An Improved Particle Filter with Applications in Ballistic Target Tracking

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1 Sensors & ransducers Vol. 72 Issue 6 June 204 pp Sensors & ransducers 204 by IFSA Publishing S. L. An Iproved Particle Filter with Applications in Ballistic arget racing Chun-Ling Wu Yong-Feng Ju 2 Chong-Zhao Han Chang an University Middle Section of an Erhuan Road Xi'an City Shaanxi P. R. China 2 Xi an Jiaotong University o.28 Xianning West Road Xi'an Shaanxi P. R. China el.: fax: E-ail: inforationway@63.co Received: 30 April 204 /Accepted: 30 May 204 /Published: 30 June 204 Abstract: In this paper we present an iproved particle filter algorith for ballistic target tracing the quadrature Kalan particle filter (QKPF. he proposed algorith uses quadrature Kalan filter (QKF for generating the proposal distribution. he QKF is a recursive nonlinear filtering algorith developed in the Kalan filtering fraewor. It linearizes the nonlinear functions using statistical linear regression ethod through a set of Gaussian-Herite quadrature points need not to copute the Jacobian atrix and is easy to ipleent. Moreover the generated proposal distribution integrates the latest easureents into syste state transition density so it can atch the posterior density well. he siulation results show that QKPF is a viable alternative for sequential estiation in nonlinear dynaic odels. Copyright 204 IFSA Publishing S. L. Keywords: Ballistic target tracing Quadrature Kalan filter Iportance density function Particle filter.. Introduction Ballistic target tracing using conventional radar easureents is a difficult proble due to the inherent nonlinearity in both the target otion odel and the radar observation odel. he ain goal is to estiate in a recursive fashion the unnown ineatic state e.g. position and velocity of the target given a sequence of noisy position easureents. In general a closed-for analytical expression for the iniu ean square error (MMSE estiate of the hidden state cannot be obtained in the ballistic target tracing proble due to the nonlinearity. We resort then to a sequential iportance sapling ethod referred to in the literature as particle filtering (PF [] to approxiate the optial MMSE estiate. Farina et al. proposed in [2] the application of a bootstrap particle filter (PF for the proble of ballistic target tracing assuing a nonlinear target otion odel and a linear observation odel with both odels specified in Cartesian coordinates. In this paper we use the sae otion odel as in [2] but assue an alternative nonlinear observation odel in polar coordinates. Furtherore instead of using the standard bootstrap filter we present an iproved particle filter to this ballistic target tracing proble. It is nown that the PF require the design of proposal distribution that can approxiate the posterior distribution reasonably well. he standard particle filter is to saple fro the probabilistic odel of the states evolution (transition prior. his strategy can however fail if the new easureents appear in the tail of the prior or if the lielihood is too peaed in coparison to the prior. o overcoe this proble several techniques based on linearization have been proposed in the literature. For exaple the extended Kalan particle filter 96

2 Sensors & ransducers Vol. 72 Issue 6 June 204 pp (EKPF [2] and the unscented particle filter (UPF [3 4] this two algoriths use EKF [3] and the unscented filter (UF [5] to generate the proposal distribution respectively. Recently in [6] the quadrature Kalan filter (QKF algorith is proposed. he QKF linearizes the nonlinear functions using statistical linear regression ethod through a set of Gaussian-Herite quadrature points. It has been proved that QKF has higher estiation accuracy than the EKF and the UF [6]. In this paper we use QKF to generate the proposal distribution and obtain a new ind particle filter i.e. the quadrature Kalan particle filter (QKPF. his paper is divided into 5 sections. Section is this introduction. In section 2 we review briefly the target otion and the radar easureent odels. In section 3 we present the proposed particle filter tracer. In section 4 we discuss the perforance of our algorith. Finally section 5 suarizes the ain results in this paper. 2. he Model he plant selected in this paper is fro literature [ 3] and the geoetry relation is illustrated in Fig.. It is assued that the body is falling vertically the effect of gravity is negligible. body x ( t = ω ( t (3 3 3 where ω ( t ω ( t and 2 ω ( t 3 are the zero-ean uncorrelated noises with covariances given by Q ( t -5 and g is the constant (5 0 that relates the air density with altitude. he range at tie t zt ( is 2 zt ( ( M [ x( t H] 2 rt ( = (4 where rt s the uncorrelated observation noise with 4 2 covariance R ( t = 0 ft. he easureents are ade with a frequency of Hz. he initial true state of the syste is x(0 = é ù êë úû and the initial estiates and covariance atrixes of these states are ˆ( x 0 0 = é êë úû é ù 6 P(0 0 = êë úû In this experient we don t introduce any process noise into the siulation Q ( = 0 for both filters. ù H r(t M x 2 (t x (t Fig.. he geoetry of the proble. We wish to estiate the position x ( t velocity x ( t and constant ballistic coefficient x ( t of a 2 3 body as it reenters the atosphere at a very high altitude at a very high velocity. he position of the body is easured at discrete points in tie using radar capable of easuring range corrupted by Gaussian easureent noise. he radar is at an altitude H (00 000ft and the horizontal range between the body and the radar M is (00 000ft. he continuous tie dynaics of this syste are x ( t =- x ( t + ω ( t ( 2 x ( t =- e x ( t x ( t + ω ( t (2 -gx ( t Quadrature Kalan Particle Filter 3.. Particle Filter he PF is a coputer-based ethod for ipleenting an optial recursive Bayesian filter by Monte Carlo siulations. he central idea is to represent the required probability density function (PDF by a set of rando saples (particles. As the nuber of particles is increased the representation of the required PDF becoes ore accurate. he PF can be described as follows. Assue that we have a set of rando saples (particles x - i= fro the posterior density { } px ( z: at tie -. he PF is an algorith for propagating and updating the set of rando saples { x - i= } to a new set of rando saples at tie { x i= } which are approxiately distributed as the posterior density px ( z :. Denote x0: = { x0 x x2 x} and z: = { z z2 z} be the states and observations up to tie step respectively. In PF algorith the choice of proposal distribution is the ey proble. In this paper we use QKF algorith to generate proposal distribution. And 97

3 Sensors & ransducers Vol. 72 Issue 6 June 204 pp next we first describe the QKF algorith and then give the detailed QKPF algorith Quadrature Kalan Filter he QKF algorith is developed fro the statistical linear regression view rather than the nuerical integration perspective. he QKF linearizes the nonlinear functions using statistical linear regression ethod through a set of Gaussian- Herite quadrature points that paraeterize the Gaussian density. It has been proved that QKF has higher estiation accuracy than the EKF and the UF [6]. he algorith of QKF is given by algorith : Algorith : Quadrature Kalan Filter ie update: Assued that at tie the posterior density function px ( z ~ ( x ; xˆ P is : nown factorize P = S S (5 2 Copute the quadrature points { X } = as: X = S ξ + xˆ (6 3 Evaluate the propagated quadrature points * X { } : = X = f( X = (7 * 4 Estiate the predicted state and the corresponding error covariance: 3 Evaluate the propagated quadrature points { Z } = as: Z = hx ( = (2 4 Estiate the predicted easureent: ˆ z ω Z = = (3 5 Estiate the innovation covariance atrix: P = R + ω ( Z ˆz ( Z ˆz zz (4 = where R is the easureent noise covariance. 6 Estiate the cross covariance atrix: P = ω ( X xˆ ( Z ˆz xz (5 = 7 Estiate the Kalan gain K = P P (6 xz zz 8 Estiate the updated state and the corresponding error covariance: 9 Factorize the P xˆ = xˆ + K ( z ˆz (7 P = P K P K (8 zz P = S S (9 * xˆ ω X = = (8 P Q X x X x (9 * * = + ω ( ˆ ( ˆ = where Q is the process noise covariance. At the end of the tie update we have the predicted density px ( z = ( xˆ P. : Measureent update step: Factorize P = S S (0 2 Evaluate the quadrature points { X } : = X = S ξ + xˆ ( At the end of the easureent update we have the posterior density px ( z = ( x; xˆ P. : About quadrature points ξ and the corresponding weights ω we give the following coputation ethod [6]. Consider a scalar rando variable x having a Gaussian probability density ( x;0. he expected value of the function g( x can be approxiated as ω ξ (20 R E( gx ( gx ( ( x;0 dx g( = = Instead of finding quadrature points using rootfinding ethods which ay be atheatically unstable a coputationally better approach is presented in [] to find the quadrature points and 98

4 Sensors & ransducers Vol. 72 Issue 6 June 204 pp weights. his approach exploits the relationship between orthogonal polynoials and tri-diagonals. Suppose J is a syetric tridiagonal atrix with zero diagonal eleents and J = 2 + ( (2 where is the quadrature point nuber in this n paper we tae 3 quadrature point. hen = 3 x n x is the state diension. he quadrature point ξ is taen to be ξ = 2λ where the λ is the -th eigenvalue of J ; and the corresponding weight ω (( 2 = η where ( η is the first eleent of the -th noralized eigenvector of J Quadrature Kalan Particle Filter (QKPF In this paper the new particle filter QKPF use the QKF to generate the proposal distribution. In section 3.2 we have given the detailed QKF algorith the QKPF algorith is now given by algorith 2. Algorith 2: he Quadrature Kalan Particle Filter Initialization: at tie = 0. For i= Draw the particles x 0 fro the prior px ( 0 and set xˆ = Ex [ ] 0 0 P = E[( x -xˆ ( x - xˆ ] * * P = Q + ω ( X xˆ ( X xˆ = 3 Measureent update: Incorporate new observation P = S ( S X = S ξ + xˆ Z ( i = h( X ( ( P R Z z Z z zz = + ω ( ( ˆ = P = ω ( X xˆ ( Z ˆz xz = K = P ( P xz zz x = x + K ( z ˆz P = P ( K P K zz 2. For i= saple x ~ qx ( x z» ( x ; x P - ˆ 0: : P = S ( S 3. For i= evaluate the noralized weights w pz ( x px ( x = - w- qx ( x0: - z: P = S ( S For each i= he initial iportant weight w0 =. Prediction and update: For each tie ³. For i= update the particles with the QKF Calculate the quadrature points X ( : X = S ξ + xˆ = ( 2 ie update: Propagate particle into future X = f( X *( i ( i ( ( * xˆ ω X = = w i= = å w w = w w where pz ( x is the lielihood function of the easureent z. 4. Resaple: 2 If eff < th ( eff = å ( w i= is saple volue th is a soe given threshold value set w = and resaple with { x w i } =. Output: the experience probability distribution of filtering distribution syste state estiate and covariance are given respectively: = d - : i= px ˆ( z å ( x x 99

5 Sensors & ransducers Vol. 72 Issue 6 June 204 pp x i = xˆ = å = ( ˆ ( ˆ å - - i = P x x x x ote: he initial quadrature points and the associated weights { } ξ ω are deterined off- = line in advance. 4. Siulation Results Ballistic Coefficient Estiation Error Seconds Fig. 4. Ballistic coefficient estiation error. PF QKPF he siulated target is traced over 60 tie steps. In the experient the nuber of the particles is = 200. Furtherore about the QKPF algorith the higher the nuber of quadrature points (per axis is the lesser the estiation error becoes. But the estiation accuracy coes with an increased coputational coplexity. In this paper we choose the nuber = 3. If we choose the nubers = 5 the estiation accuracy will be higher. he following figures give the position velocity and ballistic coefficient estiation error respectively. Fro the Fig. 2 to Fig. 4 we can see that the estiation errors of QKPF algorith are all lower than the PF. So the siulation experient iprove that the estiation accuracy of QKPF is higher than that of the PF apparently. 5. Conclusion We presented in this paper a quadrature Kalan particle filter for ballistic target tracing proble. he proposed filter adopts the quadrature Kalan filter to generate the proposal distribution. Copared with the PF algorith the QKPF algorith iproved the nuerical stability and the nuerical accuracy but at the expense of increased coputational coplexity. we conclude that the QKPF algorith is a good alternative for the ballistic target tracing proble being studied. Acnowledgeents Position Estiation Error PF QKPF he authors wish to than Yong-Feng Ju Chong- Zhao Han. his wor was supported in part by a grant fro ational atural Science Fund ( and was supported by Central Universities Fundaental Research Funds (CHD200JC09 CHD20JC065. References Velocity Estiation Error Seconds Fig. 2. Position estiation error. PF QKPF Seconds Fig. 3. Velocity estiation error. []. A. Doucet. de Freitas. J. Gordon (Eds. Sequential Monte Carlo ethods in practice Springer-Verlag ew Yor January 200. [2]. M. S. Arulapala S. Masell. Gordon and. Clapp A tutorial on particle filters for online nonlinear/non-gaussian Bayesian tracing IEEE ransactions on Signal Processing Vol. 50 Issue pp [3]. A. Farina B. Ristic D. Benvenuti racing a ballistic target: coparison of several nonlinear filters IEEE ransactions on Aerospace and Electronic Systes Vol. 38 Issue 3 July 2002 pp [4]. R. van der Merwe A. Doucet. de Freitas et al. he unscented particle filter echnical Report of the Cabridge University Engineering Departent CUED/ F-IFE G/R 380 Cabridge University Press Cabridge UK 2000 pp [5]. S. J. Julier and J. Uhlann Unscented filtering and nonlinear estiation Proceedings of the IEEE Vol. 92 Issue pp

6 Sensors & ransducers Vol. 72 Issue 6 June 204 pp [6]. I. Arasaratna S. Hayin R. J. Elliott Discrete-tie nonlinear filtering algoriths using Gauss Herite quadrature Proceedings of the IEEE Vol. 95 Issue pp Copyright International Frequency Sensor Association (IFSA Publishing S. L. All rights reserved. ( 20