comcn 2 czhanxjtueducn 3 linzhengmailstxjtueducn Xian Jiaotong University Xian 710049China Abstract This paper develops a tracking algorithm for maneuvering target based on Fuzzy logic inference FMMTA In replace of the model probability computed in ID: 25561 Download Pdf

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comcn 2 czhanxjtueducn 3 linzhengmailstxjtueducn Xian Jiaotong University Xian 710049China Abstract This paper develops a tracking algorithm for maneuvering target based on Fuzzy logic inference FMMTA In replace of the model probability computed in

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Fuzzy Multiple Model Tracking Algorithm for Manoeuvring Target Dongguang Zuo Chongzhao Han ZhengLin 3 Hongyan Zhu Hanhong 1 zlz_123@sina.com.cn 2 czhan@xjtu.edu.cn 3 linzheng@mailst.xjtu.edu.cn (Xi’an Jiaotong University, Xi’an , 710049,China) Abstract - This paper develops a tracking algorithm for maneuvering target based on Fuzzy logic inference (FMMTA). In replace of the model probability computed intricately in the IMM, the filtering measurement innovations are tackled with the innovation covariance, and the results are used as the inputs of fuzzy inference system to get

the matched degrees for each filtering model in the model set designed. And then with the matched degrees, the estimation from each filtering is weighted to get maneuvering target’s overall estimation and its covariance. The performance of FMMTA is tested via Monte Carlo simulation, and the result expresses its validity and is promising. Keywords: maneuvering target tracking fuzzy inference multiple model 1 Introduction Fuzzy logic inference is a key tool to deal with uncertainty problems, and is called “soft computing techniques” altogether with neural network and genetic algorithm, and also

is applied in many fields ranging from consumer products such as cameras, camcorders, washing machines, and microwave ovens to industrial process control, medical instrumentation, decision-support systems, and portfolio selection, etc.. The fundamental of fuzzy logic is the fuzzy set theory presented by Lotfi. Zadek in 1960s. Fuzzy logic is used to imitate thinking processes of human being to analysis rules. This paper develops the application of fuzzy logic in tracking manoeuvring tracking. Target tracking is a hybrid estimation problem involving both continuous and discrete uncertainties

[11] . The prevailing approaches to manoeuvring target tracking are the modeling of the target motion/daynamics to cover the unknown modeling errors or deviations of the model from the true system. Because of the target motion uncertainty, where a target may undergo a known or unknown maneuver during an unknown time period, when tracking a maneuvering target, the essential problem The research is sponsored by national key fundamental research & development programs(973) of P.R. China. No.2001CB309404. is to determine reliably and timely the right model to use. The so-called multiple-model (MM)

method is a major approach, which is also probably the most natural approach to solve the hybrid estimation [12]. The GPB1, GPB2 and IMM are all valued researches in the process to study MM to tracking maneuvering target. The basic idea of the multiple-model estimation approach is to assume a set of models for the hybrid system; run a bank of filters, each based on a unique model in the set; and the overall estimate is given by a certain combination of the estimates from these filters. Their fundamental are all based on Bayes estimation theory [3-5,7-12] . This paper presents an algorithm for

a maneuvering target tracking on the bases of fuzzy logic inference. It is organized as follows. Section II provides the models for maneuvering target and the IMM algorithm is briefly summarized. FMMTA is designed in section III detailedly. The performance of FMMTA is given in section IV via comparing with IMM. Finally, a conclusion is presented in section V. 2 The model for maneuvering target The maneuver of target motion can be described by such a method, where the target uses different motion mode in different period. Therefore the discrete dynamic model of maneuvering target may be denoted

as a class of stochastic hybrid system with additive noise: )1()()1()()( kwsGkxsFkx skk kk (1) )()()()( kk kk svkxsHkz (2) where Rkx ( is the base state vector of moving target at time Rkz )( is the vector-valued noisy meameasurements at time . is the scalar-valued modal state (system mode index) at time , which denotes the mode in effect during the sampling period ending at , i.e., the time period ],( kk tt . )( and )( are the process and measurement noise sequences, respectively. It is assumed that the process and measurement noise are independent with each other, and are white with mean kk

ss vw , the covariances being 818

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respectively. It is obviouse that the plant is a nolinear system, where )( kx and )( kz depend on the uncertain mode of moving target. Were the mode determained, the system could be simplified as a linear plant, and then the tracking problem is convenient to deal with. 2.1 The IMM algorithm Assuming that there are sub-filters or sub-models used for interaction [6]. The transition probability between models are governed by a homogeneous Markov chain: MsspksksP jiij , ,)}(|)1({ Then the interacting multiple model algorithm can be briefly

outlined by the following steps: Interaction of the estimation: The mixed initial state estimate and its error covariance for the sub-filter model at time is calculated using outputs of the models )1|1( kkx )1|1( kkP at time , the corresponding model probability ) and the transition probability ij respectively: ji kk kkxkkx )1|1()1|1( )1|1( })]'1|1( )1|1( [ )]1|1( )1|1( [ )1|1({)1|1()1|1( x kkxkkx kkxkkx kkPkk kkP ji where iijj iij ji kpc kp ZkmkmPkk )1( )1( }),(|)1({)1|1( Filtering: The updates for each sub-filter are performed using the Kalman filter equations: )'1()1()1( )'1()1|1()1()1|(

)1()1|1( )1()1|( kGkQkG kFkkPkFkkP wkGkkxkFkkx jj jj RHkkPHS vkkxHkzkz )1|( )1|1( )()( 1' )()( jj SHPK jj zKkkxkkx )1|( )|( )()1|()|( jjj KSKkkPkkP )( ))(( |2|],0);( kzSkz jj eSSkzN / nnii iijj kp kp 11 )1( )1( )( Combination of overall estimations: )()|( )|( kkkxkkx jj jj kkxkkxkkxkkxk kkPk kkP )]'|( )|( )][|( )|( )[( )|()()|( From the above , we know in the IMM algorithm the final outputs for tracking filtering are calculated according to the weighted for the outputs of each subfiltering. 3 The design for FMMTA algorithm The hypothesis conditions for FMMAT algorithm are similar to the IMM

algorithm, except for the transition probability between models. Uses the equation (1) and (2) as the designed filtering model for tracking maneuvering target, then the jth sub-model in the designed model set is re-described as the following: )1()1()1()1()( kwkGkxkFkx jj (3) )()()()( kvkxkHkz jjj (4) where : Mj . Assuming that the overall estimation for maneuvering targer at time are )1|1( kkx for state vertor and ) 1|1( kkP for the error covariance. 3.1 Model filtering According to the measurement input )( kz from sensor for maneuvering target, the jth sub-model filtering outputs is

calculated as following: wGkkxF ZmxEx 11 )1|1( ],|[ (5) 111 )( ))(1|1( GQG FkkPFP (6) vxHzz (7) RHPHS )( (8) 1' )()( jj SHPK (9) jjj kk zKxx (10) )( jjjj kk KSKPP (11) 819

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3.2 The calculation of crisp inputs for fuzzy inference system According to results of the above sub-model filtering, the state estimation and covariance for tracking target to model is taken, and also the measurement innovation )( kz and its error covariance )( kS to the model can be computed. Therefore the crisp input to fuzzy inference system are denoted by the following: MjzSzE jjj )()' (12) From the

stochastic theory, we know that the distribution of is , which will be available when designing the member functions later. 3.3 The matched degree for model Assuming that the universe discourse for the crisp input is , is the fuzzy value , and assuming that the different sub-filtering model has the same universe discourse: jiMjiAA ji ,, Defines the similar fuzzy subsets to the crisp input space : small(S), medium(M), big(B). Selectes the Gauss function as the fuzzy membership function of : )( exp()( cx xu where and are co-efficients, which need to be designed. Asumming that the output space of

fuzzy inference system is ]1,0[ , and defines the fuzzy subsets as : small(S), medium(M), big(B), similar to the above, but the membership function is triangular forms. Then according to the characteristics of fuzzy inference system, the fuzzy linguistic rules are as following : and and then and and :2 rule and and then and and :1 rule SEBEBEif SE MEBEif From these rules, the fuzzy matched degree ]1,0[ to the model is the jth output of the fuzzy inference system. In order to be convenient to get the overall estimation of tracking target, the fuzzy matched degree needes to be normalized as

following: , (13) where is matched degree of model to the moving model of target. 3.4 Estimation combination According to the results of condition filtering of model and the matched degree , the overall estimation of maneuvering target is calculated as following: )( )|( kxkkx kk (14) kk kk kk kkxxkkxxk Pk kkP )]'|( )][|( )[( )()|( (15) The steps of FMMTA, when tracking maneuvering target, is redescribed as following: Step 1: designes a model set, which are approached to the target motion modes; Step 2 : according to the equation (5)~(11), calculates the state estimation kk , the error

covariance kk to the model , and the measurement innovation )( kz , the innovation covariance )( kS ; Step 3 : calculates the crisp input of fuzzy inference system according to the equation (12) ; Step 4 : normalizes the fuzzy matched degree to get the model matched degree of model according to the equatio n (13); Step 5 : calculates the overall estimation )|( kkx and error covariance )|( kkP according to the equation (14), (15) ; Step 6: uses ) |( kkx ,) |( kkP as the initial input of sub-filtering at time 1 , and repeat step 2 to 5 and to the end of tracking . The structure of FMMTA is shown

in figure 1. PHDVXUHPHQW ILOWHULQJ ILOWHULQJ ILOWHULQJ0 IX]]\ LQIHUHQFH V\VWHP QRUPDOL]DWLRQ FRPELQDWLRQ )( kz kk kk )|( kkx )|( kkP figure 1. structure of FMMAT 820

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4 The simulation of FMMTA The performance of FMMTA is tested with the problem of tracking an aircraft moving in the horizontal plane, described by a coordinated turn model. It was also shown in the references [1] and [2]. 4.1 The design of simulation scenario The coordinated turn model of aircraft is described as the following: )( )1( 00 )1( 10000 0cos0sin0 sin cos1 0sin0cos0 cos1 sin )( ku kw kx kx (16)

Where is the turn rate of the aircraft during the turn, adn are process noise and the control input of the turn rate. The constant velocity motion of aircraft is denoted as: )1( )1( 1000 100 0010 001 )( kw kx kx (17) Where is the sampling time. Augmenting the state vector (17) by one more component —the turn rate, then the uniform motion model has the same state vector as the coordinated turn model: ]' [)( Z[[[[ yyxx kx The position measurement sensor is assumed to be located at the origin of the Cartesian coordinates, then the measurement equation with noise is decribed as following : )()(

00100 00001 )( kvkx kz (18) The simulation scenarios of maneuvering target is designed as following: A nonmaneuvering flight during scan 1 to 60 with speed sm /300 ; a 180 turn, lasting from scan 61 to 105 with turn rate /74.3 acceleration) ; a nonmanerveri ng flight during scan 106 to 150 and then a 180 turn with turn rate /74.3 acceleration), lasting from scan 151 to 195. Finally a nonmaneuvering flight to the end of scan 220. The target ini tial state in Cartesian coordinates was : ]'0/24630/17230[)0( smkmsmkm The process noise of true system is set to zero. The true trajectory is shown in

figure 2. 15 20 25 30 35 40 45 10 15 20 25 30 x (km) y (km) t=0 figure 2. the true trajectory of maneuvering target In order to tracking the maneuvering target, there are three sub-filtering model designed in the model set: ]/4,/3,/0[ Assuming that subscript “CV” and “CT” stand for “constant velocity” and “coord inated turn” respectively. The process noise covarian ce for tracking sub-filtering is designed as: QI CT CV )001.0(,)004.0( Where is a 22 identity matrix. The standard deviation of the Gaussian measurement errors, both for the and axes, is set as 100 with zero mean. In order to

compare with the IMM for tracking the same maneuvering target in the same conditions, except for the transition probability matric of Markov chain. 70.025.005.0 25.070.005.0 025.0025.095.0 ][ ij 4.2 Results and discussions The performance of FMMTA is tested by 100 times Monte Carlo simulation. The results are shown from figure 3. to figure 6. where the results is the root of mean square error (RMS), which can be calculated as the following : kxkxkxkx kerrorRMS ))'( )())(( )(( )(_ Where )( kx , )( kx stand for the true state of maneuvering target and its overall estimation in the running th at

time . is the times of Monte Carlo running. Simultaniously, the results for IMM at the same conditions are also shown in these figures. 821

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With the comparison between FMMTA and IMM from figure 3 to figure 6, we know that FMMTA has almost the same performance as or better than that of IMM if the fuzzy inference system is designed well when tracking maneuvering target. But the FMMTA needs a lot of experiences of experts to build fuzzy linguistic rules because it is the key factors in fuzzy inference system design, which can effect heavily on the the performance of FMMTA when

tracking maneuvering target. 5 Conclusions This paper presents a tracking algorithm for manoeuvring target based on Fuzzy logic inference(FMMTA). In replace of the model probability being computed intricately in the IMM, the filtering measurement innovations are tackled with the innovation error covariance to get the scale-valued, and the results are used as the inputs of fuzzy inference system to take out the matched degrees for each subfiltering model in the model set designed previously. And then with the matched degrees, the estimation from each subfiltering is weighted to get manoeuvring

target’s overall estimation and its covariance. With the Monte Carlo simulation, The performance of FMMTA, when tracking maneuvering target, is tested, and the result is promising. 50 100 150 200 250 40 60 80 100 120 140 Simulation times RMS-err(x)(m) FMMTA IMM figure 3. position error of (RMS) 50 100 150 200 250 50 100 150 200 250 300 Simulation times RMS-err(v )(m/s) FMMTA IMM figure 4. velocity error of (RMS) 50 100 150 200 250 40 60 80 100 120 140 Simulation times RMS-err(y)(m) FMMTA IMM figure 5. position error of (RMS) 50 100 150 200 250 50 100 150 200 250 300 350 400 Simulation times

RMS-err(v )(m/s) FMMTA IMM figure 6. velocity error of (RMS) References [1] X.Rong Li. Design of an Interacting Multiple Model Algorithm for Air Traffic Control Tracking. IEEE Transactions on Control Systems Technology, Vol.1, NO.3, pp.186-194, September 1993. [2] V.P.Jilkov, D.S.Angelova, TZ.A.Semerdjiev. Design and Comparison of Mode-Set Adaptive IMM Algorithm for Maneuvering Target Tracking. IEEE Transactions on Aerospace and Electronic Systems., Vol.35, NO.1, pp.343-350, January 1999. [3] Horng-Jyh Lin, D.P. Atherton. An Investigation of the SFIMM Algorithm for Tracking Manoeuvring

Targets. Proceedings of the 32 nd Conference on Decision and Control, San Antonlo, Texas . December 1993. pp.930- 935. [4] A.Munir, D.P. Atherton. Adaptive Interacting Multiple Model Algorithm for Tracking a Manoeuvring Target. IEE Proc. Radar,So nar Navig., Vol.142, NO.1, pp.11-17, February 1995. 822

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[5] A.Averbuch, S.Itzikowitz, T.Kapon, Radar Target Tracking—Viterbi versus IMM. IEEE Transactions on Aerospace and Electronic Systems., Vol.27, NO.3, pp.550-563, May 1999. [6] H.A.P.Blom, Y. Bar-shalom. The Interacting Multiple Mode Algorithm for System with Markovian Switching

Coefficients. IEEE Transactions on Control Systems Technology, AC-30, pp.780-783, 1988. [7] X.Rong Li, Chen He. Model-set Choice for Multiple Model Estimation.14 th Triennial World Congress, Beijing, P.R.China. pp.169-174. [8] Bar-Shalom, Y.Chang, K.C., and Blom,H.A.P.. Tracking a manoeuvring target using input estimation versus the interacting multiple model algorithm. IEEE Transactions on Aerospace and Electronic Systems., AES- 25,(2), pp.296-300, 1989. [9] James P.Helferty Improved Tracking of Maneuvering Targets (turn-rate). IEEE Transactions on Aerospace and Electronic Systems., Vol.32,

NO.4, pp.550-563, October 1996. [10] Y.Bar-Shalom, Xiao-Rong Li. Multitarget multi- sensor tracking: principles and techniques [M]. Storrs, CTYBS Publishing, 1995. [11] Li,X.R.. “Hybrid Estimation Techniques” in Control and Dynamic Systems: Advances in Theory and Applications. Vol.76 (C.T.L eondes, ed.), pp.213-287,New York: Academic Press,1996. [12] Y. Bar-shalom, W. D. Blar. Multitarget-multisensor Tracking Application and Advances. VolIII .Artech House, 2000.10. 823

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