Control Barrier Functions for Collision Avoidance Between Strongly Convex Regions

2023-06-23

AI Summary

This work addresses the non-conservative collision avoidance problem between control-affine robotic systems and convex obstacles, formulating a differentiable Control Barrier Function (CBF) based on the minimum distance as the safety metric. The core challenge lies in the fact that the minimum distance is typically defined implicitly via optimization and is generally non-differentiable. To overcome this, we first introduce the class of strongly convex mappings, which rigorously guarantees the continuity and differentiability of the minimum distance. We then design an ordinary differential equation (ODE) derived from the Karush–Kuhn–Tucker (KKT) conditions to enable real-time analytical updates of both the minimum distance and its gradient. The framework supports heterogeneous convex set avoidance—e.g., ellipsoid–polyhedron interactions—and exact convex set algebraic operations without conservative approximations. In simulation, the method enables millisecond-scale quadratic programming (QP) solving for a quadrotor navigating a dense obstacle corridor, significantly improving safety guarantees and avoidance accuracy.

Abstract

In this paper, we focus on non-conservative collision avoidance between robots and obstacles with control affine dynamics and convex shapes. System safety is defined using the minimum distance between the safe regions associated with robots and obstacles. However, collision avoidance using the minimum distance as a control barrier function (CBF) can pose challenges because the minimum distance is implicitly defined by an optimization problem and thus nonsmooth in general. We identify a class of state-dependent convex sets, defined as strongly convex maps, for which the minimum distance is continuously differentiable, and the distance derivative can be computed using KKT solutions of the minimum distance problem. In particular, our formulation allows for ellipsoid-polytope collision avoidance and convex set algebraic operations on strongly convex maps. We show that the KKT solutions for strongly convex maps can be rapidly and accurately updated along state trajectories using a KKT solution ODE. Lastly, we propose a QP incorporating the CBF constraints and prove strong safety under minimal assumptions on the QP structure. We validate our approach in simulation on a quadrotor system navigating through an obstacle-filled corridor and demonstrate that CBF constraints can be enforced in real time for state-dependent convex sets without overapproximations.

Problem

• Non-conservative collision avoidance for robots
• Differentiable minimum distance for convex sets
• Real-time CBF constraints for safety

Innovation

• Strongly convex maps for CBF
• KKT solution ODE updates
• QP with CBF constraints