Proximity Queries for Absolutely Continuous Parametric Curves

Abstract

In motion planning problems for autonomous robots, such as self-driving cars, the robot must ensure that its planned path is not in close proximity to obstacles in the environment. However, the problem of evaluating the proximity is generally non-convex and serves as a significant computational bottleneck for motion planning algorithms. In this paper, we present methods for a general class of absolutely continuous parametric curves to compute: (i) the minimum separating distance, (ii) tolerance verification, and (iii) collision detection. Our methods efficiently compute bounds on obstacle proximity by bounding the curve in a convex region. This bound is based on an upper bound on the curve arc length that can be expressed in closed form for a useful class of parametric curves including curves with trigonometric or polynomial bases. We demonstrate the computational efficiency and accuracy of our approach through numerical simulations 1 of several proximity problems.

Intent-Aware Probabilistic Trajectory Estimation for Collision Prediction with Uncertainty Quantification

“…We employ the collision prediction methods described in [17] to find the intersection between the boundary of the uncertainty region, given in Equation (6), and the agent's parametric trajectory, Equation (7). This method is computationally efficient and relies on interval optimization techniques to find the point intersection without sacrificing accuracy.…”

“…Using these basis functions, collision prediction can be quickly and analytically checked and a new plan can be computed by modifying the trajectory parameters. The method used in this paper for collision prediction is detailed in [17] and can be implemented with any absolutely continuous curve.…”