Collision-Free Encoding for Chance-Constrained Nonconvex Path Planning

The path planning methods based on nonconvex constrained optimization, such as mixed-integer linear programming (MILP), have found various important applications, ranging from unmanned aerial vehicles (UAVs) and autonomous underwater vehicles (AUVs) to space vehicles. Moreover, their stochastic extensions have enabled risk-aware path planning, which explicitly limits the probability of failure to a user-specified bound. However, a major challenge of those path planning methods is constraint violation between discrete time steps. In the existing approach, a path is represented by a sequence of waypoints and the safety constraints (e.g., obstacle avoidance) are imposed on waypoints. Therefore, the trajectory between waypoints could violate the safety constraints. A naive continuous-time extension results in unrealistic computation cost. In this paper, we propose a novel approach to ensure constraint satisfaction between waypoints without employing a continuous-time formulation. The key idea is to enforce that the same inequality constraint is satisfied on any two adjacent time steps, under assumptions of polygonal obstacles and straight line trajectory between waypoints. The resulting problem encoding is MILP, which can be solved efficiently by commercial solvers. Thus, we also introduce novel extensions to risk-allocation path planners with improved scalability for real-world scenarios and run-time performance. While the proposed encoding approach is general, the particular emphasis of this paper is placed on the chance-constrained, nonconvex path-planning problem (CNPP). We provide extensive simulation results on CNPP to demonstrate the path safety and scalability of our encoding and related path planners.