Trajectory Estimation for Tactical Ballistic Missiles in Terminal Phase Using On-line Input Estimator

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1 Proc. Natl. Sci. Counc. ROC(A) Vol. 23, No. 5, pp Trajectory Estimation for Tactical Ballistic Missiles in Terminal Phase Using On-line Input Estimator SOU-CHEN LEE, YU-CHAO HUANG, AND CHENG-YU LIU Department of System Engineering Chung Cheng Institute of Technology Taoyuan, Taiwan, R.O.C. (Receied Noember 20, 1998; Accepted March 11, 1999) ABSTRACT This work presents a noel on-line estimation approach to estimation of the trajectory of a tactical ballistic missile with simple maneuering in the terminal phase. This proposed approach incorporates the extended Kalman filter and an innoatie recursie input estimator into the hypothetical testing scheme. Relatiely simple steering is accomplished by the ballistic missile and is considered to be an abrupt lateral acceleration in the dynamic equation. Also formulated herein is a regression equation between the obsered alue of the residual sequence and the theoretical residual sequence of the extended Kalman filter with no input. In addition, a recursie least-squares estimator is proided not only to extract the magnitude of the input, but also to proide a testing criterion for detecting the onset and presence of the input. Numerical simulation has demonstrated the superior capabilities of this input approach. Moreoer, trajectory estimates of the two ariations of the ballistic coefficient can erify the robustness and computational stability within ery short time. Key Words: anti tactical ballistic missile, trajectory estimation, extended Kalman filter, input estimation I. Introduction The effectieness of the Tactical Ballistic Missile (TBM) during the Persian Gulf War had a startling effect globally. Expertise deeloped during the War led to the aailability of the Anti Tactical Ballistic Missile (ATBM) for use in conentional air defense missiles and, ultimately, to upgraded performance. The modern TBM was designed to maneuer in the terminal phase to escape tracking and interception. Therefore, on-line trajectory estimation of the TBM in the regular fire control sequence for the ATBM is highly desired in radar tracking, direct hit guidance, and early warning systems. Normally, trajectory estimation of a flying ehicle is inestigated primarily in post analysis to identify states and key parameters in aailable flight data measured using radar, satellites, and on-board sensors. For a conentional ATBM air defense missile, engagement is limited in the re-entry phase since radar is the only measurement instrument used to sense the TBM. Thus, reconstructing the states and parameters of a reentry ehicle is relatiely difficult, particularly during maneuers. Trajectory estimation of a maneuering TBM has receied little attention. Chang et al. (1977) introduced a maneuering re-entry ehicle filter with a newly defined augmented state ector. In the filter, the augmented state ector contains position, elocity, and corresponding parameters for drag and maneuer forces; the extended Kalman filter (EKF) has been proposed the task of estimation. The re-entry ehicle filter performance, howeer, is degraded if maneuering forces are absent. The related parameter estimation also relies heaily on the model and is assumed to be a constant Gaussian Marko process in the state equation. Abutaleb (1985) presented a Pontryagin filter based on the Pontryagin minimum principle, in which the unknown maneuering forces are treated as a control used to drie a ehicle s dynamic so that it will follow the noisy measured trajectory. Neertheless, the Pontryagin filter is sensitie to parameter ariation and initial conditions. The input estimation technique has been successfully applied in many engineering areas, including the tracking problem, initial leeling of strapdown inertial naigation systems, and inerse heat conduction problems. In TBM, maneuering is considered to be an extra lateral acceleration input which rapidly changes a missile s position, elocity, and heading. Therefore, deeloping an input estimation approach to identify this lateral acceleration is highly desired. In this study, we present a noel adaptie filter to proide accurate and fast trajectory estimation with low sensitiity to 644

2 Trajectory Estimation for TB Missiles an unknown key parameter, i.e., the ballistic coefficient. The proposed adaptie filter consists of the EKF and an innoatie input estimator with the input detection criterion. The input estimator proposed herein can continuously estimate all possible lateral accelerations and substitute the estimated results into the EKF in order to identify the TBM s position and elocity if the criterion is satisfied. The impact of the ballistic coefficient uncertainty can be treated as an another output drien by extra accelerations, which are accompanied by lateral accelerations that are to be estimated. Simulation results indicate that the proposed filter leads to much improement compared with the EKF without an input estimator and the augmented EKF, whose state ector contains the position, elocity and lateral acceleration. Simulation results further demonstrate that the proposed algorithm proides robustness to the ballistic coefficient. Due to the nature of data reconstruction, recommended trajectory estimation can proide further insight into TBM system behaior and possibly facilitate a more general model design and analysis inoling the ATBM strategy. The rest of this paper is organized as follows. The next section describes a maneuering TBM model in a 2 dimensional plane, where an equation of motion is introduced and the extended Kalman filter is applied to the estimation problem. In Section III, we present adaptie filtering with a noel input estimation scheme for lateral acceleration of TBM. This section also discusses the related test criterion for input detection. The extended Kalman filter is also combined with the input estimator. Next, Section IV presents two simulation examples which demonstrate the effectieness of the proposed filter and its robustness to the ballistic coefficient. Concluding remarks are finally made in the last section. II. Dynamic Model Consider a 2 dimensional TBM in the re-entry phase and in a flat, nonrotating earth with constant graity model, as illustrated in Fig. 1. The distance that the TBM traels in this phase is shorter than the distance traeled before re-entry. The setting for the TBM is a point mass and constant weight along a ballistic trajectory, where three types of significant forces act on the TBM, among which graity and drag are two types. The adanced tactical ballistic missile usually steers at the right time in the terminal plase so as to escape radar tracking and missile interception. This steering may rapidly change the heading, elocity and acceleration of the TBM. The third force is introduced in a plane which is perpendicular to the elocity ector if the TBM performs a maneuer. The force is induced by a maneuer and may be represented by components along X R and Y R, which are relatie to the acceleration terms in the equations of motion. The TBM equations with a maneuer in radar coordinates (O R,X R,Y R ) centered at a radar site can be represented as x = Dg W cosγ + a x = 2 2β gcosγ + a x, (1) y = Dg W sinγ g + a y = 2 2β gsinγ g + a y, (2) where D and W denote the drag and weight of the missile, respectiely, represents air density and, a x and a y are lateral accelerations generated by a maneuer along X R and Y R, respectiely. The flight path angle γ and ballistic coefficient β are defined as γ = tan 1 ( y x ), β = Fig. 1. Tactical ballistic missile geometry. W SC D0, where S and C D0 represent the reference area and zerolift drag coefficient, respectiely. The air density is originally a function of altitude and should be considered in Eqs. (1) and (2) because the altitude dramatically changes in flight at speeds oer 100 km. The conentionally used approximation model for air density related to altitude is = e y/30000 =0.0034e y/22000 Let the states ector be for y<30000 ft, for y ft. X=[x 1 x 2 x 3 x 4 ] T =[x y x y ] T. (3) 645

3 S.C. Lee et al. The nonlinear state equation can be deried as X =F(X,t)+ϕ u+iζ. (4) In Eq. (4), ζ denotes the process noise a ector with a ariance of Q, I stands for the identity matrix, u is the lateral acceleration term of the TBM, F(X, t)= x 3 x 4, ϕ = 2β (x x 2 4 )gcosγ 2β (x x 2 4 )gsinγ g u=[0 0 a x a y ] T =[0 0 u 3 u 4 ] T For a conentional air defense missile system, ground radar is the major instrument used to detect a TBM; it proides the position, elocity, and een the acceleration of tracked targets. The measurement equation is Z=HX+ε, (5) where ε represents the measurement noise ector and H denotes the identity matrix. Equations (4) and (5) form the dynamic equations for the TBM with a maneuer after re-entry. Once all the states at a specific time instance are precisely known, the trajectory is reconstructed using a fixed-point smoother and n-step ahead predictor. The EKF is a conentionally used state estimator for nonlinear dynamic equations and is applied straightforwardedly herein. Let ε denote white and be normally distributed with zero mean and a ariance of R. The predicted and updated state ectors from t=n t to t= (n+1) t under input u n at t=n t are gien by X n+1/n = φ n X n/n + ϕu n, (6) X n+1/n+1 = X n+1/n + K n+1 (Z n+1 HX n+1/n ), (7) where t is the sampling period, Z n+1 denotes measurements at t=(n+1) t, and the transition matrix as φ n = I + F(X, t) X X = X n/n t. A linearization of F(X,t) can be easily obtained, F(X, t) X X = X n/n = f 32 = = 60000β g(x x 4 2 ) cosγ 44000β g(x x 4 2 ) cosγ f 33 = 2β g(2x 3 cosγ x 4 sinγ), f 34 = 2β g(2x 4 cosγ + x 3 sinγ), f 42 = = 60000β g(x x 4 2 ) sinγ 44000β g(x x 4 2 ) sinγ f 43 = 2β g(2x 3 sinγ + x 4 cosγ), f 44 = 2β g(2x 4 sinγ x 3 cosγ) , 0 f 32 f 33 f 34 0 f 42 f 43 f 44 X = X n/n for x 2 <30000 ft, for x ft, for x 2 <30000 ft, for x ft, The Kalman gain K n+l and coariance matrix of X n+1/n and X n+1/n+1, i.e., P n+1/n and P n+1/n+1, respectiely, are K n+1 =P n+1/n H T (HP n+1/n H T +R) 1 (8) P n+1/n =φ n P t n/n φ n +ΓQΓ, (9) P n+1/n+1 =(I K n+l H)P n+l/n (10) and Γ=I t. The unpredictable input term u n in Eq. (6) is the dominant term regarding the accuracy in state estimation. The EKF may conerge well with long time propagation if u n is omitted in Eq. (6). Howeer, a long conergence time is unacceptable for defense considerations, which require a rapid reaction within just a few minutes. Therefore, an algorithm must be deeloped to estimate the input acceleration and achiee a rapid and accurate trajectory estimation. III. Adaptie Filtering with Input Estimation For a situation in which the input term u n in Eq. (6) remains unknown and onset is unpredictable, this section presents a noel adaptie filter scheme which consists of the EKF and a recursie least-squares estimator of input with the test criterion. Adaptie refers to a situation in which the filter sequentially 646

4 Trajectory Estimation for TB Missiles adjusts an input based on measurements. The input estimator attempts to estimate u n using a single radar measurement. Moreoer the estimated input is combined with the EKF if it satisfies the test criterion. 1. Input Estimation Let X n+1/n, X n+1/n+1 denote the predicted and updated states, respectiely, for the EKF with no input at t=(n+l) t. For simplicity, denote X n+1 = X n+1/n+1 and X n+1 = X n+1/n+1, and let M n+1 =(I K n+l H)φ n, N n+1 = (I K n+1 H)ϕ. The updated state can be represented by X n+ l =( n + l M i i = n +1 l 1 Σ l ) X n + ( M n + i ) K n + j Z n + j j =1 + K n + l Z n + l l=1, 2,... (11) Assume that the TBM maneuers with constant lateral acceleration in each axis during kdt t (k+s)dt; then, the abrupt inputs are Then, Allow l 1 Σ Z k + l = H[ ( M k + i ) N k + j u k + j 1 + N k + l u k + l 1 ] j =1 + Z k + l. l l 1 Σ Y k + l = Z k + l H ( M k + i ) N k + j u k + j 1. j =1 Thus, we hae the regression equation where l Y k + l = Φ k + l u k + l 1 + Z k + l, (17) Φ k+l =HN k+l. Therefore, the recursie least-squares estimator of input can be expressed as u = 0 t < k t, t>(k + s) t k, s >0 u k + r k t t (k + s) t r = 0, 1, 2,..., s, (12) where u k + l 1 = u k + l 2 + G k + l (Y k + l Φ k + l u k + l 2 ), (18) where u k+r denotes a constant ector. Similarly, the updated states ector of the EKF formation with input, Eq. (8), can be expressed as X k + l =( k + l M i i = k +1 l 1 Σ l )X k + ( M k + i )(K k + j Z k + j j =1 + N k + j u k + j 1 )+K k + l Z k + l + N k + l u k + l 1. (13) The EKF formations with no input and with an input yield the same results during t k t, so that X k = X k. The difference induced by the abrupt inputs between these two formations during k t t (k+s) t can then be written as X k + l = X k + l X k + l = M k + l X k + l 1 + N k + l u k + l 1. (14) Define the residuals of measurement for the two EKF formations with and without inputs, respectiely, as Z k + l = Z k + l HX k + l, (15) Z k + l = Z k + l HX k + l. (16) Y k + l = Z k + l HM k + l X k + l 1, (19) X k + l 1 = X k + l 1 u = u = M k + l X k + l 2 + N k + l u k + l 2, and the gain G i and ariance of u i and V i are G k+l =V k+l 1 Φ k+l ξ 1, (20) T V k + l 1 = V k + l 2 V k + l 2 Φ k + l T [Φ k + l V k + l 2 Φ k + l + ξ] 1 Φ k + l V k + l 2, (21) where ξ is the ariance of Z. The innoatie measurement Y k + l contains only partial information about Z k+l, leading subsequently to a biased input estimator. Howeer, this biased estimator still functions properly. Note that the estimated input is drien by X k + l and Z k + l. Therefore, it is not only a function of the inputs, but also depends on the estimation errors of the EKF. In fact, X k + l represents the estimated states obtained using the EKF formation with no input, namely, the original filter. Therefore, the actual system must be compared with the simulated system to generate Y k + l during input estimation. 647

5 S.C. Lee et al. In the preious section, we assumed that β was constant, which is not always true. Let u β be the extra acceleration ector induced by iolation of aboe assumption. Then, the state equation can be rewritten as X =F(X,t)+ϕ u+ϕ u β +Iζ =F(X,t)+ϕ U+Iζ, (22) where U=u+u β. Therefore, the augmented input ector U can be estimated in the same manner as described aboe. The partially improper assumption about β may be coered by the proposed filter and is ealuated later by means of simulation. 2. Criterion for Testing In Eq. (12), k and s represent the starting and stopping points of the system input, respectiely. As mentioned earlier, the onset of system input is unpredictable and can not be assumed in adance. Here, we will present a testing method to determine k and s. Assume that ε 1, ε 2,... are independent, identically distributed random ariables, and that ε N(0, R). (23) Since u i, i=3, 4 is biased, according to the central limit theory, we hae u i N(u bi,v ii ) i=3, 4 as t, (24) where u b denotes a certain alue which corresponds to the true input u i and is equal to zero if u i does not exist; in addition, V ii denotes the ariance of u i. Therefore, the hypotheses can be defined to test the existence of an input as H 0 : u bi =0 (absence of input) i=3, 4, (25) H 1 : u bi 0 (existence of input) i=3, 4, (26) and the existence of a normalized statistical ariable as u st = u i V ii i=3, 4. (27) Assume that the randomness of the initial alue of V can be neglected. Then, u st N(0,1) when H 0 is true. Therefore, the test of whether either H 0 or H l holds true becomes Fig. 2. The mechanism of the proposed adaptie filter scheme. u st >t st H 1 is hold, (28) u st t st H 0 is hold, (29) where [ t st,t st ] is the confidence interal, which can be determined by examining the cumulatie normal distribution table for a certain preset confidence leel α. Figure 2 illustrates the mechanism used in the proposed scheme. The estimated input is applied to the EKF when the switch is on, which denotes that H 1 is on hold. Otherwise, the switch is off, indicating that H 0 is on hold. 3. Adaptie Filtering By using the adjusted on-line input, the predicted and the updated states for the adaptie filter at time interal k t t (k+s) t can be obtained as X k+l+1/k+l = φ k + l X k+l/k+l X k+l+1/k+l+1 = X k+l+1/k+l + ϕu k+l 1, (30) + K k+l+1 (Z k+l+1 HX k+l+1/k+l ). (31) The Kalman gain becomes K k+l+1 = P k+l+1/k+l H T (HP k+l+1/k+l H T + R) 1 with the coariance matrix of the adaptie filter at k t t (k+s) t: where P k+l+1/k+l P k+l+1/k+l+1 = P k+l+1/k+l +φ k+l L k+l+1 φ T k+l +ϕv k+l ϕ T = P k+l+1/k+l + P k+l+1/k+l, (32) =(I K k+l+1 H)P k+l+1/k+l, (33) P k+l+1/k+l = φ k+l L k+l+1 φ T k+l + ϕv k+l ϕ T, l Σ l +1 L k+l+1 = ( M k+i 1 ) N k+j V k+j 1 N T T k+j ( M k+i 1 ) j =1 l+1 648

6 Trajectory Estimation for TB Missiles alues. The following simulations are all for a single flight. 1. True Value of β Ealuation T = M k+l L k+l M k+ l. P k+l+1/k+l represents the increment in coariance induced by u i, i=k, k+1,..., k+l. It is easy to proe that the P k+l+1/k+l is positie definite, implying that when u i and i=k, k+1,..., k+l are introduced to reduce the state estimation errors, the coariance of estimated states must increase. For a time beyond the interal t<k t and t>(k+s) t, the estimation states can also be based upon the original EKF. Notably, the initial states and coariance matrices at t>(k+s) t are reinitiated by X k+s/k+s and. P k+s/k+s Fig. 3. Measured position. IV. Computer Simulations This section presents two cases with a confidence leel under α=95% to ealuate the effectieness of the proposed adaptie filter scheme, as shown in Fig. 2. The first case compares the proposed method with the EKF and augmented EKF under the true alue of β to demonstrate the capability of the proposed method. The other case inestigates the robustness of the proposed scheme to β. Consider a scenario corresponding to Eqs. (1) and (2) with an initial altitude of ft, an initial downrange of 1000 ft, an initial elocity of 4500 ft/sec, a re-entry angle of 70, and β=500 1b/ft 2. The lateral accelerations with magnitude 4g and 2g in the X R and Y R axes, respectiely, are applied to the dynamic equations within 20 t 25 sec after re-entry. Figures 3 and 4 display the measured position and elocity generated by Eqs. (4) and (5) with N(0,1) measurement noises, R=0.3I, and Q=0.01I. The first data detected by radar are adopted as the initial Ealuating the adaptie filter with input estimation initially inoles performing 2 dimensional trajectory simulation of the TBM and estimating the trajectory for a gien re-entry course with β= 500 lb/ft 2. This case adopts three methods of trajectory estimation, namely, adaptie filters, the EKF, and the augmented EKF, with acceleration inputs a x and a y taken as two augmented states to be estimated by the EKF. a x and a y can be modeled using a Gauss-Marko process as follows: a x = α x a x + 2α x σ x2 ζ, (34) a y = α y a y + 2α y σ y2 ζ, (35) where 1/α x and 1/α y denote time constants, and σ x 2 and σ y 2 represent the ariance of a x and a y, respectiely. Let the augmented state ector be X =[x 1 x 2 x 3 x 4 x 5 x 6 ] T =[x y x y a x a y ] T ; then, the state equation is gien by where X = F(X,t)+I ζ, (36) Fig. 4. Measured elocity. 649

7 S.C. Lee et al. x 3 x 4 F(X,t)= 2β (x x 4 2 )gcosγ + x 5 2β (x x 4 2 )gsinγ g + x 6 α x α y I I = 2α x σ2 x α y σ2 y. Fig. 6. Detected inputs (upper) and standard deiations (below). The augmented EKF is employed when the state equation is substituted into Eqs. (6) and (7). Figures 5 and 6 show the estimated and detected accelerations and standard deiations (STD) obtained using the augmented EKF and the adaptie filter, respectiely. As mentioned in the preious section, the detected inputs of the adaptie filter are generated from actual inputs and state estimation errors. The larger inputs are required to compensate before the exact inputs are applied. The detected inputs gradually decrease when the lateral accelerations are introduced. The same phenomenon is seen in Fig. 5 for the same reason. Figures 7-9 present comparisons of the resulting estimation errors and STD in position among these Fig. 7. Estimation errors and standard deiation in position without input estimation. Fig. 5. Estimated inputs (upper) and standard deiations (below) obtained using augmented EKF. three methods. As seen in these figures, the proposed filter immediately reduces the estimation errors by more than two orders as compared to the EKF and the augmented EKF. The maximum estimation errors for the EKF are up to ft at t=20 sec and ft at t=21.65 sec in X R and Y R, respectiely. For the augmented EKF, the maximum errors reach 3000 ft and 3000 ft in X R and Y R, respectiely, at t=40 sec. When using the proposed scheme, the errors are bounded by 8.27 ft and 19.5 ft along X R and Y R, respectiely, and rapidly approach a steady state after the inputs are remoed. The same phenomenon appears in Figs , which display the estimation errors and STD in elocity. The maximum errors reach ft/ sec and 5.65 l0 3 ft/sec for the EKF and reach 500 ft/sec and 500 ft/sec for the augmented EKF in X R and 650

8 Trajectory Estimation for TB Missiles Fig. 8. Estimation errors and standard deiation in position obtained using augmented EKF. Fig. 10. Estimation errors and standard deiation in elocity without input estimation. Fig. 9. Estimation errors and standard deiation in position with input estimation. Fig. 11. Estimation errors and standard deiation in elocity obtained using augmented EKF. Y R, respectiely. Howeer, the proposed scheme s errors are limited to 11.2 ft/sec and 17.6 ft/sec and quickly tend toward a steady state. Table 1 compares the estimation errors at impact point, that is, the steady errors, of the adaptie filter with the EKF and the augmented EKF, reealing the much greater accuracy of the proposed scheme. In addition, the augmented EKF requires that α x, α y, σ x, and σ y, which are normally unknown beforehand, be assigned. This difficulty is absent from the proposed method. The contributions and feasibility of the input estimator are, therefore, obious and merit further study. 2. β Variation Ealuation In this simulation study, we present the perfor- Fig. 12. Estimation errors and standard deiation in elocity with input estimation. 651

9 S.C. Lee et al. Table 1. The Steady Errors of All States with β=500 lb/ft 2 States The adaptie The EKF The augmented filter EKF Position in X R 1.57 ft ft 3000 ft Position in Y R 1.03 ft ft 3000 ft Velocity in X R ft/sec ft/sec 325 ft/sec Velocity in Y R ft/sec ft/sec 310 ft/sec Fig. 14. Estimation errors in elocity with β=650 lb/ft 2. Fig. 13. Estimation errors in position with β=650 lb/ft 2. mance of low sensitiity to β of the proposed scheme. As mentioned earlier, β is assumed to be a constant, which is not always true. Two worse scenarios of β ariations of ±30% are inestigated while examining the robustness of the proposed method. Figures show that the estimation errors in position and elocity are under β=650 lb/ft 2 and β=350 lb/ft 2. The proposed method naturally proides better estimation results, roughly more two orders better than those obtained using the original filter. When β increases by 30%, the maximum estimation errors in position of the original filter are ft at t=21.95 sec and ft at t=24.4 sec in X R and Y R, respectiely. In addition, the peak estimation errors of the proposed method in position are 7.53 ft and ft along X R and Y R, respectiely. The errors of the original filter are much larger than those of the proposed method at any time instance, as indicated in Fig. 13. The maximum estimation errors in elocity of the original filter are significantly greater than those of the proposed method in each axis. The original filter s errors in elocity estimation are always larger than those of the proposed method, as depicted in Fig. 14. When β decreases by 30%, the maximum errors of the original filter in position are l ft and ft and in Fig. 15. Estimation errors in elocity with β=350 lb/ft 2. Fig. 16. Estimation errors in elocity with β=350 lb/ft

10 Trajectory Estimation for TB Missiles Table 2. The Steady Errors of All States with β=650 lb/ft 2 Table 3. The Steady Errors of All States with β=350 lb/ft 2 States The proposed scheme The original filter Position in X R 1.71 ft ft Position in Y R 1.59 ft ft Velocity in X R 1.1 ft/sec ft/sec Velocity in Y R 2.12 ft/sec ft/sec States The proposed scheme The original filter Position in X R 1.0 ft ft Position in Y R ft ft Velocity in X R ft/sec ft/sec Velocity in Y R 0.39 ft/sec ft/sec elocity are 5.53 l0 3 ft/sec and 9.84 l0 3 ft/sec along X R and Y R, respectiely. Howeer, the proposed method still produces a satisfactory estimate. The maximum errors are 10 ft and ft in position estimation and are 15 ft/sec and 24.9 ft/sec in elocity estimation along X R and Y R, respectiely. Tables 2 and 3 list the steady alues of errors under ariation of β. From those table, we can infer that the proposed adaptie filter scheme is much lower sensitie to β than is the EKF. V. Conclusions This study has proposed an on-line estimation algorithm composed of the EKF and an input estimator with the test criterion that can achiee accurate and fast trajectory estimation of a maneuering TBM. A recursie least-squares estimator of input estimation has been presented to identify the maneuer forces of the TBM. The method automatically detects the lateral acceleration of the maneuering TBM and also is a more accurate and faster approach than the EKF. Numerical simulation has demonstrated the superior capabilities of this input approach. Moreoer, trajectory estimation for two different alues of the ballistic coefficient has erified the robustness of the proposed scheme in spite of parameter ariation, confirming that this scheme is worth further deelopment. Acknowledgment This work was supported by the National Science Council of the Republic of China under Grant NSC D References Abutaleb, A. S. (1985) Improed trajectory estimation of maneuering reentry ehicles using a nonlinear filter based on the pontryagin minimum principle. IEEE International Radar Conference, pp Arlington, VA, U.S.A. Chang, C. B., R. H. Whiting, and M. Athan (1977) On the state and parameter estimation for maneuering reentry ehicles. IEEE Transactions on Automatic Control, AC-22(1), !"#$"%"#&'!"#$!%&'!"#$%!&'!"#$%&'()*+,-./)*# :;<=>?=@ ABC9+,DE!"#$%&'()*+,-./, :';<=>?@6ABCDBEFGHIJKL!"#$%&'()*+,-./ :;<=>?@ABCDEFGH&456IJK!"#$%&'(#)*+!,-./ *'9:;<3.=>-?@A456BCD!"#$%&'()*+,-./ :; < =>?@A&BCDEFGH9:I!"#$%&'()*+%, 653