)

119871119896119878(119894)

119871119896119870

(119894)119879

119871119896

(iii) Mode probability update (for 119894 = 1 2 119872)

(a) mode likelihood Λ(119894)

119871119896≜ 119901[119911

(119894)

119871119896| 119898

(119894)

119896

119885119871119896minus1]assume

= N(119911(119894)

119871119896 0 119878

(119894)

119871119896)

(b) mode probability 120583(119894)

119871119896= 120583

(119894)

119871119896|119896minus1Λ

(119894)

119871119896

(sum119895120583(119895)

119871119896|119896minus1Λ

(119895)

119871119896)

(iv) Combination (for 119894 = 1 2 119872)

x119871119896|119896 = sum

119894

x(119894)119871119896|119896

120583(119894)

119871119896

(119894)

119871119896|119896

= sum

119894

[119875(119894)

119871119896|119896+ (x119871119896|119896 minus x(119894)

119871119896|119896) (x119871119896|119896 minus x(119894)

119871119896|119896)

119879

] 120583(119894)

119871119896

(28)

Singerrsquos MMS of time delayed slaving data is completed bypropagating the combined estimate based on the currentmodersquos dynamic model such that

xsync119871119896+119904|119896

= Φsync119896+119904119896

x119871119896|119896

Φsync119896+119904119896

= 119865sync119896+119904minus1

119865sync119896+119904minus2

sdot sdot sdot 119865sync119896

Φsync119896119896

= 119868119899

119896 gt 1

(29)

where 119865sync119896+119904minus1

119865sync119896+119904minus2

119865sync119896

are systemmatrices dependingon a flight phase mode at current time 119896

42 Synchronization Using an IMM with a Nonlinear BallisticModel For a nonlinear ballistic model the state vector forthe propelled mode is denoted as

x119905 = [119909119905 119910119905 119911119905 119905 119905

119905 120585119905 120591119905]119879 (30)

where 120585119905 is the drag coefficient and 120591119905 is the thrust Generallythe drag coefficient varies significantlywith theMachnumberregime subsonic transonic and supersonic Therefore wetake advantage of the dynamic model considering a Machnumber-dependent multiplier [11 12] such that

[

[

[

]

]

]

=

120591

119881

[

[

[

]

]

]

thrust term

+ 120585120585119898119863[

[

[

]

]

]

drag term

+ 119892[

[

[

0

0

1

]

]

]

gravity term

+ 1

120585 = 2

= 3

(31)

The first term on the right side of (31) represents the thrust(m2

s) of the SLV in 119909 119910 and 119911 directions Two distinctmultiplemodes of a nonlinear SLVmodel can be separated bythe existence of thrust In other words the existence of thrustsignifies propelled flight mode whereas zero thrust signifiescoasting flight mode 119881 is the magnitude of the velocity V =

[ ]

119879 that is the SLV speed (ms) The second part of(31) is the drag term which is related to velocity and altitudethat is 119863 = minus120588(119911)1198812 where 120588(119911) = 1205880119890

minus119888119911 is the air density(kgm3) at an altitude 119911 (m) and 119888 is the air density constant(mminus1) [9] 120585 is the drag coefficient and 120585119898 is theMachnumber-dependent drag coefficient multiplier which is approximatedby the cubic spline curve shown in Figure 3 In this paperfor the drag characteristics of the SLV which are applied inthe subsequent simulation section drag coefficients of theSaturn119881 launch vehicle [15] are usedThe third part of (31) isa gravity term Gravity 119892 is the standard acceleration due togravity at sea level which is assumed to be constant through-out the trajectory with a value of 9812 ms2 1 2 and 3

are assumed to be continuous time zeromeanwhite Gaussiannoises The drag coefficient and thrust acceleration are rep-resented as Wiener processes with a slow variation [11 12]

8 Mathematical Problems in Engineering

0 2 4 6 8 1004

06

08

1

12

14

16

18

2

Mach number

Nor

mal

ized

dra

g co

effici

ent

Selected pointsLinear interpolationCubic spline interpolation

Figure 3 Normalized drag coefficient [15]

We can modify the dynamic equations (30) and (31) as acompact form such that

x119905 = 119891 [x119905] + 119905 (32)

where

119891 [x119905] =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119905

119905

119905

120591

119905

119881119905

+ 120585119905119863119905119905

120591

119905

119881119905

+ 120585119905119863119905119905

120591

119905

119881119905

+ 120585119905119863119905119905 minus 119892

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

119905 = [1119905 2119905 3119905]119879

(33)

The state vector equation (33) is discretized by a second-order Taylor expansion [26] Then (33) can be written as

a discretized continuous time systemwithwhite process noisesuch that

x119896+1 = x119896 + 119891 [x119896] Δ119905 + 119860119896119891 [x119896]Δ119905

2

2

+ 120596119896(34)

where 119860119896 is the Jacobian of (33) evaluated at x119896 [26] and120596119896 is the discretized continuous time process noise for thesampling interval Δ119905 The corresponding covariance matrixof the discretized process noise is

119876 =[

[

[

1198761119902V 06times1 06times101times6 Δ119905119902120585 0

01times6 0 Δ119905119902120591

]

]

]

1198761 =

[

[

[

[

Δ1199053

3

1198683

Δ1199052

2

1198683

Δ1199052

2

1198683 Δ1199051198683

]

]

]

]

(35)

where 1198683 is the 3 times 3 identity matrix and the continuous timeprocess noise intensities 119902V 119902120585 and 119902120591 are the correspondingpower spectral densities

The measurement matrix 119867NL for the nonlinear ballisticmultiple model can be depicted as

119911NL119896 = 119867NLxNL119896 + ]119896

119867NL = [1198683 0] xNL119896 = [119909119896 119910119896 119911119896 119896 119896

119896 120585119896 120591119896]119879

(36)

where ]119896 is measurement noise with error covariance119877119896120575119896minus119895 = 119864[]119896]

119879

119895] that is (14)

An IMMalgorithm for nonlinear dynamics with differentsizes of the mode state vector is summarized as follows [25]

(v) Model-conditioned reinitialization (for 119894 = 1 2

119872)

(a) predicted mode probability 120583(119894)

NL119896|119896minus1 ≜

Pr119898(119894)

119896| 119911NL119896minus1 = sum

119895120587119895119894120583

(119895)

NL119896minus1(b) mixing weight

120583119895|119894

NL119896minus1 ≜ Pr 119898(119895)

119896minus1| 119898

(119894)

119896 119911NL119896minus1 =

sum119895120587119895119894120583

(119895)

NL119896minus1

120583(119894)

NL119896minus1

(37)

(c) unbiased mixing estimate and covariance

x(119894)NL119896minus1|119896minus1 ≜ 119864 [xNL119896minus1|119896minus1 | 119898(119894)

119896 119911NL119896minus1] = sum

119895

(119895)

NL119896minus1|119896minus1120583119895|119894

NL119896minus1

where (1)

NL119896minus1|119896minus1 ≜ x(1)NL119896minus1|119896minus1 (2)

NL119896minus1|119896minus1 ≜ [x(2)119879

NL119896minus1|119896minus1 120591119896minus1]119879

119875

(119894)

NL119896minus1|119896minus1 = sum

119895

[119875(119894)

NL119896minus1|119896minus1 + (x(119894)NL119896minus1|119896minus1 minus (119895)

NL119896minus1|119896minus1) (x(119894)NL119896minus1|119896minus1 minus (119895)

NL119896minus1|119896minus1)119879

] 120583119895|119894

NL119896minus1

(38)

Mathematical Problems in Engineering 9

(vi) Model-conditioned filtering (for 119894 = 1 2 119872)

(a) predicted estimate and covariance

x(119894)NL119896|119896minus1 = x(119894)NL119896minus1|119896minus1 + 119891 [x(119894)NL119896minus1|119896minus1] Δ119905

+

(119894)

NL119896minus1119891 [x(119894)NL119896minus1|119896minus1]Δ119905

2

2

(119894)

119896=

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x=x(119894)119896

119875(119894)

NL119896|119896minus1 = Ω(119894)

119896minus1119875

(119894)

NL119896minus1|119896minus1Ω(119894)119879

119896minus1+ 119876119896minus1

Ω(119894)

119896minus1= 119868 +

(119894)

NL119896minus1

(39)

(b) measurement residual (119894)

NL119896 = 119911NL119896 minus

119867NLx(119894)

NL119896|119896minus1 minus ]119896

(c) residual covariance 119878(119894)NL119896 = 119867NL119875(119894)

NL119896|119896minus1119867119879

NL minus

119877119896(d) filter gain 119870(119894)

NL119896 = 119875(119894)

NL119896|119896minus1119867119879

NL(119878(119894)

NL119896)minus1

(e) update of state and covariance

x(119894)NL119896|119896 = x(119894)NL119896|119896minus1 + 119870(119894)

NL119896(119894)

NL119896

119875(119894)

NL119896|119896 = 119875(119894)

NL119896|119896minus1 minus 119870(119894)

NL119896119878(119894)

NL119896119870(119894)119879

NL119896

(40)

(vii) Mode probability update (for 119894 = 1 2 119872)

(a) mode likelihood

Λ(119894)

NL119896 ≜ 119901 [119911(119894)

NL119896 | 119898(119894)

119896 119885NL119896minus1]

assume= N (119911

(119894)

NL119896 0 119878(119894)

NL119896)

(41)

(b) mode probability 120583(119894)

NL119896 = 120583(119894)

NL119896|119896minus1Λ(119894)

NL119896

(sum119895120583(119895)

NL119896|119896minus1Λ(119895)

NL119896)

(viii) Combination (for 119894 = 1 2 119872)

xNL119896|119896 = sum

119894

x(119894)NL119896|119896120583(119894)

NL119896

(119894)

NL119896|119896 = sum

119894

[119875(119894)

NL119896|119896

+ (xNL119896|119896 minus x(119894)NL119896|119896) (xNL119896|119896 minus x(119894)NL119896|119896)119879

] 120583(119894)

NL119896

(42)

The nonlinearMMS of time delayed slaving data is completedby propagating the combined estimate based on the currentmodersquos estimated vector such that

xNL119896+119904|119896 = xNL119896|119896 + 119891 [xNL119896|119896] sdot 119904 + NL119896119891 [xNL119896|119896]

sdot

1199042

2

(43)

where 119904 is a lead-time for synchronization (119904 = 119899 sdot Δ119905) If theSLV is in the PFP (120583(1)

NL119896 gt 120583(2)

NL119896) the current state estimateincludes the thrust term that is the state vector xNL119896 =

[119909119896 119910119896 119911119896 119896 119896

119896 120585119896 120591119896]119879 If the SLV is not in the PFP

the state vector does not include the thrust term (120591119896 = 0)

5 Simulation Results

To demonstrate the performance of the proposed synchro-nization approaches for delayed slaving data we simulatedthe SLV tracking problem based on the nominal flight tra-jectory of the Korea Space Launch Vehicle-I (KSLV-I) In thesimulation the radar measurement noise intensities of (12)are selected as 120590119903 = 15m 120590120593 = 001 deg and 120590120579 = 001 degThe nominal flight sequence of KSLV-I is as follows First thepayload fairing separates during the first stage flight at 2154 sAfter the first stage engine shutdown at 2287 s the upperstage separates from the first stage and enters the CFP Thesecond stage continues in the CFP until the kick motor igni-tion at 395 s and the vehicle then enters the PFP At the endof kickmotor combustion (4527 s) the upper stage enters thetarget orbit in CFP as in the previous separation Finally thesatellite is inserted into the target orbit after it separates fromthe upper stage during CFP at 540 s Since delayed slavingdata from the MCS to the TSs can occur in both the PFP andCFP synchronization simulations are conducted at arbitrarypoints of the flight phases Figure 4 shows the synchroniza-tion error of the delayed slaving data with respect to thebenchmark and the proposed approaches for delays of 119904 =

01 02 1 s Figures 4(a)ndash4(c) illustrate synchronizationerrors where delayed slaving data occurred at 155 s which isthe first stage of PFP and Figures 4(d)ndash4(f) show synchro-nization errors where delayed slaving data occurred at 280 swhich is the first stage of CFP As shown in Figures 4(a)ndash4(c)the extrapolation and120572-120573-120574filter-based approach cannot findthe proper synchronized slaving data The synchronizationerrors of these approaches exponentially increase accordingto increasing delay 119904 in particular the error of the extrapo-lation approaches an out-of-figure bound (2500m) On theother hand the Kalman prediction and the proposedapproaches have stable synchronization errors even if thedelay time 119904 is larger Furthermore we can observe thatthe errors in the synchronization approach of the Kalmanprediction are relatively larger than the proposed approachesespecially in the PFP As explained in the flight sequence ofKSLV-I in general the motion of the SLV is described bymultiple dynamics (in our case PFP and CFP) rather thansingle dynamics However the dynamic model of the KF andprediction capability used in this paper is the constant accel-eration (CA) model This means that the filter is optimizedwith respect to CAmotion which is actually not occurring inour application This is why the performance of the Kalmanprediction-based synchronization is worse than the proposedapproaches Nonetheless we can see that the 119910-axis synchro-nization errors shown in Figures 4(b) and 4(e) are smallThis result is observed because the 119910-axis motion of KSLV-I in a local coordinate is relatively smaller than the otheraxis motion and the CA model approximately expresses this

10 Mathematical Problems in Engineering

01 02 03 04 05 06 07 08 09 10

500

1000

1500

2000

2500

Delay (sec)

x-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

Cubic spline extrapolation

Kalman filter

(a)

(b)

(c)

(d)

(e)

(f)

Linear MMNonlinear MM120572-120573-120574 filter 120572-120573-120574 filter

120572-120573-120574 filter120572-120573-120574 filter

120572-120573-120574 filter 120572-120573-120574 filter

01 02 03 04 05 06 07 08 09 1Delay (sec)

0

500

1000

1500

2000

2500

y-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

z-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

x-p

ositi

on ab

solu

te er

ror i

n CF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500y

-pos

ition

abso

lute

erro

r in

CFP

(m)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

z-p

ositi

on ab

solu

te er

ror i

n CF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

Figure 4 Synchronization errors at PFP and CFP

Mathematical Problems in Engineering 11

small motion A single KF prediction model cannot exactlydescribe the motion of both PFP and CFP whereas theproposed multiple model-based approaches work well Asshown in Figure 4 regardless of the delay 119904 the proposedmultiplemodel-based approaches find synchronized positionvectors In the comparison of synchronization performancebetween linear IMM and nonlinear IMM synchronizationthe nonlinear IMM-based synchronization approach showsthe best performance except for 119910-axis where CA motion isdominant In addition the difference between the proposedmultiple model-based approaches is very small as shown inFigure 4 but the complexities of the algorithms for real-timeapplications are quite dissimilar Hence the operator mayadaptively select one of the proposed approaches accordingto onersquos environment

6 Conclusions

In this paper we investigated the time synchronizationapproaches of delayed slaving data in the RSS for SLV track-ingOne of themost important roles of the RSS is to distributeslaving data to each TS for continuous tracking of the SLV Ifthere is a critical network delay resulting in time delayed slav-ing data being sent to each TS theMCS will not receive accu-rate SLV tracking data This problem can give rise to signifi-cant difficulties for the SLVmission progress and analysis Toovercome this problemwe proposedMMSapproacheswhichtake advantage of the multiple motion models of an SLVThelinear IMM-based synchronization approach was developedusing Singerrsquos model with ternary uniform mixtures and thenonlinear IMM-based synchronization approachwas derivedfrom a nonlinear ballistic model with a drag coefficient Forverification of the proposed algorithms SLV tracking simu-lations using KSLV-I and the radar measurement data gen-erated from nominal trajectory were conducted To demon-strate the superiority of time synchronization performancein these simulations we compared the proposed algorithmwith benchmark approaches for absolute error between thenominal trajectory data and the synchronized slaving datathe simulation results demonstrated that the proposed MMSapproaches performed competitively

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] R Varaprasad and V Seshagiri Rao ldquoRange safety real-timesystem for satellite launch vehicle missions-testing methodolo-giesrdquo Defence Science Journal vol 56 no 5 pp 693ndash700 2006

[2] H R Song M G Jeon T S Choi and V Shin ldquoTwo fusionpredictors for discrete-time linear systems with different typesof observationsrdquo International Journal of Control Automationand Systems vol 7 no 4 pp 651ndash658 2009

[3] H Song V Shin and M Jeon ldquoMobile node localization usingfusion prediction-based interacting multiple model in cricketsensor networkrdquo IEEE Transactions on Industrial Electronicsvol 59 no 11 pp 4349ndash4359 2012

[4] T Kirubarajan and Y Bar-Shalom ldquoKalman filter versus IMMestimator when do we need the latterrdquo IEEE Transactions onAerospace and Electronic Systems vol 39 no 4 pp 1452ndash14572003

[5] E Mazor A Averbuch Y Bar-Shalom and J Dayan ldquoInteract-ing multiple model methods in target tracking a surveyrdquo IEEETransactions on Aerospace and Electronic Systems vol 34 no 1pp 103ndash123 1998

[6] P D Hanlon and P S Maybeck ldquoMultiple-model adaptiveestimation using a residual correlation Kalman filter bankrdquoIEEE Transactions on Aerospace and Electronic Systems vol 36no 2 pp 393ndash406 2000

[7] S-J Shin ldquoRe-entry vehicle tracking with a new multiplemodel estimation applicable to highly non-linear dynamicsrdquoIET Radar Sonar amp Navigation vol 9 no 5 pp 581ndash588 2015

[8] S Conover J Kerce G Brown L Ehrman and D HardimanldquoImpact point prediction of small ballistic munitions with aninteracting multiple model estimatorrdquo in Acquisition TrackingPointing and Laser Systems Technologies XXI vol 6569 ofProceedings of SPIE Orlando Fla USA 2007

[9] V C Ravindra Y Bar-Shalom and P Willett ldquoProjectileidentification and impact point predictionrdquo IEEE Transactionson Aerospace and Electronic Systems vol 46 no 4 pp 2004ndash2021 2010

[10] W J Farrell III ldquoInteracting multiple model filter for tacticalballistic missile trackingrdquo IEEE Transactions on Aerospace andElectronic Systems vol 44 no 2 pp 418ndash426 2008

[11] T Yuan Y Bar-Shalom PWillett E Mozeson S Pollak and DHardiman ldquoA multiple IMM estimation approach with unbi-ased mixing for thrusting projectilesrdquo IEEE Transactions onAerospace and Electronic Systems vol 48 no 4 pp 3250ndash32672012

[12] T Yuan Y Bar-Shalom P Willett and D Hardiman ldquoImpactpoint prediction for thrusting projectiles in the presence ofwindrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 50 no 1 pp 102ndash119 2014

[13] X Li and V Jilkov ldquoSurvey of maneuvering target trackingpart II motion models of ballistic and space targetsrdquo IEEETransactions on Aerospace and Electronic Systems vol 46 no1 pp 96ndash119 2010

[14] J Cesar Bolzani de Campos Ferreira and JWaldmann ldquoCovari-ance intersection-based sensor fusion for sounding rockettracking and impact area predictionrdquo Control Engineering Prac-tice vol 15 no 4 pp 389ndash409 2007

[15] httpwwwbraeunigusapollosaturnVhtm[16] RA Singer ldquoEstimating optimal tracking filter performance for

manned maneuvering targetsrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 6 no 4 pp 473ndash483 1970

[17] R A Singer and KW Behnke ldquoReal-time tracking filter evalu-ation and selection for tactical applicationsrdquo IEEE Transactionson Aerospace and Electronic Systems vol AES-7 no 1 pp 100ndash110 1971

[18] W Press B Flannery S Teukolsky and W Vetterling Numer-ical Recipes in FORTRAN The Art of Scientific ComputingCambridge University New York NY USA 2nd edition 1992

[19] S Dyer and J Dyer ldquoCubic-spline interpolation part Irdquo IEEEInstrumentation amp Measurement Magazine pp 44ndash46 2001

[20] Y Bar-Shalom X Li and T Kirubarajan Estimation with Appli-cations to Tracking and Navigation John Wiley amp Sons NewYork NY USA 2001

12 Mathematical Problems in Engineering

[21] N Morrison Introduction to Sequential Smoothing and Predic-tion McGraw-Hill 1969

[22] E Brookner Tracking and Kalman Filtering Made Easy JohnWiley amp Sons 1998

[23] P Suchomski ldquoExplicit expressions for debiased statistics of 3Dconverted measurementsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 35 no 1 pp 368ndash370 1999

[24] H R Song I Y Song and V Shin ldquoMultisensory predictionfusion of nonlinear functions of the state vector in discrete-timesystemsrdquo International Journal of Distributed Sensor Networksvol 2015 Article ID 249857 13 pages 2015

[25] H Song andYHan ldquoComparison of space launch vehicle track-ing using different types of multiple modelsrdquo in Proceedingsof the International Conference on Information Fusion pp 1ndash8Heidelberg Germany July 2016

[26] R KMehra ldquoA comparison of several nonlinear filters for reen-try vehicle trackingrdquo IEEE Transactions on Automatic Controlvol 16 no 4 pp 307ndash319 1971

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

01

minus20 4minus6 0

0028

120583 =

1205902 =

p(ax)

minus6ms2

4246

a (m2s4)

m2s4

(a)

01

minus5 810

005

120583 = 1ms2

1205902 = 106

p(ay)

a (m2s4)

m2s4

(b)

01

minus40 4minus7 0

0008

a (m2s4)

120583 = minus7ms2

1205902 = 12674 m2s4

p(az)

(c)

01

005 005

minus4 40

008

120583 = 0ms2

1205902 = 183

p(ax) p(ay) p(az)

a (m2s4)

m2s4

(d)

Figure 2 Nonzero mean acceleration PDF in the PFP model (a) along 119909-axis (b) along 119910-axis and (c) along 119911-axis Zero mean accelerationPDF in the CFP Model (d) along 119909-axis 119910-axis and 119911-axis

119911119871119896 = 119867119871x119871119896 + ]119896 119867119871 =[

[

[

1 0 0

0 03times2 1 03times2 0 03times20 0 1

]

]

]

x119871119896 = [119909119896 119896 119896 119910119896 119896

119896

119911119896 119896 119896]119879 (22)

where ]119896 is measurement noise (as shown in (14)) with errorcovariance

411 IMM with Singerrsquos Linear Model From (19)ndash(22) inSection 41 we can rewrite the Markov jump linear systemswhere the 119894th model of the finite multiple model set M =

119898(1)

119898(119872)

obeys the following equations

x119871119896+1 = Ψ(119894)

119896x119871119896 + 119908

(119894)

119896

119911119871119896 = 119867119871x119871119896 + ]119896(23)

where

cov (119908(119894)

119896) = 119876

(119894)

119896

cov (]119896) = 119877119896

Ψ(119894)

119896= diag (Ψ119909119896 (120572

(119894)

119909 Δ119905) Ψ119910119896 (120572

(119894)

119910 Δ119905)

Ψ119911119896 (120572(119894)

119911 Δ119905))

119876(119894)

119896= diag (119876119909119896 (120572

(119894)

119909 Δ119905) 119876119910119896 (120572

(119894)

119910 Δ119905)

119876119911119896 (120572(119894)

119911 Δ119905))

(24)

Here 1205902

119886119899for the SLV can be represented by TUM as in

Figure 2 The PDFs of Figures 2(a)ndash2(d) are experimentallysampled from the nominal acceleration profile of the SLVThesuperscript (i) denotes quantities pertinent to model 119898(119894) inM and the jumps of the system mode are assumed to havetransition probabilities

Pr 119898(119895)

119896+1| 119898

(119894)

119896 ≜ 120587119894119895

(25)

where 119898(119894)

119896denotes the event in which model 119898

(119894) matchesthe system mode in effect at time 119896 In our application 119872 =

2 and 119898(1) and 119898

(2) are the propelled and coasting modesrespectively

Finally complete recursion of the IMM with modematched KF for the SLV tracking is summarized as follows

(i) Model-conditioned reinitialization (for 119894 = 1 2

119872)

Mathematical Problems in Engineering 7

(a) predicted mode probability 120583(119894)

119871119896|119896minus1≜ Pr119898(119894)

119896|

119911119871119896minus1 = sum119895120587119895119894120583

(119895)

119871119896minus1

(b) mixingweight 120583119895|119894

119871119896minus1≜ Pr119898(119895)

119896minus1| 119898

(119894)

119896 119911119871119896minus1 =

sum119895120587119895119894120583

(119895)

119871119896minus1120583

(119894)

119871119896minus1

(c) mixing estimate and covariance

x(119894)119871119896minus1|119896minus1

≜ 119864 [x119871119896minus1|119896minus1 | 119898(119894)

119896 119911119871119896minus1]

= sum

119895

x(119895)119871119896minus1|119896minus1

120583119895|119894

119871119896minus1

119875

(119894)

119871119896minus1|119896minus1= sum

119895

[119875(119894)

119871119896minus1|119896minus1+ (x(119894)

119871119896minus1|119896minus1minus x(119895)

119871119896minus1|119896minus1)

sdot (x(119894)119871119896minus1|119896minus1

minus x(119895)119871119896minus1|119896minus1

)

119879

] 120583119895|119894

119871119896minus1

(26)

(ii) Model-conditioned filtering (for 119894 = 1 2 119872)

(a) predicted estimate and covariance

x(119894)119871119896|119896minus1

= Ψ(119894)

119896minus1x(119894)119871119896minus1|119896minus1

+ u(119894)

119896minus1

where u(1)

119896minus1= [119906

119879

119909119896minus1119906119879

119910119896minus1119906119879

119911119896minus1]

119879

u(2)

119896minus1= 03times1

119875(119894)

119871119896|119896minus1= Ψ

(119894)

119896minus1119875

(119894)

119871119896minus1|119896minus1Ψ

(119894)119879

119896minus1+ 119876

(119894)

119896minus1

(27)

(b) measurement residual (119894)

119871119896= 119911119871119896 minus119867119871x

(119894)

119871119896|119896minus1minus

]119896(c) residual covariance 119878(119894)

119871119896= 119867119871119875

(119894)

119871119896|119896minus1119867

119879

119871minus 119877119896

(d) filter gain 119870(119894)

119871119896= 119875

(119894)

119871119896|119896minus1119867

119879

119871(119878

(119894)

119871119896)minus1

(e) update of state and covariance x(119894)119871119896|119896

= x(119894)119871119896|119896minus1

+

119870(119894)

119871119896(119894)

119871119896 119875

(119894)

119871119896|119896= 119875

(119894)

119871119896|119896minus1minus 119870

(119894)

119871119896119878(119894)

119871119896119870

(119894)119879

119871119896

(iii) Mode probability update (for 119894 = 1 2 119872)

(a) mode likelihood Λ(119894)

119871119896≜ 119901[119911

(119894)

119871119896| 119898

(119894)

119896

119885119871119896minus1]assume

= N(119911(119894)

119871119896 0 119878

(119894)

119871119896)

(b) mode probability 120583(119894)

119871119896= 120583

(119894)

119871119896|119896minus1Λ

(119894)

119871119896

(sum119895120583(119895)

119871119896|119896minus1Λ

(119895)

119871119896)

(iv) Combination (for 119894 = 1 2 119872)

x119871119896|119896 = sum

119894

x(119894)119871119896|119896

120583(119894)

119871119896

(119894)

119871119896|119896

= sum

119894

[119875(119894)

119871119896|119896+ (x119871119896|119896 minus x(119894)

119871119896|119896) (x119871119896|119896 minus x(119894)

119871119896|119896)

119879

] 120583(119894)

119871119896

(28)

Singerrsquos MMS of time delayed slaving data is completed bypropagating the combined estimate based on the currentmodersquos dynamic model such that

xsync119871119896+119904|119896

= Φsync119896+119904119896

x119871119896|119896

Φsync119896+119904119896

= 119865sync119896+119904minus1

119865sync119896+119904minus2

sdot sdot sdot 119865sync119896

Φsync119896119896

= 119868119899

119896 gt 1

(29)

where 119865sync119896+119904minus1

119865sync119896+119904minus2

119865sync119896

are systemmatrices dependingon a flight phase mode at current time 119896

42 Synchronization Using an IMM with a Nonlinear BallisticModel For a nonlinear ballistic model the state vector forthe propelled mode is denoted as

x119905 = [119909119905 119910119905 119911119905 119905 119905

119905 120585119905 120591119905]119879 (30)

where 120585119905 is the drag coefficient and 120591119905 is the thrust Generallythe drag coefficient varies significantlywith theMachnumberregime subsonic transonic and supersonic Therefore wetake advantage of the dynamic model considering a Machnumber-dependent multiplier [11 12] such that

[

[

[

]

]

]

=

120591

119881

[

[

[

]

]

]

thrust term

+ 120585120585119898119863[

[

[

]

]

]

drag term

+ 119892[

[

[

0

0

1

]

]

]

gravity term

+ 1

120585 = 2

= 3

(31)

The first term on the right side of (31) represents the thrust(m2

s) of the SLV in 119909 119910 and 119911 directions Two distinctmultiplemodes of a nonlinear SLVmodel can be separated bythe existence of thrust In other words the existence of thrustsignifies propelled flight mode whereas zero thrust signifiescoasting flight mode 119881 is the magnitude of the velocity V =

[ ]

119879 that is the SLV speed (ms) The second part of(31) is the drag term which is related to velocity and altitudethat is 119863 = minus120588(119911)1198812 where 120588(119911) = 1205880119890

minus119888119911 is the air density(kgm3) at an altitude 119911 (m) and 119888 is the air density constant(mminus1) [9] 120585 is the drag coefficient and 120585119898 is theMachnumber-dependent drag coefficient multiplier which is approximatedby the cubic spline curve shown in Figure 3 In this paperfor the drag characteristics of the SLV which are applied inthe subsequent simulation section drag coefficients of theSaturn119881 launch vehicle [15] are usedThe third part of (31) isa gravity term Gravity 119892 is the standard acceleration due togravity at sea level which is assumed to be constant through-out the trajectory with a value of 9812 ms2 1 2 and 3

are assumed to be continuous time zeromeanwhite Gaussiannoises The drag coefficient and thrust acceleration are rep-resented as Wiener processes with a slow variation [11 12]

8 Mathematical Problems in Engineering

0 2 4 6 8 1004

06

08

1

12

14

16

18

2

Mach number

Nor

mal

ized

dra

g co

effici

ent

Selected pointsLinear interpolationCubic spline interpolation

Figure 3 Normalized drag coefficient [15]

We can modify the dynamic equations (30) and (31) as acompact form such that

x119905 = 119891 [x119905] + 119905 (32)

where

119891 [x119905] =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119905

119905

119905

120591

119905

119881119905

+ 120585119905119863119905119905

120591

119905

119881119905

+ 120585119905119863119905119905

120591

119905

119881119905

+ 120585119905119863119905119905 minus 119892

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

119905 = [1119905 2119905 3119905]119879

(33)

The state vector equation (33) is discretized by a second-order Taylor expansion [26] Then (33) can be written as

a discretized continuous time systemwithwhite process noisesuch that

x119896+1 = x119896 + 119891 [x119896] Δ119905 + 119860119896119891 [x119896]Δ119905

2

2

+ 120596119896(34)

where 119860119896 is the Jacobian of (33) evaluated at x119896 [26] and120596119896 is the discretized continuous time process noise for thesampling interval Δ119905 The corresponding covariance matrixof the discretized process noise is

119876 =[

[

[

1198761119902V 06times1 06times101times6 Δ119905119902120585 0

01times6 0 Δ119905119902120591

]

]

]

1198761 =

[

[

[

[

Δ1199053

3

1198683

Δ1199052

2

1198683

Δ1199052

2

1198683 Δ1199051198683

]

]

]

]

(35)

where 1198683 is the 3 times 3 identity matrix and the continuous timeprocess noise intensities 119902V 119902120585 and 119902120591 are the correspondingpower spectral densities

The measurement matrix 119867NL for the nonlinear ballisticmultiple model can be depicted as

119911NL119896 = 119867NLxNL119896 + ]119896

119867NL = [1198683 0] xNL119896 = [119909119896 119910119896 119911119896 119896 119896

119896 120585119896 120591119896]119879

(36)

where ]119896 is measurement noise with error covariance119877119896120575119896minus119895 = 119864[]119896]

119879

119895] that is (14)

An IMMalgorithm for nonlinear dynamics with differentsizes of the mode state vector is summarized as follows [25]

(v) Model-conditioned reinitialization (for 119894 = 1 2

119872)

(a) predicted mode probability 120583(119894)

NL119896|119896minus1 ≜

Pr119898(119894)

119896| 119911NL119896minus1 = sum

119895120587119895119894120583

(119895)

NL119896minus1(b) mixing weight

120583119895|119894

NL119896minus1 ≜ Pr 119898(119895)

119896minus1| 119898

(119894)

119896 119911NL119896minus1 =

sum119895120587119895119894120583

(119895)

NL119896minus1

120583(119894)

NL119896minus1

(37)

(c) unbiased mixing estimate and covariance

x(119894)NL119896minus1|119896minus1 ≜ 119864 [xNL119896minus1|119896minus1 | 119898(119894)

119896 119911NL119896minus1] = sum

119895

(119895)

NL119896minus1|119896minus1120583119895|119894

NL119896minus1

where (1)

NL119896minus1|119896minus1 ≜ x(1)NL119896minus1|119896minus1 (2)

NL119896minus1|119896minus1 ≜ [x(2)119879

NL119896minus1|119896minus1 120591119896minus1]119879

119875

(119894)

NL119896minus1|119896minus1 = sum

119895

[119875(119894)

NL119896minus1|119896minus1 + (x(119894)NL119896minus1|119896minus1 minus (119895)

NL119896minus1|119896minus1) (x(119894)NL119896minus1|119896minus1 minus (119895)

NL119896minus1|119896minus1)119879

] 120583119895|119894

NL119896minus1

(38)

Mathematical Problems in Engineering 9

(vi) Model-conditioned filtering (for 119894 = 1 2 119872)

(a) predicted estimate and covariance

x(119894)NL119896|119896minus1 = x(119894)NL119896minus1|119896minus1 + 119891 [x(119894)NL119896minus1|119896minus1] Δ119905

+

(119894)

NL119896minus1119891 [x(119894)NL119896minus1|119896minus1]Δ119905

2

2

(119894)

119896=

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x=x(119894)119896

119875(119894)

NL119896|119896minus1 = Ω(119894)

119896minus1119875

(119894)

NL119896minus1|119896minus1Ω(119894)119879

119896minus1+ 119876119896minus1

Ω(119894)

119896minus1= 119868 +

(119894)

NL119896minus1

(39)

(b) measurement residual (119894)

NL119896 = 119911NL119896 minus

119867NLx(119894)

NL119896|119896minus1 minus ]119896

(c) residual covariance 119878(119894)NL119896 = 119867NL119875(119894)

NL119896|119896minus1119867119879

NL minus

119877119896(d) filter gain 119870(119894)

NL119896 = 119875(119894)

NL119896|119896minus1119867119879

NL(119878(119894)

NL119896)minus1

(e) update of state and covariance

x(119894)NL119896|119896 = x(119894)NL119896|119896minus1 + 119870(119894)

NL119896(119894)

NL119896

119875(119894)

NL119896|119896 = 119875(119894)

NL119896|119896minus1 minus 119870(119894)

NL119896119878(119894)

NL119896119870(119894)119879

NL119896

(40)

(vii) Mode probability update (for 119894 = 1 2 119872)

(a) mode likelihood

Λ(119894)

NL119896 ≜ 119901 [119911(119894)

NL119896 | 119898(119894)

119896 119885NL119896minus1]

assume= N (119911

(119894)

NL119896 0 119878(119894)

NL119896)

(41)

(b) mode probability 120583(119894)

NL119896 = 120583(119894)

NL119896|119896minus1Λ(119894)

NL119896

(sum119895120583(119895)

NL119896|119896minus1Λ(119895)

NL119896)

(viii) Combination (for 119894 = 1 2 119872)

xNL119896|119896 = sum

119894

x(119894)NL119896|119896120583(119894)

NL119896

(119894)

NL119896|119896 = sum

119894

[119875(119894)

NL119896|119896

+ (xNL119896|119896 minus x(119894)NL119896|119896) (xNL119896|119896 minus x(119894)NL119896|119896)119879

] 120583(119894)

NL119896

(42)

The nonlinearMMS of time delayed slaving data is completedby propagating the combined estimate based on the currentmodersquos estimated vector such that

xNL119896+119904|119896 = xNL119896|119896 + 119891 [xNL119896|119896] sdot 119904 + NL119896119891 [xNL119896|119896]

sdot

1199042

2

(43)

where 119904 is a lead-time for synchronization (119904 = 119899 sdot Δ119905) If theSLV is in the PFP (120583(1)

NL119896 gt 120583(2)

NL119896) the current state estimateincludes the thrust term that is the state vector xNL119896 =

[119909119896 119910119896 119911119896 119896 119896

119896 120585119896 120591119896]119879 If the SLV is not in the PFP

the state vector does not include the thrust term (120591119896 = 0)

5 Simulation Results

To demonstrate the performance of the proposed synchro-nization approaches for delayed slaving data we simulatedthe SLV tracking problem based on the nominal flight tra-jectory of the Korea Space Launch Vehicle-I (KSLV-I) In thesimulation the radar measurement noise intensities of (12)are selected as 120590119903 = 15m 120590120593 = 001 deg and 120590120579 = 001 degThe nominal flight sequence of KSLV-I is as follows First thepayload fairing separates during the first stage flight at 2154 sAfter the first stage engine shutdown at 2287 s the upperstage separates from the first stage and enters the CFP Thesecond stage continues in the CFP until the kick motor igni-tion at 395 s and the vehicle then enters the PFP At the endof kickmotor combustion (4527 s) the upper stage enters thetarget orbit in CFP as in the previous separation Finally thesatellite is inserted into the target orbit after it separates fromthe upper stage during CFP at 540 s Since delayed slavingdata from the MCS to the TSs can occur in both the PFP andCFP synchronization simulations are conducted at arbitrarypoints of the flight phases Figure 4 shows the synchroniza-tion error of the delayed slaving data with respect to thebenchmark and the proposed approaches for delays of 119904 =

01 02 1 s Figures 4(a)ndash4(c) illustrate synchronizationerrors where delayed slaving data occurred at 155 s which isthe first stage of PFP and Figures 4(d)ndash4(f) show synchro-nization errors where delayed slaving data occurred at 280 swhich is the first stage of CFP As shown in Figures 4(a)ndash4(c)the extrapolation and120572-120573-120574filter-based approach cannot findthe proper synchronized slaving data The synchronizationerrors of these approaches exponentially increase accordingto increasing delay 119904 in particular the error of the extrapo-lation approaches an out-of-figure bound (2500m) On theother hand the Kalman prediction and the proposedapproaches have stable synchronization errors even if thedelay time 119904 is larger Furthermore we can observe thatthe errors in the synchronization approach of the Kalmanprediction are relatively larger than the proposed approachesespecially in the PFP As explained in the flight sequence ofKSLV-I in general the motion of the SLV is described bymultiple dynamics (in our case PFP and CFP) rather thansingle dynamics However the dynamic model of the KF andprediction capability used in this paper is the constant accel-eration (CA) model This means that the filter is optimizedwith respect to CAmotion which is actually not occurring inour application This is why the performance of the Kalmanprediction-based synchronization is worse than the proposedapproaches Nonetheless we can see that the 119910-axis synchro-nization errors shown in Figures 4(b) and 4(e) are smallThis result is observed because the 119910-axis motion of KSLV-I in a local coordinate is relatively smaller than the otheraxis motion and the CA model approximately expresses this

10 Mathematical Problems in Engineering

01 02 03 04 05 06 07 08 09 10

500

1000

1500

2000

2500

Delay (sec)

x-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

Cubic spline extrapolation

Kalman filter

(a)

(b)

(c)

(d)

(e)

(f)

Linear MMNonlinear MM120572-120573-120574 filter 120572-120573-120574 filter

120572-120573-120574 filter120572-120573-120574 filter

120572-120573-120574 filter 120572-120573-120574 filter

01 02 03 04 05 06 07 08 09 1Delay (sec)

0

500

1000

1500

2000

2500

y-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

z-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

x-p

ositi

on ab

solu

te er

ror i

n CF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500y

-pos

ition

abso

lute

erro

r in

CFP

(m)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

z-p

ositi

on ab

solu

te er

ror i

n CF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

Figure 4 Synchronization errors at PFP and CFP

Mathematical Problems in Engineering 11

small motion A single KF prediction model cannot exactlydescribe the motion of both PFP and CFP whereas theproposed multiple model-based approaches work well Asshown in Figure 4 regardless of the delay 119904 the proposedmultiplemodel-based approaches find synchronized positionvectors In the comparison of synchronization performancebetween linear IMM and nonlinear IMM synchronizationthe nonlinear IMM-based synchronization approach showsthe best performance except for 119910-axis where CA motion isdominant In addition the difference between the proposedmultiple model-based approaches is very small as shown inFigure 4 but the complexities of the algorithms for real-timeapplications are quite dissimilar Hence the operator mayadaptively select one of the proposed approaches accordingto onersquos environment

6 Conclusions

In this paper we investigated the time synchronizationapproaches of delayed slaving data in the RSS for SLV track-ingOne of themost important roles of the RSS is to distributeslaving data to each TS for continuous tracking of the SLV Ifthere is a critical network delay resulting in time delayed slav-ing data being sent to each TS theMCS will not receive accu-rate SLV tracking data This problem can give rise to signifi-cant difficulties for the SLVmission progress and analysis Toovercome this problemwe proposedMMSapproacheswhichtake advantage of the multiple motion models of an SLVThelinear IMM-based synchronization approach was developedusing Singerrsquos model with ternary uniform mixtures and thenonlinear IMM-based synchronization approachwas derivedfrom a nonlinear ballistic model with a drag coefficient Forverification of the proposed algorithms SLV tracking simu-lations using KSLV-I and the radar measurement data gen-erated from nominal trajectory were conducted To demon-strate the superiority of time synchronization performancein these simulations we compared the proposed algorithmwith benchmark approaches for absolute error between thenominal trajectory data and the synchronized slaving datathe simulation results demonstrated that the proposed MMSapproaches performed competitively

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] R Varaprasad and V Seshagiri Rao ldquoRange safety real-timesystem for satellite launch vehicle missions-testing methodolo-giesrdquo Defence Science Journal vol 56 no 5 pp 693ndash700 2006

[2] H R Song M G Jeon T S Choi and V Shin ldquoTwo fusionpredictors for discrete-time linear systems with different typesof observationsrdquo International Journal of Control Automationand Systems vol 7 no 4 pp 651ndash658 2009

[3] H Song V Shin and M Jeon ldquoMobile node localization usingfusion prediction-based interacting multiple model in cricketsensor networkrdquo IEEE Transactions on Industrial Electronicsvol 59 no 11 pp 4349ndash4359 2012

[4] T Kirubarajan and Y Bar-Shalom ldquoKalman filter versus IMMestimator when do we need the latterrdquo IEEE Transactions onAerospace and Electronic Systems vol 39 no 4 pp 1452ndash14572003

[5] E Mazor A Averbuch Y Bar-Shalom and J Dayan ldquoInteract-ing multiple model methods in target tracking a surveyrdquo IEEETransactions on Aerospace and Electronic Systems vol 34 no 1pp 103ndash123 1998

[6] P D Hanlon and P S Maybeck ldquoMultiple-model adaptiveestimation using a residual correlation Kalman filter bankrdquoIEEE Transactions on Aerospace and Electronic Systems vol 36no 2 pp 393ndash406 2000

[7] S-J Shin ldquoRe-entry vehicle tracking with a new multiplemodel estimation applicable to highly non-linear dynamicsrdquoIET Radar Sonar amp Navigation vol 9 no 5 pp 581ndash588 2015

[8] S Conover J Kerce G Brown L Ehrman and D HardimanldquoImpact point prediction of small ballistic munitions with aninteracting multiple model estimatorrdquo in Acquisition TrackingPointing and Laser Systems Technologies XXI vol 6569 ofProceedings of SPIE Orlando Fla USA 2007

[9] V C Ravindra Y Bar-Shalom and P Willett ldquoProjectileidentification and impact point predictionrdquo IEEE Transactionson Aerospace and Electronic Systems vol 46 no 4 pp 2004ndash2021 2010

[10] W J Farrell III ldquoInteracting multiple model filter for tacticalballistic missile trackingrdquo IEEE Transactions on Aerospace andElectronic Systems vol 44 no 2 pp 418ndash426 2008

[11] T Yuan Y Bar-Shalom PWillett E Mozeson S Pollak and DHardiman ldquoA multiple IMM estimation approach with unbi-ased mixing for thrusting projectilesrdquo IEEE Transactions onAerospace and Electronic Systems vol 48 no 4 pp 3250ndash32672012

[12] T Yuan Y Bar-Shalom P Willett and D Hardiman ldquoImpactpoint prediction for thrusting projectiles in the presence ofwindrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 50 no 1 pp 102ndash119 2014

[13] X Li and V Jilkov ldquoSurvey of maneuvering target trackingpart II motion models of ballistic and space targetsrdquo IEEETransactions on Aerospace and Electronic Systems vol 46 no1 pp 96ndash119 2010

[14] J Cesar Bolzani de Campos Ferreira and JWaldmann ldquoCovari-ance intersection-based sensor fusion for sounding rockettracking and impact area predictionrdquo Control Engineering Prac-tice vol 15 no 4 pp 389ndash409 2007

[15] httpwwwbraeunigusapollosaturnVhtm[16] RA Singer ldquoEstimating optimal tracking filter performance for

manned maneuvering targetsrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 6 no 4 pp 473ndash483 1970

[17] R A Singer and KW Behnke ldquoReal-time tracking filter evalu-ation and selection for tactical applicationsrdquo IEEE Transactionson Aerospace and Electronic Systems vol AES-7 no 1 pp 100ndash110 1971

[18] W Press B Flannery S Teukolsky and W Vetterling Numer-ical Recipes in FORTRAN The Art of Scientific ComputingCambridge University New York NY USA 2nd edition 1992

[19] S Dyer and J Dyer ldquoCubic-spline interpolation part Irdquo IEEEInstrumentation amp Measurement Magazine pp 44ndash46 2001

[20] Y Bar-Shalom X Li and T Kirubarajan Estimation with Appli-cations to Tracking and Navigation John Wiley amp Sons NewYork NY USA 2001

12 Mathematical Problems in Engineering

[21] N Morrison Introduction to Sequential Smoothing and Predic-tion McGraw-Hill 1969

[22] E Brookner Tracking and Kalman Filtering Made Easy JohnWiley amp Sons 1998

[23] P Suchomski ldquoExplicit expressions for debiased statistics of 3Dconverted measurementsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 35 no 1 pp 368ndash370 1999

[24] H R Song I Y Song and V Shin ldquoMultisensory predictionfusion of nonlinear functions of the state vector in discrete-timesystemsrdquo International Journal of Distributed Sensor Networksvol 2015 Article ID 249857 13 pages 2015

[25] H Song andYHan ldquoComparison of space launch vehicle track-ing using different types of multiple modelsrdquo in Proceedingsof the International Conference on Information Fusion pp 1ndash8Heidelberg Germany July 2016

[26] R KMehra ldquoA comparison of several nonlinear filters for reen-try vehicle trackingrdquo IEEE Transactions on Automatic Controlvol 16 no 4 pp 307ndash319 1971

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

(a) predicted mode probability 120583(119894)

119871119896|119896minus1≜ Pr119898(119894)

119896|

119911119871119896minus1 = sum119895120587119895119894120583

(119895)

119871119896minus1

(b) mixingweight 120583119895|119894

119871119896minus1≜ Pr119898(119895)

119896minus1| 119898

(119894)

119896 119911119871119896minus1 =

sum119895120587119895119894120583

(119895)

119871119896minus1120583

(119894)

119871119896minus1

(c) mixing estimate and covariance

x(119894)119871119896minus1|119896minus1

≜ 119864 [x119871119896minus1|119896minus1 | 119898(119894)

119896 119911119871119896minus1]

= sum

119895

x(119895)119871119896minus1|119896minus1

120583119895|119894

119871119896minus1

119875

(119894)

119871119896minus1|119896minus1= sum

119895

[119875(119894)

119871119896minus1|119896minus1+ (x(119894)

119871119896minus1|119896minus1minus x(119895)

119871119896minus1|119896minus1)

sdot (x(119894)119871119896minus1|119896minus1

minus x(119895)119871119896minus1|119896minus1

)

119879

] 120583119895|119894

119871119896minus1

(26)

(ii) Model-conditioned filtering (for 119894 = 1 2 119872)

(a) predicted estimate and covariance

x(119894)119871119896|119896minus1

= Ψ(119894)

119896minus1x(119894)119871119896minus1|119896minus1

+ u(119894)

119896minus1

where u(1)

119896minus1= [119906

119879

119909119896minus1119906119879

119910119896minus1119906119879

119911119896minus1]

119879

u(2)

119896minus1= 03times1

119875(119894)

119871119896|119896minus1= Ψ

(119894)

119896minus1119875

(119894)

119871119896minus1|119896minus1Ψ

(119894)119879

119896minus1+ 119876

(119894)

119896minus1

(27)

(b) measurement residual (119894)

119871119896= 119911119871119896 minus119867119871x

(119894)

119871119896|119896minus1minus

]119896(c) residual covariance 119878(119894)

119871119896= 119867119871119875

(119894)

119871119896|119896minus1119867

119879

119871minus 119877119896

(d) filter gain 119870(119894)

119871119896= 119875

(119894)

119871119896|119896minus1119867

119879

119871(119878

(119894)

119871119896)minus1

(e) update of state and covariance x(119894)119871119896|119896

= x(119894)119871119896|119896minus1

+

119870(119894)

119871119896(119894)

119871119896 119875

(119894)

119871119896|119896= 119875

(119894)

119871119896|119896minus1minus 119870

(119894)

119871119896119878(119894)

119871119896119870

(119894)119879

119871119896

(iii) Mode probability update (for 119894 = 1 2 119872)

(a) mode likelihood Λ(119894)

119871119896≜ 119901[119911

(119894)

119871119896| 119898

(119894)

119896

119885119871119896minus1]assume

= N(119911(119894)

119871119896 0 119878

(119894)

119871119896)

(b) mode probability 120583(119894)

119871119896= 120583

(119894)

119871119896|119896minus1Λ

(119894)

119871119896

(sum119895120583(119895)

119871119896|119896minus1Λ

(119895)

119871119896)

(iv) Combination (for 119894 = 1 2 119872)

x119871119896|119896 = sum

119894

x(119894)119871119896|119896

120583(119894)

119871119896

(119894)

119871119896|119896

= sum

119894

[119875(119894)

119871119896|119896+ (x119871119896|119896 minus x(119894)

119871119896|119896) (x119871119896|119896 minus x(119894)

119871119896|119896)

119879

] 120583(119894)

119871119896

(28)

Singerrsquos MMS of time delayed slaving data is completed bypropagating the combined estimate based on the currentmodersquos dynamic model such that

xsync119871119896+119904|119896

= Φsync119896+119904119896

x119871119896|119896

Φsync119896+119904119896

= 119865sync119896+119904minus1

119865sync119896+119904minus2

sdot sdot sdot 119865sync119896

Φsync119896119896

= 119868119899

119896 gt 1

(29)

where 119865sync119896+119904minus1

119865sync119896+119904minus2

119865sync119896

are systemmatrices dependingon a flight phase mode at current time 119896

42 Synchronization Using an IMM with a Nonlinear BallisticModel For a nonlinear ballistic model the state vector forthe propelled mode is denoted as

x119905 = [119909119905 119910119905 119911119905 119905 119905

119905 120585119905 120591119905]119879 (30)

where 120585119905 is the drag coefficient and 120591119905 is the thrust Generallythe drag coefficient varies significantlywith theMachnumberregime subsonic transonic and supersonic Therefore wetake advantage of the dynamic model considering a Machnumber-dependent multiplier [11 12] such that

[

[

[

]

]

]

=

120591

119881

[

[

[

]

]

]

thrust term

+ 120585120585119898119863[

[

[

]

]

]

drag term

+ 119892[

[

[

0

0

1

]

]

]

gravity term

+ 1

120585 = 2

= 3

(31)

The first term on the right side of (31) represents the thrust(m2

s) of the SLV in 119909 119910 and 119911 directions Two distinctmultiplemodes of a nonlinear SLVmodel can be separated bythe existence of thrust In other words the existence of thrustsignifies propelled flight mode whereas zero thrust signifiescoasting flight mode 119881 is the magnitude of the velocity V =

[ ]

119879 that is the SLV speed (ms) The second part of(31) is the drag term which is related to velocity and altitudethat is 119863 = minus120588(119911)1198812 where 120588(119911) = 1205880119890

minus119888119911 is the air density(kgm3) at an altitude 119911 (m) and 119888 is the air density constant(mminus1) [9] 120585 is the drag coefficient and 120585119898 is theMachnumber-dependent drag coefficient multiplier which is approximatedby the cubic spline curve shown in Figure 3 In this paperfor the drag characteristics of the SLV which are applied inthe subsequent simulation section drag coefficients of theSaturn119881 launch vehicle [15] are usedThe third part of (31) isa gravity term Gravity 119892 is the standard acceleration due togravity at sea level which is assumed to be constant through-out the trajectory with a value of 9812 ms2 1 2 and 3

are assumed to be continuous time zeromeanwhite Gaussiannoises The drag coefficient and thrust acceleration are rep-resented as Wiener processes with a slow variation [11 12]

8 Mathematical Problems in Engineering

0 2 4 6 8 1004

06

08

1

12

14

16

18

2

Mach number

Nor

mal

ized

dra

g co

effici

ent

Selected pointsLinear interpolationCubic spline interpolation

Figure 3 Normalized drag coefficient [15]

We can modify the dynamic equations (30) and (31) as acompact form such that

x119905 = 119891 [x119905] + 119905 (32)

where

119891 [x119905] =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119905

119905

119905

120591

119905

119881119905

+ 120585119905119863119905119905

120591

119905

119881119905

+ 120585119905119863119905119905

120591

119905

119881119905

+ 120585119905119863119905119905 minus 119892

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

119905 = [1119905 2119905 3119905]119879

(33)

The state vector equation (33) is discretized by a second-order Taylor expansion [26] Then (33) can be written as

a discretized continuous time systemwithwhite process noisesuch that

x119896+1 = x119896 + 119891 [x119896] Δ119905 + 119860119896119891 [x119896]Δ119905

2

2

+ 120596119896(34)

where 119860119896 is the Jacobian of (33) evaluated at x119896 [26] and120596119896 is the discretized continuous time process noise for thesampling interval Δ119905 The corresponding covariance matrixof the discretized process noise is

119876 =[

[

[

1198761119902V 06times1 06times101times6 Δ119905119902120585 0

01times6 0 Δ119905119902120591

]

]

]

1198761 =

[

[

[

[

Δ1199053

3

1198683

Δ1199052

2

1198683

Δ1199052

2

1198683 Δ1199051198683

]

]

]

]

(35)

where 1198683 is the 3 times 3 identity matrix and the continuous timeprocess noise intensities 119902V 119902120585 and 119902120591 are the correspondingpower spectral densities

The measurement matrix 119867NL for the nonlinear ballisticmultiple model can be depicted as

119911NL119896 = 119867NLxNL119896 + ]119896

119867NL = [1198683 0] xNL119896 = [119909119896 119910119896 119911119896 119896 119896

119896 120585119896 120591119896]119879

(36)

where ]119896 is measurement noise with error covariance119877119896120575119896minus119895 = 119864[]119896]

119879

119895] that is (14)

An IMMalgorithm for nonlinear dynamics with differentsizes of the mode state vector is summarized as follows [25]

(v) Model-conditioned reinitialization (for 119894 = 1 2

119872)

(a) predicted mode probability 120583(119894)

NL119896|119896minus1 ≜

Pr119898(119894)

119896| 119911NL119896minus1 = sum

119895120587119895119894120583

(119895)

NL119896minus1(b) mixing weight

120583119895|119894

NL119896minus1 ≜ Pr 119898(119895)

119896minus1| 119898

(119894)

119896 119911NL119896minus1 =

sum119895120587119895119894120583

(119895)

NL119896minus1

120583(119894)

NL119896minus1

(37)

(c) unbiased mixing estimate and covariance

x(119894)NL119896minus1|119896minus1 ≜ 119864 [xNL119896minus1|119896minus1 | 119898(119894)

119896 119911NL119896minus1] = sum

119895

(119895)

NL119896minus1|119896minus1120583119895|119894

NL119896minus1

where (1)

NL119896minus1|119896minus1 ≜ x(1)NL119896minus1|119896minus1 (2)

NL119896minus1|119896minus1 ≜ [x(2)119879

NL119896minus1|119896minus1 120591119896minus1]119879

119875

(119894)

NL119896minus1|119896minus1 = sum

119895

[119875(119894)

NL119896minus1|119896minus1 + (x(119894)NL119896minus1|119896minus1 minus (119895)

NL119896minus1|119896minus1) (x(119894)NL119896minus1|119896minus1 minus (119895)

NL119896minus1|119896minus1)119879

] 120583119895|119894

NL119896minus1

(38)

Mathematical Problems in Engineering 9

(vi) Model-conditioned filtering (for 119894 = 1 2 119872)

(a) predicted estimate and covariance

x(119894)NL119896|119896minus1 = x(119894)NL119896minus1|119896minus1 + 119891 [x(119894)NL119896minus1|119896minus1] Δ119905

+

(119894)

NL119896minus1119891 [x(119894)NL119896minus1|119896minus1]Δ119905

2

2

(119894)

119896=

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x=x(119894)119896

119875(119894)

NL119896|119896minus1 = Ω(119894)

119896minus1119875

(119894)

NL119896minus1|119896minus1Ω(119894)119879

119896minus1+ 119876119896minus1

Ω(119894)

119896minus1= 119868 +

(119894)

NL119896minus1

(39)

(b) measurement residual (119894)

NL119896 = 119911NL119896 minus

119867NLx(119894)

NL119896|119896minus1 minus ]119896

(c) residual covariance 119878(119894)NL119896 = 119867NL119875(119894)

NL119896|119896minus1119867119879

NL minus

119877119896(d) filter gain 119870(119894)

NL119896 = 119875(119894)

NL119896|119896minus1119867119879

NL(119878(119894)

NL119896)minus1

(e) update of state and covariance

x(119894)NL119896|119896 = x(119894)NL119896|119896minus1 + 119870(119894)

NL119896(119894)

NL119896

119875(119894)

NL119896|119896 = 119875(119894)

NL119896|119896minus1 minus 119870(119894)

NL119896119878(119894)

NL119896119870(119894)119879

NL119896

(40)

(vii) Mode probability update (for 119894 = 1 2 119872)

(a) mode likelihood

Λ(119894)

NL119896 ≜ 119901 [119911(119894)

NL119896 | 119898(119894)

119896 119885NL119896minus1]

assume= N (119911

(119894)

NL119896 0 119878(119894)

NL119896)

(41)

(b) mode probability 120583(119894)

NL119896 = 120583(119894)

NL119896|119896minus1Λ(119894)

NL119896

(sum119895120583(119895)

NL119896|119896minus1Λ(119895)

NL119896)

(viii) Combination (for 119894 = 1 2 119872)

xNL119896|119896 = sum

119894

x(119894)NL119896|119896120583(119894)

NL119896

(119894)

NL119896|119896 = sum

119894

[119875(119894)

NL119896|119896

+ (xNL119896|119896 minus x(119894)NL119896|119896) (xNL119896|119896 minus x(119894)NL119896|119896)119879

] 120583(119894)

NL119896

(42)

The nonlinearMMS of time delayed slaving data is completedby propagating the combined estimate based on the currentmodersquos estimated vector such that

xNL119896+119904|119896 = xNL119896|119896 + 119891 [xNL119896|119896] sdot 119904 + NL119896119891 [xNL119896|119896]

sdot

1199042

2

(43)

where 119904 is a lead-time for synchronization (119904 = 119899 sdot Δ119905) If theSLV is in the PFP (120583(1)

NL119896 gt 120583(2)

NL119896) the current state estimateincludes the thrust term that is the state vector xNL119896 =

[119909119896 119910119896 119911119896 119896 119896

119896 120585119896 120591119896]119879 If the SLV is not in the PFP

the state vector does not include the thrust term (120591119896 = 0)

5 Simulation Results

To demonstrate the performance of the proposed synchro-nization approaches for delayed slaving data we simulatedthe SLV tracking problem based on the nominal flight tra-jectory of the Korea Space Launch Vehicle-I (KSLV-I) In thesimulation the radar measurement noise intensities of (12)are selected as 120590119903 = 15m 120590120593 = 001 deg and 120590120579 = 001 degThe nominal flight sequence of KSLV-I is as follows First thepayload fairing separates during the first stage flight at 2154 sAfter the first stage engine shutdown at 2287 s the upperstage separates from the first stage and enters the CFP Thesecond stage continues in the CFP until the kick motor igni-tion at 395 s and the vehicle then enters the PFP At the endof kickmotor combustion (4527 s) the upper stage enters thetarget orbit in CFP as in the previous separation Finally thesatellite is inserted into the target orbit after it separates fromthe upper stage during CFP at 540 s Since delayed slavingdata from the MCS to the TSs can occur in both the PFP andCFP synchronization simulations are conducted at arbitrarypoints of the flight phases Figure 4 shows the synchroniza-tion error of the delayed slaving data with respect to thebenchmark and the proposed approaches for delays of 119904 =

01 02 1 s Figures 4(a)ndash4(c) illustrate synchronizationerrors where delayed slaving data occurred at 155 s which isthe first stage of PFP and Figures 4(d)ndash4(f) show synchro-nization errors where delayed slaving data occurred at 280 swhich is the first stage of CFP As shown in Figures 4(a)ndash4(c)the extrapolation and120572-120573-120574filter-based approach cannot findthe proper synchronized slaving data The synchronizationerrors of these approaches exponentially increase accordingto increasing delay 119904 in particular the error of the extrapo-lation approaches an out-of-figure bound (2500m) On theother hand the Kalman prediction and the proposedapproaches have stable synchronization errors even if thedelay time 119904 is larger Furthermore we can observe thatthe errors in the synchronization approach of the Kalmanprediction are relatively larger than the proposed approachesespecially in the PFP As explained in the flight sequence ofKSLV-I in general the motion of the SLV is described bymultiple dynamics (in our case PFP and CFP) rather thansingle dynamics However the dynamic model of the KF andprediction capability used in this paper is the constant accel-eration (CA) model This means that the filter is optimizedwith respect to CAmotion which is actually not occurring inour application This is why the performance of the Kalmanprediction-based synchronization is worse than the proposedapproaches Nonetheless we can see that the 119910-axis synchro-nization errors shown in Figures 4(b) and 4(e) are smallThis result is observed because the 119910-axis motion of KSLV-I in a local coordinate is relatively smaller than the otheraxis motion and the CA model approximately expresses this

10 Mathematical Problems in Engineering

01 02 03 04 05 06 07 08 09 10

500

1000

1500

2000

2500

Delay (sec)

x-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

Cubic spline extrapolation

Kalman filter

(a)

(b)

(c)

(d)

(e)

(f)

Linear MMNonlinear MM120572-120573-120574 filter 120572-120573-120574 filter

120572-120573-120574 filter120572-120573-120574 filter

120572-120573-120574 filter 120572-120573-120574 filter

01 02 03 04 05 06 07 08 09 1Delay (sec)

0

500

1000

1500

2000

2500

y-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

z-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

x-p

ositi

on ab

solu

te er

ror i

n CF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500y

-pos

ition

abso

lute

erro

r in

CFP

(m)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

z-p

ositi

on ab

solu

te er

ror i

n CF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

Figure 4 Synchronization errors at PFP and CFP

Mathematical Problems in Engineering 11

small motion A single KF prediction model cannot exactlydescribe the motion of both PFP and CFP whereas theproposed multiple model-based approaches work well Asshown in Figure 4 regardless of the delay 119904 the proposedmultiplemodel-based approaches find synchronized positionvectors In the comparison of synchronization performancebetween linear IMM and nonlinear IMM synchronizationthe nonlinear IMM-based synchronization approach showsthe best performance except for 119910-axis where CA motion isdominant In addition the difference between the proposedmultiple model-based approaches is very small as shown inFigure 4 but the complexities of the algorithms for real-timeapplications are quite dissimilar Hence the operator mayadaptively select one of the proposed approaches accordingto onersquos environment

6 Conclusions

In this paper we investigated the time synchronizationapproaches of delayed slaving data in the RSS for SLV track-ingOne of themost important roles of the RSS is to distributeslaving data to each TS for continuous tracking of the SLV Ifthere is a critical network delay resulting in time delayed slav-ing data being sent to each TS theMCS will not receive accu-rate SLV tracking data This problem can give rise to signifi-cant difficulties for the SLVmission progress and analysis Toovercome this problemwe proposedMMSapproacheswhichtake advantage of the multiple motion models of an SLVThelinear IMM-based synchronization approach was developedusing Singerrsquos model with ternary uniform mixtures and thenonlinear IMM-based synchronization approachwas derivedfrom a nonlinear ballistic model with a drag coefficient Forverification of the proposed algorithms SLV tracking simu-lations using KSLV-I and the radar measurement data gen-erated from nominal trajectory were conducted To demon-strate the superiority of time synchronization performancein these simulations we compared the proposed algorithmwith benchmark approaches for absolute error between thenominal trajectory data and the synchronized slaving datathe simulation results demonstrated that the proposed MMSapproaches performed competitively

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] R Varaprasad and V Seshagiri Rao ldquoRange safety real-timesystem for satellite launch vehicle missions-testing methodolo-giesrdquo Defence Science Journal vol 56 no 5 pp 693ndash700 2006

[2] H R Song M G Jeon T S Choi and V Shin ldquoTwo fusionpredictors for discrete-time linear systems with different typesof observationsrdquo International Journal of Control Automationand Systems vol 7 no 4 pp 651ndash658 2009

[3] H Song V Shin and M Jeon ldquoMobile node localization usingfusion prediction-based interacting multiple model in cricketsensor networkrdquo IEEE Transactions on Industrial Electronicsvol 59 no 11 pp 4349ndash4359 2012

[4] T Kirubarajan and Y Bar-Shalom ldquoKalman filter versus IMMestimator when do we need the latterrdquo IEEE Transactions onAerospace and Electronic Systems vol 39 no 4 pp 1452ndash14572003

[5] E Mazor A Averbuch Y Bar-Shalom and J Dayan ldquoInteract-ing multiple model methods in target tracking a surveyrdquo IEEETransactions on Aerospace and Electronic Systems vol 34 no 1pp 103ndash123 1998

[6] P D Hanlon and P S Maybeck ldquoMultiple-model adaptiveestimation using a residual correlation Kalman filter bankrdquoIEEE Transactions on Aerospace and Electronic Systems vol 36no 2 pp 393ndash406 2000

[7] S-J Shin ldquoRe-entry vehicle tracking with a new multiplemodel estimation applicable to highly non-linear dynamicsrdquoIET Radar Sonar amp Navigation vol 9 no 5 pp 581ndash588 2015

[8] S Conover J Kerce G Brown L Ehrman and D HardimanldquoImpact point prediction of small ballistic munitions with aninteracting multiple model estimatorrdquo in Acquisition TrackingPointing and Laser Systems Technologies XXI vol 6569 ofProceedings of SPIE Orlando Fla USA 2007

[9] V C Ravindra Y Bar-Shalom and P Willett ldquoProjectileidentification and impact point predictionrdquo IEEE Transactionson Aerospace and Electronic Systems vol 46 no 4 pp 2004ndash2021 2010

[10] W J Farrell III ldquoInteracting multiple model filter for tacticalballistic missile trackingrdquo IEEE Transactions on Aerospace andElectronic Systems vol 44 no 2 pp 418ndash426 2008

[11] T Yuan Y Bar-Shalom PWillett E Mozeson S Pollak and DHardiman ldquoA multiple IMM estimation approach with unbi-ased mixing for thrusting projectilesrdquo IEEE Transactions onAerospace and Electronic Systems vol 48 no 4 pp 3250ndash32672012

[12] T Yuan Y Bar-Shalom P Willett and D Hardiman ldquoImpactpoint prediction for thrusting projectiles in the presence ofwindrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 50 no 1 pp 102ndash119 2014

[13] X Li and V Jilkov ldquoSurvey of maneuvering target trackingpart II motion models of ballistic and space targetsrdquo IEEETransactions on Aerospace and Electronic Systems vol 46 no1 pp 96ndash119 2010

[14] J Cesar Bolzani de Campos Ferreira and JWaldmann ldquoCovari-ance intersection-based sensor fusion for sounding rockettracking and impact area predictionrdquo Control Engineering Prac-tice vol 15 no 4 pp 389ndash409 2007

[15] httpwwwbraeunigusapollosaturnVhtm[16] RA Singer ldquoEstimating optimal tracking filter performance for

manned maneuvering targetsrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 6 no 4 pp 473ndash483 1970

[17] R A Singer and KW Behnke ldquoReal-time tracking filter evalu-ation and selection for tactical applicationsrdquo IEEE Transactionson Aerospace and Electronic Systems vol AES-7 no 1 pp 100ndash110 1971

[18] W Press B Flannery S Teukolsky and W Vetterling Numer-ical Recipes in FORTRAN The Art of Scientific ComputingCambridge University New York NY USA 2nd edition 1992

[19] S Dyer and J Dyer ldquoCubic-spline interpolation part Irdquo IEEEInstrumentation amp Measurement Magazine pp 44ndash46 2001

[20] Y Bar-Shalom X Li and T Kirubarajan Estimation with Appli-cations to Tracking and Navigation John Wiley amp Sons NewYork NY USA 2001

12 Mathematical Problems in Engineering

[21] N Morrison Introduction to Sequential Smoothing and Predic-tion McGraw-Hill 1969

[22] E Brookner Tracking and Kalman Filtering Made Easy JohnWiley amp Sons 1998

[23] P Suchomski ldquoExplicit expressions for debiased statistics of 3Dconverted measurementsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 35 no 1 pp 368ndash370 1999

[24] H R Song I Y Song and V Shin ldquoMultisensory predictionfusion of nonlinear functions of the state vector in discrete-timesystemsrdquo International Journal of Distributed Sensor Networksvol 2015 Article ID 249857 13 pages 2015

[25] H Song andYHan ldquoComparison of space launch vehicle track-ing using different types of multiple modelsrdquo in Proceedingsof the International Conference on Information Fusion pp 1ndash8Heidelberg Germany July 2016

[26] R KMehra ldquoA comparison of several nonlinear filters for reen-try vehicle trackingrdquo IEEE Transactions on Automatic Controlvol 16 no 4 pp 307ndash319 1971

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0 2 4 6 8 1004

06

08

1

12

14

16

18

2

Mach number

Nor

mal

ized

dra

g co

effici

ent

Selected pointsLinear interpolationCubic spline interpolation

Figure 3 Normalized drag coefficient [15]

We can modify the dynamic equations (30) and (31) as acompact form such that

x119905 = 119891 [x119905] + 119905 (32)

where

119891 [x119905] =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119905

119905

119905

120591

119905

119881119905

+ 120585119905119863119905119905

120591

119905

119881119905

+ 120585119905119863119905119905

120591

119905

119881119905

+ 120585119905119863119905119905 minus 119892

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

119905 = [1119905 2119905 3119905]119879

(33)

The state vector equation (33) is discretized by a second-order Taylor expansion [26] Then (33) can be written as

a discretized continuous time systemwithwhite process noisesuch that

x119896+1 = x119896 + 119891 [x119896] Δ119905 + 119860119896119891 [x119896]Δ119905

2

2

+ 120596119896(34)

where 119860119896 is the Jacobian of (33) evaluated at x119896 [26] and120596119896 is the discretized continuous time process noise for thesampling interval Δ119905 The corresponding covariance matrixof the discretized process noise is

119876 =[

[

[

1198761119902V 06times1 06times101times6 Δ119905119902120585 0

01times6 0 Δ119905119902120591

]

]

]

1198761 =

[

[

[

[

Δ1199053

3

1198683

Δ1199052

2

1198683

Δ1199052

2

1198683 Δ1199051198683

]

]

]

]

(35)

where 1198683 is the 3 times 3 identity matrix and the continuous timeprocess noise intensities 119902V 119902120585 and 119902120591 are the correspondingpower spectral densities

The measurement matrix 119867NL for the nonlinear ballisticmultiple model can be depicted as

119911NL119896 = 119867NLxNL119896 + ]119896

119867NL = [1198683 0] xNL119896 = [119909119896 119910119896 119911119896 119896 119896

119896 120585119896 120591119896]119879

(36)

where ]119896 is measurement noise with error covariance119877119896120575119896minus119895 = 119864[]119896]

119879

119895] that is (14)

An IMMalgorithm for nonlinear dynamics with differentsizes of the mode state vector is summarized as follows [25]

(v) Model-conditioned reinitialization (for 119894 = 1 2

119872)

(a) predicted mode probability 120583(119894)

NL119896|119896minus1 ≜

Pr119898(119894)

119896| 119911NL119896minus1 = sum

119895120587119895119894120583

(119895)

NL119896minus1(b) mixing weight

120583119895|119894

NL119896minus1 ≜ Pr 119898(119895)

119896minus1| 119898

(119894)

119896 119911NL119896minus1 =

sum119895120587119895119894120583

(119895)

NL119896minus1

120583(119894)

NL119896minus1

(37)

(c) unbiased mixing estimate and covariance

x(119894)NL119896minus1|119896minus1 ≜ 119864 [xNL119896minus1|119896minus1 | 119898(119894)

119896 119911NL119896minus1] = sum

119895

(119895)

NL119896minus1|119896minus1120583119895|119894

NL119896minus1

where (1)

NL119896minus1|119896minus1 ≜ x(1)NL119896minus1|119896minus1 (2)

NL119896minus1|119896minus1 ≜ [x(2)119879

NL119896minus1|119896minus1 120591119896minus1]119879

119875

(119894)

NL119896minus1|119896minus1 = sum

119895

[119875(119894)

NL119896minus1|119896minus1 + (x(119894)NL119896minus1|119896minus1 minus (119895)

NL119896minus1|119896minus1) (x(119894)NL119896minus1|119896minus1 minus (119895)

NL119896minus1|119896minus1)119879

] 120583119895|119894

NL119896minus1

(38)

Mathematical Problems in Engineering 9

(vi) Model-conditioned filtering (for 119894 = 1 2 119872)

(a) predicted estimate and covariance

x(119894)NL119896|119896minus1 = x(119894)NL119896minus1|119896minus1 + 119891 [x(119894)NL119896minus1|119896minus1] Δ119905

+

(119894)

NL119896minus1119891 [x(119894)NL119896minus1|119896minus1]Δ119905

2

2

(119894)

119896=

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x=x(119894)119896

119875(119894)

NL119896|119896minus1 = Ω(119894)

119896minus1119875

(119894)

NL119896minus1|119896minus1Ω(119894)119879

119896minus1+ 119876119896minus1

Ω(119894)

119896minus1= 119868 +

(119894)

NL119896minus1

(39)

(b) measurement residual (119894)

NL119896 = 119911NL119896 minus

119867NLx(119894)

NL119896|119896minus1 minus ]119896

(c) residual covariance 119878(119894)NL119896 = 119867NL119875(119894)

NL119896|119896minus1119867119879

NL minus

119877119896(d) filter gain 119870(119894)

NL119896 = 119875(119894)

NL119896|119896minus1119867119879

NL(119878(119894)

NL119896)minus1

(e) update of state and covariance

x(119894)NL119896|119896 = x(119894)NL119896|119896minus1 + 119870(119894)

NL119896(119894)

NL119896

119875(119894)

NL119896|119896 = 119875(119894)

NL119896|119896minus1 minus 119870(119894)

NL119896119878(119894)

NL119896119870(119894)119879

NL119896

(40)

(vii) Mode probability update (for 119894 = 1 2 119872)

(a) mode likelihood

Λ(119894)

NL119896 ≜ 119901 [119911(119894)

NL119896 | 119898(119894)

119896 119885NL119896minus1]

assume= N (119911

(119894)

NL119896 0 119878(119894)

NL119896)

(41)

(b) mode probability 120583(119894)

NL119896 = 120583(119894)

NL119896|119896minus1Λ(119894)

NL119896

(sum119895120583(119895)

NL119896|119896minus1Λ(119895)

NL119896)

(viii) Combination (for 119894 = 1 2 119872)

xNL119896|119896 = sum

119894

x(119894)NL119896|119896120583(119894)

NL119896

(119894)

NL119896|119896 = sum

119894

[119875(119894)

NL119896|119896

+ (xNL119896|119896 minus x(119894)NL119896|119896) (xNL119896|119896 minus x(119894)NL119896|119896)119879

] 120583(119894)

NL119896

(42)

The nonlinearMMS of time delayed slaving data is completedby propagating the combined estimate based on the currentmodersquos estimated vector such that

xNL119896+119904|119896 = xNL119896|119896 + 119891 [xNL119896|119896] sdot 119904 + NL119896119891 [xNL119896|119896]

sdot

1199042

2

(43)

where 119904 is a lead-time for synchronization (119904 = 119899 sdot Δ119905) If theSLV is in the PFP (120583(1)

NL119896 gt 120583(2)

NL119896) the current state estimateincludes the thrust term that is the state vector xNL119896 =

[119909119896 119910119896 119911119896 119896 119896

119896 120585119896 120591119896]119879 If the SLV is not in the PFP

the state vector does not include the thrust term (120591119896 = 0)

5 Simulation Results

To demonstrate the performance of the proposed synchro-nization approaches for delayed slaving data we simulatedthe SLV tracking problem based on the nominal flight tra-jectory of the Korea Space Launch Vehicle-I (KSLV-I) In thesimulation the radar measurement noise intensities of (12)are selected as 120590119903 = 15m 120590120593 = 001 deg and 120590120579 = 001 degThe nominal flight sequence of KSLV-I is as follows First thepayload fairing separates during the first stage flight at 2154 sAfter the first stage engine shutdown at 2287 s the upperstage separates from the first stage and enters the CFP Thesecond stage continues in the CFP until the kick motor igni-tion at 395 s and the vehicle then enters the PFP At the endof kickmotor combustion (4527 s) the upper stage enters thetarget orbit in CFP as in the previous separation Finally thesatellite is inserted into the target orbit after it separates fromthe upper stage during CFP at 540 s Since delayed slavingdata from the MCS to the TSs can occur in both the PFP andCFP synchronization simulations are conducted at arbitrarypoints of the flight phases Figure 4 shows the synchroniza-tion error of the delayed slaving data with respect to thebenchmark and the proposed approaches for delays of 119904 =

01 02 1 s Figures 4(a)ndash4(c) illustrate synchronizationerrors where delayed slaving data occurred at 155 s which isthe first stage of PFP and Figures 4(d)ndash4(f) show synchro-nization errors where delayed slaving data occurred at 280 swhich is the first stage of CFP As shown in Figures 4(a)ndash4(c)the extrapolation and120572-120573-120574filter-based approach cannot findthe proper synchronized slaving data The synchronizationerrors of these approaches exponentially increase accordingto increasing delay 119904 in particular the error of the extrapo-lation approaches an out-of-figure bound (2500m) On theother hand the Kalman prediction and the proposedapproaches have stable synchronization errors even if thedelay time 119904 is larger Furthermore we can observe thatthe errors in the synchronization approach of the Kalmanprediction are relatively larger than the proposed approachesespecially in the PFP As explained in the flight sequence ofKSLV-I in general the motion of the SLV is described bymultiple dynamics (in our case PFP and CFP) rather thansingle dynamics However the dynamic model of the KF andprediction capability used in this paper is the constant accel-eration (CA) model This means that the filter is optimizedwith respect to CAmotion which is actually not occurring inour application This is why the performance of the Kalmanprediction-based synchronization is worse than the proposedapproaches Nonetheless we can see that the 119910-axis synchro-nization errors shown in Figures 4(b) and 4(e) are smallThis result is observed because the 119910-axis motion of KSLV-I in a local coordinate is relatively smaller than the otheraxis motion and the CA model approximately expresses this

10 Mathematical Problems in Engineering

01 02 03 04 05 06 07 08 09 10

500

1000

1500

2000

2500

Delay (sec)

x-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

Cubic spline extrapolation

Kalman filter

(a)

(b)

(c)

(d)

(e)

(f)

Linear MMNonlinear MM120572-120573-120574 filter 120572-120573-120574 filter

120572-120573-120574 filter120572-120573-120574 filter

120572-120573-120574 filter 120572-120573-120574 filter

01 02 03 04 05 06 07 08 09 1Delay (sec)

0

500

1000

1500

2000

2500

y-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

z-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

x-p

ositi

on ab

solu

te er

ror i

n CF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500y

-pos

ition

abso

lute

erro

r in

CFP

(m)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

z-p

ositi

on ab

solu

te er

ror i

n CF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

Figure 4 Synchronization errors at PFP and CFP

Mathematical Problems in Engineering 11

small motion A single KF prediction model cannot exactlydescribe the motion of both PFP and CFP whereas theproposed multiple model-based approaches work well Asshown in Figure 4 regardless of the delay 119904 the proposedmultiplemodel-based approaches find synchronized positionvectors In the comparison of synchronization performancebetween linear IMM and nonlinear IMM synchronizationthe nonlinear IMM-based synchronization approach showsthe best performance except for 119910-axis where CA motion isdominant In addition the difference between the proposedmultiple model-based approaches is very small as shown inFigure 4 but the complexities of the algorithms for real-timeapplications are quite dissimilar Hence the operator mayadaptively select one of the proposed approaches accordingto onersquos environment

6 Conclusions

In this paper we investigated the time synchronizationapproaches of delayed slaving data in the RSS for SLV track-ingOne of themost important roles of the RSS is to distributeslaving data to each TS for continuous tracking of the SLV Ifthere is a critical network delay resulting in time delayed slav-ing data being sent to each TS theMCS will not receive accu-rate SLV tracking data This problem can give rise to signifi-cant difficulties for the SLVmission progress and analysis Toovercome this problemwe proposedMMSapproacheswhichtake advantage of the multiple motion models of an SLVThelinear IMM-based synchronization approach was developedusing Singerrsquos model with ternary uniform mixtures and thenonlinear IMM-based synchronization approachwas derivedfrom a nonlinear ballistic model with a drag coefficient Forverification of the proposed algorithms SLV tracking simu-lations using KSLV-I and the radar measurement data gen-erated from nominal trajectory were conducted To demon-strate the superiority of time synchronization performancein these simulations we compared the proposed algorithmwith benchmark approaches for absolute error between thenominal trajectory data and the synchronized slaving datathe simulation results demonstrated that the proposed MMSapproaches performed competitively

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] R Varaprasad and V Seshagiri Rao ldquoRange safety real-timesystem for satellite launch vehicle missions-testing methodolo-giesrdquo Defence Science Journal vol 56 no 5 pp 693ndash700 2006

[2] H R Song M G Jeon T S Choi and V Shin ldquoTwo fusionpredictors for discrete-time linear systems with different typesof observationsrdquo International Journal of Control Automationand Systems vol 7 no 4 pp 651ndash658 2009

[3] H Song V Shin and M Jeon ldquoMobile node localization usingfusion prediction-based interacting multiple model in cricketsensor networkrdquo IEEE Transactions on Industrial Electronicsvol 59 no 11 pp 4349ndash4359 2012

[4] T Kirubarajan and Y Bar-Shalom ldquoKalman filter versus IMMestimator when do we need the latterrdquo IEEE Transactions onAerospace and Electronic Systems vol 39 no 4 pp 1452ndash14572003

[5] E Mazor A Averbuch Y Bar-Shalom and J Dayan ldquoInteract-ing multiple model methods in target tracking a surveyrdquo IEEETransactions on Aerospace and Electronic Systems vol 34 no 1pp 103ndash123 1998

[6] P D Hanlon and P S Maybeck ldquoMultiple-model adaptiveestimation using a residual correlation Kalman filter bankrdquoIEEE Transactions on Aerospace and Electronic Systems vol 36no 2 pp 393ndash406 2000

[7] S-J Shin ldquoRe-entry vehicle tracking with a new multiplemodel estimation applicable to highly non-linear dynamicsrdquoIET Radar Sonar amp Navigation vol 9 no 5 pp 581ndash588 2015

[8] S Conover J Kerce G Brown L Ehrman and D HardimanldquoImpact point prediction of small ballistic munitions with aninteracting multiple model estimatorrdquo in Acquisition TrackingPointing and Laser Systems Technologies XXI vol 6569 ofProceedings of SPIE Orlando Fla USA 2007

[9] V C Ravindra Y Bar-Shalom and P Willett ldquoProjectileidentification and impact point predictionrdquo IEEE Transactionson Aerospace and Electronic Systems vol 46 no 4 pp 2004ndash2021 2010

[10] W J Farrell III ldquoInteracting multiple model filter for tacticalballistic missile trackingrdquo IEEE Transactions on Aerospace andElectronic Systems vol 44 no 2 pp 418ndash426 2008

[11] T Yuan Y Bar-Shalom PWillett E Mozeson S Pollak and DHardiman ldquoA multiple IMM estimation approach with unbi-ased mixing for thrusting projectilesrdquo IEEE Transactions onAerospace and Electronic Systems vol 48 no 4 pp 3250ndash32672012

[12] T Yuan Y Bar-Shalom P Willett and D Hardiman ldquoImpactpoint prediction for thrusting projectiles in the presence ofwindrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 50 no 1 pp 102ndash119 2014

[13] X Li and V Jilkov ldquoSurvey of maneuvering target trackingpart II motion models of ballistic and space targetsrdquo IEEETransactions on Aerospace and Electronic Systems vol 46 no1 pp 96ndash119 2010

[14] J Cesar Bolzani de Campos Ferreira and JWaldmann ldquoCovari-ance intersection-based sensor fusion for sounding rockettracking and impact area predictionrdquo Control Engineering Prac-tice vol 15 no 4 pp 389ndash409 2007

[15] httpwwwbraeunigusapollosaturnVhtm[16] RA Singer ldquoEstimating optimal tracking filter performance for

manned maneuvering targetsrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 6 no 4 pp 473ndash483 1970

[17] R A Singer and KW Behnke ldquoReal-time tracking filter evalu-ation and selection for tactical applicationsrdquo IEEE Transactionson Aerospace and Electronic Systems vol AES-7 no 1 pp 100ndash110 1971

[18] W Press B Flannery S Teukolsky and W Vetterling Numer-ical Recipes in FORTRAN The Art of Scientific ComputingCambridge University New York NY USA 2nd edition 1992

[19] S Dyer and J Dyer ldquoCubic-spline interpolation part Irdquo IEEEInstrumentation amp Measurement Magazine pp 44ndash46 2001

[20] Y Bar-Shalom X Li and T Kirubarajan Estimation with Appli-cations to Tracking and Navigation John Wiley amp Sons NewYork NY USA 2001

12 Mathematical Problems in Engineering

[21] N Morrison Introduction to Sequential Smoothing and Predic-tion McGraw-Hill 1969

[22] E Brookner Tracking and Kalman Filtering Made Easy JohnWiley amp Sons 1998

[23] P Suchomski ldquoExplicit expressions for debiased statistics of 3Dconverted measurementsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 35 no 1 pp 368ndash370 1999

[24] H R Song I Y Song and V Shin ldquoMultisensory predictionfusion of nonlinear functions of the state vector in discrete-timesystemsrdquo International Journal of Distributed Sensor Networksvol 2015 Article ID 249857 13 pages 2015

[25] H Song andYHan ldquoComparison of space launch vehicle track-ing using different types of multiple modelsrdquo in Proceedingsof the International Conference on Information Fusion pp 1ndash8Heidelberg Germany July 2016

[26] R KMehra ldquoA comparison of several nonlinear filters for reen-try vehicle trackingrdquo IEEE Transactions on Automatic Controlvol 16 no 4 pp 307ndash319 1971

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

(vi) Model-conditioned filtering (for 119894 = 1 2 119872)

(a) predicted estimate and covariance

x(119894)NL119896|119896minus1 = x(119894)NL119896minus1|119896minus1 + 119891 [x(119894)NL119896minus1|119896minus1] Δ119905

+

(119894)

NL119896minus1119891 [x(119894)NL119896minus1|119896minus1]Δ119905

2

2

(119894)

119896=

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x=x(119894)119896

119875(119894)

NL119896|119896minus1 = Ω(119894)

119896minus1119875

(119894)

NL119896minus1|119896minus1Ω(119894)119879

119896minus1+ 119876119896minus1

Ω(119894)

119896minus1= 119868 +

(119894)

NL119896minus1

(39)

(b) measurement residual (119894)

NL119896 = 119911NL119896 minus

119867NLx(119894)

NL119896|119896minus1 minus ]119896

(c) residual covariance 119878(119894)NL119896 = 119867NL119875(119894)

NL119896|119896minus1119867119879

NL minus

119877119896(d) filter gain 119870(119894)

NL119896 = 119875(119894)

NL119896|119896minus1119867119879

NL(119878(119894)

NL119896)minus1

(e) update of state and covariance

x(119894)NL119896|119896 = x(119894)NL119896|119896minus1 + 119870(119894)

NL119896(119894)

NL119896

119875(119894)

NL119896|119896 = 119875(119894)

NL119896|119896minus1 minus 119870(119894)

NL119896119878(119894)

NL119896119870(119894)119879

NL119896

(40)

(vii) Mode probability update (for 119894 = 1 2 119872)

(a) mode likelihood

Λ(119894)

NL119896 ≜ 119901 [119911(119894)

NL119896 | 119898(119894)

119896 119885NL119896minus1]

assume= N (119911

(119894)

NL119896 0 119878(119894)

NL119896)

(41)

(b) mode probability 120583(119894)

NL119896 = 120583(119894)

NL119896|119896minus1Λ(119894)

NL119896

(sum119895120583(119895)

NL119896|119896minus1Λ(119895)

NL119896)

(viii) Combination (for 119894 = 1 2 119872)

xNL119896|119896 = sum

119894

x(119894)NL119896|119896120583(119894)

NL119896

(119894)

NL119896|119896 = sum

119894

[119875(119894)

NL119896|119896

+ (xNL119896|119896 minus x(119894)NL119896|119896) (xNL119896|119896 minus x(119894)NL119896|119896)119879

] 120583(119894)

NL119896

(42)

The nonlinearMMS of time delayed slaving data is completedby propagating the combined estimate based on the currentmodersquos estimated vector such that

xNL119896+119904|119896 = xNL119896|119896 + 119891 [xNL119896|119896] sdot 119904 + NL119896119891 [xNL119896|119896]

sdot

1199042

2

(43)

where 119904 is a lead-time for synchronization (119904 = 119899 sdot Δ119905) If theSLV is in the PFP (120583(1)

NL119896 gt 120583(2)

NL119896) the current state estimateincludes the thrust term that is the state vector xNL119896 =

[119909119896 119910119896 119911119896 119896 119896

119896 120585119896 120591119896]119879 If the SLV is not in the PFP

the state vector does not include the thrust term (120591119896 = 0)

5 Simulation Results

To demonstrate the performance of the proposed synchro-nization approaches for delayed slaving data we simulatedthe SLV tracking problem based on the nominal flight tra-jectory of the Korea Space Launch Vehicle-I (KSLV-I) In thesimulation the radar measurement noise intensities of (12)are selected as 120590119903 = 15m 120590120593 = 001 deg and 120590120579 = 001 degThe nominal flight sequence of KSLV-I is as follows First thepayload fairing separates during the first stage flight at 2154 sAfter the first stage engine shutdown at 2287 s the upperstage separates from the first stage and enters the CFP Thesecond stage continues in the CFP until the kick motor igni-tion at 395 s and the vehicle then enters the PFP At the endof kickmotor combustion (4527 s) the upper stage enters thetarget orbit in CFP as in the previous separation Finally thesatellite is inserted into the target orbit after it separates fromthe upper stage during CFP at 540 s Since delayed slavingdata from the MCS to the TSs can occur in both the PFP andCFP synchronization simulations are conducted at arbitrarypoints of the flight phases Figure 4 shows the synchroniza-tion error of the delayed slaving data with respect to thebenchmark and the proposed approaches for delays of 119904 =

01 02 1 s Figures 4(a)ndash4(c) illustrate synchronizationerrors where delayed slaving data occurred at 155 s which isthe first stage of PFP and Figures 4(d)ndash4(f) show synchro-nization errors where delayed slaving data occurred at 280 swhich is the first stage of CFP As shown in Figures 4(a)ndash4(c)the extrapolation and120572-120573-120574filter-based approach cannot findthe proper synchronized slaving data The synchronizationerrors of these approaches exponentially increase accordingto increasing delay 119904 in particular the error of the extrapo-lation approaches an out-of-figure bound (2500m) On theother hand the Kalman prediction and the proposedapproaches have stable synchronization errors even if thedelay time 119904 is larger Furthermore we can observe thatthe errors in the synchronization approach of the Kalmanprediction are relatively larger than the proposed approachesespecially in the PFP As explained in the flight sequence ofKSLV-I in general the motion of the SLV is described bymultiple dynamics (in our case PFP and CFP) rather thansingle dynamics However the dynamic model of the KF andprediction capability used in this paper is the constant accel-eration (CA) model This means that the filter is optimizedwith respect to CAmotion which is actually not occurring inour application This is why the performance of the Kalmanprediction-based synchronization is worse than the proposedapproaches Nonetheless we can see that the 119910-axis synchro-nization errors shown in Figures 4(b) and 4(e) are smallThis result is observed because the 119910-axis motion of KSLV-I in a local coordinate is relatively smaller than the otheraxis motion and the CA model approximately expresses this

10 Mathematical Problems in Engineering

01 02 03 04 05 06 07 08 09 10

500

1000

1500

2000

2500

Delay (sec)

x-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

Cubic spline extrapolation

Kalman filter

(a)

(b)

(c)

(d)

(e)

(f)

Linear MMNonlinear MM120572-120573-120574 filter 120572-120573-120574 filter

120572-120573-120574 filter120572-120573-120574 filter

120572-120573-120574 filter 120572-120573-120574 filter

01 02 03 04 05 06 07 08 09 1Delay (sec)

0

500

1000

1500

2000

2500

y-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

z-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

x-p

ositi

on ab

solu

te er

ror i

n CF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500y

-pos

ition

abso

lute

erro

r in

CFP

(m)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

z-p

ositi

on ab

solu

te er

ror i

n CF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

Figure 4 Synchronization errors at PFP and CFP

Mathematical Problems in Engineering 11

small motion A single KF prediction model cannot exactlydescribe the motion of both PFP and CFP whereas theproposed multiple model-based approaches work well Asshown in Figure 4 regardless of the delay 119904 the proposedmultiplemodel-based approaches find synchronized positionvectors In the comparison of synchronization performancebetween linear IMM and nonlinear IMM synchronizationthe nonlinear IMM-based synchronization approach showsthe best performance except for 119910-axis where CA motion isdominant In addition the difference between the proposedmultiple model-based approaches is very small as shown inFigure 4 but the complexities of the algorithms for real-timeapplications are quite dissimilar Hence the operator mayadaptively select one of the proposed approaches accordingto onersquos environment

6 Conclusions

In this paper we investigated the time synchronizationapproaches of delayed slaving data in the RSS for SLV track-ingOne of themost important roles of the RSS is to distributeslaving data to each TS for continuous tracking of the SLV Ifthere is a critical network delay resulting in time delayed slav-ing data being sent to each TS theMCS will not receive accu-rate SLV tracking data This problem can give rise to signifi-cant difficulties for the SLVmission progress and analysis Toovercome this problemwe proposedMMSapproacheswhichtake advantage of the multiple motion models of an SLVThelinear IMM-based synchronization approach was developedusing Singerrsquos model with ternary uniform mixtures and thenonlinear IMM-based synchronization approachwas derivedfrom a nonlinear ballistic model with a drag coefficient Forverification of the proposed algorithms SLV tracking simu-lations using KSLV-I and the radar measurement data gen-erated from nominal trajectory were conducted To demon-strate the superiority of time synchronization performancein these simulations we compared the proposed algorithmwith benchmark approaches for absolute error between thenominal trajectory data and the synchronized slaving datathe simulation results demonstrated that the proposed MMSapproaches performed competitively

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] R Varaprasad and V Seshagiri Rao ldquoRange safety real-timesystem for satellite launch vehicle missions-testing methodolo-giesrdquo Defence Science Journal vol 56 no 5 pp 693ndash700 2006

[2] H R Song M G Jeon T S Choi and V Shin ldquoTwo fusionpredictors for discrete-time linear systems with different typesof observationsrdquo International Journal of Control Automationand Systems vol 7 no 4 pp 651ndash658 2009

[3] H Song V Shin and M Jeon ldquoMobile node localization usingfusion prediction-based interacting multiple model in cricketsensor networkrdquo IEEE Transactions on Industrial Electronicsvol 59 no 11 pp 4349ndash4359 2012

[4] T Kirubarajan and Y Bar-Shalom ldquoKalman filter versus IMMestimator when do we need the latterrdquo IEEE Transactions onAerospace and Electronic Systems vol 39 no 4 pp 1452ndash14572003

[5] E Mazor A Averbuch Y Bar-Shalom and J Dayan ldquoInteract-ing multiple model methods in target tracking a surveyrdquo IEEETransactions on Aerospace and Electronic Systems vol 34 no 1pp 103ndash123 1998

[6] P D Hanlon and P S Maybeck ldquoMultiple-model adaptiveestimation using a residual correlation Kalman filter bankrdquoIEEE Transactions on Aerospace and Electronic Systems vol 36no 2 pp 393ndash406 2000

[7] S-J Shin ldquoRe-entry vehicle tracking with a new multiplemodel estimation applicable to highly non-linear dynamicsrdquoIET Radar Sonar amp Navigation vol 9 no 5 pp 581ndash588 2015

[8] S Conover J Kerce G Brown L Ehrman and D HardimanldquoImpact point prediction of small ballistic munitions with aninteracting multiple model estimatorrdquo in Acquisition TrackingPointing and Laser Systems Technologies XXI vol 6569 ofProceedings of SPIE Orlando Fla USA 2007

[9] V C Ravindra Y Bar-Shalom and P Willett ldquoProjectileidentification and impact point predictionrdquo IEEE Transactionson Aerospace and Electronic Systems vol 46 no 4 pp 2004ndash2021 2010

[10] W J Farrell III ldquoInteracting multiple model filter for tacticalballistic missile trackingrdquo IEEE Transactions on Aerospace andElectronic Systems vol 44 no 2 pp 418ndash426 2008

[11] T Yuan Y Bar-Shalom PWillett E Mozeson S Pollak and DHardiman ldquoA multiple IMM estimation approach with unbi-ased mixing for thrusting projectilesrdquo IEEE Transactions onAerospace and Electronic Systems vol 48 no 4 pp 3250ndash32672012

[12] T Yuan Y Bar-Shalom P Willett and D Hardiman ldquoImpactpoint prediction for thrusting projectiles in the presence ofwindrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 50 no 1 pp 102ndash119 2014

[13] X Li and V Jilkov ldquoSurvey of maneuvering target trackingpart II motion models of ballistic and space targetsrdquo IEEETransactions on Aerospace and Electronic Systems vol 46 no1 pp 96ndash119 2010

[14] J Cesar Bolzani de Campos Ferreira and JWaldmann ldquoCovari-ance intersection-based sensor fusion for sounding rockettracking and impact area predictionrdquo Control Engineering Prac-tice vol 15 no 4 pp 389ndash409 2007

[15] httpwwwbraeunigusapollosaturnVhtm[16] RA Singer ldquoEstimating optimal tracking filter performance for

manned maneuvering targetsrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 6 no 4 pp 473ndash483 1970

[17] R A Singer and KW Behnke ldquoReal-time tracking filter evalu-ation and selection for tactical applicationsrdquo IEEE Transactionson Aerospace and Electronic Systems vol AES-7 no 1 pp 100ndash110 1971

[18] W Press B Flannery S Teukolsky and W Vetterling Numer-ical Recipes in FORTRAN The Art of Scientific ComputingCambridge University New York NY USA 2nd edition 1992

[19] S Dyer and J Dyer ldquoCubic-spline interpolation part Irdquo IEEEInstrumentation amp Measurement Magazine pp 44ndash46 2001

[20] Y Bar-Shalom X Li and T Kirubarajan Estimation with Appli-cations to Tracking and Navigation John Wiley amp Sons NewYork NY USA 2001

12 Mathematical Problems in Engineering

[21] N Morrison Introduction to Sequential Smoothing and Predic-tion McGraw-Hill 1969

[22] E Brookner Tracking and Kalman Filtering Made Easy JohnWiley amp Sons 1998

[23] P Suchomski ldquoExplicit expressions for debiased statistics of 3Dconverted measurementsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 35 no 1 pp 368ndash370 1999

[24] H R Song I Y Song and V Shin ldquoMultisensory predictionfusion of nonlinear functions of the state vector in discrete-timesystemsrdquo International Journal of Distributed Sensor Networksvol 2015 Article ID 249857 13 pages 2015

[25] H Song andYHan ldquoComparison of space launch vehicle track-ing using different types of multiple modelsrdquo in Proceedingsof the International Conference on Information Fusion pp 1ndash8Heidelberg Germany July 2016

[26] R KMehra ldquoA comparison of several nonlinear filters for reen-try vehicle trackingrdquo IEEE Transactions on Automatic Controlvol 16 no 4 pp 307ndash319 1971

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

01 02 03 04 05 06 07 08 09 10

500

1000

1500

2000

2500

Delay (sec)

x-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

Cubic spline extrapolation

Kalman filter

(a)

(b)

(c)

(d)

(e)

(f)

Linear MMNonlinear MM120572-120573-120574 filter 120572-120573-120574 filter

120572-120573-120574 filter120572-120573-120574 filter

120572-120573-120574 filter 120572-120573-120574 filter

01 02 03 04 05 06 07 08 09 1Delay (sec)

0

500

1000

1500

2000

2500

y-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

z-p

ositi

on ab

solu

te er

ror i

n PF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

x-p

ositi

on ab

solu

te er

ror i

n CF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500y

-pos

ition

abso

lute

erro

r in

CFP

(m)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

0

500

1000

1500

2000

2500

z-p

ositi

on ab

solu

te er

ror i

n CF

P (m

)

01 02 03 04 05 06 07 08 09 1Delay (sec)

Cubic spline extrapolation

Kalman filter

Linear MMNonlinear MM

Figure 4 Synchronization errors at PFP and CFP

Mathematical Problems in Engineering 11

small motion A single KF prediction model cannot exactlydescribe the motion of both PFP and CFP whereas theproposed multiple model-based approaches work well Asshown in Figure 4 regardless of the delay 119904 the proposedmultiplemodel-based approaches find synchronized positionvectors In the comparison of synchronization performancebetween linear IMM and nonlinear IMM synchronizationthe nonlinear IMM-based synchronization approach showsthe best performance except for 119910-axis where CA motion isdominant In addition the difference between the proposedmultiple model-based approaches is very small as shown inFigure 4 but the complexities of the algorithms for real-timeapplications are quite dissimilar Hence the operator mayadaptively select one of the proposed approaches accordingto onersquos environment

6 Conclusions

In this paper we investigated the time synchronizationapproaches of delayed slaving data in the RSS for SLV track-ingOne of themost important roles of the RSS is to distributeslaving data to each TS for continuous tracking of the SLV Ifthere is a critical network delay resulting in time delayed slav-ing data being sent to each TS theMCS will not receive accu-rate SLV tracking data This problem can give rise to signifi-cant difficulties for the SLVmission progress and analysis Toovercome this problemwe proposedMMSapproacheswhichtake advantage of the multiple motion models of an SLVThelinear IMM-based synchronization approach was developedusing Singerrsquos model with ternary uniform mixtures and thenonlinear IMM-based synchronization approachwas derivedfrom a nonlinear ballistic model with a drag coefficient Forverification of the proposed algorithms SLV tracking simu-lations using KSLV-I and the radar measurement data gen-erated from nominal trajectory were conducted To demon-strate the superiority of time synchronization performancein these simulations we compared the proposed algorithmwith benchmark approaches for absolute error between thenominal trajectory data and the synchronized slaving datathe simulation results demonstrated that the proposed MMSapproaches performed competitively

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] R Varaprasad and V Seshagiri Rao ldquoRange safety real-timesystem for satellite launch vehicle missions-testing methodolo-giesrdquo Defence Science Journal vol 56 no 5 pp 693ndash700 2006

[2] H R Song M G Jeon T S Choi and V Shin ldquoTwo fusionpredictors for discrete-time linear systems with different typesof observationsrdquo International Journal of Control Automationand Systems vol 7 no 4 pp 651ndash658 2009

[3] H Song V Shin and M Jeon ldquoMobile node localization usingfusion prediction-based interacting multiple model in cricketsensor networkrdquo IEEE Transactions on Industrial Electronicsvol 59 no 11 pp 4349ndash4359 2012

[4] T Kirubarajan and Y Bar-Shalom ldquoKalman filter versus IMMestimator when do we need the latterrdquo IEEE Transactions onAerospace and Electronic Systems vol 39 no 4 pp 1452ndash14572003

[5] E Mazor A Averbuch Y Bar-Shalom and J Dayan ldquoInteract-ing multiple model methods in target tracking a surveyrdquo IEEETransactions on Aerospace and Electronic Systems vol 34 no 1pp 103ndash123 1998

[6] P D Hanlon and P S Maybeck ldquoMultiple-model adaptiveestimation using a residual correlation Kalman filter bankrdquoIEEE Transactions on Aerospace and Electronic Systems vol 36no 2 pp 393ndash406 2000

[7] S-J Shin ldquoRe-entry vehicle tracking with a new multiplemodel estimation applicable to highly non-linear dynamicsrdquoIET Radar Sonar amp Navigation vol 9 no 5 pp 581ndash588 2015

[8] S Conover J Kerce G Brown L Ehrman and D HardimanldquoImpact point prediction of small ballistic munitions with aninteracting multiple model estimatorrdquo in Acquisition TrackingPointing and Laser Systems Technologies XXI vol 6569 ofProceedings of SPIE Orlando Fla USA 2007

[9] V C Ravindra Y Bar-Shalom and P Willett ldquoProjectileidentification and impact point predictionrdquo IEEE Transactionson Aerospace and Electronic Systems vol 46 no 4 pp 2004ndash2021 2010

[10] W J Farrell III ldquoInteracting multiple model filter for tacticalballistic missile trackingrdquo IEEE Transactions on Aerospace andElectronic Systems vol 44 no 2 pp 418ndash426 2008

[11] T Yuan Y Bar-Shalom PWillett E Mozeson S Pollak and DHardiman ldquoA multiple IMM estimation approach with unbi-ased mixing for thrusting projectilesrdquo IEEE Transactions onAerospace and Electronic Systems vol 48 no 4 pp 3250ndash32672012

[12] T Yuan Y Bar-Shalom P Willett and D Hardiman ldquoImpactpoint prediction for thrusting projectiles in the presence ofwindrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 50 no 1 pp 102ndash119 2014

[13] X Li and V Jilkov ldquoSurvey of maneuvering target trackingpart II motion models of ballistic and space targetsrdquo IEEETransactions on Aerospace and Electronic Systems vol 46 no1 pp 96ndash119 2010

[14] J Cesar Bolzani de Campos Ferreira and JWaldmann ldquoCovari-ance intersection-based sensor fusion for sounding rockettracking and impact area predictionrdquo Control Engineering Prac-tice vol 15 no 4 pp 389ndash409 2007

[15] httpwwwbraeunigusapollosaturnVhtm[16] RA Singer ldquoEstimating optimal tracking filter performance for

manned maneuvering targetsrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 6 no 4 pp 473ndash483 1970

[17] R A Singer and KW Behnke ldquoReal-time tracking filter evalu-ation and selection for tactical applicationsrdquo IEEE Transactionson Aerospace and Electronic Systems vol AES-7 no 1 pp 100ndash110 1971

[18] W Press B Flannery S Teukolsky and W Vetterling Numer-ical Recipes in FORTRAN The Art of Scientific ComputingCambridge University New York NY USA 2nd edition 1992

[19] S Dyer and J Dyer ldquoCubic-spline interpolation part Irdquo IEEEInstrumentation amp Measurement Magazine pp 44ndash46 2001

[20] Y Bar-Shalom X Li and T Kirubarajan Estimation with Appli-cations to Tracking and Navigation John Wiley amp Sons NewYork NY USA 2001

12 Mathematical Problems in Engineering

[21] N Morrison Introduction to Sequential Smoothing and Predic-tion McGraw-Hill 1969

[22] E Brookner Tracking and Kalman Filtering Made Easy JohnWiley amp Sons 1998

[23] P Suchomski ldquoExplicit expressions for debiased statistics of 3Dconverted measurementsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 35 no 1 pp 368ndash370 1999

[24] H R Song I Y Song and V Shin ldquoMultisensory predictionfusion of nonlinear functions of the state vector in discrete-timesystemsrdquo International Journal of Distributed Sensor Networksvol 2015 Article ID 249857 13 pages 2015

[25] H Song andYHan ldquoComparison of space launch vehicle track-ing using different types of multiple modelsrdquo in Proceedingsof the International Conference on Information Fusion pp 1ndash8Heidelberg Germany July 2016

[26] R KMehra ldquoA comparison of several nonlinear filters for reen-try vehicle trackingrdquo IEEE Transactions on Automatic Controlvol 16 no 4 pp 307ndash319 1971

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

small motion A single KF prediction model cannot exactlydescribe the motion of both PFP and CFP whereas theproposed multiple model-based approaches work well Asshown in Figure 4 regardless of the delay 119904 the proposedmultiplemodel-based approaches find synchronized positionvectors In the comparison of synchronization performancebetween linear IMM and nonlinear IMM synchronizationthe nonlinear IMM-based synchronization approach showsthe best performance except for 119910-axis where CA motion isdominant In addition the difference between the proposedmultiple model-based approaches is very small as shown inFigure 4 but the complexities of the algorithms for real-timeapplications are quite dissimilar Hence the operator mayadaptively select one of the proposed approaches accordingto onersquos environment

6 Conclusions

In this paper we investigated the time synchronizationapproaches of delayed slaving data in the RSS for SLV track-ingOne of themost important roles of the RSS is to distributeslaving data to each TS for continuous tracking of the SLV Ifthere is a critical network delay resulting in time delayed slav-ing data being sent to each TS theMCS will not receive accu-rate SLV tracking data This problem can give rise to signifi-cant difficulties for the SLVmission progress and analysis Toovercome this problemwe proposedMMSapproacheswhichtake advantage of the multiple motion models of an SLVThelinear IMM-based synchronization approach was developedusing Singerrsquos model with ternary uniform mixtures and thenonlinear IMM-based synchronization approachwas derivedfrom a nonlinear ballistic model with a drag coefficient Forverification of the proposed algorithms SLV tracking simu-lations using KSLV-I and the radar measurement data gen-erated from nominal trajectory were conducted To demon-strate the superiority of time synchronization performancein these simulations we compared the proposed algorithmwith benchmark approaches for absolute error between thenominal trajectory data and the synchronized slaving datathe simulation results demonstrated that the proposed MMSapproaches performed competitively

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] R Varaprasad and V Seshagiri Rao ldquoRange safety real-timesystem for satellite launch vehicle missions-testing methodolo-giesrdquo Defence Science Journal vol 56 no 5 pp 693ndash700 2006

[2] H R Song M G Jeon T S Choi and V Shin ldquoTwo fusionpredictors for discrete-time linear systems with different typesof observationsrdquo International Journal of Control Automationand Systems vol 7 no 4 pp 651ndash658 2009

[3] H Song V Shin and M Jeon ldquoMobile node localization usingfusion prediction-based interacting multiple model in cricketsensor networkrdquo IEEE Transactions on Industrial Electronicsvol 59 no 11 pp 4349ndash4359 2012

[4] T Kirubarajan and Y Bar-Shalom ldquoKalman filter versus IMMestimator when do we need the latterrdquo IEEE Transactions onAerospace and Electronic Systems vol 39 no 4 pp 1452ndash14572003

[5] E Mazor A Averbuch Y Bar-Shalom and J Dayan ldquoInteract-ing multiple model methods in target tracking a surveyrdquo IEEETransactions on Aerospace and Electronic Systems vol 34 no 1pp 103ndash123 1998

[6] P D Hanlon and P S Maybeck ldquoMultiple-model adaptiveestimation using a residual correlation Kalman filter bankrdquoIEEE Transactions on Aerospace and Electronic Systems vol 36no 2 pp 393ndash406 2000

[7] S-J Shin ldquoRe-entry vehicle tracking with a new multiplemodel estimation applicable to highly non-linear dynamicsrdquoIET Radar Sonar amp Navigation vol 9 no 5 pp 581ndash588 2015

[8] S Conover J Kerce G Brown L Ehrman and D HardimanldquoImpact point prediction of small ballistic munitions with aninteracting multiple model estimatorrdquo in Acquisition TrackingPointing and Laser Systems Technologies XXI vol 6569 ofProceedings of SPIE Orlando Fla USA 2007

[9] V C Ravindra Y Bar-Shalom and P Willett ldquoProjectileidentification and impact point predictionrdquo IEEE Transactionson Aerospace and Electronic Systems vol 46 no 4 pp 2004ndash2021 2010

[10] W J Farrell III ldquoInteracting multiple model filter for tacticalballistic missile trackingrdquo IEEE Transactions on Aerospace andElectronic Systems vol 44 no 2 pp 418ndash426 2008

[11] T Yuan Y Bar-Shalom PWillett E Mozeson S Pollak and DHardiman ldquoA multiple IMM estimation approach with unbi-ased mixing for thrusting projectilesrdquo IEEE Transactions onAerospace and Electronic Systems vol 48 no 4 pp 3250ndash32672012

[12] T Yuan Y Bar-Shalom P Willett and D Hardiman ldquoImpactpoint prediction for thrusting projectiles in the presence ofwindrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 50 no 1 pp 102ndash119 2014

[13] X Li and V Jilkov ldquoSurvey of maneuvering target trackingpart II motion models of ballistic and space targetsrdquo IEEETransactions on Aerospace and Electronic Systems vol 46 no1 pp 96ndash119 2010

[14] J Cesar Bolzani de Campos Ferreira and JWaldmann ldquoCovari-ance intersection-based sensor fusion for sounding rockettracking and impact area predictionrdquo Control Engineering Prac-tice vol 15 no 4 pp 389ndash409 2007

[15] httpwwwbraeunigusapollosaturnVhtm[16] RA Singer ldquoEstimating optimal tracking filter performance for

manned maneuvering targetsrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 6 no 4 pp 473ndash483 1970

[17] R A Singer and KW Behnke ldquoReal-time tracking filter evalu-ation and selection for tactical applicationsrdquo IEEE Transactionson Aerospace and Electronic Systems vol AES-7 no 1 pp 100ndash110 1971

[18] W Press B Flannery S Teukolsky and W Vetterling Numer-ical Recipes in FORTRAN The Art of Scientific ComputingCambridge University New York NY USA 2nd edition 1992

[19] S Dyer and J Dyer ldquoCubic-spline interpolation part Irdquo IEEEInstrumentation amp Measurement Magazine pp 44ndash46 2001

[20] Y Bar-Shalom X Li and T Kirubarajan Estimation with Appli-cations to Tracking and Navigation John Wiley amp Sons NewYork NY USA 2001

12 Mathematical Problems in Engineering

[21] N Morrison Introduction to Sequential Smoothing and Predic-tion McGraw-Hill 1969

[22] E Brookner Tracking and Kalman Filtering Made Easy JohnWiley amp Sons 1998

[23] P Suchomski ldquoExplicit expressions for debiased statistics of 3Dconverted measurementsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 35 no 1 pp 368ndash370 1999

[24] H R Song I Y Song and V Shin ldquoMultisensory predictionfusion of nonlinear functions of the state vector in discrete-timesystemsrdquo International Journal of Distributed Sensor Networksvol 2015 Article ID 249857 13 pages 2015

[25] H Song andYHan ldquoComparison of space launch vehicle track-ing using different types of multiple modelsrdquo in Proceedingsof the International Conference on Information Fusion pp 1ndash8Heidelberg Germany July 2016

[26] R KMehra ldquoA comparison of several nonlinear filters for reen-try vehicle trackingrdquo IEEE Transactions on Automatic Controlvol 16 no 4 pp 307ndash319 1971

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

[21] N Morrison Introduction to Sequential Smoothing and Predic-tion McGraw-Hill 1969

[22] E Brookner Tracking and Kalman Filtering Made Easy JohnWiley amp Sons 1998

[23] P Suchomski ldquoExplicit expressions for debiased statistics of 3Dconverted measurementsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 35 no 1 pp 368ndash370 1999

[24] H R Song I Y Song and V Shin ldquoMultisensory predictionfusion of nonlinear functions of the state vector in discrete-timesystemsrdquo International Journal of Distributed Sensor Networksvol 2015 Article ID 249857 13 pages 2015

[25] H Song andYHan ldquoComparison of space launch vehicle track-ing using different types of multiple modelsrdquo in Proceedingsof the International Conference on Information Fusion pp 1ndash8Heidelberg Germany July 2016

[26] R KMehra ldquoA comparison of several nonlinear filters for reen-try vehicle trackingrdquo IEEE Transactions on Automatic Controlvol 16 no 4 pp 307ndash319 1971

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of