quadrature filters for maneuvering target tracking filters for maneuvering target tracking ... and...

Quadrature Filters for Maneuvering Target TrackingAbhinoy Kumar Singh

Department of Electrical Engineering,Indian Institute of Technology Patna,

Bihar 800013, India,Email: [email protected]

Shovan BhaumikDepartment of Electrical Engineering,Indian Institute of Technology Patna,

Bihar 800013, India,Email: [email protected]

Abstract—In this paper, a maneuvering target track-ing problem has been solved by using the Guss-Hermite filter (GHF) and sparse-grid Gauss-Hermitefilter (SGHF). Univariate Gauss-Hermite quadraturerule is extended for multidimensional systems by usingthe product rule and the Smolyak’s rule in GHF andSGHF respectively. The SGHF, which is an alternativeof GHF reduces the computational burden considerably.The performance of the quadrature filters have beencompared with the cubature Kalman filter (CKF), andthe unscented Kalman filter (UKF) for the maneuveringtarget tracking problem. The simulation results exhibitthe improvement of performance with the quadraturefilters 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 recursiveestimation of dynamic parameters of a maneuvering target.This kind of tracking problem is common in several real-life problems like underwater target tracking of enemyships [1], air traffic control [2], [3] for military and civilapplications etc. This problem becomes challenging due tothe high nonlinearity of the systems and non-availability ofthe optimal solution.

The literature about nonlinear filtering begin with ex-tended Kalman filter (EKF)[4], until recently which hasbeen the natural choice of the designers, has been exten-sively used to solve the tracking problem of a maneuveringtarget. The EKF uses local linearization technique to ap-proximate the mean and the covariance of the non Gaussianprobability density function. Due to such crude approxi-mation and severely nonlinear nature of the problem, thefilter looses track in several times. The non satisfactoryresults of the EKF forces the researchers to search for moreadvanced filters to solve the maneuvering target trackingproblem. Post EKF, several nonlinear filters like unscentedkalman 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 theintractable integrals are approximated numerically.

In this paper, the quadrature filters GHF and SGHFare used to solve the maneuvering target tracking, as the

accuracy of these filters is highest among all the nonlinearfiltering algorithms. In these filters, the intractable integralsare approximated numerically by using Gauss-Hermitequadrature rule which is defined for the single dimensionintegral. The GHF utilizes product rule to extend thesingle dimensional rule to the multidimensional rule, but itscomputational cost increases exponentially with increasingdimension and hence it suffers from the curse of dimen-sionality problem. The SGHF is an extension of GHF,which utilizes the Smolyak’s rule [10], [11] to extend thesingle dimensional quadrature rule for the multidimensionalsystems. It reduces the computational cost considerably.

II. Problem Formulation

The maneuver of a civilian aircraft generally follows aprototype, characterized by constant velocity and constantturn rate. Knowledge about the speed and the turn rateduring maneuver is extremely important for air trafficcontrol. In this section, a problem of maneuvering targettracking with constant but unknown turn rate has beenformulated. However, to some extent the model could alsobe used for varying turn rate as the noise is incorporated tocapture the variability. The target, assumed to be maneu-vering with constant turn rate, is popularly known as co-ordinated turn in avionics vocabulary [4]. The coordinatedturn model, adopted for target motion is summarized in[12] and well described in [4]. In recent years, Arasaratnamet.al. [6] and Bin jia et.al. [13] have adopted this problemto compare the accuracy of their proposed algorithms withexisting methods.

To formulate the problem, we assume an object is ma-neuvering with a constant turn rate in a plane parallel tothe ground i.e. during maneuver the hight of the vehicleremains constant. If the turn rate is a known constant,the process model remains linear. However, constant butunknown turn rate, which needs to be estimated, forces theprocess model to a set of nonlinear equations. The equationof motion of an object in plane (x, y) following coordinatedturn model could be describe as

x = −Ωy (1)

y = Ωx (2)[978-1-4799-4040-0/14/$31.00 c© 2014 IEEE]

IEEE International Conference on Recent Advances and Innovations in Engineering (ICRAIE-2014), May 09-11, 2014, Jaipur, India

and, Ω = 0, (3)

Where x, y represent the position in x and y directionrespectively. Ω is the angular rate which is a constant. Innavigation convention, Ω < 0 implies a counter clockwiseturn. State space representation of the above equations is

x = Ax + w, (4)

Where x is a state vector defined as x = [x x y y Ω]T . Theprocess noise is added to incorporate the uncertainties inthe process equation, arising due to wind speed, variationin turn rate, change in velocity etc.

The target dynamics is discretized to obtain the discreteprocess equation as

xk+1 = φkxk + wk, (5)

Where

φk =

1 sin(Ωk−1T )

Ωk−10 −1 − cos(Ωk−1T )

Ωk−10

0 cos(Ωk−1T ) 0 −sin(Ωk−1T ) 0

0 1 − cos(Ωk−1T )Ωk−1

1 sin(Ωk−1T )Ωk−1

00 sin(Ωk−1T ) 0 cos(Ωk−1T ) 00 0 0 0 1

.

In general, the nonlinear measurement equation could bewritten as

zk = γ(xk). (6)

In this problem, we assume the range and the bearing angleboth are available from the measurement. So the nonlinearfunction γ(.) becomes

γ(xk) =[ √

x2k + y2

k

atan2(yk, xk)

]+ vk, (7)

where atan2 is the four quadrant inverse tangent function.Both wk and vk are white Gaussian noise of zero mean andQ and R covariance respectively and T is sampling time.

III. Evaluation of multi-dimensional integralwith quadrature rule and different quadrature

filterThe basic principle involved in the quadrature filters

GHF and SGHF is Gauss-Hermite quadrature rule of in-tegration which provides an approximate way to solve theintractable integrals encountered in the nonlinear Bayesianfiltering framework. Although the Gauss-Hermite rule ofintegration is available in literature [14][15] for more thanfifty years, the same has been incorporated in the es-timation very recently, mainly due to the work of Itoand Xiong [16]. In these filters, the unknown probabilitydensity function (pdf) has been approximated as Gaussianusing a set of Gauss-Hermite quadrature points and theirrespective weights. But the Gauss-Hermite quadrature ruleis defined for the single dimensional system. This single

dimensional numerical integration rule could be extendedfor the multidimensional system by utilizing the productrule in GHF and the same could be achieved by utilizingthe Smolyak rule in SGHF. The use of product rule causesthe computational load to increase exponentially and henceGHF suffers from the curse of dimensionality problem.The Smolyak rule could improve the computational efficacysharply and hence could reduce the curse of dimensionalityproblem.

A. single dimensional Gauss-Hermite quadrature ruleConsider any integral of the form

I =∫ α

−αf(x)W (x)dx, (8)

where x is a single dimensional variable, f(x) is a non-linear function of x and W (x) is the weight function. Inthis paper, the weight function is Gaussian distributionfunction.

Gauss-Hermite quadrature rule states that the integral Ican be approximated numerically as

I ≈m∑i=1

f(qi)wi (9)

where qi are the quadrature point and wi are the cor-responding weights. For the m-point quadrature rule, thisrule is exact for polynomials having degree upto (2m− 1).

There are several methods available in literature forselecting univariate quadrature points and correspondingweights. A commonly used method is moment matchingmethod [7], [9], where these are evaluated by solving a setof moment equations

1 1 · · · 1q1 q2 · · · qm...

.... . .

...qm−11 qm−1

2 · · · qm−1m

w1w2...wm

=

M0M1

...Mm−1

.

As for m number of quadrature points, we have 2mnumbers of unknown including m number of each quadra-ture points and corresponding weights, but for the samecase only m number of moments and hence equationsare available. So the designers suffer through the lack ofequations while using this method. Some authors selectthe quadrature points arbitrarily and calculate the cor-responding weights by using moment equations[9], whilesome author chose the quadrature points as the zeros ofthe Hermite polynomial[17], which may suffer from themathematical unstability[7]. In this paper, the method usedfor selecting the quadrature points and their correspondingweights was first introduced by Golub et al in [18] and laterutilized by Arasaratnam et al in filtering literature for thefirst time to develop GHF. This method is described belowfor finding the m points quadrature rule.


Let us consider a symmetric tridiagonal matrix J havingzero diagonal elements and Ji,i+1 =

√i/2; 1 ≤ i ≤ (m−1).

The quadrature points are at qi =√

2ψi, where ψi is the itheigenvalue of the matrix J. The ith weight, wi, is chosen aswi = κ2

i1, where κi1 is the first element of the ith normalizedeigenvector of J [7][8].

B. Gauss-Hermite filterIn GHF, the single dimensional quadrature points are

generated by utilizing the Golub’s technique discribed inprevious section. The single dimensional quadrature rulecould be extended to the multidimensional quadrature ruleby applying the product rule.

Let us consider a multidimensional random variable xand the weight function as the standard normal distribu-tion, hence the integral of interest will be

IN =∫ ∞−∞

f(x)ℵ(x, 0, In)dx (10)

By applying product rule the integral IN could be ap-proximated as

IN ≈m∑i1

...

m∑in

f(qi1 , qi2 , ..., qin)wi1wi2 ...win (11)

To evaluation the expected value of an n dimensionalintegral with m-point GHF, mn number of multivariatequadrature points and corresponding weights are required.For an example, for a three dimensional system and threepoint GHF, twenty-seven quadrature points and weightsare required which may be expressed as qi, qj , qk andwiwjwk respectively for i = 1, 2, 3; j = 1, 2, 3; andk = 1, 2, 3. As the number of quadrature points increasesexponentially with increasing dimension, the GHF suffersfrom the curse of dimensionality problem.

C. Sparse-grid Gauss-Hermite filterIn SGHF, the single dimensional quadrature rule is ex-

tended to multidimensional rule by using the Smolyak rule.The Smolyak rule is introduced in mathematical literaturein sixties only [10], but in filtering literature it is usedvery recently to derive SGHF [9]. It could reduce thecomputational load sharply encountered with product rule.

1) Smolyak rule: Any integral of the form

In,L(f) =∫<n

f(x)ℵ(x; 0, In)

can be approximated numericaaly as

In,L(f) ≈L−1∑

q=L−n(−1)L−1−qCn−1

L−1−q

∑Ξ∈Nn

q

(Il1⊗Il2⊗...⊗Iln),

(12)where In,L represents the numerical evaluation of n-dimensional integral with the accuracy level L which meansthat the approximation is exact for all the polynomialshaving degree upto (2L − 1), C stands for the binomial

coefficient i.e. Cnk = n!/k!(n − k)!, Ilj is the single di-mensional quadrature rule with accuracy level lj ∈ Ξ i.e.Ξ , (l1, l2, ..., ln), ⊗ stands for the tensor product and Nn

q

is set of possible values of lj given as

Nnq =

Ξ :n∑j=1

lj = n+ q

for q ≥ 0

= ∅ for q < 0.

(13)

Equation (12) can be written as

In,L(f) ≈L−1∑

q=L−n(−1)L−1−qCn−1

L−1−q

∑Ξ∈Nn

q

∑qs1∈Xl1

∑qs2∈Xl2

...∑

qsn∈Xln

f(qs1 , qs2 , ..., qsn)ws1ws2 ...wsn ,

(14)

where Xlj is the set of quadrature points for the single di-mensional quadrature rule Ilj , [qs1 , qs2 , ..., qsn

]T is a Sparse-grid quadrature (SGQ) point i.e. qsj

∈ Xlj and wsjis

the weight associated with qsj . Some SGQ points occuremultiple times, that could be counted once by adding theirweight.

The final set of the SGQ poits are

Xn,L =L−1⋃

q=L−n

⋃Ξ∈Nn

q

(Xl1 ⊗Xl2 ⊗ ...⊗Xln), (15)

where⋃

represents union of the individual SGQ points.Note: The accuracy of the SGHF increases with in-

creasing the accuracy level L, but at the same time thecomputational load also increases.

Note: The number of elements in Xlj should be higherthan or equal to lj , which is chosen as (2lj − 1) in thispaper, similar to [9].

Note: As from equation (13), the values of lj variesbetween 1 to L, hence single dimensional quadrature pointsare generated for the accuracy level of 1 to L.

Note: While evaluating the multidimensional quadra-ture points, several points appears repeatedly. These pointsare considered once and there weights are added for everyrepeatation.

IV. Simulation Result

The maneuvering target tracking problem formulated insection II, has been solved in MATLAB environment byusing the UKF, CKF, GHF and SGHF. Experimentationhas been done by considering κ = −2 for the UKF, 3-pointsGHF and 3rd-degree of accuracy for the SGHF.

As already discussed in section-II, the process and mea-surement noises are normally distributed with zero mean


Fig. 1. Truth and estimated values of position in x-direction

Fig. 2. Truth and estimated values of velocity in x-direction

and covariance Q and R respectively. We consider

Q =

gT 3

3 gT 2

2 0 0 0

gT 2

2 gT 0 0 0

0 0 gT 3

3 gT 2

2 0

0 0 gT 2

2 gT 00 0 0 0 0.009T

, (16)

where T is the sampling time which is taken as 0.5 secondand g is some constant given as g = 0.1. R is considered as

Fig. 3. Truth and estimated values of position in y-direction

Fig. 4. Truth and estimated values of velocity in y-direction

diag([σ2r σ2

t ]) where σr = 120m and σt =√

70mrad.The initial truth value is considered as

x0 = [1000m 30m/s 1000m 0m/s − 3/s], whilethe initial estimate of the covariance is P0|0 =diag([200m2 20m2/s2 200m2 20m2/s2 100mrad2/s2]).The initial estimate is considered to be normallydistributed random number with mean x0 and covarianceP0|0. The simulation is performed for 50 seconds andthe result is insured by evaluating the RMSE in terms ofposition, velocity and turn rate for 50 independent MonteCarlo runs.

The truth and estimate for different filters are shown


Fig. 5. Truth and estimated values of turn rate in degree

Fig. 6. RMSE plot of position for 50 seconds

in Fig-1 to 5 and the RMSE plots are shown in Fig-6to Fig-8. The RMSE plots show that the accuracy of thequadrature filters are better than the UKF and the CKF.The computational time for the CKF, GHF and SGHF arenoticed as 1.32, 11.45 and 10.16 time higher, relative to thesame for the UKF.

V. Discussions and ConclusionsThe 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 unknownand modeled with the Gaussian noise. The RMSE of theposition, velocity and turn rate have been evaluated usingthe UKF, the CKF and the quadrature filters GHF and

Fig. 7. RMSE plot of velocity for 50 seconds

Fig. 8. RMSE plot of Turn rate in degree for 50 seconds

SGHF. The quadrature filters show higher accuracy thanthe UKF and the CKF. Both the quadrature filters havesimilar accuracy, but SGHF filter shows relatively lesscomputational cost and hence can be the best option forsolving the maneuvering target tracking problem.

References[1] S.K. Katsikas, A. K. Lerosb, and D. G. Lainiotis, “Underwater

tracking of a maneuvering target using time delay measure-ments,” Signal Processing, vol. 41, no. 1, pp. 17-29, January 1995.

[2] N. Ikoma, N. Ichimura, T. Higuchi, and H. Maeda, “ManeuveringTarget Tracking by Using Particle Filter,” Proceedings, IFSAWorld Congress and 20th NAFIPS International Conference, vol.4, pp. 2223 - 2228, 25-28 July 2001.


[3] Murat Efe, and D.P.Atherton, “Maneuvering Target TrackingUsing Adaptive Turn Rate Models in the Interacting MultipleModel Algorithm,” Proceedings, 35th conference on Decision andControl, pp. 3151-56, Kobe, Japan, December 1996.

[4] Y. Bar-Shalom and X. R. Li, “Estimation with Applicationto Tracking and Navigation,” A Wiley-Interscience Publication,John Wiley and Sons, INC., New York

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[6] Ienkaran Arasaratnam, and Simon Haykin, “Cubature KalmanFilter,” IEEE Trans. Autom. Control, vol. 54,no. 6, pp.1254-1269,June 2009.

[7] I. Arasaratnam, S. Haykin and R.J. Elliott, “Discrete-time non-linear filtering algorithms using Gauss-Hermite quadrature,”Proc. IEEE, vol. 95,no. 5, pp. 953-977, July 2007.

[8] A. K. singh snd S. Bhaumik, “Nonlinear estimation using trans-formed Gauss-Hermite quadrature points,” IEEE intern. conf.signal process., comput. control, Solan, India, 2013, pp. 1-4

[9] Bin Jia , Ming Xin, and Yang Cheng, “Sparse-grid quadraturenonlinear filtering,” Automatica, vol. 48, no. 2, pp. 327-341, Feb.2012.

[10] S.A. Smolyak., “Quadrature and interpolation formulas for ten-

sor products of certain classes of functions,” 1963, Soviet Math-ematics Doklady, vol. 4, pp. 240-243

[11] F. Heiss and V. Winschel “Likelihood approximation by numer-ical integration on sparse grids,” , Jr. Econ., vol. 144,no. 1, pp.62âĂŞ80 , May 2008

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[13] Bin Jia, Ming Xin, and Yang Cheng, “High-degree CubatureKalman Filter,” Automatica, vol. 49, no. 5, pp.510-518, Feb.2013.

[14] F.B. Hildebrand, “Approximate calculation of integrals,” 2nd ed.,New York, Dover Publication, 2008.

[15] V.I. Krylov, “Approximate calculation of integrals,” NC Mineola,New York, Dover Publication, 2005.

[16] Ito, K., Xiong, K., “Gaussian filters for nonlinear filtering prob-lems,” IEEE Trans. Autom. Control, vol. 45, no. 5, pp.910-927,2000

[17] W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery,“Numerical Recipes in C”, 2nd ed., Cambridge, U.K., CambridgeUniv. Press,1992.

[18] G.H. Golub and C.F. Van Loan,: “Calculation of Gauss quadra-ture rule,” maths. comput., vol. 23,no. 106, pp. 221-230, 1969

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