1998/09/06 by R. W. R. Darling, Darling, R. W. R.
Computer Science · Mathematics · #58G32 #60G35 #93E11 #FOS: Mathematics #Optimization and Control (math.OC) #Probability (math.PR) #Target Tracking and Data Fusion in Sensor Networks #math.OC #math.PR #msc:58G32 #msc:60G35 #msc:93E11
paper · pdf · doi:10.48550/arxiv.math/9809028
22 pages, 4 figures
arxiv created 1998/09/06 · openalex publication_date 1998/09/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Geometrically Intrinsic Nonlinear Recursive Filter, or GI Filter, is designed to estimate an arbitrary continuous-time Markov diffusion process X subject to nonlinear discrete-time observations. The GI Filter is fundamentally different from the much-used Extended Kalman Filter (EKF), and its second-order variants, even in the simplest nonlinear case, in that: (i) It uses a quadratic function of a vector observation to update the state, instead of the linear function used by the EKF. (ii) It is based on deeper geometric principles, which make the GI Filter coordinate-invariant. This implies, for example, that if a linear system were subjected to a nonlinear transformation f of the state-space and analyzed using the GI Filter, the resulting state estimates and conditional variances would be the push-forward under f of the Kalman Filter estimates for the untransformed system - a property which is not shared by the EKF or its second-order variants. The noise covariance of X and the observation covariance themselves induce geometries on state space and observation space, respectively, and associated canonical connections. A sequel to this paper develops stochastic differential geometry results - based on "intrinsic location parameters", a notion derived from the heat flow of harmonic mappings - from which we derive the coordinate-free filter update formula. The present article presents the algorithm with reference to a specific example - the problem of tracking and intercepting a target, using sensors based on a moving missile. Computational experiments show that, when the observation function is highly nonlinear, there exist choices of the noise parameters at which the GI Filter significantly outperforms the EKF.