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Unformatted text preview: DOMAIN PURSUIT METHOD FOR TRACKING BALLISTIC TARGETS TECHNICAL REPORT CAMS 98.9.2 Chuanxia Rao, Boris Rozovsky and Alexander Tartakovsky CENTER FOR APPLIED MATHEMATICAL SCIENCES University of Southern California Los Angeles, CA 90089 1113 SEPTEMBER 1998 Approved for public release; distribution unlimited Contents 1 Introduction 2 Continuous-Discrete Filtering 3 5 2.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Fokker-Planck equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Unnormalized ltering densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Adaptive Domain Pursuit Method 8 3.1 Splitting of convection and di usion processes . . . . . . . . . . . . . . . . . . . . 8 3.2 Domain pursuit tracker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4 Application to the Problem of Ballistic Target Tracking 11 4.1 The tracking problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.2 Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5 Conclusion 6 Acknowledgement 17 18 2 CAMS Report 98.9.2: Domain Pursuit Method for Tracking Ballistic Targets 3 Abstract In this report a fast optimal nonlinear lter is developed. This fast lter is based on two techniques: i splitting of the convection and di usion operators and ii tracking of the important" domains windows with iterative reduction of the window size a domain pursuit technique. As a result, the domain of interest is determined adaptively. The developed nonlinear ltering algorithm is then applied to a real problem of tracking ballistic targets in six dimensions. Simulation results for this problem demonstrate the fairly high statistical accuracy, e ciency, and real-time performance of the proposed algorithm. These results also show that the developed method is much more accurate compared to the traditional extended Kalman lter. 1 Introduction In applications the target tracking problem can be naturally formulated as a ltering problem for hidden Markov models: given an unobserved hidden state or signal or system process, which is usually assumed Markovian, and an observation or measurement process, which provides noisy information about the state process, one needs to estimate the state or a function of the state at a given time moment by using all the observational information available up to that time moment. Ideally, the involved stochastic dynamics are linear and Gaussian and in this case the ltering problem is solved by the Kalman and Kalman-Bucy lters. For linear and Gaussian models the Kalman lter is optimal in the mean-square sense and had a big success in a wide variety of applications. However, many real-world problems do not t well with linear dynamic models. Sometimes one can explicitly describe the distribution of the state given measurements posterior distribution but, outside the realm of the linear theory, only a very few examples have explicitly described posterior distributions. Since most real problems are nonlinear, this creates a fundamental problem. Successive linearization in short time intervals, the Extended Kalman Filtering procedure, may be applied but its serious disadvantage is that it often gives erroneous answers and re ning of computational e ort can increase them. Theoretical study of the general nonlinear ltering problem has also gone through more than three decades of e orts of mathematicians, statisticians, and engineers and has now become more or less mature as a research eld. See the books by Stratonovich 28 , Jazwinski 11 , Liptser and Shiryayev 16 , Kallianpur 13 , Rozovskii 26 , Pardoux 22 , Bensoussan 2 , Tanizaki 29 , Elliott, 4 CAMS Report 98.9.2: Domain Pursuit Method for Tracking Ballistic Targets Aggoun, and Moore 4 . In contrast to the linear case where there exists a nite dimensional statistic, a general nonlinear problem is in nite-dimensional in nature. This was the main reason why optimal nonlinear lters have not been widely used in real applications. Approximations to the optimal nonlinear lter must be adopted. The simplest and most widely used approximation is the extended Kalman lter EKF, which is basically the Kalman-Bucy lter applied to a dynamically linearized system. The EKF has been modi ed for di erent purposes and has many versions such as the second order approximation 21 and the iterative extended Kalman lter 1, 11, 12 . In addition to the EKF and its modi cations, there are two major approaches to nonlinear lter approximation. One approach assumes that the ltering densities belong to a certain class of functions such as Gaussian or exponential or some combinations. Actually EKF is an example of an assumed-density lter and is perhaps the simplest. Other examples are the Gaussian mixture lter see 27 and more recently the projection lter see 3 . The other approach is to use a direct analytical or numerical approximation to the optimal nonlinear lter. One recent advance in this direction is the Wiener chaos decomposition or spectral separation scheme S for nonlinear ltering in continuous time 17, 18 and similar related algorithms that also use the o -line on-line separation" idea 19, 20 . A direct numerical approximation to the optimal nonlinear lter is based on computing the convolution integral in the discrete ltering model 15, 29 , on using fast solvers for the Fokker-Planck equation in the continuous-discrete ltering model 14, 19, 24 , or on solving the Zakai equation in the case of continuous time 6, 8, 9, 10 . 3 Both approaches encounter computational di culties in practical applications. The problem with EKF and its modi cations or, in general, with the assumed density lters, is that they do not work well if the posterior distribution di ers from the assumed form. For example, if one assumes a Gaussian distribution and uses EKF while in reality it is multi-peak far from Gaussian, then the lter completely fails. Assumed density lters including EKF fail, for example, in many important situations such as angle-only target-tracking due to divergence, instability, inaccuracy, etc. The reason is that the prespeci ed density class is too restrictive in the general nonlinear case. Direct approximation is much better in this class of situations but has another important limitation the curse of dimensionality". If, for instance, we have a six-dimensional model, which is typical for radar applications, and 100 points are used in each component of the state in many cases 100 points could even be too few for a satisfactory estimate of the state, then the total number of spatial points is N = 10 . If one uses a fast solver with FFT which has complexity Cd N log N d, for dimension d, then one needs to perform 10 C ops at each 12 2 1 20 6 CAMS Report 98.9.2: Domain Pursuit Method for Tracking Ballistic Targets 12 5 time step, which is simply unacceptable. Even if the Fokker-Planck solver has optimal linear complexity like ADI, we still need to perform 10 C ops each step. Note that the constant Cd increases with dimension d and C 8. 6 1 What do we do then? From the one hand to avoid the di culties that EKF faces, it is desirable to use a direct approximation rather than an assumed density approach to the optimal estimate. From the other hand in order to implement direct approximations to high-dimensional real problems, we need 1 to develop fast Fokker-Planck equation solvers with linear complexity, and 2 to reduce the number N of spatial points substantially without reducing the accuracy. The former can be achieved by the so-called operator-splitting method. The latter goal is achieved by reducing adaptively at each time step the size of the spatial domain in which the equation is solved the domain should be reduced in each direction. This idea leads to the adaptive domain pursuit technique DPT that is developed below. It is worth mentioning that the proposed domain pursuit approach has some similarity with the EKF. In fact, at each time step the DPT tracks a domain window in which the target is located and then proceeds the nonlinear ltering in this moving window or multi-windows, whereas the EKF tracks only two parameters at each time step the mean and variance of the target state and then linearizes the nonlinear dynamics around the estimated mean. 1 The remainder of the report is organized as follows. In Section 2 we outline the basic facts from the theory of optimal nonlinear stochastic ltering for continuous-discrete time model. Section 3 is devoted to the development of the fast algorithm which is based on the convectiondi usion splitting and the domain pursuit technique. In Section 4 we apply the developed general algorithm to a practically important radar target tracking problem in six dimensions. The numerical results obtained in computational experiments are given in Section 4.2. These results demonstrate fairly high accuracy and e ciency of the method and show that it is much more accurate compared to the EKF. 2 Continuous-Discrete Filtering 2.1 Statement of the problem We are interested in the continuous-discrete ltering model, since this is perhaps the most appropriate model in real target tracking problems. Speci cally, the dynamics of target trajectories 1 This domain may be multiply connected multi-windows. 6 CAMS Report 98.9.2: Domain Pursuit Method for Tracking Ballistic Targets is naturally continuous while the observations are usually taken at discrete time moments. Consider the dynamic system described by the stochastic di erential equation dXt = bXt dt + Xt dWt; t 0 ; X : 0 0 1 The discrete-time noisy observations are given by Yk = hk Xtk + Rk Xtk Vk ; k = 0; 1; : : : ; 0 0 2 where b : IRd ! IRd is a vector-valued function, : IRd ! IRdd is a matrix with function entries, fWt gt is a standard d-dimensional Brownian motion Wiener process, tk = k 0, is a prior distribution of the initial condition, hk ; Rk are given functions, and Vk are i.i.d. random variables. Without loss of generality, X , fWtg and fVk g are assumed to be independent, and the following regularity conditions on the parameters of the model are also assumed: 1 the functions bk ; hk ; Qk ; Rk ; and have bounded derivatives up to an appropriate order, and 2 all the derivatives of decay at in nity faster than any power of jxj. 0 0 0 For simplicity, we only consider the case where the functions b and are time-independent. But the discussions that follow can be easily generalized to cover the time-dependent case. In fact, only the coe cients in the Fokker-Planck equation 3 below and the associated semigroup will need to be accordingly modi ed. 2.2 Fokker-Planck equation The theory of stochastic di erential equations tells us see 5, 7, 25 that under certain conditions on b and , there exists a unique solution Xt of 1 in the sense of Ito and that the probability density ut; x of this di usion process Xt satis es the Fokker-Planck equation also known as the Kolmogorov forward equation d d @ut; x = 1 X @ a xut; x , X @ b xut; x @t 2 ; @x @x @x 2 =1 =1 3 where a x is the -th row and -th column entry of the product matrix x x , and b x is the -th component of the vector bx. Let T t denote the semigroup associated with the above Fokker-Planck equation. Then its solution ut; x with the initial value u0; x = gx is T tg x. CAMS Report 98.9.2: Domain Pursuit Method for Tracking Ballistic Targets 1 7 We now note that since there is no observation available between tk, and tk , the prior transition density is the only available information on tk, ; tk and so one has 1 pXtk = x j Y k, = T pXtk,1 = j Y k, x : 1 1 h i 2.3 Unnormalized ltering densities By using the Bayes formula and the Fokker-Planck equation, one can obtain the conditional densities recursively as follows h i pXtk = x j Y k = c1 k x T pXtk,1 = j Y k, x; k 1 k 1 x x pX = x j Y = c0 1 0 0 0 0 where ck is the normalizing constant to make the integral of pXtk = x j Y k to be one, and the correction term" k x related to the observation is given by 1 , 4 k x = exp , Yk , hk x Rk xRk x Yk , hk x : 2 1 The calculations are simpli ed if in place of the usual normalized ltering density one uses the unnormalized ltering densities UFD. We de ne the UFD by 2 pk x = k x T pk, x; k 1 p x = x x: 1 0 0 0 5 It is a standard fact that for any function g such that IE jgXtj 1, the conditional expectation of gXtk given Y k = Y ; Y ; ; Yk can be obtained by 0 1 Z IE gXtk j Y k = IZ Rd gxpk xdx : pk xdx I d R 2 6 This conditional expectation is the best mean-square estimate of gXtk if IE jgXtk j 1. d This is the case if g satis es jgxj K 1 + jxj, 8x 2 IR , for some and K 0; see 16 . When Rk does not depend on x, our de nition here is slightly di erent from the usually de ned UFD. The di erence is in k x, where we keep the term ,Yk Rk Rk ,1 Yk =2. This is done to avoid computational instability: from 4-5 we have kpk k1 kT pk,1 k1 ; 8k. In fact, the Fokker-Planck equation for this UFD can also be modi ed for the purpose of stability. See Section 2.4.3 and Section 2.5.2 of 23 for details. 2 8 CAMS Report 98.9.2: Domain Pursuit Method for Tracking Ballistic Targets 3 Adaptive Domain Pursuit Method In this section, we present a windowing technique which is based on the framework of splitting the convection drift and di usion noise operators. 3.1 Splitting of convection and di usion processes To compute the unnormalized ltering density, a fast Fokker-Planck solver is needed. Our method is based on the operator-splitting technique. Assuming that in the noise term of 1 the covariance matrix is constant and diagonal, the Fokker-Planck equation 3 becomes n n @ut; x = 1 X @ a ut; x , X @ b xut; x: @t 2 @x @x 2 =1 2 =1 To proceed the splitting of convection and di usion terms, denote by Tct and Tdt the solution operators of the equations n @vt; x = , X @ b xvt; x; @t @x =1 and respectively. Then it can be proved that the following approximation formulae hold see 23 : n @wt; x = 1 X @ a wt; x; @t 2 @x 2 =1 T n ' = Td Tc n' + O ; n = Tc 2 Td Tc 2 ' + O : 2 7 8 Therefore, instead of solving the original Fokker-Planck equation, we only need to solve two simpler equations, for which methods with linear computational complexity exist. We remark that a big part of computation in solving the two simpler equations can be performed before the observations become available. This pre-calculation substantially speeds up the on-line part of the algorithm. CAMS Report 98.9.2: Domain Pursuit Method for Tracking Ballistic Targets 9 3.2 Domain pursuit tracker Even though we have developed optimal solvers for the Fokker-Planck equation, the nonlinear ltering problem can still hardly be solved in real time especially for large state dimensions. The following natural step is to narrow down the size of the domain in which the Fokker-Planck equation is solved. This can be achieved by a windowing technique based on the convectiondi usion splitting framework. A numerical approximation to the optimal lter can be expressed in the following form pik = k xi k 0 0 pi = xi0 xi 0 ; 0 j 2J X i;j j p k k,1 ; k 1; i;j where f k g is a possibly indirect approximation to the fundamental solution T at step k. Note that we use the notation xik instead of xik for the spatial points because these are vectors in IRd and xi k will denote the -th component. We remark that in real implementation, the approximate solution S hck, or k pk, of the Fokker-Planck equation is not necessarily computed directly from the above summation. For example, if an implicit scheme is used, then S h contains an inverse matrix which is not inversed directly. Roughly, if the nal S h is sparse, then the summation can be used directly; otherwise the summation will take ON FLOPS per time step and so some indirect technique such as ADI should be used. 1 1 2 Assume we are given an initial set D = fx 0; ; xN0 0g 0 1 of important" points. This set is chosen according to the initial ltering density. For example, these points can be related to the largest values of p x or with the most important information on p x. Below we describe how to e ciently compute the important" points in all the subsequent time steps and also the corresponding values of the unnormalized ltering densities at those points. 0 0 If the number N is too large, it can be reduced in the rst several ltering steps. Let K 0 be a small integer and fNk ; k 1g be a decreasing sequence of integers with Nk = NK for k K , i.e. N N NK = NK = Let L be a positive integer. We will rst construct an enlarged set of LNk, candidates for the important points at each step and then choose from them the best" Nk points according to the correction term k . 0 0 1 +1 1 10 CAMS Report 98.9.2: Domain Pursuit Method for Tracking Ballistic Targets is constant and diagonal. Our For simplicity, we assume again that the covariance matrix algorithm can be described as follows. The DPT Algorithm Initialization: start with time k = 0, domain D of size N , and pi = p xi0; 1 i N . 0 0 0 0 0 Iteration: for k K , run algorithm DPTk see below with reduction of domain size Nk . For k K , run algorithm DPTk with moving domain Dk of xed size Nk = Nk, . 1 The algorithm DPTk proceeds as follows 1 for i = 1; ; Nk, , solve the stochastic di erential integral equation 1 Xt = xik , 1 + Zt 0 bXsds + Zt 0 dWs; t 2 0; t ; with L di erent sample paths of the Wiener process Wt , including the trivial case Wt 0, and denote the solution X at time t = with the j -th sample path of Wt by i;j , j = 1; ; L; 2 determine the set Dk of Nk important points x k; ; xNk k as those i;j with the largest values of k i;j for all i and j , or, for each i, xi k has the largest value of k i;j for j = 1; ; L; 1 3 for i = 1; ; Nk , compute pik = k xik j2 X i;j j p i Jk k k,1 ; i;j where k is computed according to one of the following two formulae which follow from the operator-splitting schemes discussed in the previous section: i;j k = exp , d X d X =1 or xi k , xj k , 1 , b xj k , 1 2 2 2 2 2 ; i j j i = exp , x k , x k , 1 , r b x k + x k , 1 ; 2 2 i;j i and Jk = fj : 1 j Nk, ; min k ; pj , g, being a thresholding tolerance. k i;j k =1 1 1 CAMS Report 98.9.2: Domain Pursuit Method for Tracking Ballistic Targets 11 We stress that in general the domains Dk may be multiply connected, i.e. may contain multiple windows pieces. The block diagram of the domain pursuit tracker is shown in Figure 1. Generate D k +1 from D k by using state equation and by splitting drift and noise Determine Dk+1 Dk+1 according to correction term k +1 ( Dk +1 ) Solve FPE on D D +1 with fast solver k k (ADI or by splitting convection and dif.) k +1 k Compute pk+1(D+1) k and estimate X(tk+1) Figure 1: Domain pursuit tracker The remarkable feature of the proposed ltering algorithm is that the domain of interest adaptively changes and, as a result, the number of spatial points in the domain is usually reduced to a relatively small number. Hence the computational complexity is reduced tremendously when compared with the case where a xed size domain is used for the whole computation. In general, the number N of spatial points is exponential in d, i.e., N nd, but we are reducing this number for each n of the d dimensions. Therefore, the computational cost is exponentially reduced in our algorithms in high dimensions. 4 Application to the Problem of Ballistic Target Tracking 4.1 The tracking problem To illustrate the performance of the proposed algorithm, let us consider a real RADAR tracking problem, which is well-known to be di cult despite it does not contain perturbations in dynamics. In principle we can also handle a similar problem with infra-red or other kinds of angle-only measurements and the model that would include dynamics noise. 12 CAMS Report 98.9.2: Domain Pursuit Method for Tracking Ballistic Targets There is a radar with geodetic latitude , longitude , and height h, observing a ballistic missile and generating radar range R, azimuth A, and elevation E every seconds see Figure 2. R u E n A e ( , ,h) Earth Center Figure 2: Radar geometry The missile is assumed to be in unpowered ballistic ight whose six dimensional dynamic equations of motion are dX t = V t ; dV t = , X t ; t 0 ; 9 dt dt kX tk 3 where = 3:986012 10 , X t = X t; X t; X t and V t = V t; V t; V t are the position and velocity of the missile at time moment t, and X 0 and V 0 are Gaussian random vectors with known mean 14 1 2 3 1 2 3 IE X 0 = 0; 0; 7:45005724 10 T m; IE V 0 = ,3:96745 10 ; ,2:37208 10 ; 2:15685 10 T m sec; 6 3 3 3 and covariance CovX 0; V 0 = diag 4 10 ; 4 10 ; 4 10 ; 10 =3; 10 =3; 10 =3 : 6 6 6 4 4 4 The computation of the radar measured range, azimuth, and elevation from the missile's true CAMS Report 98.9.2: Domain Pursuit Method for Tracking Ballistic Targets inertial position at time tk = k proceeds as follows: 13 Rk = kXLtk k + " v k ; Ak = tan, XL; tk + " v k ; XL; tk E k = sin, kXL; ttkk + " v k ; XL k p where " = 16; " = " = 3:04617 10, ; and 1 1 1 1 2 2 2 1 3 3 3 1 2 3 6 10 XLt = T LOC,ECEF T ECEF,ECI t X t , XS : Here LOC, ECEF, and ECI stand for the three relevant coordinate systems: ECI for Earth Centered Inertial True-of-Date, ECEF for Earth Centered Earth Fixed, and LOC for Local East-North-Up. The coordinate transformation matrices T and T t and the radar site location XS are given by LOC,ECEF ECEF,ECI T LOC,ECEF 0 cos sin 2 3 cos!et sin!et 0 7 6 TECEF,ECIt = 6 , sin!et cos!et 0 7 ; 6 7 4 5 3 7 7; 7 5 2 6 =6 6 4 , sin , sin cos cos cos 7 7 cos , sin sin cos sin 7 ; 5 3 2 6 XS = 6 6 4 + h cos cos + h cos sin + h sin 2 0 0 1 =p ae ; 1 , e sin 2 2 3 = pae1 , e ; 1 , e sin 2 2 2 5 where = 1:12032684685 rad; = ,2:60246044764 rad; !e = 7:272205216229610, rad sec; ae = 6:37815 10 m; e = 6:69342162296 10, ; h = 0 m: 6 4.2 Monte Carlo simulations Below we present the results of computational experiments. L = 1 was taken for Algorithm 1. Assume the observations are available at every = 1 second. Since there is no noise in the state dynamics 9, we do not need the di usion smoothing as described for the general model in the algorithm. If the noise is added in the dynamics, then the general algorithm should be applied. In our simulation, we ran our lter 200 times with random initial conditions and random observations. We used two di erent values of N , the number of spatial points in the moving domain 14 CAMS Report 98.9.2: Domain Pursuit Method for Tracking Ballistic Targets Table 1: Computational complexity of the algorithm Simulation N ops cpu1 cpu2 Simulation 1 729 72,182 0.14 0.04 Simulation 2 15,625 1,546,886 0.64 0.48 window. Speci cally, N = 3 = 729 and N = 5 = 15; 625. The average computational cost for the two experiments in terms of CPU seconds and FLOPS per time step are given in Table 1, where cpu1 and cpu2 are the CPU time for one step of calculations with and without graphics, respectively. The computation was performed with Matlab on a Sun Ultra Enterprise 4000 at the University of Southern California. From this table, it is clear that real-time performance has been achieved even for tracking 10 to 20 targets. 6 6 For both values of N , we tested three di erent situations. In the rst case we assume that the true initial location of the target and its velocity are exactly on the grid; in the second case it is assumed only that the initial velocity is exactly on the grid; and in the third case both the initial location and velocity are not on the grid the most realistic case. Of course, in either case, the lter does not know the true initial location of the target. In the rst case performance is perfect: with N = 729, the average errors for both X t and V t became close to zero after t = 140 seconds. The average errors, de ned as 1 X 1 X ^ ^ Xerr; t = 200 jXn t , Xn tj and Verr; t = 200 jV n t , V n tj n n = 1; 2; 3 are shown in Figure 4 for the two simulations. Note that because the initial error in X 0 is too large, it is not shown in the picture. See Figure 3 for the average errors including the initial errors according to the state dynamics without any observed information. 200 200 =1 =1 In the second and third cases, when the initial variance for the true initial state is relatively small, we also obtained good results. For example, in the second case, with the initial variance CovX 0; V 0 = diag 10 ; 10 ; 10 ; 10; 10; 10 ; 4 4 4 the average errors in V go to zero and the average errors in X begin to decrease after t = 90 seconds. These errors are shown in Figure 5. Again the initial error in X 0 is not shown in the picture. In the third case, with the initial variance CovX 0; V 0 = diag 10 ; 10 ; 10 ; 1; 1; 1 ; 4 4 4 the average errors behave similarly except that they decrease at a slower rate. CAMS Report 98.9.2: Domain Pursuit Method for Tracking Ballistic Targets 15 The real initial errors used in experiments are listed in Table 2. Typical errors at t = 100 seconds with the original large error are shown in Table 3. This table contains data for DPT and EKF. It may be seen that the proposed method is much more accurate. Table 2: Real initial errors Variance Xerr; 0 Xerr; 0 Xerr; 0 Verr; 0 Verr; 0 Verr; 0 Initial Errors 1509.2912 1565.5626 1608.2539 43.0280 50.5172 43.7596 1 2 3 1 2 3 Table 3: Comparison with EKF Method kXerr 100k kVerr100k EKF 100 m 0.5 m sec DPT 15 m 0.1 m sec Ratio 6.7 5.0 M.C. Errors in X 15000 10000 5000 0 15000 10000 5000 0 15000 10000 5000 0 Error in V3 (m/s) Error in X3 (m) Error in V2 (m/s) Error in X2 (m) Error in V1 (m/s) Error in X1 (m) 60 40 20 0 60 40 20 0 60 40 20 0 M.C. Errors in V 0 100 200 300 0 100 200 300 0 100 200 300 0 100 200 300 0 100 200 Time t (sec) 300 0 100 200 Time t (sec) 300 Figure 3: Errors without observations 16 150 100 50 0 200 150 100 50 0 150 100 50 0 0 CAMS Report 98.9.2: Domain Pursuit Method for Tracking Ballistic Targets M.C. Errors in X 60 Error in V1 (m/s) 40 20 0 60 Error in V2 (m/s) 40 20 0 60 Error in V3 (m/s) 40 20 0 Error in X1 (m) M.C. Errors in V 0 20 40 60 80 100 0 20 40 60 80 100 Error in X2 (m) 20 40 60 80 100 0 20 40 60 80 100 Error in X3 (m) 0 20 40 60 Time t (sec) M.C. Errors in X 80 100 0 20 40 60 Time t (sec) M.C. Errors in V 80 100 400 300 200 100 0 600 400 200 0 400 300 200 100 0 0 20 40 60 Time t (sec) 80 100 Error in V3 (m/s) Error in X3 (m) Error in V2 (m/s) Error in X2 (m) 0 20 40 60 80 100 Error in V1 (m/s) Error in X1 (m) 60 40 20 0 60 40 20 0 60 40 20 0 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 Time t (sec) 80 100 Figure 4: Errors in Case 1. Left simulation 1 N = 3 ; Right simulation 2 N = 5 6 6 CAMS Report 98.9.2: Domain Pursuit Method for Tracking Ballistic Targets M.C. Errors in X 60 40 20 0 100 Error in V2 (m/s) Error in X2 (m) Error in V1 (m/s) Error in X1 (m) 3 2 1 0 3 2 1 0 3 Error in V3 (m/s) 2 1 0 M.C. Errors in V 17 0 100 200 300 400 500 0 100 200 300 400 500 50 0 80 Error in X3 (m) 60 40 20 0 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 Time t (sec) 400 500 0 100 200 300 Time t (sec) 400 500 Figure 5: Errors in Case 2, N = 3 6 5 Conclusion 1. In this report we described the developed nonlinear ltering algorithm that is based on the domain pursuit method". This method is directed towards obtaining robust nonlinear tracking algorithms with manageable complexity and high statistical performance close to the optimal level. 2. The algorithm is applied to a realistic problem that is typical for tracking ballistic missiles by radar. The considered scenario includes targets with hard" trajectories that should be localized in 100-150 seconds. The results of simulation show that the developed algorithm substantially outperforms the conventional EKF tracker it terms of mean-squared tracking error and at the same time has satisfactory computational complexity may be applied in real time. 18 CAMS Report 98.9.2: Domain Pursuit Method for Tracking Ballistic Targets 6 Acknowledgement We are grateful to Roy Danchick of TRW for useful discussions and for providing the data and realistic parameters for analysis. References 1 R. L. Bellaire, E. W. Kamen, and S. M. Zabin. A new nonlinear iterated lter with applications to target tracking. In SPIE Proc. Signal & Data Processing of Small Targets, San Diego, 1995. 2 A. Bensoussan. Stochastic Control of Partially Observable Systems. Cambridge University Press, 1992. 3 D. Brigo, B. Hanzon, and F. LeGland. A di erential geometric approach to nonlinear ltering: the projection lter. Publication Interne No. 914, IRISA, 1995. 4 R. J. Elliott, L. Aggoun, and J. B. Moore. 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