A New Nonlinear Filtering Method for Ballistic Target Tracking

Transcription

1 th International Conference on Information Fusion Seattle, WA, USA, July 6-9, 9 A New Nonlinear Filtering Method for Ballistic arget racing Chunling Wu Institute of Electronic & Information Engineering Xi an Jiaotong University Xi an, Shaanxi, P.R.China informationway@63.com Chongzhao Han Institute of Electronic & Information Engineering Xi an Jiaotong University Xi an, Shaanxi, P.R.China czhan@mail.xjtu.edu.cn Zengguo Sun Institute of Electronic & Information Engineering Xi an Jiaotong University Xi an, Shaanxi, P.R.China duffer@63.com Abstract -racing a ballistic re-entry target from radar observations is a highly complex problem in nonlinear filtering. he paper adopts a one-dimensional vertical motion model with unnown ballistic coefficient, we present a square-root quadrature Kalman filter (SRQKF) algorithm for this ballistic target tracing problem. he proposed algorithm is the square-root implementation of the quadrature Kalman filter (QKF). he quadrature Kalman filter is a recursive, nonlinear filtering algorithm developed in the Kalman filtering framewor and computes the mean and covariance of all conditional densities using the Gauss- Hermite quadrature rule. he square-root quadrature Kalman filter propagates the mean and the square root of the covariance. It guarantees the symmetry and positive semi-definiteness of the covariance matrix, improved numerical stability and the numerical accuracy, but at the expense of increased computational complexity slightly. Keywords: Re-entry, ballistic target tracing, ballistic coefficient, quadrature Kalman filter, Gauss Hermite quadrature. Introduction racing a ballistic target in the re-entry phase is a highly complex nonlinear filtering problem. he problem is important both for missile defence and safety against ageing satellites. he main goal is to estimate on-line inematic parameters (position and velocity) and ballistic coefficient, which is unnown using available measurements. he ballistic target tracing problem has been studied in many papers [-4]. Both the first and second moment-based filtering methods and density-based filtering methods have been proposed to handle this problem including extended Kalman filter (EKF) [,4,5], unscented filter (UF) [,4,6,7] ( also called unscented Kalman filter), and various particle filters(pf) [,3]. he extended Kalman filter (EKF) is based on the principle of using aylor series expansion to linearize the measurement and process models. But its accuracy depends heavily on the severity of the nonlinearities; when the nonlinearities become severe, or they cannot be well approximated by a linear function, the EKF gives a divergent estimate. Moreover, the EKF requires calculation of Jacobians (first order partial derivatives). he unscented filter (UF) is another very popular nonlinear filter developed in recent years. he basic idea is to choose deterministic sample (sigma) points that capture the mean and covariance of a Gaussian density. When propagated through a nonlinear function, these points capture the true mean and covariance up to a second-order of the nonlinear function [6, 7]. Moreover, when the state dimension is high, the computational cost of UF is increased highly. PF algorithm has high estimation accuracy considerably, but its implementation is computationally expensive. Recently, the reference [9] proposed a quadrature Kalman filter (QKF) based on the algorithm in [8]. he QKF linearizes the nonlinear functions using statistical linear regression method through a set of Gaussian-Hermite quadrature points that parameterize the Gaussian density. It has been proved that QKF has higher estimation accuracy than the EKF and the UF [9]. In this paper, based on the QKF algorithm, we proposed the square-root quadrature Kalman filter (SRQKF) for tracing the ballistic target in re-entry phase, because of its excellent numerical characteristics. his paper is divided into 5 sections. Section is this introduction. In Section, we review briefly the target motion model and the radar measurement model. In Section 3, we present the proposed square-root quadrature Kalman filter. In Section 4, we give the simulation experiment about the new filter. Finally, Section 5 summarizes the main results. he Model his paper considers a ballistic target falling vertically towards the ground observed by a surface-based tracing radar as show in Figure. In general, after re-entering into the atmosphere, a ballistic target has high speed and it taes very short time to reach the ground. he main goal is to estimate the position, velocity and ballistic coefficient of the target given a sequence of noisy position measurements generated by a conventional radar ISIF 6

2 h Drag arget Gravity Radar Figure. Ballistic target tracing scenario. arget Motion Model Under the assumption that drag and gravity are the only forces acting on the object, the inematics of target is given as [] : () h= v ρ( h) g v v= + g β () (3) β = where h is the altitude, v is the velocity, ρ( h) density, g = 9.8m/s is the air is the acceleration due to gravity and β is the ballistic coefficient. Air density, measured 3 in g/m, is exponentially related to the altitude h, ρ = γe ηh 4 with γ =.754 and η =.49. β depends on the object mass, shape and cross-sectional area. arget inematic state is determined by height and velocity, but since β is unnown, it needs to be included in the state vector. he corresponding discrete tate space model for the inematic target equations ()-(3) is obtained by the Euler approximation with small integration step τ and is given by x = f( x) + + w (4) where f ( x ) denotes the nonlinear function of states given by with matrices f ( x ) =Φx G[ D( x ) g] (5) x [ ] = hvβ τ Φ= G = [ τ ] he superscript denotes matrix transposition, and drag Dx ( ) g ρ x x x [3] ( []) [] = (6) w is the process noise, independent of measurement noise, and is modelled by a zero-mean Gaussian process with covariance matrix Parameters 3 qτ 3 qτ Q = qτ qτ qτ q 3 (m /s ) and q (g m s 5 ), control the amount of process noise in target dynamics and the ballistic coefficient, respectively.. arget Measurement Model he radar is assumed to measure target range (in this case object height) at regular intervals of seconds. he measurement equation is then z = Hx + u (7) where is the discrete time index, H = [ ] is the observation vector, and u is zero-mean white Gaussian measurement noise with variance R = σ r..3 ypical arget rajectory he position, velocity and ballistic coefficient of a typical target trajectory is shown in Figure.. he parameters of this typical target trajectory are: initial height is 696m, initial velocity is 348m/s, ballistic coefficient is 96g/m s, τ =.s and q=q=. altitude m 7 x a) Altitude 63

3 velocity m/s ballistic coefficient g/ms x b) Velocity he SRQKF c) Ballistic coefficient Figure. ypical target trajectory he SRQKF is the square-root extension of the quadrature Kalman filter. he heart of the QKF is the Gauss Hermite quadrature rule that computes the first two order moments of all conditional densities. he Gauss Hermite rule does not approximate the nonlinear function explicitly; rather, it approximates the integral using a weighted quadrature point set. Moreover, the QKF is derivative free, thereby broadening its practical applicability. For the detailed QKF algorithm, please refer to [9]. A review of square-root implementations of the Kalman filter can be found in []. he proposed SRQKF algorithm was inspired by [3]. he SRQKF propagates the state estimate and its corresponding square root covariance. Because of using the square root form, the SRQKF exhibits many desirable numerical characteristics. he square root formulation guarantees the symmetry and positive semidefiniteness of the covariance matrix, improved numerical accuracy and the numerical stability. Before descirbing the SRQKF algorithm, we introduce some notations involved in the square root formulation. m { ξi, ω i} i = is a set of quadrature points and associated weights for a weight function being the standard Gaussian density in the Gaussian quadrature. About its calculation method, please refer to [8, ]. α is the number of quadrature points; m is the total number of quadrature points in the n x dimension nx state-space and is related to α by m = α. We use notation qr{} to denote a QR decomposition of a matrix. For a matrix =, A L N, QR gives a decomposition A QR where Q N N is orthogonal, R N L is upper triangular and N L. he upper triangular part of R, R, is the transpose of the Cholesy factor of P = AA, i.e., R = S. Cholesy factor updating can be available in Matlab as Cholupdate. If S is the original Cholesy factor of P, then the Cholesy factor of the update P ± vuu is denoted as S Cholupdate { S, u, v} = ±. A diagonal matrix W S containing all square roots of the quadrature weights on the diagonal (note that quadrature weights are positive and sum to unity) is written as ω WS = ωm S and S, for square roots of the We shall use Q, R process and measurement noise covariance Q and R, respectively. he SRQKF includes two step: ⅰ) time update, at the end of which the predicted density px ( z: ) is computed, and ⅱ) measurement update, at the end of which the posterior density p( x z : ) is computed. he detailed SRQKF algorithm is summarized below: SRQKF Algorithm: ime update step ) Assumed that at time the posterior density function px ( z: )~ N ( x ; x, P ) is nown, factorize S = [chol( P )] (8) ) Compute the quadrature points Xi, = S ξ i+ x i =,, m (9) 3) Evaluate the propagated quadrature points 64

4 * Xi, = f( Xi, ), i =,, m () 4) Estimate the predicted state m * x = ω i ix = i, () 5) Estimate the square root factor of the predicted error covariance * S = qr{ χ S Q, } () where the weighted centered (prior mean is subtracted off) matrix * * * χ = [ X, x X, x * X x ] W m, S SRQKF Algorithm: Measurement update step ) Evaluate the quadrature points Xi, = S ξi + x i =,, m (3) ) Evaluate the propagated quadrature points Zi, = hx ( i, ), i=,, m (4) 3) Estimate the predicted measurement z m ω iz i= i, = (5) 4) Estimate the square root of the innovation covariance Szz, = qr{ Z S R, } (6) where the matrix Z is below, Z = [ Z, z Z, z Z m, z ] WS 5) Estimate the cross-covariance matrix Pxz, = X Z (7) where, X = [ X, x X, x Xm, x ] WS 6) Estimate the Kalman gain K = ( Pxz, Szz, ) Szz, (8) 7) Estimate the updated state x = x + K( z z ) (9) 8) Estimate the square root factor of the corresponding error covariance matrix U = KSzz, () S = cholupdate{ S, U, } = P S S In this paper, the measurement model is linear, so the measurement update step is simplified as bellow: ) Estimate the predicted measurement z = H x () ) Estimate the Kalman gain K = S H ( HS H R) + () 3) Estimate the updated state x = x + K( z z ) (3) 4) Estimate the corresponding error covariance P = ( I KH) S (4) 4 Simulation his section describes the error performance of nonlinear filters. Altitude and velocity of filters are initialized from true position and velocity as in [8]. he third component (ballistic coefficient) is initialized with a Beta distribution [3]. Its pdf is given by λ Γ ( λ+ λ) β β L p( β) = ( βu βl) Γ( λ) Γ( λ) βu βl (5) λ β β L βu βl Here λ, λ are shape parameters, and symbol Γ denotes the Gamma function. λ = λ =., β = Kg/ms, L β = 63 Kg/ms, q U = q = 5. For simulation, the third component of filter is initialized as x [3] = E[ β ], P [3,3] = σ = ( std[ β ]). he initial height of the β target is 696m, velocity is 34m/s and with randomised initial value of ballistic coefficient. he simulated target is traced over steps. he performance is analyzed with 5 MC runs. We assume that target detection and false alarm probabilities of radar equal to and, respectively. We compared the SRQKF algorithm with the EKF and the QKF in position RMSE, velocity RMSE, ballistic coefficient RMSE, and the computational cost, respectively altitude RMSE EKF QKF SRQKF a) Altitude 65

5 velocity RMSE EKF QKF SRQKF fitler for tracing a ballistic target on re-entry phase with an unnown ballistic coefficient. hrough our experimental study we found that the estimation accuracy of the QKF and the SRQKF are all superior to that of the EKF. Compared with the QKF algorithm, the SRQKF algorithm improved the numerical stability and the numerical accuracy, but at the expense of increased computational complexity. we conclude that the SRQKF algorithm is a good alternative for the ballistic target tracing problem being studied b) Velocity ballistic coeffient RMSE EKF QKF SRQKF c) Ballistic coefficient Figure 3. Comparison of RMSE about three filters able I Computational Costs of hree Fitlers Filters Computation time EKF.96 QKF.94 SRQKF.9 From Figure 3, we can see that the SRQKF algorithm outperforms EKF and QKF in position, velocity and ballistic coefficient RMSE. From able I, we can see that the QKF and SRQKF are all have higher computational cost than that of the EKF. he SRQKF has slightly higher computational cost than that of the QKF, although it has higher estimation accuracy. At last, we give the conclusion of this paper. 5 Conclusion We presented in this paper a square-root quadrature Kalman 6 Acnowledgment his wor was supported by the National Key Fundamental Research & Development Programs (973) of China, No. 7CB36. References [] X. R. Li and V. P. Jilov, A survey of maneuvering target tracing-part II: ballistic target models, Proc. Of the SPIE,, 4473: [] X. R. Li and V. P. Jilov, A Survey of Maneuvering arget racing Approximation echniques for Nonlinear Filtering, In Proc. 4 SPIE Conf. on Signal and Data Processing of Small argets, vol. 548, Orlando, FL, USA, Apr. 4. [3] B. Ristic, A. Farina, D. Benvenuti, M. S. Arulampalam, Performance Bounds and Comparison of Nonlinear Filters for racing a Ballistic Object on Reentry, IEE Proc. Radar, Sonar Navig., Vol.5, No., April 3. [4] Farina, D. Benvenuti and B. Ristic, racing a ballistic target: comparison of several nonlinear filters, IEEE rans. On Aerospace and Electronic Systems, AES- 38: , July. [5] P. Costa, Adaptive model architecture and extended Kalman-Bucy filters, IEEE rans. On Aerospace and Electronic Systems, AES-3:55 533, April 994. [6] S. J. Julier, J. K. Uhlmann and H. F. Durrant-Whyte, A new method for the nonlinear transformation of means and covariances in filters and estimators, IEEE rans. Autom. Control, Vol. 45, pp. 477, Mar.. [7] S. J. Julier and J. Uhlmann, Unscented Filtering and Nonlinear Estimation, Proceedings of the IEEE, 9(3):4 4, Mar. 4. [8] K. Ito and K. Xiong, Gaussian filters for nonlinear filtering problems, IEEE rans. Autom. Control, Vol. 45, No. 5, pp. 9 97, May. [9] Arasaratnam. I. Hayin.S. and Elliott. R.J, Discrete- ime Nonlinear Filtering Algorithms Using Gauss Hermite 66

6 Quadrature, Proceedings of the IEEE, 7, Vol: 95(5): [] ZARCHAN, P., actical and strategic missile guidance, Progress in astronautics and aeronautics, Vol. 57 (AIAA, 994, nd edn.). [] G. H. Golub and J. H. Welsch, Calculation of Gauss Quadrature rules, Math. Comput. Vol. 3, No. 6, pp. 3, 969. [] P. Kaminsi, A. Bryson, and S. Schmidt, Discrete square root filtering: A survey of current techniques, IEEE rans. Autom. Control, Vol. AC-6, No. 6, pp , Dec. 97. [3] R. van der Merwe and E. Wan, he square-root unscented Kalman filter for state and parameter estimation, in Proc. Int. Conf. Acoustic, Speech, Signal Process. (ICASSP), May, Vol. 6, pp