he B-spline function representation is commonly used for data approximation and trajectory definition but filter-based methods for nonlinear weighted least squares (NWLS) approximation are restricted to a bounded definition range. We present an algorithm termed nonlinear recursive B-spline approximation (NRBA) for iterative NWLS approximation of an unbounded set of data points by a B-spline function. NRBA is based on a marginalized particle ﬁlter (MPF), in which a Kalman ﬁlter (KF) solves the linear subproblem optimally while a particle filter (PF) deals with nonlinear approximation goals. NRBA can adjust the bounded definition range of the approximating B-spline function during run-time such that regardless of the initially chosen definition range all data points can be processed. In numerical experiments NRBA achieves approximation results close to those of the Levenberg-Marquardt algorithm. A NWLS approximation problem is a nonlinear optimization problem. The direct trajectory optimization approach also leads to a nonlinear problem. The computational effort of most solution methods grows exponentionally with the trajectory length. We demonstrate how NRBA can be applied for multiobjective trajectory optimization for a battery electric vehicle in order to determine energy-efficient velocity trajectories. With NRBA the effort increases only linearly with the processed data points and the trajectory length.
An Iterative Method Based on the Marginalized Particle Filter for Nonlinear B-spline Data Approximation and Trajectory Optimization
Mathematics 2019, 7(4), 355