Interval-wise Collision Avoidance for Autonomous Vehicles in Polygonal Domains

Journal article, 2026

This paper tailors the inter-collocation collision problem in between collocation points in optimization-based motion planning problems for autonomous vehicles. Both the vehicle and surrounding obstacles are modeled as convex polygons, explicitly defined by their vertex representations. The swept region occupied by the vehicle during a sampling interval is represented as the convex hull formed by the vehicle's vertices at the start and end poses. Leveraging these defined representations, the studied collision avoidance problem is reformulated as the separation between a convex hull and a convex polygon. From a vertex-based perspective, we propose three different methods: the hyperplane-based method, the dual method, and the support function method, to achieve separations. The first method applies the hyperplane separation theorem to directly determine hyperplanes that separate a convex hull from a convex polygon. The second constructs a dual problem calculating the distance between them and uses the strong duality of convex optimization problem to approach the separation. Based on the signed distance, the third obtains the explicit solutions to the separation by leveraging the support function. Finally, the effectiveness of the proposed formulations is demonstrated in two challenging scenarios: moose tests in confined environments and autonomous parking involving dynamic obstacles. Comparative results highlight their advantages in handling sharp obstacle corners and thin obstacles while achieving improved computational efficiency.