Gaussian Process Implicit Surfaces as Control Barrier Functions for Safe Robot Navigation

Abstract

Level set methods underpin modern safety techniques such as control barrier functions (CBFs), while also serving as implicit surface representations for geometric shapes via distance fields. Inspired by these two paradigms, we propose a unified framework where the implicit surface itself acts as a CBF. We leverage Gaussian process (GP) implicit surface (GPIS) to represent the safety boundaries, using safety samples which are derived from sensor measurements to condition the GP. The GP posterior mean defines the implicit safety surface (safety belief), while the posterior variance provides a robust safety margin. Although GPs have favorable properties such as uncertainty estimation and analytical tractability, they scale cubically with data. To alleviate this issue, we develop a sparse solution called sparse Gaussian CBFs. To the best of our knowledge, GPIS have not been explicitly used to synthesize CBFs. We validate the approach on collision avoidance tasks in two settings: a simulated 7-DOF manipulator operating around the Stanford bunny, and a quadrotor navigating in 3D around a physical chair. In both cases, Gaussian CBFs (with and without sparsity) enable safe interaction and collision-free execution of trajectories that would otherwise intersect the objects.