Ballistic Missile Trajectory

ISTANBUL UNIVERSITY – JOURNAL OF ELECTRICAL & ELECTRONICS ENGINEERING

YEAR VOLUME NUMBER

: 2005 : 5 : 2

(1463-1467)

Received Date : 13.5.2004 Accepted Date: 1.6.2005

DEVELOPMENT OF A TACTICAL BALLISTIC MISSILE TRAJECTORY PREDICTION TOOL

1Ali AKGÜL 2Sartuk KARASOY

1,2Engineering Development Directorate ROKETSAN Missiles Industries Inc. P.K.:30, 06780, Elmadağ-Ankara, Turkey

1E-mail: [email protected] 2E-mail: [email protected]

ABSTRACT

A trajectory prediction program was developed to predict the full trajectory (partially tracked by a satellite or a radar) of a tactical ballistic missile. In order to improve the accuracy of the ballistic coefficient an adaptive CD model is developed. Dynamic model, filtering of measurements, geocentric earth model and atmosphere model were applied and the details of the models and test results are presented. Input to the program is coordinates of the tracked trajectory portion and output is the full trajectory. In this way the launch and target point can be obtained within seconds of execution time.

Keywords: Trajectory prediction, application of satellite measurements, ballistic model, target tracking, missile launch surveillance 1. INTRODUCTION It is important to detect ballistic missile launches in time and accurately for effective counter measures. Trajectory estimation plays an important role in early warning system and anti-tactical ballistic missile systems for tracking and guidance. Early detection could allow for warnings to be dispatched to the predicted impact areas and would also permit interdiction of the missile launchers. The detection of launches is also important in alerting defensive systems, such as anti-tactical ballistic missile batteries, against these attacks, as early as possible [1], [2], [3]. For these purposes missile surveillance system is needed to complete national security and to

strengthen existing defense systems. Such systems may consist of satellite and/or aircraft borne radar systems. The primary purposes of missile launch surveillance are: 1) to provide a timely report of each occurrence

of a missile launch, 2) to estimate launch/trajectory parameters, and 3) to estimate present and future missile

trajectories as a function of time during flight.

In the present work, the main objective is to predict launch and target points. After the missile position and velocity vectors are obtained, the prediction starts with the time integration of equations of motion using a six-parameter-state vector, which is composed of the position and

Development Of A Tactical Ballistic Missile Trajectory Prediction Tool

Ali Akgül, Sartuk Karasoy

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velocity vectors. Backward time integration is used for predicting the launch point on the other hand, forward time integration is used for predicting the target point. The integration procedure continues until the criterion of intersection of the missile with the Earth surface is satisfied. 2. DYNAMIC MODEL The Ballistic Missile flies along a ballistic trajectory above the rotating Earth. Constructing a mathematical model for this system with some simplifying assumptions to describe its motion is the first step for identification and analysis. Consider a vehicle in the free-flight phase over an elliptic [4], non-rotating Earth, as illustrated in Fig. (1). In this study, the Ballistic Missile is taken as a point mass with constant weight following a ballistic trajectory. Significant forces acting on the Ballistic Missile are Drag and gravity. The drag force acts in the opposite direction to the velocity vector with a magnitude proportional to the air density (which is a function of the altitude [6]).

Fig. 1 Dynamic Model

The ballistic trajectory of a Ballistic Missile can be evaluated by the flight simulation programs which are using the thrust and aerodynamic data obtained from tests or analysis. While tracking a Ballistic Missile, it is considered to be difficult to estimate rocket motor size and thrust, aerodynamic and external geometry of the Ballistic Missile. The full trajectory of a tactical ballistic missile can be predicted from a tracked portion of the trajectory. In active phase of the flight, trajectory between the current location and launch point can be predicted by backward

integration in time. However target point and the trajectory directed to this point cannot be predicted properly. On the other hand, in the free-flight phase launch and target points and full trajectory can be predicted. Trajectory Equations of Motion

The trajectory differential equations are

xx mamgD =−− 21 coscos γγ (1)

yy mamgD =−− 1sin γ (2)

zz mamgD =−− 21 sincos γγ (3)

where D is the drag force; ax, ay and az denote acceleration as components along x, y and z respectively; gx, gy and gz refer gravity components along x, y and z respectively; m is the mass of the Ballistic Missile and γ1 - γ2 are the flight path angles.

+−= −

22

11 tan

zx

y

vv

vγ (4)

= −

x

z

vv1

2 tanγ (5)

replacing D, equation (1) becomes,

xxD ag

mSCV =−− 21

02 coscos21 γγρ (6)

In equation (6), ρ is the air density; V denotes missile velocity, S and CD0 represent the reference area and zero lift drag coefficient, respectively.

In equation (6), all parameters can be

estimated from the ballistic trajectory data, except S, CD0 and m. Ballistic coefficient β is defined as,

mSCD0=β (7)

Instead of using S, CD0 and m, the ballistic coefficient (equation (8)) can be used to predict ballistic flight trajectory. Ballistic coefficient can then be redefined as

212 coscos

21 γγρ

βV

ga xx

−

+= (8)



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Prediction Algorithm

While tracking a Ballistic Missile, real time trajectory, target and launch points are desired.

The trajectory estimator needs to characterize the position and velocity of the missile. Usually for this purpose, inertial Cartesian coordinates of {x,y,z} and { zyx ,, } are used as the estimation parameters. The launch and impact points can be predicted using state vector measured at different time points in the missile trajectory. The six-parameter-state vector, which is composed of the position and velocity vectors, are numerically integrated either forward or backward in time until the Earth is intersected.

To start integration and reconstruction of the ballistic trajectory, ballistic coefficient, initial position, velocity and acceleration components of the Ballistic Missile must be known. These parameters can be estimated from the known trajectory portion.

Let the state vector be,

=

=Χ

6

5

4

3

2

1

xxxxxx

vvvzyx

z

y

x

(9) (9)

The state equation can be written as,

),( Χ=Χ tF (10) (10)

or

−++−−++−

−++−=Χ

z

y

x

gxxxgxxx

gxxxxxx

tF

2126

25

24

126

25

24

2126

25

24

6

5

4

sincos))(2/(sin))(2/(

coscos))(2/(),(

γγβργβρ

γγβρ

(11) (11)

State equation can be numerically integrated either forward or backward in time until the Earth is intersected by equation (12).

∫ ΧΧ=Χt

ttt dttF

00

),(∓ (12)

Numerical Integration The trajectory parameters at any time step can be predicted with the integration of the known state vector from the previous time step. Acceleration components at time step (k-1) are calculated

using equation (13). Position and velocity components at time step (k) are calculated using equation (14). This integration is continued recursively either backward or forward in time until the altitude reaches to zero. Intersection points are the launch and the target points respectively. The predicted launch and target points and the trajectory are shown in Fig. (2).

121222

1222

21222

1sincos))(2/(

sin))(2/(coscos))(2/(

−−

−++−−++−

−++−=

kzzyx

yzyx

xzyx

kz

y

x

gvvvgvvv

gvvv

aaa

γγβργβρ

γγβρ (13)

state vector at step k,

1

001000000001000000001000000001000000001000000001

−

=

kz

y

x

z

y

x

kz

y

x

aaavvvzyx

dtdt

dtdt

dtdt

vvvzyx

(14)

dt is the time step.

Range (m)

Altitude(m)

-50000 0 50000 100000 150000 200000 250000 300000 350000 400000 450000 500000 550000 600000 650000 700000 750000

-40000

-20000

0

20000

40000

60000

80000

100000

Range (m)

Altitude(m)

25000 50000 75000 100000 125000 15000010000

20000

30000

40000

50000

60000

Predicted T

rajectory

Measured

Data

Predicted

Trajectory

Fig. 2 Launch and Target Point Prediction

Filtering Since the data obtained from satellite or radar is noisy, it needs to be filtered by appropriate algorithm. In this present work least-squares (LSE) filtering is applied. While tracking the Ballistic Missile, position and/or velocity and/or acceleration components are obtained from the satellite or radar. When this is not possible (a problem in communication link or due to insufficient capability) velocity and/or acceleration components can be calculated by



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using the known ballistic trajectory position data (whose existence is a must to make trajectory prediction). LSE is applied by polynomial fitting to the discrete position data in order to obtain a curve, which is a function of time. Velocity and acceleration vs. time curves can be obtained by calculating first and second derivatives respectively (equation (15)). Obviously the predicted trajectory is more accurate when it is possible to measure all trajectory parameters (position, velocity and acceleration).

fttt

ft

txtxtx

txLSEtttx

,,

0

0)( )()(

)(,, ,

…

…

=

⇒⇒⇒= (15) (15)

Ballistic Coefficient Update Model In the literature, the same ballistic coefficient value, which is calculated for the tracked portion of trajectory, is used in the trajectory reconstruction. However, since the ballistic coefficient is a function of CD0, S and m (equation (7)) and due to fact that CD0 changes with the flight conditions, the ballistic coefficient is no longer a constant value in the reality (Fig. (3)). Drag coefficient is the only factor that alters the ballistic coefficient during the flight. In order to update the ballistic coefficient for each estimated trajectory location and to use an adaptive CD model, a database that consists of CD0 against varying Mach number and altitude is prepared previously. For this purpose, a typical tactical ballistic missile geometry with an axisymmetric body only (without any protuberances and lifting surfaces) is used in the calculations done with Missile DATCOM [5]. Below equation, ( )SDCβ is a scale parameter that is obtained from the tracked trajectory portion, ∗

DC denotes the current estimated drag coefficient calculated by adaptive CD model and

∗β is the updated ballistic coefficient for the current estimated trajectory location.

∗∗ ×

= D

S

D

CCββ (16)

Time (sec)

BallisticCoefficient

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 1600.4

0.5

0.6

0.7

0.8

0.9

1.0

TrajectoryPrediction

Fig. 3 Ballistic Coefficient Update Model

3. TEST CASES

Five different ballistic missile trajectories were reconstructed by using the flight simulation program. For each of these appropriate noises were added to the trajectory parameters in order to induce the measurement errors [7]. To simulate the real time tracking of a Ballistic Missile from a satellite or a radar, each test trajectory is split into small trajectory portions and input to the trajectory prediction program. Then the program reconstructs the trajectories corresponding to the each of the tracked trajectory portions.

Maximum ballistic trajectory parameters of

these different ballistic test missiles are shown below,

Max. Range (km) 173 - 1410 Max. Altitude (km) 37 - 142 Flight Time (s) 190 - 382 Max. Velocity (m/s) 1150 - 4450 Having the longest range, Ballistic Test

Missile-5 trajectory is presented. The test trajectory and a portion of it (as modeling the satellite measurement) and corresponding predicted trajectory by the program are shown in Fig. (4). The predicted trajectories around target point and launch point corresponding to each tracked portion are shown in Fig. (5) and Fig. (6) respectively. The program predicts the launch and target points within seconds of execution time. Launch and target points were predicted



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accurately within the 1% error of its maximum range.

x (m)

y(m)

0 250000 500000 750000 1E+06 1.25E+06 1.5E+06-200000

-150000

-100000

-50000

0

50000

100000

Satellite MeasurementPredicted TrajectoryEarthReal Trajectory

Fig. 4 Trajectory Prediction

x (m)

y(m)

1.4125E+06 1.415E+06 1.4175E+06 1.42E+06-163000

-162500

-162000

-161500

-161000

-160500

-160000Real Trajectory

Fig. 5 Target Point Prediction

x (m)

y(m)

-10000 -8000 -6000 -4000 -2000 0 2000 4000-1000

-500

0

500

1000

1500

2000Real Trajectory

Fig. 6 Launch Point Prediction

4. CONCLUSION In this work, a trajectory prediction program was developed to predict the full trajectory (partially tracked by satellite or radar) of a tactical ballistic missile. In order to improve the accuracy of the ballistic coefficient an adaptive CD model is used. The prediction of launch and target point can be obtained within seconds. Several different test cases were tested with the present program. The accuracy of predicting the launch and target points is high. Acknowledgement This work was supported by the Turkish Ministry of Defense under the lead of Turkish Air Force (TuAF). The authors gratefully acknowledge this support. The authors would like to thank Koray Taylan for programming the graphical user interface, Alp Marangoz, Ercan

Örücü and Ayça Yetere for programming support and preparing the test trajectories and finally to Erhan Tarhan for reading and commenting the manuscript. REFERENCES [1] Lee, S.C., Huang, C.Y., Liu, C.Y.,

“Trajectory Estimation for Tactical Ballistic Missiles in Terminal Phase Using On-Line Input Estimator”, Proc. Natl. Sci. Counc. ROC(A), Vol. 23, No. 5, 1999, pp.644-653.

[2] Lee, S.C., Liu, C.Y., “Trajectory Estimation of Reentry Vehicles by Use of On-Line Input Estimator”, Journal of Guidance Control and Dynamics, Vol. 22, No. 6, Nov.-Dec. 1999,pp.808-815.

[3] Crumpton, K.S., “The Effects of Filter and Coordinate Frame Selection on the Accuracy and Efficiency of Tactical Ballistic Missile Parameter Estimation and Prediction”, Ph.D. Thesis, University of Colorado, 2000.

[4] Robert M. Rogers, “Applied Mathematics in Integrated Navigation Systems”, AIAA Education Series.

[5] Buns, K. A., Stoy, S. L., Vukelich, S. R., “Missile DATCOM: User Manual – 1997 Fortran 90 Revision”, 1998.

[6] “U.S. Standard Atmosphere, 1976”, U.S. Government Printing Office, Washington, D.C., 1976.

[7] William H. Press, “Numerical Recipes”, Cambridge University Press.

Ballistic Missile Trajectory

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