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

Mouhyemen Khan, Tatsuya Ibuki, Abhijit Chatterjee

TL;DR

A unified framework where the implicit surface itself acts as a CBF, using Gaussian process (GP) implicit surface (GPIS) to represent the safety boundaries, using safety samples which are derived from sensor measurements to condition the GP.

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.

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

TL;DR

A unified framework where the implicit surface itself acts as a CBF, using Gaussian process (GP) implicit surface (GPIS) to represent the safety boundaries, using safety samples which are derived from sensor measurements to condition the GP.

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.
Paper Structure (28 sections, 25 equations, 7 figures, 1 algorithm)

This paper contains 28 sections, 25 equations, 7 figures, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. BACKGROUND PRELIMINARIES
  3. Control Barrier Function
  4. Implicit Surfaces
  5. Gaussian Process Regression
  6. PROBLEM STATEMENT
  7. PROPOSED METHODOLOGY
  8. Data Processing
  9. Sparse Gaussian Control Barrier Function
  10. Lie Derivatives of Sparse Gaussian CBF
  11. Safe Control using Sparse Gaussian CBF
  12. TEST CASE I : MANIPULATOR SIMULATION
  13. Stanford Bunny Implicit Surface Modeling
  14. Offline Training
  15. Evaluation Metric
  16. ...and 13 more sections

Figures (7)

  • Figure 1: The safety surface of a volumetric object of interest is modeled as a sparse Gaussian CBF using safety samples.
  • Figure 2: The Stanford bunny defines the safe set boundary via Gaussian CBFs: (left) point cloud with surface normals, (middle) Gaussian CBF, (right) sparse Gaussian CBF.
  • Figure 3: The reference trajectory passes through the bunny's back and ear regions for both synthesized CBFs without (left) and with (right) sparsity.
  • Figure 4: The training times per local dataset is shown for both the CBFs (top). The time plots (bottom) are non-negative, indicating no safety violations occurred.
  • Figure 5: (Left) Gaussian CBFs trained online from local point cloud samples (white discs). (Right) Sparse Gaussian CBFs using pseudo-inputs (black discs) with local data; both show $0$-isosurfaces modeling the bunny surface.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Definition 1: Control Barrier Function cbf_Ames2017
  • Remark 1
  • Remark 2
  • Remark 3