The precise autonomous orbit keeping experiment on the PRISMA mission

The Precise AutonomousOrbit Keeping Experiment

on the PRISMA MissionSergio De Florio1 and Simone DAmico1

Abstract

This paper analyzes the problem of autonomous control of the longitude of the ascendingnode (LAN) for a satellite in low Earth orbit (LEO) by means of along-track and anti-along-track velocity increments which adjust the semimajor axis. The problems related to the pos-sibility of generating the reference orbit (RO) on-board and with the estimation of theatmospheric drag are considered. The Autonomous Orbit Keeping (AOK) experiment ofthe PRISMA formation flying mission will be the test platform of the control strategy hereexposed. The AOK on-board software shall demonstrate autonomous orbit control using aguidance law for the orbits LAN and shall implement a deterministic control algorithmusing along-track and anti-along-track velocity increments. Using GPS-based absolutenavigation data, AOK shall command thruster activations in the orbital frame to auton-omously control the orbit within a predefined window. The AOK experiment paves the wayto the accurate and autonomous orbit control of LEO satellites on a routine basis. The mainrequirement of the experiment is to demonstrate an orbit control accuracy of the osculatingascending node of 10 m . The paper shows results from real-world software simulationswhere the accuracy of the reference orbit is limited and GPS sensors and hydrazine actuatorsare accurately modeled. The fundamental approach on which the software design, validationand testing is based, is also explained.

Introduction

Autonomous Orbit Control

Autonomous navigation and orbit control is acquiring increasing interest as it canenhance mission performance and provide significant operations cost reduction. Bycontrolling the spacecraft orbit to match a chosen reference, the fulfillment of strictrequirements on different orbit parameters can be achieved in real time and with asignificant relieving of ground operations. The use of autonomous navigation andcontrol can afford cost benefits by relieving of ground station operational burden andreduction of planning and scheduling costs (reference [1]). The benefits of this typeof control can be fully exploited in remote sensing space missions where the satel-lites are generally placed in Sun-synchronous, phased and frozen orbits: examplesof missions of this type are TerraSAR-X (references [2], [3] and [4]) and ERS-1,

1

The Journal of the Astronautical Sciences, Vol. 56, No. 4, OctoberDecember 2008, pp. 477494

477

1Space Flight Technology Department, German Aerospace Center (DLR), D-82230 Wessling, Germany.

ERS-2 (references [5] and [6]). This paper analyzes the problem of autonomouscontrol of the longitude of ascending node for a satellite in LEO by means of ad-justments of the semimajor axis. Establishing a reference orbit and the way of gen-erating it is a key element of the orbit control system. The on-board generation bya numerical routine can be severely constrained by the limitation of the on-boardcomputer. On the other hand the on-board generation of the reference orbit basedon an analytical model, though less intensive from a computational point of view,can be precluded by the accuracy requirements of the grid that defines the nominallongitudes of the ascending node during the satellite life. Uploading a referenceorbit from the ground to the satellite keeps one from facing all the problems con-cerned with on-board generation but reduces the autonomy and flexibility of thecontrol system. As this type of orbit control is based on along-track and anti-along-track velocity increments which adjust the semimajor axis, a correct estimation ofthe atmospheric drag, the main non-gravitational perturbation in LEO, is also es-sential to the end of a fine control. Along-track is here defined as the directionforming a right-handed orbital frame with the directions of the radius and angularmomentum of the orbit.

The AOK Experiment

The autonomous orbit keeping experiment is the secondary objective of the Ger-man Aerospace Center (DLR) contributions to the PRISMA mission (references [7]and [8]). PRISMA is a micro-satellite mission created by the Swedish NationalSpace Board (SNSB) and Swedish Space Corporation (SSC), which serves as a testplatform for autonomous formation flying and rendezvous of spacecraft. PRISMAcomprises a fully maneuverable micro-satellite (MANGO) as well as a smaller sub-satellite (TANGO) which are launched together in a clamped configuration andseparated in orbit after completion of all checkout operations (Fig. 1). The designorbit of the satellites is Sun-synchronous, near-polar at approximately 700 km alti-tude with local time of the ascending node at 6.00 or 18.00. The mission scheduleforesees a launch in 2009 of the two spacecraft with a targeted lifetime of at leasteight months. The PRISMA mission primary objective is to demonstrate in-flighttechnology experiments related to autonomous formation flying, homing and ren-dezvous scenarios, precision close range 3D proximity operations, soft and smoothfinal approach and recede maneuvers, as well as to test instruments and unit devel-opments related to formation flying. The backbone navigation sensor is based onGPS receivers on both satellites (reference [9]). At the end of the mission, after theformation release, the AOK on-board software shall demonstrate MANGO satelliteautonomous orbit control using a guidance law for the orbits LAN that implementsan analytical feedback control algorithm using along-track and anti-along-trackvelocity increments. Using GPS-based absolute navigation data, AOK shall com-mand thruster activations in the orbital frame to autonomously control the orbitwithin a predefined window. The main requirement of the experiment is to demon-strate a control accuracy of the osculating ascending node of 10 m . The maindifference with respect to similar experiments (references [10] and [11]) are the ex-tremely strict required control accuracy and the full autonomy enhanced by the pos-sibility of on-board reference orbit propagation. The design of the AOK, as that ofthe entire on-board PRISMA flight software, follows a model-based approach. Itmakes use of Matlab/Simulink to describe the data flow, functionality and schedulingproperties of individual software components. Following the conceptual design and

1

478 De Florio and DAmico

the validation in a non real-time environment, a real-time executable of the flightsoftware for the XPC platform (reference [12]) or the LEON3 target processor (ref-erence [13]) is generated using the automatic code generator of Real Time Work-shop, a MATLAB toolbox. The full consistency of the flight software generationchain is verified by comparing relevant outputs obtained from application runs ona host standard laptop PC in a Matlab/Simulink environment and on the targetLEON3 board under the Real-Time Executive for Multiprocessor Systems(RTEMS, reference [14]).

The Problem of LAN Control

In this section the general problem of LAN control is reviewed. The free motionof the real orbit with respect to the reference orbit is analyzed in order to identifywhich orbit perturbations are relevant in the scope of this specific control problem.The relation between the evolution in time of the LAN, an Earth-fixed reference pa-rameter, and the semimajor axis of the orbit is then considered as this is the basisof the control algorithm that is to be implemented. Finally the problems of on-boardatmospheric drag estimation and reference orbit generation are considered as theseare key factors in the realization of a fine autonomous orbit control.

Deviation of the Real Orbit from the Reference

Table 1 shows the MANGO satellite nominal mean Keplerian elements. Themean orbital elements of a satellite in LEO deviate from their nominal values underthe action of perturbing forces. Orbital elements values are dictated by specific mis-sion requirements. In the case of MANGO, the orbit is required to be Sun-synchronous (in general orbits of remote sensing satellites, for which the AOKexperiment is of special interest, are imposed to be Sun-synchronous, phased andfrozen at the same time).

The Precise Autonomous Orbit Keeping Experiment on the PRISMA Mission 479

FIG. 1. PRISMA Formation.

Sun-synchronism is imposed through the relationship between and obtained bysetting the secular angular rotation of the line of nodes, due to the nonsphericity ofthe Earth, equal to the known angular rotation of the meridian plane containing themean Sun. The Sun-synchronism condition is thus represented by equation (1)where the averaged variation of due to the first three zonal terms , and (reference [15]) is represented by equation (2) as

(1)

(2)

where is the mean apparent revolution speed ofthe Sun around the Earth, is the equatorial radius of the terres-trial spheroid, is the parameter of the orbit, and is themean motion (with ).

A reference orbit is an orbit representing the mean nominal motion of the satel-lite over a long time interval. Thus the reference orbit model, in the case consideredhere, should at least consider the nonspherical terms of the Earth gravity potentialwhich cause the Sun-synchronous secular motion of the LAN. Other secular, longperiodic and short periodic terms can be included in the reference orbit dependingon the flight control accuracy requirements. Deviations of real from referenceorbital elements, leading to a violation of flight control requirements, have to becorrected by orbit maneuvers. Figures 2 and 3 show the computed contributions ofthe different perturbations to the deviation of the LAN (controlled parameter)and the semimajor axis (control parameter) from the reference orbit correspondingvalues. The analysis was performed by numerical orbit propagation over 35 days(the scheduled AOK experiment duration) and using orbit propagation model andsatellite physical parameters given in Table 2.

3.9860064 1014 m3 s2n a3p a1 e2

RE 6378,140 mSa 1.99099299 107 rad s1

4 7 sin2 i1 32 e2 3 7 sin2 ie2 cos 2 1516 nJ4 REp 4 cos i15 sin2 i 4e cot i sin

15

16 nJ4 REp 4 cos i 38 n J3 REp 3

e2

8 7 15 sin2 i cos 2 e24 1 54 sin2 i

sin2 i 52 94 1 e212cos i 94 32 1 e2 12 32 nJ2 2 REp 4

3

2n J2 REp 2 cos i

Sa

J4J3J2

ia


TABLE 1. MANGO Satellite Nominal Orbital Elements

Keplerian Element Value

Semimajor axis ( ) 7,087,297 (m)Eccentricity ( ) 0.0015 ()Inclination ( ) 98.1877 (deg)Right Ascension of Ascending Node ( ) 189.8914 (deg)Argument of perigee ( ) 1.0938 (deg)Mean anomaly at epoch ( ) 1.09369 (deg)M

ie

a

At the altitude of MANGO satellite (about 700 km) the major nonconservative per-turbation force is the atmospheric drag whose main effect is the secular decrease of thesemimajor axis and, as a consequence, the fast deviation of the LAN from its nominalvalue. The third-body perturbation has a noticeable influence on the LAN deviationbut a periodic and not appreciable effect on the semimajor axis. The solar radiationpressure (SRP) has a negligible influence on both the LAN and the semimajor axis.


FIG. 2. Deviation of LAN from Reference Orbit Value.

FIG. 3. Deviation of Semimajor Axis from Reference Orbit Value.

Relationship Between LAN and Semimajor Axis Evolution

A preliminary study of the relationship between the LAN time evolution (i.e. thephase difference at the equator) and the semimajor axis is fundamental for the orbitcontrol strategy definition. The phase difference is defined as the differencebetween the real orbit track and the reference orbit track measured along the equa-tor. Considering only the geopotential term and linearizing around the nominalorbit values, the general analytical expression for the evolution of with the orbital elements is (reference [17])

(3)

where is the right ascension of the ascending node of the reference orbit and is the sidereal revolution speed of the Sun around the Earth. The angular speed

is tied to the mean apparent revolution speed of the Sun around the Earth and to the angular rotation speed of the Earth by the equation

(4)

with where is the Sun period (one year), and where is the mean period of solar day (86,400 s). If the

orbit is Sun-synchronous , , and considering equation (4), equation (3)becomes

R SaSyTE7292115 1011 rad s1

E 2TE TSaSa 2TSa

Si Sa E

ESaSi

SiR

6J2REa 2 Si R sin 2i didt RER tan i

7

3 J2REa 24 cos2 i 1 dadt1 73 RSi R

d2

dt2 LAN

3

2Si R REa

LANJ2

LAN


TABLE 2. Propagation Parameters

MANGO Physical Properties Value

Mass 154.4 (kg)Drag area 1.3 Drag coefficient 2.5 SRP effective area 2.5 SRP coefficient 1.3

Real Orbit Propagation Value

Earth static gravity field GRACE GGM01S Tidal perturbations Solid Earth ( diurnal), Polar (IERS/IGS),

Ocean (UT/CSR 3.0 + TEG2B)Atmospheric density Harris-PriesterSun and Moon ephemerides Simplified analytical formulas (reference [16])Solar Radiation Pressure Canon-Ball model (conical Earth shadow)Numerical integration method RK4R: Runge-Kutta 4th order extended by

Richardson extrapolation

Reference Orbit Propagation Value

Earth gravity field GRACE GGM01S Numerical integration method RK4R: Runge-Kutta 4th order extended by

Richardson extrapolation

90 90

4 490 90

m2m2

(5)

with

and and for near-polar orbits.

Basic Orbit Keeping Strategy

Referring to equation (3), if and are taken as constants, is foundto have a near-parabolic variation as its second derivative is constant. This means thatif it is assumed that the semimajor axis has a linear decay under the influence of theatmospheric drag, the real orbit LAN will move parabolically eastwards of the refer-ence orbit LAN. Based on this consideration a simple maneuver cycle can be definedfor the ideal case: when the deviation between the real and nominal LAN exceeds apre-defined upper bound a correcting impulse in the along-track direction of thesatellites orbit is applied as much as twice as necessary for recovering the initial semi-major axis value. As a result, just after the impulse, the satellites LAN begins to movewestward, reaches the lower bound of the allowable band for the LAN and drifts backto the upper limit, where the next correction maneuver is made. The theoretical ma-neuver cycle and the semimajor increments to be applied in the form of equa-tion (6) can thus be obtained from equation (5) assuming also that, as the referenceorbit is Sun-synchronous, can be taken as negligible compared to .

(6)

with .The velocity increment to be applied is

(7)

where v is the mean velocity.Figure 4 shows an example of a guidance process in the ideal case in which the

atmospheric density has been fixed at the value (the average ofthe Harris-Priester model at 700 km altitude) and the atmospheric drag is the onlyorbit perturbation. At each maneuver cycle of about 1.3 days the evolution of theLAN deviation is evenly parabolic while that of the semimajor is linear. The simu-lation is performed over 35 days. Figure 5 shows the guidance process in a test sce-nario in which the atmospheric density is held constant and Sun-Moon, radiationpressure and tides perturbations are considered.

12 1014 kgm3

vc ac2a vKc 163TELMAX

ac Kc aREdadt Tc Kc aREdadt 1dadtdidt

acTc

LMAX

LANdidtdadt

1 1

12 J2 TSaTE REa 2

cos2 i

7

3 TETSa 72 J2 REa

2

4 cos2 i 1

dadiSy 23 a TETSa 1 1 tan i

d2

dt 2 LAN

3

REa 1 dadt didt dadiSy


Atmospheric Drag Estimation

By considering equation (6) it is straightforward to conclude that a correct andfine estimation of the correction maneuver is strictly connected to a correct estima-tion of which, as already outlined (Fig. 3), is mainly determined by the at-mospheric drag. As at the altitude of MANGO the atmospheric density has aremarkable variability with time, mainly due to the variability of the solar flux, anatmospheric model cannot give the required accuracy. On the other hand the ac-celerometers on-board the MANGO satellite are not suitable for the estimation ofthe atmospheric drag as the typical bias is , much larger than typicalatmospheric drag acceleration values at 700 km . The problem hasbeen solved starting by the simple consideration that all the information requiredfor the estimation of is available on-board the spacecraft: the knowledge ofthe actual orbit estimated by the on-board navigation filter with an accuracy of 2 m1 (reference [9]) and the reference orbit (uploaded or computed on-board).

The value of is then found with the following steps:

1. Filter the actual and reference orbit semimajor axis data by taking their values atthe ascending node passes.

2. Compute the differences between the actual and reference orbit semimajoraxis values.

3. Smoothing of data if required.4. Find as the angular coefficient of the line representing the linear fit over

the time of a sample of data.aANdadt

aAN

dadt

dadt

105 ms24 102 ms2

dadt


FIG. 4. Semimajor axis deviation (top), LAN deviation (middle), magnitude of along-track maneuvers (bottom) computed by AOK during the guidance process.

Test scenario: constant drag, no other perturbation.

Figure 6 shows this procedure applied in the case in which the real orbit is gener-ated by numerical orbit propagation including the aspheric Earth gravity field(GGM01S model) through an expansion in spherical harmonics up to degree andorder 90, tides and relativity gravity field perturbations, the Sun and Moon thirdbody forces, the atmospheric drag (atmospheric density model Harris-Priester) andsolar radiation pressure. The reference trajectory has been generated using the samegravity field model used to generate the real orbit. Navigation errors are not included.

Figure 7 shows the corresponding guidance process in which a sample buffer of10 s is renewed at each ascending node passage for the estimation of .The buffer is reset after each maneuver. It can be noticed that the first maneuver isoverestimated because at this point only five samples were available and, as no fit-ting was possible, the has been estimated using the first of the Gauss equations (equation (8)) approximated for circular orbits (reference [17])

(8)

where is the atmospheric density from the Harris-Priester model, is the ballis-tic coefficient of the satellite, and .

Reference Orbit Generation

The reference orbit can be generated either on-ground and uploaded on-board thesatellite during a ground station pass or directly on-board. The great advantage of

3.9860064 1014 m3 s2B

da

dt

a

dadt

dadtaAN


FIG. 5. Semimajor axis deviation (top), LAN deviation (middle), magnitude of along-track maneuvers(bottom) computed by AOK during the guidance process. Test scenario: constant drag, Sun-Moon, solar

radiation pressure, tides and relativity gravity field perturbations.


FIG. 7. Semimajor axis deviation (top), LAN deviation (middle), magnitude of along-track maneuvers(bottom) computed by AOK during the guidance process (external uploaded reference orbit). Test scenario:

drag (Harris-Priester model), Sun-Moon, solar radiation pressure, tides and relativity gravity fieldperturbations.

FIG. 6. Data Fitting Process.aAN

the on-ground generation is that a numerical orbit propagation including all theavailable harmonics of the gravity field model can be used thus achieving the bestcontrol performance. The disadvantage is that the reference orbit has to be uploadedperiodically (three days in the case of MANGO) imposing a short time limit on theautonomy of the orbit control. On the other end while an on-board reference orbitgeneration allows much longer autonomous control periods, the performance is de-graded by the fact that only limited gravity field models can be exploited due to thelimitation imposed by available on-board CPU resources.

In the case that a poor on-board generated orbit is used, the estimation of turns to be difficult. The short period oscillation of cannot be captured due tothe lack of short period perturbations in the gravity field model. Figure 8 shows thecomparison of the curves obtained with a reference orbit generated with a

gravity field model and one generated with a model. In the case ofthe orbit model a short period (for example 10 points) linear fitting is totallyunreliable for estimating the atmospheric drag. A long period (50 points) linear fit-ting would allow an accurate estimation of , but is not useful at the end ofthe orbit control because it would require a minimum maneuver cycle period of50 orbits (i.e. more than three days). Nevertheless an efficient estimation of is still possible using samples of 10 values but this time applying a propersmoothing process. The smoothing filter takes as input the points marked by circlesin Fig. 8 and flattens them on a line keeping unaltered the ratios of their distancesfrom it. Line is defined by point and angular coefficient m where is the x-coordinate of the first point, is the mean of the y-coordinates of all the pointsof the sample and m is the estimated . A guess value of is used for thevery first smoothing process. The output of the smooth filter are the points markedby triangles.

dadtdadty

x1x1, y

aANdadt

dadt

5 55 590 90

aAN

aANdadt


FIG. 8. Estimation of for a Reference Orbit Generated with a Gravity Field model.5 5dadt

Autonomous Orbit Keeping

The AOK experiment is a minimal target control problem as it has to demonstratethe autonomous fine control of the LAN with respect to a predefined reference orbit.The control accuracy with respect to a predefined phasing grid fixed to a given ref-erence frame on Earth (for example a given longitude) can then be assessed byanalyzing the relative movement of the reference orbit and the phasing grid. AOKoperations will commence after the end of the PRISMA nominal operations. The testcampaign foresees one week of initial configuration, calibration and open-loop testsand four weeks of closed-loop operations. In this section the AOK controller logicand the software development and testing environment are described.

AOK Features

The AOK control cycle is computed from equation (6) with estimatedthrough the process described in the preceding section. The main control logic isthe following:

1. An along-track velocity increment (equation (7)) is commanded if the timeelapsed from the last maneuver is larger than a predefined time interval , theLAN deviation exceeds a predefined upper limit , and isgrowing with time . Constants and are uploaded to thespacecraft via Telecommand (TC).

2. After the first along-track maneuver in order to calibrate (equation (6)) onthe actual atmospheric drag value, it is recomputed using equation (9) as

(9)

where is estimated by the long period fitting (with the number of pointsimposed by TC) if possible or by the ordinary short period fitting, is the cal-ibrated value of and is the actual cycle period i.e. the time elapsed fromthe actuation of the last maneuver.

3. An anti-along-track velocity increment (equation (7)) is commanded ifthis possibility is enabled by TC, the LAN deviation exceeds a predefined lowerlimit ( where is a real constant), and the timeelapsed from the last maneuver is larger than a predefined time interval .Constants and are uploaded to the spacecraft via TC. In this case

(10)

where is computed with equation (6) and is an empirical correction factorimposed by TC.

Figure 7 provides an example of the AOK guidance process in the case of a reference orbit being uploaded from ground. The LAN deviation upper limit,

is 10 m, the anti-along-track maneuvers are not enabled, and is zero. Asthe reference orbit is the best available and neither navigation nor sensors and actu-ators errors are introduced in the simulation, has a smooth parabolic evolution.The mean of is 2.6 m and (standard deviation) is 7.4 m after the controlhas reached a steady state (around the fourth day). The control accuracy of the os-culating ascending node is then compliant with the main requirement (10 m 1 ). In

LANLAN

tv+LMAX,

Tc

acal dadt ac Tctvn

tvdLANdt 0nLAN n LMAX

v

Tacac,acal

dadtac

acal dadt ac Tac2 ac2

ac

tv+LMAXdLANdt 0LANLAN LMAXtv+

v

dadt


contrast Fig. 9 gives an example of AOK guidance with an on-board propagatedorbit (gravitational field model GGM01S ). The upper limit is 50 mand is five orbital periods (about eight hours). The enlarged control window andwaiting times are chosen to compensate the presence of oscillations in the evolu-tion of which are due to the incompleteness of the reference orbit. Theenlargement of these two parameters allows AOK to correctly estimate . Aparabolic but noisy evolution of can still be observed. This time the mean of

is and is 53 m and the control accuracy is not compliant with themain requirement. Figure 10 shows the same test case as Fig. 9 but with the enablingof anti-along-track maneuvers. The control window is this time bounded between

and ( = 50 m, = 2). In this case the performance of AOKis better (the mean of of is 22 m and is 45 m) but at the cost of a largernumber of maneuvers which impose a negative most of the time.

Software Development and Testing Environment

Figure 11 shows the simplified architecture of the PRISMA on-board software(OBS) architecture for the MANGO satellite and the simulation environment (seealso reference [18]).

The PRISMA OBS has a layered structure with a Basic Software (BSW) leveland an Application Software (ASW) level communicating with each other throughdedicated message queues. While the BSW includes basic applications, devicedrivers and I/O-utilities, the ASW encapsulates all top-level applications likespacecraft navigation, control, telecommand and telemetry. The GPS-based navi-gation system is split into three modules located in different OBS levels and run-ning at different sample rates. The GPS interface (GIF) is part of the BSW, runs at1 s sample time and is directly fed with GPS messages issued by the Phoenix GPS

vLAN

nLMAX50 m100 m

50 mLANLAN

dadtLAN

tv+LMAX5 5


FIG. 9. Semimajor Axis Deviation (Top), LAN Deviation (Middle), Magnitude of Along-TrackManeuvers (Bottom) Computed by AOK During the Guidance Process: On-Board Generated Reference

Orbit, , .tv+ 0LMAX 50 m

receivers emulator (PEM). GIF handles GPS raw data formats and ephemerides,and performs data sampling as well as coarse editing prior to the GPS-based orbitdetermination. The GPS-based Orbit Determination (GOD) and GPS-based OrbitPrediction (GOP) are embedded in the ASW layer as part of the ORB core (30 ssample time) and the GNC core (1 s sample time), respectively. GOD implementsan extended Kalman filter to process GRAPHIC observables as well as single dif-ference carrier phase measurements from MANGO satellite. Attitude data fromboth spacecraft are applied to correct for the GPS receivers antenna offset with


FIG. 11. Simplified Scheme of the Software Architecture.

FIG. 10. Semimajor Axis Deviation (Top), LAN Deviation (Middle), Magnitude of Along-TrackManeuvers (Bottom) Computed by AOK During the Guidance Process: On-Board Generated Reference

Orbit, .LMAX 50 m 100 m, tv+ 8 h, tv 0

respect to the spacecraft center of mass. Furthermore, a history of maneuver datais provided to GOD and taken into account in the orbit determination task. GODperforms a numerical orbit propagation which is invoked after the measurementupdate and provides orbit coefficients for interpolation to GOP for both spacecraft.The GOP module interpolates the orbit coefficients provided by GOD and finallysupplies the various GNC core functions as well as the PRISMA payload withcontinuous position and velocity data. Due to the different data rates of the GPS-based navigation modules, orbit maneuver data have to be taken into account inboth GOD and GOP (cf. Fig. 11). In particular at each GNC step, the GOP taskaccounts for maneuvers which have not been considered by GOD in the last orbitdetermination/prediction process. The AOK and Autonomous Formation Control(AFC) modules are located in the ORB and GNC cores respectively. Finally theorbit propagator (PROP) generates the MANGO simulated orbit. The software isimplemented in Matlab/Simulink as well as C/C++. In particular, C++ low levelroutines, which are interfaced with Matlab/Simulink high level structures, repre-sent the computational layer of the software system including for example nu-merical integrations, data filtering, etc. Matlab/Simulink provides instead thecommunication layer, including for example input/output interfaces, complex log-ics, time synchronization. The prototyping of the software as well as the relatedanalysis and simulations are performed first on a host standard laptop PC in aMatlab/Simulink environment. All the functions can then be auto-coded using theMatlab tool Real Time Workshop and executed under the operating system RealTime Executive for Multiprocessor Systems (RTEMS) on the target LEON3processor (the PRISMA satellites on board computer). The full consistency of theflight software generation chain can thus be verified by comparing relevant out-puts generated by the software running on the host and target computer. One of thegreatest advantages of this straightforward and cheap approach is that it allowsthe test of the flight software performance directly on the on-board spacecraft com-puter. The analysis of the software execution times has special interest because itallows verification if the different software modules comply with the maximal al-located CPU loads. Analysis of heap and stack allocation and software profilingcan also be made for the detection of eventual software execution bottlenecks.

Table 3 collates the execution times and CPU loads of AOK running on the LEON3board in three modes: with an external uploaded RO, a RO generated on-board by apropagation performed in one call or in several calls. Only the one-step propagationis not compliant with the CPU load constraints. External uploaded and multi-step on-board propagated RO modes are fully compliant with the CPU load constraints be-cause AOK is located in the ORB core that is a low priority task. The possibility ofusing AOK in both external and internal propagated RO modes, enhances the robust-ness and the autonomy of the control as AOK can automatically switch to on-boardgenerated RO in case the external uploaded RO is corrupted or expires.


TABLE 3. AOK Execution Times and CPU Loads on LEON3 Board

AOK Mode Execution Time (ms) CPU Load (%)

External uploaded RO 63 0.2On-board generated RO (multi-step) 596 2On-board generated RO (one-step) 49623 165

Numerical Simulation

Figure 12 shows how the guidance process of AOK in the test scenario of Fig. 7changes if anti-along-track maneuvers are enabled. The navigation errors are includedand sensors and actuators are modeled by applying realistic errors to measurementsand control actions respectively. The reference orbit is generated on-board using aGGM01S gravity field model.

Table 4 shows mean and standard deviation values of the LAN deviation and thedelta-v budget for the simulations of Fig. 12.

The propulsion system is characterized by a Minimum Impulse Value (MIV) of0.7 and a Minimum Impulse Bit (MIB) of 0.07 . Consequently, thethrusters can only realize s which are larger than MIV and integer multiples ofMIB. The performance error of the thruster system is quantified by equation (11) as

(11)

where is the planned velocity increment computed by the AOK controller andis the actual velocity increment realized by the propulsion system. Finally the

attitude control error, which causes deviations from the desired thrust direction, istreated as Gaussian noise with zero bias and standard deviation in three axes.Table 5 collects the navigation accuracy and the parameters used for the modelingof sensors and actuators.

From an analysis of the simulation results shown in Fig. 12 and Table 4 it can beconcluded that the inclusion of navigation errors causes an oscillating evolution of

and consequently a different evaluation of the LAN deviation with respect to thea

0.2

vrvp

vr vpvp 100v

mmsmms

20 20


FIG. 12. Semimajor Axis Deviation (Top), LAN Deviation (Middle), Magnitude of Along-TrackManeuvers (Bottom) Computed by AOK During the Guidance Process.

case in which no errors are included. It follows that the maneuver planning is dif-ferent in the two cases. The inclusion of navigation errors decreases the estimationaccuracy of causing a reduction in the efficiency of the commanded maneu-vers and consequently the increase of required thrust pulses in the guidanceprocess. This is also confirmed by a 10 % increase of the delta-v budget. The in-clusion of sensors and actuators errors further reduces the efficiency of the com-manded maneuvers and this is reflected in the control accuracy. In the case in whichall the errors are included the mean of is 1.3 m and is 8.7 m. This resultis compliant with the main requirement of 10 m (1 ).

Conclusion

The AOK experiment is the secondary objective of the DLR contributions to thePRISMA formation flying mission. It has to demonstrate autonomous orbit controlof the longitude of the ascending node of a single spacecraft with an accuracy of10 m (1 ). The software implements an analytical feedback control algorithm usingalong-track and anti-along-track velocity increments. In the realization of the AOKcontroller the problem of a correct estimation of the atmospheric drag has beensolved. The software applies a filtering, linear fitting and smoothing of the differ-ence between the reference and real orbits semimajor axis values at the ascendingnode passes. Advantages and disadvantages of using an on-ground or on-board gen-erated reference orbit have been analyzed in terms of performance and operations.The results of the simulation performed are encouraging in showing that the ful-fillment of the strict control accuracy requirement is feasible. Execution time testson a representative PRISMA on-board computer indicate that at the present statethe software is fully compliant with the CPU load requirements.

LAN

dadt


TABLE 4. LAN Control Accuracy and dv Budget

Test Case Mean [m] [m] dv budget [ms]

Ideal 0.6 8.3 0.09Only navigation errors 0.9 8.3 0.1With navigation and actuators errors 1.3 8.7 0.1

TABLE 5. Navigation, Sensors and Actuators Accuracy Modeling

Absolute Navigation Accuracy Value

Mean 0 (m)2 (m)

Attitude Control Accuracy Value

Mean 0 ( )0.2 ( )

Propulsion System Accuracy Value

MIVMIB

27 105 ms7 104 ms

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The precise autonomous orbit keeping experiment on the PRISMA mission

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