Nonconvex Obstacle Avoidance using Efficient Sampling-Based Distance Functions

TL;DR

This work tackles nonconvex obstacle avoidance for robots with nonlinear dynamics and nonconvex shapes by introducing sampling-based distance functions to efficiently estimate robot-obstacle proximity. It formulates obstacle avoidance as a (nonsmooth) control barrier function (CBF) problem and derives a practical, possibly nonsmooth, CBF framework that remains robust to disturbances. A key contribution is the development of a sampling-based distance surrogate, with a generalized-gradient analysis that ensures safety guarantees through Filippov solutions, circumventing the computational bottleneck of exact distance computations. The method is demonstrated on an omnidirectional robot navigating nonconvex obstacles, and the authors analyze how performance scales with the number of samples, offering insights into computational efficiency and safety margins in real-time settings.

Abstract

We consider nonconvex obstacle avoidance where a robot described by nonlinear dynamics and a nonconvex shape has to avoid nonconvex obstacles. Obstacle avoidance is a fundamental problem in robotics and well studied in control. However, existing solutions are computationally expensive (e.g., model predictive controllers), neglect nonlinear dynamics (e.g., graph-based planners), use diffeomorphic transformations into convex domains (e.g., for star shapes), or are conservative due to convex overapproximations. The key challenge here is that the computation of the distance between the shapes of the robot and the obstacles is a nonconvex problem. We propose efficient computation of this distance via sampling-based distance functions. We quantify the sampling error and show that, for certain systems, such sampling-based distance functions are valid nonsmooth control barrier functions. We also study how to deal with disturbances on the robot dynamics in our setting. Finally, we illustrate our method on a robot navigation task involving an omnidirectional robot and nonconvex obstacles. We also analyze performance and computational efficiency of our controller as a function of the number of samples.

Figures (2)

• Figure 1: The IGT robot with its rotational joints $q_1$, $q_2$ and $q_3$, the end-effector position and the link names. Source: Lotte Rassaerts.
• Figure 2: Geometric representation with spheres. Source: Lotte Rassaerts.