Collision avoidance norms in trajectory planning

This paper studies norms that quantify safety of robotic vehicle trajectories. The main motivation behind this work is to automate the process of selecting safe motions in complex state spaces such as ones arising from environments cluttered with obstacles or when trajectories lie close to the permitted boundary of configuration space. An autonomous vehicle can plan obstacle-free trajectories in a known environment but the inherent uncertainty in sensing and motion could render these trajectories unsafe during execution. Therefore, in the presence of uncertainty it is crucial to predict in real-time the safety of planned trajectories through appropriate metrics or norms. There are a number of standard methods to weigh risks associated with vehicle behavior, for instance, based on the minimum distance to the closest obstacles, or on the average distance to obstacles along the trajectory. In this paper we study generalization of such norms based on the theory of Sobolev spaces. In particular, H^k Sobolev norms applied to the distance-to-obstacle function along a trajectory are based not only on its spatial properties but also on its time variation or frequency components. We show that this extra information renders an H¹ norm more effective for quantifying risk compared to the standard L^p norms. This is demonstrated for an analytical example as well as for a semi-realistic helicopter flying through an obstacle terrain.