Synchronized Multi-sensor Tracks Association and Fusion

Transcription

1 Synchronzed Mult-sensor Tracks Assocaton and Fuson Dongguang Zuo Chongzhao an School of Electronc and nformaton Engneerng X an Jaotong Unversty Xan 749, P.R. Chna Zlz_3@sna.com.cn czhan@jtu.edu.cn Abstract - Ths paper develops the track assocaton and fuson technque of synchronzed mult-sensor trackng system. Va the dscretzaton of lner contnuous tme system, the correlaton among common process nose and measurement noses, but also the measurement noses correlaton, are proved. And then n the lght of theoretc dervaton, the synchronzed mult-sensor track assocaton and fuson algorthm s presented. The smulaton results epress ts valdty. Keyords: correlated nose target trackng assocaton fuson. Introducton Because of the lmtatons of usng sngle sensor for trackng targets (such as accuracy, measurement space, etc.) and some other dsadvantages, most of trackng systems are beng desgned th multple sensors to mprove the system s trackng performance. Snce the concept of data fuson as addressed, the mult-sensor nformaton fuson problem s beng notced by a lot of researchers, and multsensor target trackng problem has been nvestgated n lots of lteratures ([]-[]), rangng from the dstrbuted and central or hybrd trackng systems, nvolvng the problems about noses ndependence mutually and also the dependence of correspondng estmaton errors. The lteratures [4] [6]-[8] all consder the track-to-track assocaton problem hen dealng th the mult-sensor data fuson. The [8] takes the effect of common nose on to sensors track fuson nto account frstly, but fe of them dscuss the correlaton cases beteen the system s process nose and measurement noses mutually. Ths s unconfrmed to the real orld. Ths paper, begnnng th the dscretzaton of lner contnuous tme system, has a talk about the reasons of the correlaton beteen process nose and measurement noses mutually. And then the algorthm of track-to-track assocaton and fuson s derved n the case of correlated Ruuan We ZhengLn School of Electronc and nformaton Engneerng X an Jaotong Unversty Xan 749, P.R. Chna re@chnaren.net lnzheng@sna.com process nose and measurement noses mutually on the bases of [6]-[7]. Fnally, ts valdty s proved va the Monte Carlo smulaton. Formulaton of the correlated problems Frstly, let us consder the follong lner contnuous tme system th Wener process nose: d( t) A( t) ( t) dt + σ ( t) dξ( t), t ( ) here A( t), σ ( t) are coeffcent matrces th approprate dmensons, n ( t) R s system state vector, ξ (t) s a zero-mean Wener process th untary ncrement covarance. The system ntal state () s a random vector th mean th covarance P. Assumes there are multple sensors detectng the above lner contnuous tme system, the correspondng equatons are: dy ( t) C ( t) ( t) dt + χ ( t) dη ( t) () ( ) y here ( t) s the th sensor s measurement vector, η ( t) s also a Wener process th zero-mean, untary ncrement covarance. C ( t), χ ( t) are coeffcent matrces th approprate dmensons. (), ξ (t), η ( t) are ndependent th each other. Let Φ ( t, s) be the transton matr correspondng to A (t), T be the sample tme nterval. To dscrete the above lner contnuous tme equatons( ) and( ), e can obtan: ( k + ) (( k + ) T ) z,, L, t F( k ) ( k ) + v( k ) ( k + ) y (( k + ) T ) y ( ) ( k + ) ( k + ) + ( k + ) ( 3) ( 4)

2 In (3) and (4), the tme nde s omtted for smp lcty. here F( Φ(( k + ) T, ) v( h ζ θ ) T ( k + ) h Φ(( k + ) T, τ ) σ ( τ ) dξ( τ ) ( k + ) F ) T ( ( k + ) C ( τ ) Φ( τ, ) ( k + ) ζ ( k + ) + ( k + ) ) T θ ( k + ) T χ ( τ ) dη ) T s ( k + ) v( k ) (( k + ) T, s) dξ( s) ( τ ) ( k + ) C ( τ ) Φ( τ, s) σ ( s) In the above equatons( 3) (4), v(, ( k + ) can be veed as the dscrete process nose and the dscrete measurement nose respectvely. Accordng to the features of stochastc ntegral, one can obtan the follong results easly: E( v( ), E( ( ).The system process and measurement nose s mean, that s to say, are mantaned as before, but the correspondng covarance have been changed as: Q( cov( v(, v( ) ( k ) T + (5) Φ(( k + ) T, τ) σ ( τ ) σ ( τ ) Φ (( k + ) T, τ) j ( k ) ( ) + T [ θ + δ ( k + ) r R ( cov( j χ ( τ )( χ (, ( k + ) Q( ( j ( ) (( k + ) T, τ ))( θ ( τ )) ] + ( k + ) ( ( k + )) (( k + ) T, τ )) ( k + ) r here δ s Kronecker delta functon: j j δ j ( k + )) r ( k + ) cov(( v, ζ ( k + )) ( k ) T + (7) Φ(( k + ) T, τ) σ( τ )( θ (( k + ) T, τ)) From the above equatons (6) and( 7), e can dra a concluson: due to the effect of the common process nose, the sensors measurement noses are becomng correlated mutually after beng processed dscretely, although they are uncorrelated n the lner contnuous tme system. Ths s also concdent th the real orld. The correlated stuaton beteen the dscrete process nose and measurement nose can be obtaned by the follong dervaton: B ( k + ) cov( v(, r ( k + ) Q( ( ( k + )) ( k + )) (6) ( 8) That s to say, n a ord, after the dscretzaton of lner contnuous tme system, due to the effect of common process nose, the dscrete process nose and measurement nose are correlated mutually. From the above descrpton, e kno that, hen one merges multple sensors tracks, consderng the ndependence beteen process nose and measurement nose just only or the correlaton of tracks estmaton errors s unsuffcent. In the follong secton, e present the algorthm about track-to-track assocaton and fuson through theoretc dervaton n the case of correlaton beteen process nose and measurement noses. 3.Track-to-track assocaton and fuson For smplcty, e consder the follong fgure (fgure ). The trackng system conssts of to sensors, every one has ts on nformaton processng devce to form local target track estmaton, and then these local tracks are communcated to processng center to form a better target track estmate. target Sensor Sensor z ( ˆ P k ) z ( processor processor Assocaton &Fusson ˆ k P ) Fgure. The structure of trackng system ˆ P( Assumng the samplng tme s synchronous strctly, the ntal states of the to tracks are ˆ ( ), P ( ), ˆ ( ), P ( ) respectvely, and they are ndependent of each other. The dynamc model of trackng target s: ( k + ) F ( k ) ( k ) + v( k ) ( 9) The process nose s a mean zero hte nose th covarance Q (. The measurement equatons from to sensors are: z ( ( k ) ( k ) + (, The measurement nose ( ( ) s a hte nose th mean zero, covarance R (? The covarance of the to sensors measurement noses s: R ( cov( (, ( k )) The covarance beteen process nose and measurement noses s respectvely: B ( k ) cov( v( k ), ( k )), B ( k ) cov( v( k ), ( k ))

3 3. Track-to-track assocaton Let ˆ ( k ), ˆ ( be the estmated target tracks obtaned from the to local estmator respectvely. The state estmate errors based on the local estmator are respectvely. ~ ( ( ˆ ( () ~ ( ( ˆ ( () here ( k ) and ( are the correspondng true target states. Assumng the state estmate errors to be Gaussan, Denote the dfference of the to local track estmates as ˆ ( ˆ ( ˆ ( ( 3) From the above equaton, e kno obvously that ths s the estmate of the dfference of the true states ( ( ( ( 4) If ˆ ( and ˆ ( are the to track estmates comng from the dentcal target, the hypothess test can be denoted by: : ( ( 5) hle the dfferent target, the correspondng hypothess test s : ( k ) ( 6) In order to test the equaton (5) s rght, e can carry out the follong orks. Let ~ ( ( ˆ ( ( 7) ~ Obvously, ( k ) s mean s zero f (5) s rght, and ts covarance can be obtaned by ~ ~ T( E( ( ( ) ~ E(( ~ ( ( )( ~ ( ~ P ( k + P ( k P ( ) ) ( k ( P ( k ) ( 8) here, P s cross-covarance term, ts presentaton s ( ~ ( ) ~ P E k ( ) ( 9) The state estmate based on the local sensor s measurement nformaton, can be calculated by Kalman flterng. ˆ ) ˆ F( k ( k k ) + ( ) K ( ( z( ( ˆ ( k k )) here K (k ) s the correspondng Kalman flterng gan,. Combne () th the state estmate errors and rearrange the every terms ~ ˆ ( ( [ ( ) ( )]{ ( ) ~ k k F k ( k k ) ( ) + v( k )} K ( ( Take nto account the effects of process nose and measurement noses etc. Substtutes () nto equaton (9) ( ~ ( ) ( )( ~ P k k E k ( )) [ ( ( ]{ F( k ) P + Q( k )}[ ( ( ] + K ( R ( ( K ( ) [ ( ( ] B ( ( K ( ) K ( ( B ( )[ ( k k ) F( k ) ( ( ] ( ) Ths s a lnear recurson equaton n the ntal condton [] P ( ), assumng the ntal errors to be uncorrelated. Defne the dstance beteen the state estmate errors ~ ~ D ( T ( ( ( 3) From the prevous assumpton, e kno the dstance D s ch-square dstrbuton th freedom n ( state vector dmenson). Therefore usng hypothess test theory, the hypothess test. vs. can be solved by the follong method. Acceptng f D D P D D } α ( 4) α { α here the α s a constant value. Gven the value α, the threshold D s obtaned through the equaton( 5) easly. α n D χ ( α) ( 5) ε 3. Tracks fuson If the hypothess s accepted, then one can carry out the tracks fuson (combnaton) of the to estmates. Consder the statc lnear estmaton that yelds the posteror mean ˆ n terms of the pror mean and the measurement z ˆ( + P P z zz ( z z ) (6) When carryng out tracks fuson, ˆ ( k k ) can be veed as the trackng target pror mean, but ˆ ( k k ) be the measurement. ~ ~ { ( )( ( ) ~ E k k ( )} P z P zz P ( k ( I K + Q( k )}( I K ( I K ( ( ( ( ) B ( ( K K ( ( B ( )( I K P + P P ( ){ F( k ) P ( ) + K ( R ( ( )) ( P ( ) {( ~ ~ ))( ~ ( ) ( ( ) ~ E k k k ( ) } ) ( k k ) F ( k ) ( ( K ( 8) ( ) (7) If substtutng (7) (8) nto equaton (6), e can get the tracks fuson equatons

4 ˆ( ˆ + PzPzz ( ˆ ˆ k )) ( 9) P( P PzPzz ( Pz ) ( 3) here P,, P z P zz ( k ) can be calculated by the equatons (7) (8) and () respectvely. Compared th the lterature [] algorthm, t s dfferent obvously hen calculatng the cross covarance term. The reason s that the lterature [] doesn t take nto the effects of process nose and measurement mutually. So, n order to get better state estmate hen combnng local sensors tracks, e should better consder the effects of process nose and measurement noses. 4.Smulaton results Consder the follong constant-velocty dynamc system T T ( k+ ) ( + v( T ( 3) T here ( k ) [ poston, velocty] and the sample tme nterval T 5s. v ( s a zero mean hte nose th covarance Q.4.The ntal state ~ N(, P) s random th [, ] and covarance P dag(, ). The ntal state s ndependent of process nose. Let to sensors detect the poston of the movng target. The measurement equaton s denoted by z ( k ) ( + (, ( 3) here ( ther covarance are [ ] s random hte nose th mean zero, but R ( 8, R ( respectvely. For smplcty, the correlated cases among the process nose and measurement nose are descrbed by the correlated coeffcents. The dfferent correlated coeffcents among them are consdered by the follong. In the follong, va the Monte Carlo smulaton runs, the valdty of the fuson algorthm s proved n ve of the correlated relatonshp beteen process nose and measurement noses and compared th proposed algorthm by [] [6]. All the results s gven by the root mean square( RMS) error n poston or velocty from M runs. M e( k ) [ ~ ~ ] ( 33) M j k k here ~ k k denotes the estmate error from run j at tme k, M5. Frstly, consder the mutual coeffcents among v (, ( k ), ( are ρ v. 3, ρ 3 v., ρ. 9 respectvely n the case of the correlated noses. The results gven by fgure. and fgure.3 epress that, hen the correlated relatonshp are closer, the algorthm gven by ths paper s better than the others hch beteen process nose and measurement nose are ndependent and only the state estmate error s correlated. RMS Poston error RMS Velocty error Fgure. RMS poston error over 5 runs Fgure 3. RMS velocty error over 5 runs When the mutual coeffcents among v (, ( ), ( k ) are ρ v. 5, ρ 5 v., ρ. 5 respectvely. The smulaton results are gven by fgure 4 and fgure 5. But hen the mutual coeffcents among v (, ( k ), ( are ρ v., ρ v., ρ. respectvely, the results are gven by fgure 6 and fgure 7. From the fgure to fgure 7, e can dra a basc concept: hen the mutual relatonshp among the process nose and the measurement noses s closer, the algorthm proposed by ths paper s better than the others. But as the loer of ths relatonshp, the smulaton results of three algorthms are becomng close th each other. k

5 RMS Velocty error RMS Poston error Fgure 4. RMS poston error over 5 runs RMS Poston error Fgure 5. RMS velocty error over 5 runs Fgure 6. RMS poston error over 5 runs RMS Velocty error Concluson Fgure 7. RMS velocty error over 5 runs Ths paper proposes an algorthm of track-to-track assocaton and fuson based on the local track estmates n the case of noses correlated. Va Monte Carlo smulaton tests and compared th other algorthm, the valdty of ths algorthm s proved. References [] Y. Bar-Shalom, Xao-Rong L. Multtarget-multsensor trackng:prncples and technques [M].Storrs,CTYBS Publshng,995. [] Carl G.Looney, Yaakov L.Varol, ShengTang. Multsensor Multtarget Trackng th Central-to-Local Feedback. Proc. Internatonal Conf. On Informaton Fuson. Wed-. [3] Chee-Yee Chong, Shozo Mor. Archtectures and Algorthms for Track Assocaton and Fuson[A]. Proc.999 Internatonal Conf. On Informaton Fuson.C- 34. [4] Saha, R.K. Trac-to-Track Fuson Wth Dssmlar Sensors. IEEE Transactons on Aerospace and Electronc Systems. Vol.3, NO.3 July 996, -9 [5] A.T.Alouan, T.R.Rce and R.E.elmck. On Sensor Track Fuson. Proceedngs of the Amercan Control Conference. Baltmore, 994,4-46.

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