This paper discusses the application of four nonlinear estimation techniques on two benchmark target tracking problems. The first problem is a generic air traffic control (ATC) scenario, which involves nonlinear system equations with linear measurements. The second study is a classical ground surveillance problem, where a moving airborne platform with a sensor is used to track a moving target. The tracking scenario is set in two dimensions, with the measurement providing nonlinear bearing-only observations. These two target tracking problems provide a good benchmark for comparing the following nonlinear estimation techniques: the common extended and unscented Kalman filters (EKF/UKF), the particle filter (PF), and the relatively new smooth variable structure filter (SVSF). The results of applying the SVSF on the two target tracking problems demonstrate its stability and robustness. Both of these attributes make use of the SVSF advantageous over other popular methods. The filters performances are quantified in terms of robustness, resilience to poor initial conditions and measurement outliers, and tracking accuracy and computational complexity. The purpose of this paper is to demonstrate the effectiveness of applying the SVSF on nonlinear target tracking problems, which in the past have typically been solved by Kalman or particle filters.
Issue Section:
Technical Briefs
References
1.
Kalman
, R. E.
, 1960, “A New Approach to Linear Filtering and Prediction Problems
,” Trans. ASME J. Basic Eng.
, 82
, pp. 35
–45
.2.
Grewal
, M. S.
, and Andrews
, A. P.
, 2008, Kalman Filtering: Theory and Practice Using MATLAB
, 3rd ed., Wiley
, New York
.3.
Bar-Shalom
, Y.
, Rong Li
, X.
, and Kirubarajan
, T.
, 2001, Estimation With Applications to Tracking and Navigation
, Wiley
, New York
.4.
Ristic
, B.
, Arulampalam
, S.
, and Gordon
, N.
, 2004, Beyond the Kalman Filter: Particle Filters for Tracking Applications
, Artech House
, Boston, MA
.5.
Habibi
, S. R.
, Burton
, R.
, and Chinniah
, Y.
, 2002, “Estimation Using a New Variable Structure Filter
,” American Control Conference
, Anchorage, AK, pp. 2937
–2942
.6.
Habibi
, S. R.
, and Burton
, R.
, 2003, “The Variable Structure Filter
,” ASME J. Dyn. Syst. Meas. Control
, 125
, pp. 287
–293
.7.
Nise
, N.
, 2011, Control Systems Engineering
, 6th ed., Wiley
, New York
.8.
Gadsden
, S. A.
, 2011, “Smooth Variable Structure Filtering: Theory and Applications
,” Ph.D. thesis, Department of Mechanical Engineering, McMaster University, Hamilton, ON.9.
Anderson
, B. D. O.
, and Moore
, J. B.
, 1979, Optimal Filtering
, Prentice-Hall
, Englewood Cliffs, NJ
.10.
Gelb
, A.
, 1974, Applied Optimal Estimation
, MIT Press
, Cambridge, MA
.11.
Welch
, G.
, and Bishop
, G.
, 2006, “An Introduction to the Kalman Filter
,” Department of Computer Science
, University of North Carolina
, Chapel Hill, NC
.12.
Simon
, D.
, 2006, Optimal State Estimation: Kalman, H-Infinity, and Nonlinear Approaches
, Wiley-Interscience
, Hoboken, NJ
.13.
Van der Merwe
, R.
, and Wan
, E.
, 2004, “Sigma-Point Kalman Filters for Integrated Navigation
,” Proceedings of the Institute of Navigation Annual Meeting
, Dayton, OH
, pp. 641
–654
.14.
Van der Merwe
, R.
, 2004, “Sigma-Point Kalman Filters for Probabilistic Inference in Dynamic State Space Models
,” Ph.D. thesis, OGI School of Science and Engineering, Oregon Health and Science University, Portland, OR.15.
Al-Shabi
, M.
, 2011, “The General Toeplitz/Observability SVSF
,”, Ph.D. thesis, Department of Mechanical Engineering, McMaster University, Hamilton.16.
Ambadan
, J.
, and Tang
, Y.
, 2009, “Sigma-Point Kalman Filter Data Assimilation Methods for Strongly Nonlinear Systems
,” J. Atmos. Sci.
, 66
(2
), pp. 261
–285
.17.
Tang
, X.
, Zhao
, X.
, and Zhang
, X.
, 2008, “The Square-Root Spherical Simplex Unscented Kalman Filter for State and Parameter Estimation
,” Proceedings of the International Conference on Signal Processing
, Beijing
, pp. 260
–263
.18.
Julier
, S. J.
, Ulhmann
, J. K.
, and Durrant-Whyte
, H. F.
, 2000, “A New Method for Nonlinear Transformation of Means and Covariances in Filters and Estimators
,” IEEE Trans. Autom. Control
, 45
, pp. 472
–482
.19.
Julier
, S. J.
, and Uhlmann
, J. K.
, 2004, “Unscented Filtering and Nonlinear Estimation
,” Proc. IEEE
, 92
(3
), pp. 401
–422
.20.
Kim
, S.
, and Han
, W.
, 2005, “Joint Estimation of Time Delay and Channel Amplitude by Simplex Unscented Filter Without Assisted Pilot in CDMA Systems
,” The 7th International Conference on Advanced Communications Technology
, Phoenix Park, South Korea
, pp. 233
–240
.21.
Julier
, S.
, 2003, “The Spherical Simplex Unscented Transformation
,” Proceedings of the American Conference Conference
, Denver
, pp. 2430
–2434
.22.
Haykin
, S.
, 2001, Kalman Filtering and Neural Networks
, Wiley
, New York
.23.
Moral
, D.
, 1998, “Measure Valued Processes and Interacting Particle Systems: Application to Non-Linear Filtering Problems
,” Ann. Appl. Prob.
, 8
(2
), pp. 438
–495
.24.
Gordon
, N. J.
, Salmond
, D. J.
, and Smith
, A. F. M.
, 1993, “Novel Approach to Nonlinear/Non-Gaussian Bayesian State Estimation
,” IEEE Proc.-F
, 140
(2
), pp. 107
–113
.25.
MacCormick
, J.
, and Blake
, A.
, 1999, “A Probabilistic Exclusion Principle for Tracking Multiple Objects
,” Proceedings of International Conference on Computer Vision
, Kerkyra, Greece
, pp. 572
–578
.26.
Kanazawa
, K.
, Koller
, D.
, and Russell
, S. J.
, 1995, “Stochastic Simulation Algorithms for Dynamic Probabilistic Networks
,” Proceedings of the Eleventh Annual Conference on Uncertainty in Artificial Intelligence
, Montreal
, pp. 346
–351
.27.
Hammersley
, J. M.
, and Morton
, K. W.
, 1954, “Poor Man’s Monte Carlo
,” J. Roy. Stat. Soc. B
, 16
, pp. 23
–28
.28.
Handschin
, J. E.
, and Mayne
, D. Q.
, 1969, “Monte Carlo Techniques to Estimate the Conditional Expectation in Multi-Stage Non-Linear Filtering
,” Int. J. Control
, 9
(5
), pp. 547
–559
.29.
Akashi
, H.
, and Kumamoto
, H.
, 1977, “Random Sampling Approach to State Estimation in Switching Environments
,” Automatica
, 13
, pp. 429
–434
.30.
Doucet
, A.
, de Freitas
, N.
, and Gordon
, N. J.
, 2001, Sequential Monte Carlo Methods in Practice
, Springer
, New York
.31.
Djuric
, P. M.
, and Goodsill
, S. J.
, 2002, “Guest Editorial Special Issue on Monte Carlo Methods for Statistical Signal Processing
,” IEEE Trans. Signal Process.
, 50
(2
), p. 173
.32.
Kong
, A.
, Liu
, J. S.
, and Wong
, W. H.
, 1994, “Sequential Imputations and Bayesian Missing Data Problems
,” J. Am. Stat. Assoc.
, 89
(425
), pp. 278
–288
.33.
Habibi
, S. R.
, 2007, “The Smooth Variable Structure Filter
,” Proc. IEEE
, 95
(5
), pp. 1026
–1059
.34.
Wang
, H.
, Kirubarajan
, T.
, and Bar-Shalom
, Y.
, 1999, “Precision Large Scale Air Traffic Surveillance Using an IMM Estimator With Assignment
,” Trans. IEEE Aerosp. Electron. Syst. J.
, 35
(1
), pp. 255
–266
.35.
Lin
, X.
, Kirubarajan
, T.
, Bar-Shalom
, Y.
, and Maskell
, S.
, 2002, “Comparison of EKF, Pseudomeasurement Filter and Particle Filter for a Bearings-only Target Tracking Problem
,” Proceedings of the SPIE Conference of Signal and Data Processing of Small Targets
, Orlando, FL.36.
Hernandez
, M. L.
, Kirubarajan
, T.
, and Bar-Shalom
, Y.
, 2004, “Multisensor Resource Management in the Presence of Clutter Using Cramér-Rao Lower Bounds
,” IEEE Trans. Aerosp. Electron. Syst.
, 40
(2
), pp. 399
–416
.37.
Zhang
, X.
, and Willett
, P.
, 2001, “Cramér-Rao Bounds for Discrete-Time Linear Filtering With Measurement Origin Uncertainties
,” Proceedings of the Workshop on Estimation
, Tracking and Fusion: A Tribute to Yaakov Bar-Shalom, CA.38.
Tichavsky
, P.
, 1998, “Posterior Cramér-Rao Bounds for Discrete-Time Nonlinear Filtering
,” IEEE Trans. Signal Process.
, 46
(5
), pp. 1386
–1396
.39.
Arasaratnam
, I.
, Haykin
, S.
, and Elliott
, R. J.
, 2007, “Discrete-Time Nonlinear Filtering Algorithms Using Gauss-Hermite Quadrature
,” Proc. IEEE.
, 95
(5)
, pp. 953
–977
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