Koopman Operator Method for Chance-Constrained Motion Primitive Planning

Abstract

The use of motion primitives to plan trajectories has received significant attention in the robotics literature. This work considers the application of motion primitives to path planning and obstacle avoidance problems in which the system is subject to significant parametric and/or initial condition uncertainty. In problems involving parametric uncertainty, optimal path planning is achieved by minimizing the expected value of a cost function subject to probabilistic (chance) constraints on vehicle-obstacle collisions. The Koopman operator provides an efficient means to compute expected values for systems under parametric uncertainty. In the context of motion planning, these include both the expected cost function and chance constraints. This work describes a maneuver-based planning method that leverages the Koopman operator to minimize an expected cost while satisfying user-imposed risk tolerances. The developed method is illustrated in two separate examples using a Dubins car model subject to parametric uncertainty in its dynamics or environment. Prediction of constraint violation probability is compared with a Monte Carlo method to demonstrate the advantages of the Koopman-based calculation.