Improvements for Sparse Grid Tiles (STile) Filter

In this paper we present an advanced version of the Sparse-Grid-Tiles (STile) Filter. With increasing calculation power of modern systems, the focus of Stochastic Filtering turns to nonlinear effects. Sophisticated methods have to be investigated in application areas, where linearized methods like the Extended Kalman Filter tend to suboptimality or even divergence and conservative Particle Filters are insufficient due to the curse of dimension. The STile Filter algorithm approximates the probability distribution on an adaptive set of sparse grid tiles. The motivation of sparse grids is to achieve the approximation order of a regular grid by a specific subset of grid points. The numerical improvements of this paper enhance the speed and accuracy of the algorithm. Finally, robustness against measurement errors is achieved by Integrity Monitoring. We compare and analyze three methods in a Monte Carlo setting.

[1]  X. R. Li,et al.  Survey of maneuvering target tracking: II. Ballistic target models , 2001 .

[2]  Antonio F. Gómez-Skarmeta,et al.  High-Integrity IMM-EKF-Based Road Vehicle Navigation With Low-Cost GPS/SBAS/INS , 2007, IEEE Transactions on Intelligent Transportation Systems.

[3]  Carolyn Kalender,et al.  Sparse Grid-Based Nonlinear Filtering , 2013, IEEE Transactions on Aerospace and Electronic Systems.

[4]  Guannan Zhang,et al.  A Hybrid Sparse-Grid Approach for Nonlinear Filtering Problems Based on Adaptive-Domain of the Zakai Equation Approximations , 2014, SIAM/ASA J. Uncertain. Quantification.

[5]  G. Kallianpur.  Stochastic Filtering Theory , 1980 .

[6]  Gertjan Looye,et al.  Advances in Aerospace Guidance, Navigation and Control , 2013 .

[7]  Florian Nadel,et al.  Stochastic Processes And Filtering Theory , 2016 .

[8]  H. Bungartz,et al.  Sparse grids , 2004, Acta Numerica.

[9]  Ming Xin,et al.  Sparse-grid quadrature nonlinear filtering , 2012, Autom..

[10]  S. Shreve,et al.  Stochastic differential equations , 1955, Mathematical Proceedings of the Cambridge Philosophical Society.

[11]  Fred Daum,et al.  Generalized particle flow for nonlinear filters , 2010, Defense + Commercial Sensing.

[12]  Yang Cheng,et al.  Sparse Gauss-Hermite Quadrature Filter with Application to Spacecraft Attitude Estimation , 2011 .

[13]  D. Crisan,et al.  Fundamentals of Stochastic Filtering , 2008 .

[14]  L. Bergman,et al.  Solution of the Four Dimensional Fokker-Planck Equation: Still a Challenge , 2005 .

[15]  D. Laneuville,et al.  Grid Based Solution of Zakai Equation with Adaptive Local Refinement for Bearings-only Tracking , 2008, 2008 IEEE Aerospace Conference.

[16]  Doyoon Kim,et al.  Integrity Monitoring Algorithms Using Filtering Approaches for Higher Navigation Performance: Consideration of the Non-Gaussian GNSS Measurements , 2007 .

[17]  J. Strikwerda.  Finite Difference Schemes and Partial Differential Equations , 1989 .

[18]  Fred Daum,et al.  Particle flow for nonlinear filters with log-homotopy , 2008, SPIE Defense + Commercial Sensing.

[19]  Jeffrey K. Uhlmann,et al.  Unscented filtering and nonlinear estimation , 2004, Proceedings of the IEEE.

[20]  John B. Moore,et al.  Optimal State Estimation , 2006 .

[21]  X. Rong Li,et al.  A Survey of Maneuvering Target Tracking — Part II : Ballistic Target Models , 2001 .

[22]  Carolyn Kalender,et al.  Nonlinear Filtering Using Sparse Grids , 2011 .

[23]  X. Rong Li,et al.  A survey of maneuvering target tracking-part VIa: density-based exact nonlinear filtering , 2010, Defense + Commercial Sensing.