Resident Space Object Tracking Using an Interacting Multiple Model Mixing Scheme Quang M. Lam1 LexerdTek Corporation Clifton, VA 20124
A multiple model estimation scheme is proposed to enhance the robustness of a resident space object (RSO) tracker subject to its maneuverability uncertainties (unplanned or unknown jet firing activities) and other system variations. The concept is based on the Interacting Multiple Model (IMM) estimation scheme. Within the IMM framework, two Extended Kalman Filter (EKF) models: (i) a 6 State (Position and Velocity of a constant orbiting RSO) EKF and (ii) a 9 state (Position, Velocity, and Acceleration of a maneuvering RSO) EKF are designed and implemented to achieve RSO maneuvering detection and enhanced tracking accuracy. The IMM estimation scheme is capable of providing enhanced state vector estimation accuracy and consistent prediction of the RSO maneuvering status, thus offering an attractive design feature for future Space Situational Awareness (SSA) missions. The design concept is illustrated using the Matlab/Based Simulation testing environment. I.
Introduction
There are several error sources that can cause the target state estimator to produce a poor performance characteristic. Two primary ones, which concern designers the most, are: (1) the mismatch between the design filter model and system model (a precise model of the system is difficult to obtain) and (2) the dynamic variations of the state equation due to target maneuverability or environmental torque disturbances. The bigger the mismatch between the system model and the filter design model is, the more degraded performance the tracking system will exhibit. Therefore, the key factor to the successful development of an enhanced RSO tracker is to design a robust estimation scheme which has less susceptibility to system variations and RSO’s unpredictable maneuvering plan. Performance expectations of future tracker may be required to possess a very robust performance characteristic so that it can address situations wherein the design filter model is inadequate to represent or reproduce the system responses collected by a set of sensors. In this case, the robust estimation scheme is expected to successfully “reconstruct” the signal or adequately provide the target state estimate vector at an accuracy level which is significantly better than the results produced by current estimation approaches. It is obvious that the multiple model estimation scheme is the most attractive scheme to address the filter model mismatch of an unpredictable and unknown maneuvering plan.
1
[email protected], Ph.D. and CTO
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The problems of detecting and tracking maneuvering targets for terrestrial applications have been extensively studied during the past three decades [1-6]. Methods for detecting and tracking maneuvering targets are dictated by the operating conditions and environmental variations. In other words, the maneuvering target tracking solutions will vary from clutter-less environments to high-clutter, false- alarm, or closely-spaced-targets environments. In the former, the new measurement is used to update an existing track and help to detect a maneuver. In the latter, measurement collected in this “noisy” environment could lead to an erroneous conclusion about the target maneuverability because of the difference in the interpretation of a measurement in the vicinity of an existing track. For space based applications (i.e., Space Situational Awareness, SSA, see [7-11]), RSO maneuvering detection and tracking still remain as an active research topic and are currently considered as challenging subjects, especially for the front end measurement to RSO's state vector association in the context of multiple RSOs multiple sensors detection and tracking. In this effort, we assume the front end measurement to individual RSO's state vector association is accomplished via [11] and the reference list cited therein and the investigation mainly focuses on RSO's state vector accuracy improvement and maneuvering detection using the IMM scheme. This paper investigates the IMM based design architecture as a solution to improve RSO’s state vector estimate accuracy subject to maneuvering uncertainty. Its benefits are illustrated using Matlab/Simulink based environment. In section II, we present the problem formulation of a six state EKF tracking an RSO. Section III presents the performance evaluation of the proposed IMM along with some discussions. Section IV concludes the paper with some summary remarks. II. Problem Formulation A. Extended Kalman Filter Formulation The position and velocity vectors of the target relative to the sensor platform, i.e., X(t)=[r t(t) vt(t)]T, expressed in an ECI frame are written as follows [7]: d rt (t ) rt (t ) vt (t ) dt (2.1) rt (t ) d vt (t ) vt (t ) 3 w(t ) dt rt where rt(t) and vt(t) are target or RSO position and velocity, respectively and is the gravity constant and w(t) is the state process noise accounting for both gravitational and nongravitational acceleration uncertainties. Rewriting equation (2.1) in terms of state space continuous dynamic expression as follows,
rt 03 x 3 d X (t ) dt vt f (r , t ))
I 3 x 3 rt 03 x 3 w(t ) (2.2) 03 x 3 vt I 3 x 3 The term f(r,t), expressed in equation (2.2), is a nonlinear function per axis and exists only for the diagonal terms. Equation (2.2) will be used as the dynamic predicted model for the EKF when we linearize the nonlinear function f about its current state vector as first order Taylor 2
Series expansion. It will be carried out in the complete EKF formulation once we introduce the nonlinear measurement model as follows,
h 1 (X) 1 (t ) (t ) z m (t ) (t ) h( X (t )) (t ) h 2 (X) 2 (t ) (t ) el (t ) h 3 (X) 3
or
2 2 2 E Z s 1 (t ) 1 - s z m (t ) tan ( ) 2 (t ) E (t ) 1 z 3 sin ( )
(2.3)
where h1() is the range ((t)), h2() is the azimuth angle ((t)), and h3() is the elevation angle (el(t)). The continuous dynamic state space system model of equations (2.2) and (2.3) is now written in a compact form so that later it can be “linearized” and “discretized” as a block transformation into the discrete EKF process, d X (t ) f (t , X ) X (t ) L w(t ) dt z m (t ) h( X ) (t )
(2.4)
The continuous EKF model (2.4), linearized about its state vector for the dynamic process f(t,X) (i.e., only for the velocity term) and the measurement process h(X), are written as follows, d X (t ) F (t , X ) | ˆ X (t ) L w(t ) X X dt z m (t ) H ( X ) | ˆ (t )
(2.5)
X X
where the partial linearized block F and H matrices are computed as follows
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0 0 0 f1 F rtx f 2 r tx f 3 r tx
0 0
0 0
1 0
0 1
0 f1 rty
0 f1 rtz
0
0
0
0
f 2 rty
f 2 rtz
f 3 rty
f 3 rtz
0
0
0
0
0 0 1 0 0 0 X Xˆ
where
f1 3X 2 1 f1 3XY 3XZ f1 = [ 5 3 ] ; = [ 5 ] ; = [ 5 ] rty r r r r rtz rtx 3Y 2 1 3XY f 2 3YZ f 2 f 2 = ; = = [ 5 ] ; [ 3] ; 5 5 rty r r r r rtx rtz 3Z 2 1 f 3 f 3 XZ f 3 YZ = 3 5 ; = 3 5 ; = [ 5 3 ] ; rty r r r r rtx rtz r rtx2 rty2 rtz2 Define X=rtx; Y=rty; Z=rtz
h1 rtx h H 2 rtx h3 rtx
h1 rty
h1 rtz
h1 vtx
h1 vty
h2 rty
h2 rtz
h2 vtx
h2 vty
h3 rty
h3 rtz
h3 vtx
h3 vty
h1 rtx h H 2 rtx h3 rtx
h1 rty
0
0
0
h2 rty
h2 rtz
0
0
h3 rty
h3 rtz
0
0
h1 vtz h2 vtz h3 vtz ˆ X X 0 0 0 ˆ X X
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Where
h1 X Y Z h1 h1 = ; = ; = ; 2 2 2 0.5 2 2 2 0.5 2 rty ( X Y Z ) rtx ( X Y Z ) rtz ( X Y 2 Z 2 )0.5
h2 Y X h2 = 2 ; = 2 ; 2 rty X Y 2 rtx X Y
h3 2XZ 2YZ h3 = ; = 2 2 2 2 2 0.5 2 2 rty (X Y )( X 2 Y 2 Z 2 ) 0.5 rtx (X Y )( X Y Z ) (X 2 Y 2 Z 2 ) 2Z 2 h3 = ; rtz (X 2 Y 2 )( X 2 Y 2 Z 2 ) 2 Note that the predicted position (i.e., X, Y, and Z) in the ECI frame estimated by the EKF will be used to compute the partial derivatives. Extended Kalman Filter (EKF) Processing Flow: The OD processing is accomplished on the ground to avoid computational difficulties with on-board processing. We compute the discrete pair [,Qk] of the state transition matrix and process noise using the algorithm described in Ref. 4 in real time as shown in the processing flow of Figure 1. The initial orbit state vector (i.e., 6 state position and velocity elements) and the 6x6 initial state error covariance matrix are usually obtained from an IOD process, and the entire processing flow illustrates the typical EKF data processing. The left side of the diagram presents the EKF state processing which consists of two main steps of state prediction via dynamic propagation and state update via sensor measurement processing. The right hand side of the diagram represents the covariance matrix processing which also consists of two stages, prediction and update. The processing cycle is then repeated with new measurements supplied from the sensors and the prediction step will be propagated via the dynamic process between the measurement times. B. IMM Architecture & Algorithm Description The two EKFs (i.e., [P V] & [P V A]) are implemented using the IMM scheme to perform precision RSO tracking subject to maneuvering uncertainties (see Figure 2.) Extending from two EKFs to three or more with each EKF to be scheduled with a different set of [Q, R] to reflect the capability difference of RSO maneuvering signature (i.e., Q) and the respective sensor detecting them (i.e., R) is quite straight forward and attractive as a design option. The detailed IMM algorithm, step by step for one cycle, is captured in Table 1.
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Initial State STState
^ X(k-1|k-1)
Initial Covariance Covariance
State Transition Matrix & Measurement Matrix Update
P(k-1|k-1)
^ ^ (X), H(X)
^ ^ ^ X(k|k-1)=(X)X(k-1|k-1)
EKF Predicted Stage Predicted Measurem
• •
^ ^ ^ Z(k|k-1)=H(X)X(k|k-1)
Zmi(k), i=1:m Measurem ent
^ ^ P(k|k-1)=(X)P(k-1|k-1)T(X)+LQLT
Measurement or Output Covariance Calculation
i(k)=Zmi(k)-Z(k}k-1)
Gt Gate Threshold, One Satisfied Residual (within threshold)
R
^ ^ S(k)=H(x)P(k|k-1)H(x)+R
Calculation of Multiple Measurement Residuals ^
(Residuals from All Measurements)
Q
Matrix Inversion
S-1 di2(k)=iT(k)S-1i(k)
•
EKF Gain Calculation
^ T·S-1 K(k)=P(k|k-1)H(X)
Select Zm for min(di2
