A Theoretical Analysis of the Velocity Obstacle Method for Nonholonomic Vehicles and Underactuated Surface Vessels

Collision avoidance (CA) systems are pivotal for enabling vehicles to autonomously complete tasks in environments containing obstacles. With its low computational burden and underlying flexibility, the velocity obstacle (VO) algorithm presents a favorable method to avoid collisions, which is based on representing obstacles in the velocity space. In this study, we use the VO principle to form a reactive strategy for vehicles to avoid collisions with dynamic obstacles, which is applied to two different classes of systems, specifically nonholonomic vehicles and underactuated surface vessels. Instead of producing velocity references, the algorithm outputs collision-free directions, thus circumventing the need for explicitly controlling the vehicle speed. Moreover, the majority of existing VO approaches are only supported by simulations and experiments of specific CA scenarios, and the few studies that include some theoretical assurance are based on assumptions that cannot be guaranteed in the general case. In this article, we consider factors such as vehicle dynamics and constraints in a rigorous analysis of the algorithm. We analytically derive conditions ensuring feasibility of the avoidance maneuvers and overall safety of the vehicle, which provide intuitive requirements on the parameters of the algorithm. The theoretical results are supported through simulations and experiments of the strategy applied to a nonholonomic vehicle and an underactuated marine vessel.