A Sufficient Condition for Convex Hull Property in General Convex Spatio-Temporal Corridors

TL;DR

This paper establishes a sufficient condition ensuring the convex hull property in complex, time-dependent spatio-temporal corridors, enabling safer and more optimal trajectory planning for autonomous vehicles.

Contribution

It provides a theoretical proof for convex hull property in general convex spatio-temporal corridors, supporting complex shapes in motion planning.

Findings

Guarantees convex hull property in complex corridors

Reduces search space to O(1/n^2)

Results in smoother trajectories with less harsh braking

Abstract

Motion planning is one of the key modules in autonomous driving systems to generate trajectories for self-driving vehicles to follow. A common motion planning approach is to generate trajectories within semantic safe corridors. The trajectories are generated by optimizing parametric curves (e.g. Bezier curves) according to an objective function. To guarantee safety, the curves are required to satisfy the convex hull property, and be contained within the safety corridors. The convex hull property however does not necessary hold for time-dependent corridors, and depends on the shape of corridors. The existing approaches only support simple shape corridors, which is restrictive in real-world, complex scenarios. In this paper, we provide a sufficient condition for general convex, spatio-temporal corridors with theoretical proof of guaranteed convex hull property. The theorem allows for using more complicated shapes to generate spatio-temporal corridors and minimizing the uncovered search space to $O(\frac{1}{n^2})$ compared to $O(1)$ of trapezoidal corridors, which can improve the optimality of the solution. Simulation results show that using general convex corridors yields less harsh brakes, hence improving the overall smoothness of the resulting trajectories.