adar/ racing of Constant Veocity arget : Comparison of Batch (LE) and EKF Performance I. Leibowicz homson-csf Deteis/IISA La cef de Saint-Pierre 1 Bd Jean ouin 7885 Eancourt Cede France Isabee.Leibowicz @ deteis.thomson-csf.com P. Nicoas homson-csf Deteis La cef de Saint-Pierre 1 Bd Jean ouin 7885 Eancourt Cede France Phiippe.Nicoas @ deteis.thomson-csf.com L. atton homson-csf Deteis La cef de Saint-Pierre 1 Bd Jean ouin 7885 Eancourt Cede France Laurent.ratton @ deteis.thomson-csf.com Abstract - In this paper we compare performance of batch (aimum Lieihood Estimate LE) and iterative (Etended Kaman Fiter) techniques in the case of joint adar/ tracing. wo issues are addressed. racing accuracy of both methods are evauated by simuations and compared to the Cramer-ao Lower Bound (CLB). he adar scheduing is optimized in order to increase tracing accuracy. Keywords : data fusion, batch method, LE, Kaman fiter, mutisensor tracing, sensor scheduing, Cramer- ao Lower Bound. 1 Introduction his paper deas with the surface target tracing issue using adar and sensors. Athough mutipe sensors fusion may be usefu for tracing accuracy improvement, our primariy motivation is reated to tracing under discretion constraint. As a matter of fact, it is often required to imit adar activity to avoid aerting the enemy to one s presence and position. wo issues have been investigated : How does tracing system perform during period when adar measurements are not avaiabe? How shoud adar activity be managed so that tracing accuracy is optimized? hese two points are addressed in the paper in the foowing way : We compare an iterative mutisensor adar/ tracing method ie Kaman Fiter to the LE batch method with a sma adar measurement sampe size. Effects of measurement error eve are considered. We propose a radar measurement scheduing in order to improve tracing accuracy. In this paper the probem of measurement association is not discussed since it is assumed that measurements and adar measurements are aready associated. he remainder of this paper is organized as foows. Section deas with the mode of measurement data provided by both sensors. he estimation techniques are described in section 3. hen the performance of the methods is derived in section 4. In this section eperimenta resuts derived from simuations are presented to iustrate the performance. he section 5 deas with the adar scheduing optimization probem. Data mode In this section the notations used in the remainder of this paper are defined. A singe target is assumed to maintain a constant veocity vector a aong the observation period. Observer and target state vectors are defined in Cartesian coordinates in a erence space whose origin, fied on earth surface is arbitrary. In the remainder of the paper, the atitude of the observer is assumed to be constant a aong the observation period. he observer state is : o ) [ o ), y o ), v,o ), v y,o )] X =. (.1) oreover the target is assumed to be a surface target so the atitude of the target is nu. he target state is : c ) [ c ), y c ), v,c, v y,c ] X =. (.) he reative target state vector X with respect to the observer is the system state vector defined by :
X ) Xc ) Xo ) = [ ),y ),v ),vy )] = (.3) he system state vector is unnown and deterministic..1 State equation he state equation of the system is inear : where Φ ( t, t ) ( t, t ). X ) + v ) X + 1) = Φ + 1 (.4) ( t t ) I I 1 = I = (.5) I 1 and v( t ) is a sequence of zero-mean, white, Gaussian process noise. In the remainder of the paper v( t ) is assumed to be nu.. easurement equations he measurement equation is noninear : with where [ X )] + w ) Z ) = (.6) Z ) = Θ ) + w ) ) = ) ) + z ), ), (.7) -1 ) Θ ) = tan y ), (.8) and ˆ -1 ) Z ) = Θ ) = tan + w ). y) (.9) w ) and w ) are respectivey the additive adar and measurement errors supposed to be zeromean white noises and independent with respective variance σ ), σ ) Θ and σ ) Θ. Let Θ ˆ = [ Θˆ ˆ 1),..., Θ )] be the set of the bearings provided by the adar during the observation period. Let ˆ = [ˆ 1),..., ˆ )] be the set of the ranges provided by the adar during the observation period. Let Θ ˆ = [ Θˆ ˆ 1),..., Θ N )] be the set of N bearings provided by the during the observation period. Let Θ ˆ = ( Θ ˆ, ˆ Θ, ˆ ) be the joint measurement vector. he error covariance matrices Σ Θ, Σ Θ and Σ associated respectivey to bearings, adar bearings and adar ranges are defined by: Θ = diag[ )] Θ = 1,...,N Σ σ, (.1) Θ = diag[ )] Θ = 1,..., Σ σ, (.11) = diag[ = 1,..., Σ σ )]. (.1) 3 Estimation techniques 3.1 Batch method LE he aimum Lieihood Estimation method (LE)[1] estimates the reative target state vector. he aim is to find the state vector components which maimize the ieihood function. In the case of joint adar/ measurements, the estimation probem is observabe as soon as the first adar measurement is received under constant veocity vector hypothesis. Observations provided by both sensors are assumed independent and the ieihood function defined as P( Θ ˆ / X) is equa to the product of the respective ieihood functions : = P( Θ ˆ / X) P( Θ ˆ / X) P(ˆ / X). (3.1) Under the Gaussian hypothesis, the function og can be written : og = [ Θˆ + [ Θˆ + [ˆ Σ Σ Σ Θ Θ [ Θˆ [ Θˆ [ˆ (3.) he LE is the soution to the ieihood equation : ( og ) =. (3.3) Θ he partia derivatives and X = Xˆ and equa to : Θ X= Xˆ X= Xˆ = = h (Xˆ )A( Θ(Xˆ (Xˆ ), t )B((Xˆ ), t are evauated at ), (3.4) ), (3.5) where t is the erence time of the state estimate. he observation matri A( Θ (Xˆ ),t ) for the bearing is defined
by : A( Xˆ ), t ) Xˆ ) cos cos cos = - sin t, Xˆ ) - sin tn, Xˆ ) - sin t( t1) cos t, Xˆ )) t( t ) cos tn, Xˆ )) t( tn ) cos Xˆ )) Xˆ ) - t( t1) sin t, Xˆ ) - t( t ) sin tn, Xˆ ) - t( tn ) sin Xˆ ) t, Xˆ ) t, Xˆ ) N (3.6) where t ) is the time deay between the measurement time t and the erence time. In the foowing, the matrices A( Θ (Xˆ ), t ) computed respectivey with bearings and adar bearings are denoted by A ( Θ (Xˆ ), t ) and A ( Θ (Xˆ ), t ). he diagona matri h (Xˆ ) is the reative horizonta range from target to the observer computed at estimated state vector. he observation matri B((Xˆ ), t ) for the adar range is defined by : B((Xˆ ), t ) 1) y1) t 1 ) 1 ) t 1 )y 1 ) = ) y ) t ) ) t )y ). (3.7) ) y ) t ) ) t )y ) he soution of equation (3.3) is cassicay obtained with an iterative agorithm ie Gauss-Newton defined by : X +1 = X s d where d is defined by : d = [ ˆ h ΣΘ ˆ h  +  ˆ Σ ˆ h Â Θ + Bˆ ˆ Σ ˆ Bˆ ].[ ˆ ( ˆ h ΣΘ Θ (Xˆ )) + Bˆ ˆ Σ +  ˆ h ΣΘ (ˆ (Xˆ ))]. ( Θˆ (Xˆ )) he matrices  = A ( Θ(Xˆ ), t ), (3.8) (3.9)  = A( Θ(Xˆ ),t ), Bˆ = B((Xˆ ), t ), ˆ = (Xˆ ) and ˆ h = h (Xˆ ) are evauated at the estimated state vector Xˆ = X computed at the th iteration. he step size s is seected at each iteration to ensure convergence. 3. ecursive methods - EKF he state propagation equation, defined by (.4) is inear whie the observation equation, defined by (.6), is not. Standard Etended Kaman Fiter equations [3] have been used. Nevertheess in the case of joint adar/ tracing, the Jacobian H of, computed at the prediction step, taes either of the two foowing epressions : In case of adar measurement, H = - y + z y In case of measurement, H = - y + z, (3.1), (3.11) with, y standing for the predicted reative coordinates of the target and z the observer atitude at the prediction time. 4 Performance comparison A onte Caro simuation eperiment was conducted to demonstrate and to compare the performance of the two mutisensor tracing agorithms to the CLB. 4.1 Derivation of the Cramer-ao ower bound (CLB) he Cramer-ao bound is a ower bound for the mean square error corresponding to an unbiased estimator []. In case of adar/ fusion the Fisher Information atri (FI), evauated at erence time and computed at the true state, is defined by : FI ) = [A + B h ΣΘ Σ h B]. A + A h ΣΘ h A (4.1) he CLB for any unbiased estimator of the target state [ ), y ), v, v is : c c,c y,c ] CLB 4. Simuation resuts ) [FI )] 4..1 Description of the scenario =. (4.) Estimation performance is evauated in a singe target environment. he observation duration is 1 s. arget and observer trajectories have constant veocity vectors. he observer speed is 1 m/s. he observer heading defined with respect to abscissa ais is 6 degrees. he initia target position in Cartesian coordinates is (4 m;
4 m). he target heading defined with respect to abscissa ais is 45 degrees. he target speed is 1 m/s. Performance is evauated in the foowing situation : measurements are avaiabe a aong the observation period. he adar transmits during the interva [ ; 3 s] (31 measurements) with a constant measurement accuracy. For both sensors the measurement rate is 1 s. 4.. Description of the simuations Using the scenario described above, the target state vector is estimated, at erence time t, by : - the LE method, - the Cartesian EKF. he figures 1 and represent the eperimenta mean square error of estimated bearing and range and the corresponding CLB as a function of the standard deviation of bearing measurement error. easurements were generated syntheticay and the measurement noise was additive, zero-mean, independent and Gaussian. adians eters,8,7,6,5,4,3,,1 Comparison S error/clb Bearing erence time = 5s 1 3 4 5 6 7 8 9 1 11 CLB S LE S Kaman Comparison S error/clb ange erence time = 5s 1 1 8 6 CLB 4 S LE S Kaman 1 3 4 5 6 7 8 9 1 11 Figure 1 : Comparison of estimated Bearings and ange S error ( erence time = 5 s, 31 adar measurements) eters Figure : Comparison of estimated Bearings and ange S error ( erence time = 1 s, 31 adar measurements) 4.3 Discussion At ow vaues of the measurement error ( t ) σ Θ ( ess than 1 degree), the performance of both methods is neary identica and approimates the CLB. As σ Θ adians,8,6,4, Comparison S error/clb Bearing erence time =1s 1 3 4 5 6 7 8 9 1 11 CLB S LE S Kaman Comparison S error/clb ange erence time =1s 35 3 5 15 CLB 1 S LE 5 S Kaman 1 3 4 5 6 7 8 9 1 11 ( t ) increases, the abiity to estimate the target state through the measurements decreases and the EKF departs appreciaby from the LE which remains near the CLB. hose resuts give an indication of the method which shoud be used according to the type of sensor: if measurements are provided by an phase direction finder, the EKF can be used whie if the measurements are provided by an ampitude direction finder the LE is the most adequate method. he poor performance of the EKF in case of high measurement error eve is due to the noninearity of the observation equation : a poor estimation of the target state, due to arge measurement error, eads to significant errors when the observation equation is inearized. Nevertheess this drawbac of the EKF can be aeviated by increasing the adar measurement sampe size : the EKF then approimates the CLB. In case of sma adar measurement sampe size due to discretion constraint, a way to improve the performance accuracy is to optimize the adar scheduing.
5 adar scheduing Performance obviousy depends on adar activity. A way to improve the tracing performance is to increase adar activity. In fact, it is often necessary to restrict active transmission in order to avoid aerting the enemy to one s presence and position. he probem is then to find the best way to use the adar. Assume that ony adar observations are avaiabe during the observation period. he objective is to find the adar measurement times {,..., τ i,..., τ } which, for an arbitrary erence time t, maimize the determinant of the Fisher information matri (FI) : { τ,..., τ } arg ma (det(fi ))) { },..., i = (5.1) τ,..., τ,..., 1 i τ 5.1 Approimation of the FI determinant Let consider radar measurements and N measurements. Using the computation method described in the erence [4], the determinant of the FI, can be approimated by the foowing epression : ( FI) = det (FI( τ, τ,)) + det 3 det (FI( τ,)). τ < τ < τ < τ < τ = τ τ (5.) det (FI( τ, )) and det 3 (FI( τ, τ, ) are respectivey defined by : det and det (FI( τ, )) τ = 4 σ 4 σ 4 tpt t ) [1 + θ ( tt + τ + t )) ] 1, 3 (FI( τ, τ, ) = [ τ τ ( τ τ ) θ ] 6 σ σ (5.3) (5.4) where : - θ is the bearing rate, - is the distance between the observer and the target at the erence time t, - det (FI( τ, )) is the contribution of term computed using gradient vectors of two radar measurements and two measurements, - det 3 (FI( τ, τ,) is the contribution of term computed using gradient vectors of three radar measurements and one measurement. Note that the determinant of the FI is independent of the erence time [1]. So the above approimations have been computed for t =. emar : the notation t is used for measurement time and τ is used for adar measurement time. 5. Optima adar scheduing he foowing tabe indicates the optima radar scheduing according to the number of adar measurements (measurement rate is assumed to be 1 s). Number of adar measurements Optima scheduing (seconds) = ; τ. 3 = ; τ or = ; τ =Τ-1 ; τ 3 4 = ; τ -1 ; τ 4 5 = ; τ = ; τ 4-1; τ 5 or = ; τ - ; τ 4-1 ; τ 5 abe 1 : Optima adar scheduing he resuts presented above are obtained by soving equation (5.1) with the approimation of the det(fi) given by equation (5.). hose resuts are independent of the geometry of the scenario since the target is assumed to maintain a constant veocity vector. 6 Concusions In this paper, joint adar/ tracing techniques (LE and EKF) for a singe constant veocity target have been considered and their performance has been evauated and compared to the CLB. he infuence of the measurement error eve with a sma adar measurement sampe size have been studied and it has been shown that batch method is more robust to arge measurement error. Such method is recommended when rea-time estimation is not required (for eampe in maritime
surveiance mission) and when targets are non maneuvering. A radar scheduing optimization method has aso been presented which increases tracing accuracy. Further investigations incude : he infuence of coordinates system, he etension of the joint radar/ tracing method and adar measurement scheduing optimization in the presence of maneuvering target. 7 eferences [1] S.C. Nardone, A.G. Lindgren, K.F.Gong, Fundamenta Properties and Performance of conventiona Bearing-ony arget otion Anaysis, IEEE ransaction on automatic contro, Vo AC-9, No 9, September 1984. [] H. L. Van rees, Detection, Estimation and oduation heory, Part 1, Wiey, 1998. [3] S. Bacman.Popoi, Design and Anaysis of odern racing Systems, Artech House 1999 [4] J.P. Le Cadre, Properties of Estimabiity Criteria for A, IEE Proc. adar, Sonar, Navig. Vo 145, no 3, Apri 99