A t-Distribution-Based Particle Filter for Bearings-Only Tracking
Keywords:
bearing-only tracking, expectation maximisation, particle filter, student-t distribution
Abstract
Bearings-only target tracking is a nonlinear estimation problem often addressed by linearised filters where the uncertainty in the sensor and motion models is typically modeled by Gaussian densities. In this paper, a particle filter or sequential Monte Carlo method is developed, based on student-t distribution, which is heavier tailed than Gaussians and hence more robust. The t-distribution-based particle filter provides an approximate solution to nonlinear non-Gaussian estimation problems. To estimate the target state based on samples, an expectation maximisation (EM)-type algorithm was developed and embedded in a student-t particle filter. The expectation step was implemented by the particle filter. In this step, the distribution of the states and the state vector were estimated. Consequently, in the maximisation step, the nonlinear observation equation was approximated as a mixture of the Gaussian and student-t models. A bearings-only tracking problem was simulated to present the implementation of the particle filter algorithm based on both the mixture of the Gaussian model and student-t. Simulations and real life data taken from the digital global system for mobile communications (GSM) real-time data-logging tracking system showed that the student-t-based particle filter significantly outperformed the Gaussian mixture filter and successfully accommodated a nonlinear model for a target-tracking scenario.
References
Aidala, V. J. (1979). Kalman filter behavior in bearings-only tracking applications. Aerospace and Electronic Systems, 15(1): 29–39.
Aidala V. J. and Nardone, S. C. (1982). Biased estimation properties of the pseudolinear tracking filter. Aerospace and Electronic Systems, 18(4): 432–441.
Aidala, V. and Hammel, S. (1983). Utilization of modified polar coordinates for bearings-only tracking. Automatic Control, 28(3): 283–294.
Bar-Shalom, Y. and Fortmann, T. E. (1988). Tracking and data association, Academic Press, New York, p. 138–231.
Bar-Shalom, Y., Thiagalingam K. and X.-Rong Li. (2002). Estimation with applications to tracking and navigation. John Wiley & Sons, Inc., New York, USA, p. 421–490.
Bar-Shalom, Y., Li, X., and Kirubarajan, T. (2001). Estimation with Applications to Tracking and Navigation. John Wiley & Sons, Inc, 547pp.
Brehard, T. and Le Cadre, J. P. (2006). Closed-form posterior cramer-Rao bounds for bearing-only tracking. Aerospace and Electronic Systems, 42(4): 1198–1223.
Clark, J. M. C., Vinter, R. B. and Yaqoob, M. M. (2007). Shifted rayleigh filter: a new algorithm for bearings-only tracking. Aerospace and Electronic Systems, 43(4): 1373–1384.
Godsill, S., Doucet, A. and West, M. (2004). Monte carlo smoothing for non-linear time series. Journal of the American Statistical Association, 199: 156–168.
Gordon, N. J., Salmond, D. J. and Smith, A. (1993). Novel approach to nonlinear/non-gaussian bayesian state estimation, In IEEE proceedings on Radar signal process, 140(2): 107–113.
Goshen-Meskin, D. and Bar-Itzhack, I. Y. (1992). Observability analysis of piece-wise constant systems. Aerospace and Electronic Systems, 28(4): 1056–1067.
Dosso, S. E, Wilmut M. J. (2013): Bayesian tracking of multiple acoustic sources in an uncertain ocean environment. The Journal of the Acoustical Society of America, 133(4): EL274–280.
Doucet, A., De Freitas, J. F. G. and Gordon, N. (2001). Sequential monte carlo methods in Practice, Springer-Verlag: New York, 17–41.
Gustafsson, F. (2010). Particle filter theory and practice with positioning applications. Aerospace and Electronic Systems, 25(7): 53–81.
Haluk, E. B. (2011). Bearings-only tracking. A thesis submitted to the graduate school of natural and applied sciences of middle east technical university, 116pp.
Huang, G. P., Zhou, K. X., Trawny, N. and Roumeliotis, S. I. (2010). A bank of MAP estimators for single-sensor range-only target tracking, In Proceeding of American Control Conference, p. 6974–6980.
Kim, J. and Stoffer, D. S. (2008). Fitting stochastic volatility models in the presence of irregular sampling via particle methods and the EM Algorithm, Journal of Time Series Analysis, 29(5): 811–833.
Kitagawa, G. (1996). Monte Carlo filter and smoother for non-Gaussian nonlinear state space models, Journal of Computational and Graphical Statistics, 5: 1–25.
Kitagawa, G. and Sato, S. (2001). Monte carlo smoothing and self-organising state space model. In: Sequential Monte Carlo Methods in Practice, Doucet, A. de Freitas, N. and Gordon, N., (Eds). Springer-Verlag, New York, p. 177–195.
Kronhamn, T. (1998). Bearings-only target motion analysis based on a multi hypothesis Kalman filter and adaptive ownship motion control. IEEE Proceedings – Radar, Sonar and Navigation, 145(4): 247–252.
La Scala, B. F., Mallick, M. and Arulampalam, S. (2007). Differential geometry measures of nonlinearity for filtering with nonlinear dynamic and linear measurement models. In SPIE Conference on Signal and Data Processing of Small Targets, vol. 6699.
Le Cadre J. P. and Tremois, O. (1998). Bearings-only tracking for maneuvering sources,” IEEE Transactions on Aerospace and Electronic Systems, 34(1): 179–193.
Li J, Zhou H. (2013). Tracking of time-evolving sound speed profiles in shallow water using an ensemble Kalman-particle filter. The Journal of the Acoustical Society of America, 133(3): 1377–1386.
Li X, Sun H, Jiang L, Shi Y, Wu Y. (2015): Modified particle filtering algorithm for single acoustic vector sensor DOA tracking. Sensors, 15(10): 26198–26211.
Lindgren A. G. and Kai F. Gong. (1978). Position and velocity estimation via bearing observations. Aerospace and Electronic Systems, 14(4): 564–577.
Liu, X., Kirubarajan, T., Bar-Shalom, Y. and Maskell, S. (2002). Comparison of EKF, pseudo-measurement and particle filters for bearing-only target tracking problem. In Proc.SPIE Conference on Image and Signal Processing for Small Targets, 4728: 240–250.
Liu, B. and Hao, C. (2013). Sequential bearings-only-tracking initiation with particle filtering method. The Scientific World Journal, 2013: 1–7.
Musicki, D., and Evans, R. J., (2006). Measurement gaussian sum mixture target tracking. In 9th International Conference on Information Fusion, p. 1–8.
Musicki, D. (2007). Bearings only single-sensor target tracking using gaussian sum measurement presentation. IEEE Transactions on Aerospace and Electronic Systems, 15(1): 29–39.
Musicki, D. (2009). Bearings only single-sensor target tracking using gaussian mixtures. Automatica, 45(9): 2088–2092.
Nardone, S., Lindgren, A. and Kai Gong (1984). Fundamental properties and performance of conventional bearings-only target motion analysis. Automatic Control, 29(9): 775–787.
Nerurkar, E. D. Roumeliotis, S. I. and Martinelli, A. (2009). Distributed maximum a posteriori estimation for multi-robot cooperative localization. In Proceeding of IEEE International Conference on Robotics and Automation, p. 1402–1409.
Radhakrishnan, K., Unnikrishnan, A. and Balakrishnan, K. G. (2010). Bearing only tracking of maneuvering targets using a single coordinated turn model. International Journal of Computer Applications, 1(1): 25–33.
Ristic, B., Arulampalam, M. and Gordon, A. (2004). Beyond kalman filters: particle filters for target tracking application. Artech House, London, UK, p. 35–57.
Robinson, P. and Yin, M. (1994). Modified spherical coordinates for radar In Proceeding AIAA Guidance, Navigation and Control Conference, p. 55–64.
Sanjeev, A. M., Ristic, B., Gordon, N. and Mansell, T. (2004). Bearings-only tracking for manoeuvring targets using particle filters, Journal on Applied Signal Processing, 15: 2351–2365.
Tirri, A. E., Fasano, G., Accardo, D. and Moccia, A. (2014). Particle filtering for Obstacle Tracking in UAS Sense and Avoid Applications. The Scientific World Journal, 2014, Article ID 280478, 1–12.
Warner, G. A, Dosso, S.E, Dettmer J, and Hannay DE (2015). Bayesian environmental inversion of airgun modal dispersion using a single hydrophone in the Chukchi Sea. The Journal of the Acoustical Society of America, 137(6): 3009–3023.
Aidala V. J. and Nardone, S. C. (1982). Biased estimation properties of the pseudolinear tracking filter. Aerospace and Electronic Systems, 18(4): 432–441.
Aidala, V. and Hammel, S. (1983). Utilization of modified polar coordinates for bearings-only tracking. Automatic Control, 28(3): 283–294.
Bar-Shalom, Y. and Fortmann, T. E. (1988). Tracking and data association, Academic Press, New York, p. 138–231.
Bar-Shalom, Y., Thiagalingam K. and X.-Rong Li. (2002). Estimation with applications to tracking and navigation. John Wiley & Sons, Inc., New York, USA, p. 421–490.
Bar-Shalom, Y., Li, X., and Kirubarajan, T. (2001). Estimation with Applications to Tracking and Navigation. John Wiley & Sons, Inc, 547pp.
Brehard, T. and Le Cadre, J. P. (2006). Closed-form posterior cramer-Rao bounds for bearing-only tracking. Aerospace and Electronic Systems, 42(4): 1198–1223.
Clark, J. M. C., Vinter, R. B. and Yaqoob, M. M. (2007). Shifted rayleigh filter: a new algorithm for bearings-only tracking. Aerospace and Electronic Systems, 43(4): 1373–1384.
Godsill, S., Doucet, A. and West, M. (2004). Monte carlo smoothing for non-linear time series. Journal of the American Statistical Association, 199: 156–168.
Gordon, N. J., Salmond, D. J. and Smith, A. (1993). Novel approach to nonlinear/non-gaussian bayesian state estimation, In IEEE proceedings on Radar signal process, 140(2): 107–113.
Goshen-Meskin, D. and Bar-Itzhack, I. Y. (1992). Observability analysis of piece-wise constant systems. Aerospace and Electronic Systems, 28(4): 1056–1067.
Dosso, S. E, Wilmut M. J. (2013): Bayesian tracking of multiple acoustic sources in an uncertain ocean environment. The Journal of the Acoustical Society of America, 133(4): EL274–280.
Doucet, A., De Freitas, J. F. G. and Gordon, N. (2001). Sequential monte carlo methods in Practice, Springer-Verlag: New York, 17–41.
Gustafsson, F. (2010). Particle filter theory and practice with positioning applications. Aerospace and Electronic Systems, 25(7): 53–81.
Haluk, E. B. (2011). Bearings-only tracking. A thesis submitted to the graduate school of natural and applied sciences of middle east technical university, 116pp.
Huang, G. P., Zhou, K. X., Trawny, N. and Roumeliotis, S. I. (2010). A bank of MAP estimators for single-sensor range-only target tracking, In Proceeding of American Control Conference, p. 6974–6980.
Kim, J. and Stoffer, D. S. (2008). Fitting stochastic volatility models in the presence of irregular sampling via particle methods and the EM Algorithm, Journal of Time Series Analysis, 29(5): 811–833.
Kitagawa, G. (1996). Monte Carlo filter and smoother for non-Gaussian nonlinear state space models, Journal of Computational and Graphical Statistics, 5: 1–25.
Kitagawa, G. and Sato, S. (2001). Monte carlo smoothing and self-organising state space model. In: Sequential Monte Carlo Methods in Practice, Doucet, A. de Freitas, N. and Gordon, N., (Eds). Springer-Verlag, New York, p. 177–195.
Kronhamn, T. (1998). Bearings-only target motion analysis based on a multi hypothesis Kalman filter and adaptive ownship motion control. IEEE Proceedings – Radar, Sonar and Navigation, 145(4): 247–252.
La Scala, B. F., Mallick, M. and Arulampalam, S. (2007). Differential geometry measures of nonlinearity for filtering with nonlinear dynamic and linear measurement models. In SPIE Conference on Signal and Data Processing of Small Targets, vol. 6699.
Le Cadre J. P. and Tremois, O. (1998). Bearings-only tracking for maneuvering sources,” IEEE Transactions on Aerospace and Electronic Systems, 34(1): 179–193.
Li J, Zhou H. (2013). Tracking of time-evolving sound speed profiles in shallow water using an ensemble Kalman-particle filter. The Journal of the Acoustical Society of America, 133(3): 1377–1386.
Li X, Sun H, Jiang L, Shi Y, Wu Y. (2015): Modified particle filtering algorithm for single acoustic vector sensor DOA tracking. Sensors, 15(10): 26198–26211.
Lindgren A. G. and Kai F. Gong. (1978). Position and velocity estimation via bearing observations. Aerospace and Electronic Systems, 14(4): 564–577.
Liu, X., Kirubarajan, T., Bar-Shalom, Y. and Maskell, S. (2002). Comparison of EKF, pseudo-measurement and particle filters for bearing-only target tracking problem. In Proc.SPIE Conference on Image and Signal Processing for Small Targets, 4728: 240–250.
Liu, B. and Hao, C. (2013). Sequential bearings-only-tracking initiation with particle filtering method. The Scientific World Journal, 2013: 1–7.
Musicki, D., and Evans, R. J., (2006). Measurement gaussian sum mixture target tracking. In 9th International Conference on Information Fusion, p. 1–8.
Musicki, D. (2007). Bearings only single-sensor target tracking using gaussian sum measurement presentation. IEEE Transactions on Aerospace and Electronic Systems, 15(1): 29–39.
Musicki, D. (2009). Bearings only single-sensor target tracking using gaussian mixtures. Automatica, 45(9): 2088–2092.
Nardone, S., Lindgren, A. and Kai Gong (1984). Fundamental properties and performance of conventional bearings-only target motion analysis. Automatic Control, 29(9): 775–787.
Nerurkar, E. D. Roumeliotis, S. I. and Martinelli, A. (2009). Distributed maximum a posteriori estimation for multi-robot cooperative localization. In Proceeding of IEEE International Conference on Robotics and Automation, p. 1402–1409.
Radhakrishnan, K., Unnikrishnan, A. and Balakrishnan, K. G. (2010). Bearing only tracking of maneuvering targets using a single coordinated turn model. International Journal of Computer Applications, 1(1): 25–33.
Ristic, B., Arulampalam, M. and Gordon, A. (2004). Beyond kalman filters: particle filters for target tracking application. Artech House, London, UK, p. 35–57.
Robinson, P. and Yin, M. (1994). Modified spherical coordinates for radar In Proceeding AIAA Guidance, Navigation and Control Conference, p. 55–64.
Sanjeev, A. M., Ristic, B., Gordon, N. and Mansell, T. (2004). Bearings-only tracking for manoeuvring targets using particle filters, Journal on Applied Signal Processing, 15: 2351–2365.
Tirri, A. E., Fasano, G., Accardo, D. and Moccia, A. (2014). Particle filtering for Obstacle Tracking in UAS Sense and Avoid Applications. The Scientific World Journal, 2014, Article ID 280478, 1–12.
Warner, G. A, Dosso, S.E, Dettmer J, and Hannay DE (2015). Bayesian environmental inversion of airgun modal dispersion using a single hydrophone in the Chukchi Sea. The Journal of the Acoustical Society of America, 137(6): 3009–3023.
How to Cite
Nkemnole, E. B., & Abass, O. (1). A t-Distribution-Based Particle Filter for Bearings-Only Tracking. Journal of Scientific Research and Development, 17(1), 57-64. Retrieved from http://jsrd.unilag.edu.ng/article/view/43
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