Quadrature Filters for Maneuvering Target Tracking

Transcription

1 Quadrature Filters for Maneuvering Target Tracking Abhinoy Kumar Singh Department of Electrical Engineering, Indian Institute of Technology Patna, Bihar 813, India, Shovan Bhaumik Department of Electrical Engineering, Indian Institute of Technology Patna, Bihar 813, India, Abstract In this paper, a maneuvering target tracking problem has been solved by using the Guss- Hermite filter (GHF) and sparse-grid Gauss-Hermite filter (SGHF) Univariate Gauss-Hermite quadrature rule is extended for multidimensional systems by using the product rule and the Smolyak s rule in GHF and SGHF respectively The SGHF, which is an alternative of GHF reduces the computational burden considerably The performance of the quadrature filters have been compared with the cubature Kalman filter (CKF), and the unscented Kalman filter (UKF) for the maneuvering target tracking problem The simulation results exhibit the improvement of performance with the quadrature filters compared to the CKF and the UKF Index Terms Maneuvering target tracking; Gauss- Hermite quadrature rule; Product rule; Smolyak s rule I INTRODUCTION Maneuvering target tracking is a process of recursive estimation of dynamic parameters of a maneuvering target This kind of tracking problem is common in several reallife problems like underwater target tracking of enemy ships [1], air traffic control [2], [3] for military and civil applications etc This problem becomes challenging due to the high nonlinearity of the systems and non-availability of the optimal solution The literature about nonlinear filtering begin with extended Kalman filter (EKF)[4], until recently which has been the natural choice of the designers, has been extensively used to solve the tracking problem of a maneuvering target The EKF uses local linearization technique to approximate the mean and the covariance of the non Gaussian probability density function Due to such crude approximation and severely nonlinear nature of the problem, the filter looses track in several times The non satisfactory results of the EKF forces the researchers to search for more advanced filters to solve the maneuvering target tracking problem Post EKF, several nonlinear filters like unscented kalman filter(ukf)[5], the cubature Kalman filter (CKF) [6], Gauss-Hermit filter(ghf)[7][8], the sparse-grid Gauss- Hermite filter (SGHF)[9] etc are developed, where the intractable integrals are approximated numerically In this paper, the quadrature filters GHF and SGHF are used to solve the maneuvering target tracking, as the [ /14/$31 c 214 IEEE] accuracy of these filters is highest among all the nonlinear filtering algorithms In these filters, the intractable integrals are approximated numerically by using Gauss-Hermite quadrature rule which is defined for the single dimension integral The GHF utilizes product rule to extend the single dimensional rule to the multidimensional rule, but its computational cost increases exponentially with increasing dimension and hence it suffers from the curse of dimensionality problem The SGHF is an extension of GHF, which utilizes the Smolyak s rule [1], [11] to extend the single dimensional quadrature rule for the multidimensional systems It reduces the computational cost considerably II Problem Formulation The maneuver of a civilian aircraft generally follows a prototype, characterized by constant velocity and constant turn rate Knowledge about the speed and the turn rate during maneuver is extremely important for air traffic control In this section, a problem of maneuvering target tracking with constant but unknown turn rate has been formulated However, to some extent the model could also be used for varying turn rate as the noise is incorporated to capture the variability The target, assumed to be maneuvering with constant turn rate, is popularly known as coordinated turn in avionics vocabulary [4] The coordinated turn model, adopted for target motion is summarized in [12] and well described in [4] In recent years, Arasaratnam etal [6] and Bin jia etal [13] have adopted this problem to compare the accuracy of their proposed algorithms with existing methods To formulate the problem, we assume an object is maneuvering with a constant turn rate in a plane parallel to the ground ie during maneuver the hight of the vehicle remains constant If the turn rate is a known constant, the process model remains linear However, constant but unknown turn rate, which needs to be estimated, forces the process model to a set of nonlinear equations The equation of motion of an object in plane (x, y) following coordinated turn model could be describe as ẍ = Ωẏ (1) ÿ = Ωẋ (2)

2 and, Ω =, (3) Where x, y represent the position in x and y direction respectively Ω is the angular rate which is a constant In navigation convention, Ω < implies a counter clockwise turn State space representation of the above equations is ẋ = Ax + w, (4) Where x is a state vector defined as x = [x ẋ y ẏ Ω] T The process noise is added to incorporate the uncertainties in the process equation, arising due to wind speed, variation in turn rate, change in velocity etc The target dynamics is discretized to obtain the discrete process equation as Where x k+1 = φ k x k + w k, (5) sin(ω k 1 T ) 1 1 cos(ω k 1T ) Ω k 1 Ω k 1 cos(ω k 1 T ) sin(ω k 1 T ) φ k = 1 cos(ω k 1 T ) sin(ω k 1 T ) 1 Ω k 1 Ω k 1 sin(ω k 1 T ) cos(ω k 1 T ) 1 In general, the nonlinear measurement equation could be written as z k = γ(x k ) (6) In this problem, we assume the range and the bearing angle both are available from the measurement So the nonlinear function γ() becomes [ ] x 2 γ(x k ) = k + yk 2 + v atan2(y k, x k ) k, (7) where atan2 is the four quadrant inverse tangent function Both w k and v k are white Gaussian noise of zero mean and Q and R covariance respectively and T is sampling time III Evaluation of multi-dimensional integral with quadrature rule and different quadrature filter The basic principle involved in the quadrature filters GHF and SGHF is Gauss-Hermite quadrature rule of integration which provides an approximate way to solve the intractable integrals encountered in the nonlinear Bayesian filtering framework Although the Gauss-Hermite rule of integration is available in literature [14][15] for more than fifty years, the same has been incorporated in the estimation very recently, mainly due to the work of Ito and Xiong [16] In these filters, the unknown probability density function (pdf) has been approximated as Gaussian using a set of Gauss-Hermite quadrature points and their respective weights But the Gauss-Hermite quadrature rule is defined for the single dimensional system This single dimensional numerical integration rule could be extended for the multidimensional system by utilizing the product rule in GHF and the same could be achieved by utilizing the Smolyak rule in SGHF The use of product rule causes the computational load to increase exponentially and hence GHF suffers from the curse of dimensionality problem The Smolyak rule could improve the computational efficacy sharply and hence could reduce the curse of dimensionality problem A single dimensional Gauss-Hermite quadrature rule Consider any integral of the form I = α α f(x)w (x)dx, (8) where x is a single dimensional variable, f(x) is a nonlinear function of x and W (x) is the weight function In this paper, the weight function is Gaussian distribution function Gauss-Hermite quadrature rule states that the integral I can be approximated numerically as I m f(q i )w i (9) i=1 where q i are the quadrature point and w i are the corresponding weights For the m-point quadrature rule, this rule is exact for polynomials having degree upto (2m 1) There are several methods available in literature for selecting univariate quadrature points and corresponding weights A commonly used method is moment matching method [7], [9], where these are evaluated by solving a set of moment equations q 1 q 2 q m q1 m 1 q2 m 1 qm m 1 w 1 w 2 w m = M M 1 M m 1 As for m number of quadrature points, we have 2m numbers of unknown including m number of each quadrature points and corresponding weights, but for the same case only m number of moments and hence equations are available So the designers suffer through the lack of equations while using this method Some authors select the quadrature points arbitrarily and calculate the corresponding weights by using moment equations[9], while some author chose the quadrature points as the zeros of the Hermite polynomial[17], which may suffer from the mathematical unstability[7] In this paper, the method used for selecting the quadrature points and their corresponding weights was first introduced by Golub et al in [18] and later utilized by Arasaratnam et al in filtering literature for the first time to develop GHF This method is described below for finding the m points quadrature rule

3 Let us consider a symmetric tridiagonal matrix J having zero diagonal elements and J i,i+1 = i/2; 1 i (m 1) The quadrature points are at q i = 2ψ i, where ψ i is the i th eigenvalue of the matrix J The i th weight, w i, is chosen as w i = κ 2 i1, where κ i1 is the first element of the i th normalized eigenvector of J [7][8] B Gauss-Hermite filter In GHF, the single dimensional quadrature points are generated by utilizing the Golub s technique discribed in previous section The single dimensional quadrature rule could be extended to the multidimensional quadrature rule by applying the product rule Let us consider a multidimensional random variable x and the weight function as the standard normal distribution, hence the integral of interest will be I N = f(x)ℵ(x,, I n )dx (1) By applying product rule the integral I N could be approximated as m I N i 1 m i n f(q i1, q i2,, q in )w i1 w i2 w in (11) To evaluation the expected value of an n dimensional integral with m-point GHF, m n number of multivariate quadrature points and corresponding weights are required For an example, for a three dimensional system and three point GHF, twenty-seven quadrature points and weights are required which may be expressed as {q i, q j, q k } and {w i w j w k } respectively for i = 1, 2, 3; j = 1, 2, 3; and k = 1, 2, 3 As the number of quadrature points increases exponentially with increasing dimension, the GHF suffers from the curse of dimensionality problem C Sparse-grid Gauss-Hermite filter In SGHF, the single dimensional quadrature rule is extended to multidimensional rule by using the Smolyak rule The Smolyak rule is introduced in mathematical literature in sixties only [1], but in filtering literature it is used very recently to derive SGHF [9] It could reduce the computational load sharply encountered with product rule 1) Smolyak rule: Any integral of the form I n,l (f) = f(x)ℵ(x;, I n ) R n can be approximated numericaaly as I n,l (f) L 1 q=l n ( 1) L 1 q C n 1 L 1 q Ξ N n q (I l1 I l2 I ln ), (12) where I n,l represents the numerical evaluation of n- dimensional integral with the accuracy level L which means that the approximation is exact for all the polynomials having degree upto (2L 1), C stands for the binomial coefficient ie Ck n = n!/k!(n k)!, I l j is the single dimensional quadrature rule with accuracy level l j Ξ ie Ξ (l 1, l 2,, l n ), stands for the tensor product and Nq n is set of possible values of l j given as n Nq n = Ξ : l j = n + q for q (13) j=1 = for q < Equation (12) can be written as I n,l (f) L 1 q=l n ( 1) L 1 q C n 1 L 1 q Ξ N n q q s1 X l1 q s2 X l2 q sn X ln f(q s1, q s2,, q sn )w s1 w s2 w sn, (14) where X lj is the set of quadrature points for the single dimensional quadrature rule I lj, [q s1, q s2,, q sn ] T is a Sparsegrid quadrature (SGQ) point ie q sj X lj and w sj is the weight associated with q sj Some SGQ points occure multiple times, that could be counted once by adding their weight The final set of the SGQ poits are X n,l = L 1 q=l n Ξ N n q (X l1 X l2 X ln ), (15) where represents union of the individual SGQ points Note: The accuracy of the SGHF increases with increasing the accuracy level L, but at the same time the computational load also increases Note: The number of elements in X lj should be higher than or equal to l j, which is chosen as (2l j 1) in this paper, similar to [9] Note: As from equation (13), the values of l j varies between 1 to L, hence single dimensional quadrature points are generated for the accuracy level of 1 to L Note: While evaluating the multidimensional quadrature points, several points appears repeatedly These points are considered once and there weights are added for every repeatation IV Simulation Result The maneuvering target tracking problem formulated in section II, has been solved in MATLAB environment by using the UKF, CKF, GHF and SGHF Experimentation has been done by considering κ = 2 for the UKF, 3-points GHF and 3 rd -degree of accuracy for the SGHF As already discussed in section-ii, the process and measurement noises are normally distributed with zero mean

4 Fig 1 Truth and estimated values of position in x-direction Fig 3 Truth and estimated values of position in y-direction Fig 2 Truth and estimated values of velocity in x-direction Fig 4 Truth and estimated values of velocity in y-direction and covariance Q and R respectively We consider g T 3 g T g T 2 gt 2 Q = g T 3 g T 2, (16) 3 2 g T 2 gt 2 9T where T is the sampling time which is taken as 5 second and g is some constant given as g = 1 R is considered as diag([σr 2 σt 2 ]) where σ r = 12m and σ t = 7mrad The initial truth value is considered as x = [1m 3m/s 1m m/s 3 /s], while the initial estimate of the covariance is P = diag([2m 2 2m 2 /s 2 2m 2 2m 2 /s 2 1mrad 2 /s 2 ]) The initial estimate is considered to be normally distributed random number with mean x and covariance P The simulation is performed for 5 seconds and the result is insured by evaluating the RMSE in terms of position, velocity and turn rate for 5 independent Monte Carlo runs The truth and estimate for different filters are shown

5 Fig 5 Truth and estimated values of turn rate in degree Fig 7 RMSE plot of velocity for 5 seconds Fig 6 RMSE plot of position for 5 seconds in Fig-1 to 5 and the RMSE plots are shown in Fig-6 to Fig-8 The RMSE plots show that the accuracy of the quadrature filters are better than the UKF and the CKF The computational time for the CKF, GHF and SGHF are noticed as 132, 1145 and 116 time higher, relative to the same for the UKF V Discussions and Conclusions The quadrature filters GHF and SGHF are applied to track a maneuvering target with coordinated turn model The turn rate of the target is assumed to be unknown and modeled with the Gaussian noise The RMSE of the position, velocity and turn rate have been evaluated using the UKF, the CKF and the quadrature filters GHF and Fig 8 RMSE plot of Turn rate in degree for 5 seconds SGHF The quadrature filters show higher accuracy than the UKF and the CKF Both the quadrature filters have similar accuracy, but SGHF filter shows relatively less computational cost and hence can be the best option for solving the maneuvering target tracking problem References [1] SK Katsikas, A K Lerosb, and D G Lainiotis, Underwater tracking of a maneuvering target using time delay measurements, Signal Processing, vol 41, no 1, pp 17-29, January 1995 [2] N Ikoma, N Ichimura, T Higuchi, and H Maeda, Maneuvering Target Tracking by Using Particle Filter, Proceedings, IFSA World Congress and 2th NAFIPS International Conference, vol 4, pp , July 21

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