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1 Sensors 007, 7, Full Paper sensors ISSN by MDPI An Improved Partcle Flter for Target Tracng n Sensor Systems Xue Wang *, Sheng Wang and Jun-Je Ma State Key Laboratory of Precson Measurement Technology and Instrument, Tsnghua Unversty, Bejng 00084, P. R. Chna; E-mals: wang_sheng00@mals.tsnghua.edu.cn; mjj@mals.tsnghua.edu.cn * Author to whom correspondence should be addressed; E-mal: wangxue@mal.tsnghua.edu.cn Receved: 3 December 006 / Accepted: 7 January 007 / Publshed: 9 January 007 Abstract: Sensor systems are not always equpped wth the ablty to trac targets. Sudden maneuvers of a target can have a great mpact on the sensor system, whch wll ncrease the mss rate and rate of false target detecton. The use of the generc partcle flter (PF) algorthm s well nown for target tracng, but t can not overcome the degeneracy of partcles and cumulaton of estmaton errors. In ths paper, we propose an mproved PF algorthm called PF-RBF. Ths algorthm uses the radal-bass functon networ (RBFN) n the samplng step for dynamcally constructng the process model from observatons and updatng the value of each partcle. Wth the RBFN samplng step, PF-RBF can gve an accurate proposal dstrbuton and mantan the convergence of a sensor system. Smulaton results verfy that PF-RBF performs better than the Unscented Kalman Flter (UKF), PF and Unscented Partcle Flter (UPF) n both robustness and accuracy whether the observaton model used for the sensor system s lnear or nonlnear. Moreover, the ntrnsc property of PF-RBF determnes that, when the partcle number exceeds a certan amount, the executon tme of PF-RBF s less than UPF. Ths maes PF-RBF a better canddate for the sensor systems whch need many partcles for target tracng. Keywords: Partcle flter, radal-bass functon networ, target tracng, sensor system

2 Sensors 007, Introducton For survellance, a sensor system s nstalled to search for targets and provde relable detecton wthn the gven regon. The sensor system can measure the range and bearng of the targets, but t can not trac them. In spte of the recent advances n sensor technology, there are no devces that can detect the manned maneuvers of a traced target n survellance and gudance systems []. Flterng s used for estmatng and tracng the state of a target as a set of observatons becomes avalable onlne []. When a target moves, acceleraton may be unexpected, varyng over tme and follow an unnown profle, whch has great mpact on the sensor system. Even a short-term acceleraton wll cause an error n the measurement sequence and result n dvergence, f compensatons are not used n tme. For mplementng target tracng n a sensor system, the extended Kalman flter (EKF) was ntroduced [3], but because EKF only uses the frst order terms of the Taylor seres expanson of nonlnear functons, t often ntroduces large errors n the estmated statstcs of the posteror dstrbutons of the states. Ths s especally evdent when the models are hghly nonlnear and the local lnearty assumpton breas down []. In these cases the Unscented Kalman flter (UKF) [4] wth true nonlnear models was proposed. Unle EKF, UKF uses true nonlnear models and nstead approxmates the dstrbuton of the state random varable. In UKF, the state dstrbuton s stll represented by a Gaussan random varable, but t s specfed usng a mnmal set of determnstcally chosen sample ponts. These sample ponts completely capture the true mean and covarance of the Gaussan random varable, and when propagated through the true nonlnear system, captures the posteror mean and covarance accurately to the nd order for any nonlnearty, wth errors only ntroduced n the 3rd and hgher orders. However, UKF s lmted n that t does not apply to general non-gaussan dstrbutons []. Then generc PF [5] was presented for handlng multmodal probablty densty functons [6] and solvng nonlnear non-gaussan problems [7], whch are wdely adopted n maneuverng target tracng [5,6,7,8]. However, degeneracy wll lmt the ablty of generc PF to search for lower mnma n other regons of the error surface []. If the process nose ncreases, the estmaton error wll accumulate, whch wll lead to the dvergence of the sensor system []. To solve ths problem, the mproved partcle flters have always used varous methods to propagate the mean and covarance of the Gaussan approxmaton to the state dstrbuton, such as unscented partcle flter (UPF) whch resulted from usng a UKF and Marov chan Monte Carlo (MCMC) step wthn a partcle flter framewor, but they also ncreased the executon tme sharply []. They are computatonally ntensve and dffcult to mplement n real tme n a sensor system. Ths paper proposes a PF-RBF algorthm for target tracng n sensor systems. It s an mproved PF algorthm whch uses the radal-bass functon networ (RBFN) n samplng. Accordng to the observatons, PF-RBF uses RBFN to approxmate the movng trajectory, construct the process model, perform samplng and decrease the cumulated effect of errors. We compare PF-RBF wth UKF, PF and UPF. The results show that PF-RBF can trac targets effectvely, especally when the observaton model of the sensor system s nonlnear, and t also dsplays good real tme performance. The remander of ths paper s organzed as follows: target moton and observaton model of sensor systems are ntroduced n secton. The proposed PF-RBF algorthm s presented n secton 3. The expermental results are descrbed n secton 4. Fnally, the conclusons are gven n secton 5.

3 Sensors 007, Target Moton and Observaton Model of Sensor Systems In a sensor system, the measurement sequences are n polar or Cartesan coordnates, but the target dynamcs are best descrbed n Cartesan coordnates, so we assume that measurement sequences descrbed n Cartesan coordnates are avalable. The target tracng s always formulated as a dynamc state space model. The general state space model can be broen down nto a state transton and state measurement model: x (, ) (, = f x u y = h x r) () where s the tme ndex. The state varable x whch refers to the poston and velocty of the target s propagated by the process model f over tme. The observaton model h s used to map the state varable x to the correspondng observaton y ; u and r are the process nose and the observaton nose respectvely. It s assumed that u and r are whte Gaussan nose wth zero means and uncorrelated wth each other. It s always assumed that the states follow a frst order Marov process and the observatons are assumed to be ndependent gven the states p x x () ( : ) ( ) In target tracng, the posteror densty p( x y : ), where y: { y, y,, y} p y x (3) =, consttutes the complete soluton to the sequental estmaton problem []. So the estmaton accuracy of posteror densty s used as the metrc n tracng. Because of the measured data and computatonal loadng, most sensor systems use a poston and velocty two-state process model [], a maneuver-drvng nput model [, 8] or use two process models and swtch between them n the system [9]. However, these models are only approprate for one of two stuatons, the presence or absence of target maneuvers. For robust target tracng, a dynamc process model should be constructed. 3. Prncple of the PF-RBF Algorthm The generc PF algorthm s a sequental mportance samplng method whch s based on Monte Carlo smulaton and Bayesan samplng estmaton theores. Varous PFs all contan three mportant steps: samplng current value of each partcle, evaluaton of the recursve mportant weghts and resamplng. Recent research has focused almost exclusvely on the weghtng and resamplng for mprovng the tracng accuracy [0], [], whle PF-RBF algorthm focuses on the samplng step. 3.. Generc Partcle Flter In Bayesan samplng estmaton theory, the posteror densty ( : ) densty p( x y: ) p( y x) p( x y ) ( ) ( ) : p x y: p y y: where p x y can be nferred from pror = (4)

4 Sensors 007, 7 47 and ( ) ( ) ( ) p y y = p y x p x y x (5) d : : ( ) ( ) ( ) p x y = p x x p x y dx (6) : : Then PF uses the Monte Carlo smulaton method to approxmate the posteror densty by N partcles wth the assocated weght N p( x y: ) ω δ( x x ) (7) = For solvng the dffculty of samplng from the posteror densty functon, the sequental mportance samplng method s used, whch samples from a nown, easy-to-sample, proposal dstrbuton q( x0: y : ), where x 0: s the hstorcal state varables and y : s the correspondng observaton. The recursve estmate for the mportance weghts of partcle can be derved as follows: ( ) ( ) ( ) q x x0:, y: p y x p x x ω = ω (8) Then the estmated state can be approxmated by N xˆ ω x (9) = 3.. Trajectory Approxmaton wth RBFN Radal bass functons are a specal class of functon. Ther responses decrease (or ncrease) monotoncally wth the dstance from a central pont []. Centre, dstance scale and precse shape of the radal functon are parameters of the model, all fxed f t s lnear [9]. In prncple, they could be employed n any sort of lnear or nonlnear model and sngle-layer or mult-layer networ. RBFN s a three-layer feed-forward neural networ whch s embedded wth several radal-bass functons. Such a networ s characterzed by an nput layer, a sngle layer of nonlnear processng neurons, and an output layer. The output of RBFN [9] s calculated as: ( ) M τφ j j j (0) j= y = z c where z s an nput vector, φ j s a bass functon, denotes the Eucldean norm, τ j s the weght n the output layer, M s the number of neurons n the hdden layer, and c j s the center of RBF n the nput vector space. The functonal form of φ j s always assumed as follows: φ z = exp z / σ () ( ) ( ) Fgure llustrates the shape of functon φ ( z) wth σ =.

5 Sensors 007, 7 48 Fgure. The shape of functon φ ( z) wth σ = Ph(Z) Z RBFNs present good approxmaton propertes. The RBFN famly s broad enough to unformly approxmate any contnuous functon on a compact set. For any contnuous nput-output functon f ( z ) M there s an RBFN wth a set of centers { t } = mappng functon F ( z ) realzed by the RBFN s close to f ( ) and a common wdth σ > 0 such that the nput-output L norm, p [, ] z n the p []. Here the nput-output mappng s represented by M z t F( z) = τ G = σ () m0 where σ > 0, ω j R, and G s an contnuous ntegrable bounded functon n a subset R R and G z dz 0 (3) R m 0 Because the trajectory of target s a typcal contnuous functon, RBFN can approxmate t and construct the dynamc process model for state estmaton n PF algorthm well. In PF-RBF, we use prevous observatons and current predcton to tran RBFN; the detals wll be dscussed below. ( ) 3.3. Improved PF Algorthm Combned wth RBFN The process model of the generc PF algorthm always uses a sngle moton model whch s statc and cannot offer a dynamc and consstent approxmaton of state varables for target tracng [,,8]. The mproved PF algorthm based on an nteractng mult-model (IMM) flter [3] can solve t, whch allows for several modes combned wth a weghted estmate [4], but the state varable must be estmated and updated n each tme step for each model separately, so the executon tme wll ncrease sharply. The objectve of our algorthm s to perform a robust and accurate approxmaton of state varables and decrease the executon tme. Our algorthm contans fve steps: () constructng RBFN based on prevous observatons y 0: and current predcton x for approxmatng the trajectory of target. It should be noted that the observatons y 0: do not equal to the states of target, whch just have relatonshp to the state varable x0: accordng to the observaton model h. So, for tranng the RBFN, the state varable x 0: should be nferred from observatons y 0:. Because of the observaton nose, we can only get the approxmate state varable x 0: nstead of x0: ; () samplng new value of each partcle based on RBFN; (3) evaluatng the recursve mportance weghts; (4) resamplng; (5) output. The pseudo-code for our algorthm s outlned n Algorthm :

6 Sensors 007, 7 49 Algorthm. Algorthm. Intalzaton: = 0 For =,, N x0 ~ sample( px ( 0)) / draw the states from the pror. For =,, (a) Constructng RBFN Use the prevous observatons y0: to nfer the prevous approxmate state varable x0: ; Predct the current state by nematc theory wth the gven tme nterval T: x = x + v T + 0.5a T (4) x x v = T (5) x + x 3 x a = v v = T (6) Construct RBFN wth the prevous real state of target x0: and current state predcton x. (b) Samplng For =,, N xˆ ~ q x x, y by the constructed RBFN Sample ( 0: : ) ( ˆ0: ) M = τφ j j j j= xˆ x c (c) Evaluatng For =,, N Evaluate the recursve mportance weghts by (8) For =,, N Normalze the mportance weghts = N j ω j = ω ω (7) (d) Resamplng Multply / suppress samples x ˆ wth hgh / low mportance weghts ω, respectvely, to obtan N random samples 0: p x y. (e) Output Output the estmated state Set ω = ω =, x ˆ = x =,, N N x approxmately dstrbuted accordng to ( 0: : ) xˆ N ˆ = ωx = As llustrated n Algorthm, n the PF-RBF algorthm, the RBFN s dynamcally traned as the process model for the samplng step. For approxmatng the state of the target, we use the prevous

7 Sensors 007, 7 50 observatons and current predcton based on observatons as the samples to tran the RBFN. Then, each partcle uses the estmated state sequence xˆ 0: as nput vectors to sample the new value of the state varable x ˆ. Then PF-RBF contnues the process of evaluatng and normalzng mportance weghts, resamplng, and output as generc PF algorthm. Because RBFN s traned wth the observatons and predcton based on observatons, t wll not be mpacted by the posteror dstrbuton error of each partcle, and can approxmate the real trajectory of target effectvely. Wth the gudance of RBFN, each partcle can estmate the probablty dstrbuton more effectvely and eep the multmodalty at the same tme. Although the computaton complexty of RBFN s O(n 3 ) [0], where n s the number of tranng, at each tme nstant there s only one tranng process and the tme s rrelatve wth the partcle number, so the change rate of executon tme of PF-RBF should nearly equals to PF. Furthermore, because the randomness of the movement of target, we can just adopt the nearest m samples of approxmate state varable x m: to tran the RBFN, whch can sgnfcantly reduce the computaton complexty. 4. Expermental Results 4.. Experment Setup The estmaton mprovement obtaned by PF-RBF algorthm s llustrated n ths secton. The UKF s more sutable than the EKF for proposal dstrbuton generaton wthn the partcle flter framewor [6], so we only compare the state tracng results of UKF, PF, UPF and PF-RBF. For smplfyng and generalzng the problem, we just performed one-dmenson tracng. Two-dmenson tracng can be easly derved. The used assumptons and parameters are lsted below. For smulaton, a tme-seres s generated by the followng process model x = x + v T + a T + u (8) where x + v T presents the state transton model, a T + u presents the process nose, and T s the tme nterval. Here, for smulatng the crcuty of the maneuverng target, the related factors are gven by v = sn( βπ ( ) ) (9) a = v = βπ cos βπ ( ( )) u s a term of process nose whch s drawn from a Gaussan dstrbuton N ( 0,0.). In target tracng, the whole process nose s consdered as the degree of maneuverng. β = 0.04 s a scalar parameter. For comparng tracng performance, a non-statonary observaton model s used as n []: φ x + r 30 y = (0) φ x + r > 30 wth φ = 0. and φ = 0.5. The observaton nose, r s drawn from a Gaussan dstrbuton N, whch s the same as the parameter proposed by []. Gven only the nosy ( 0,0.0000) observatons y, the dfferent flters are used to estmate the underlyng clean state sequence x for =,,, 60. The experment s repeated 00 tmes wth random re-ntalzaton for each run. All of the partcle flters use 0 to 00 partcles and resdual resamplng. The pont scalng parameter of UKF

8 Sensors 007, 7 5 s set to α =, whch s used to control the sze of the sgma pont dstrbuton and should deally be a small number to avod samplng non-local effects when the nonlneartes are strong, scalng parameter for hgher order terms of Taylor seres expanson s set to β = 0, whch s a non-negatve weghtng term to be used to ncorporate nowledge of the hgher order moments of the dstrbuton, and sgma pont selecton scalng parameter s set to κ =, ths parameter s used to guarantee postve semdefnteness of the covarance matrx. Three parameters are optmal for the scalar cases, and they are also adopted n []. The number of the samples of approxmate state varable x m: used for tranng the RBFN s 0. The tme nterval s s. The root-mean-square-errors (RMSEs) of the estmaton values are acqured for comparson. 4.. Tracng results In Fgure the estmated results generated from a sngle run of the dfferent flters are compared, where the partcle number s 00 n each partcle flter. Fgure 3 shows the correspondng errors of dfferent flters versus the process noses. Fgure. Plot of estmated results generated from a sngle run of the dfferent flters wth non-statonary observaton model. Flter estmates (posteror means) True x UKF estmate PF estmate UPF estmate PF RBF estmate Tme (s) As llustrated n Fgure 3, the errors generated by all algorthms are strongly correlatve wth process nose,.e., the tendences of the errors are coherent wth the tendency of the process nose. The error of UKF algorthm ncreases sharply because the estmaton error cumulates and results n system dvergence. Although the errors of estmated states n four algorthms are nearly equal when the observaton model s lnear ( > 30 ), our algorthm stands out by ts lower error performance compared to the other three algorthms when the observaton model s nonlnear ( 30 ).

9 Sensors 007, 7 5 Fgure 3. Errors of dfferent flters versus the process noses n a sngle run Flter errors UKF PF UPF PF RBF Process Nose Tme (s) The RMSEs of PF-RBF are all small whether the observaton model s lnear or nonlnear. Ths means that the PF-RBF s robust and effectve for both lnear and nonlnear observaton models. When a maneuver occurs ( 0 < < 30 ), the RMSEs of PF-RBF are stll smaller than 0.0, whle the RMSEs of the other three algorthms ncrease sharply. Ths confrms verfes that PF-RBF s also sutable for tracng maneuverng targets. Then we compared the mean and varance of RMSEs generated by each algorthm wth 00 partcles over 00 ndependent runs. As shown n Table, the superor performance of the PF-RBF s clearly evdent for both lnear and nonlnear observaton models. Table. Estmated results generated by each algorthm where the mean and varance of RMSEs are calculated over 00 ndependent runs. The partcle number s 00. Algorthm Mean of RMSEs Nonlnear Lnear UKF Generc PF UPF PF-RBF

10 Sensors 007, Executon tme and accuracy In each PF algorthm, the executon tme and accuracy are partally determned by the partcle number. In ths secton, we compare the average RMSE and executon tme of PF, UPF and PF-RBF. The partcle number ncreases from 0 to 00. The average RMSE calculated over 00 ndependent runs, whch decreases when the partcle number ncreases, are shown n Fgure 4,. When the observaton model s lnear, PF-RBF performs best. And for the nonlnear observaton model, the average RMSE of PF-RBF s bggest when the partcle number s 0, but the average RMSE of PF-RBF decreases rapdly and become lowest when the partcle number s larger than 0. As a whole, PF-RBF has superor performance n target tracng and t s robust for both lnear and nonlnear observaton models. Fgure 4. The average RMSEs of three algorthms calculated over 00 ndependent runs wth the lnear and non-lnear observaton model. Average Root Mean Square Error (Posteror) PF Nonlnear UPF Nonlnear PF RBF Nonlnear PF Lnear UPF Lnear PF RBF Lnear Partcle Number Fgure 5 shows that the executon tme of three algorthms for 60 step tracng all ncrease lnearly wth partcle number and UPF taes more tme than PF-RBF when the partcle number exceeds 50. Although PF-RBF ncludes the RBFN tranng step whch taes most tme, for each tme pont there s only one tranng process and the tme s ndependent of the partcle number, so the rate of change of executon tme of PF-RBF nearly equals the PF one, and because UPF contans UKF and MCMC steps for each partcle, the change rate s hgher than PF-RBF. As llustrated n Fgure 5, the change rate of UPF s about 7 tmes greater than the one of PF-RBF. Ths maes the PF-RBF a better canddate for the scenaros whch requre many partcles.

11 Sensors 007, 7 54 Fgure 5. The average executon tme of three partcle flters for 60 steps tracng over 00 ndependent runs. 4 Average Executon Tme (s) PF UPF PF RBF Partcle Number 5. Conclusons In ths paper, we focused on target tracng n sensor systems. The PF-RBF algorthm was proposed, whch trans RBFN to approxmate the trajectory of target and constructs dynamc process model accordng to the prevous observatons and current predctons. The traned RBFN s used to perform samplng steps for each partcle, nstead of the classcal process model. Wth the gudance of RBFN, each partcle can gve an accurate proposal dstrbuton, the cumulated effect of errors can be decreased, and the sensor system remans convergent even f the target maneuvers. The target tracng experment results verfy that when the observaton model s lnear, PF-RBF perform better than UKF, PF and UPF, and for a nonlnear observaton model, PF-RBF s most robust and accurate, except when the partcle number s less than 0. Moreover, the rate of change of executon tme of PF-RBF s about seven tmes less than that of UPF, t ths maes the executon tme of PF-RBF less than that of UPF when the partcle number exceeds a certan number, whch n ths paper s 50. It should be noted that the crtcal partcle number thresholds depend on the gven problem, and may be dfferent n other problems. However, t stll mples that PF-RBF s sutable for dealng wth sensor systems whch need many partcles for mult-target tracng. Acnowledgements Ths paper s sponsored Natonal Grand Research 973 Program of Chna (No. 006CB303000) and Natonal Natural Scence Foundaton of Chna (No , No , No ).

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