This work presents an efficient method to solve a class of continuous-time, continuous-space stochastic optimal control problems of robot motion in a cluttered environment. The method builds upon a path integral representation of the stochastic optimal control problem that allows computation of the optimal solution through sampling and estimation process. As this sampling process often leads to a local minimum especially when the state space is highly non-convex due to the obstacle field, we present an efficient method to alleviate this issue by devising a proposed topological motion planning algorithm. Combined with a receding-horizon scheme in execution of the optimal control solution, the proposed method can generate a dynamically feasible and collision-free trajectory while reducing concern about local optima. Illustrative numerical examples are presented to demonstrate the applicability and validity of the proposed approach.
The RRT* algorithm has efficiently extended Rapidly-exploring Random Trees (RRTs) to endow it with asymptotic optimality. We propose Goal-Rooted Feedback Motion Trees (GR-FMTs) that honor state input constraints and generate collision-free feedback policies. Given analytic solutions for optimal local steering, GR-FMTs obtain and realize safe, dynamically feasible, and asymptotically optimal trajectories toward goals. Second, for controllable linear systems with linear state input constraints, we propose a fast method for local steering, based on polynomial basis functions and segmentation. GR-FMTs with the method obtain and realize trajectories that are collision-free, dynamically feasible under constraints, and asymptotically optimal within a set we define. The formulation includes linear or quadratic programming of small sizes, where constraints are identified by root-finding in low or medium order of polynomials and added progressively.