A spline filter for multidimensional nonlinear state estimation

The problem of nonlinear/non-Gaussian filtering has generated significant interest in the literature. The effectiveness of a nonlinear/non-Gaussian filtering algorithm depends on the accurate representation of the probability density function of the system state. The Extended Kalman Filter (EKF), the Approximate Grid Based methods (AGBM), and particle based filters have been developed to solve nonlinear/non-Gaussian problems. However, the EKF may not perform well with high nonlinearity due to its linear approximation; the AGBM often incurs extremely high storage/computational cost; particle filters provide weighted samples at discrete points in the state space instead of a continuous estimate of the probability density function of the system state, but most of them may suffer from the well-known degeneracy problem. In this paper, a comprehensive solution for nonlinear/non-Gaussian state estimation that can provide a continuous estimate of the probability density function of the system state is developed based on B-splines. The proposed spline filter is capable of modeling any arbitrary probability density function of the system state, and it is able to provide statistically the same estimation accuracy as the particle filter, but with only a few knots. It is also important to emphasize that, unlike most of the particle based algorithms (e.g., Sequential Important Sampling (SIS), Generic Particle Filter (GPF), and Sequential Monte Carlo (SMC) filter), the spline filter is free from degeneracy-like problems due to its continuous nature. In addition, by moving the knots dynamically, it ensures that the splines cover, and only cover, the regions where the probability of system state is significant so that the high efficiency of the spline filter is maintained. To make it applicable in common tracking applications, the spline filter is further extended to a multiple model one with the capability to handle systems with multiple maneuvering models. Besides theoretical derivations, simulation results are provided to verify the effectiveness of the proposed spline filter.