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A Learning-Based Control Barrier Function for Car-Like Robots: Toward Less Conservative Collision Avoidance

Jianye Xu, Bassam Alrifaee

TL;DR

The work tackles conservatism in collision avoidance for car-like robots by replacing circle-based safety margins with an MTV-based margin that accounts for actual geometry and heading. A relative-dynamics framework enables learning a differentiable neural network surrogate, $h_\theta(\boldsymbol{x}^{j\underline{i}})$, to approximate the non-differentiable MTV margin, yielding a CBF $h_{MTV}(\boldsymbol{x}^{j\underline{i}})=h_\theta(\boldsymbol{x}^{j\underline{i}})-e_{\max}$ with relative degree $r=2$. The authors provide a theoretical foundation for applying this margin to the nonlinear kinematic bicycle model, train the margin on a large dataset (≈80k points) with an ~2.8% width error, and implement a CBF-QP that minimally adjusts a nominal RL controller to enforce safety. Case studies on overtaking and bypassing demonstrate that the MTV-based margin reduces conservatism (e.g., 33.5% less lateral space for bypassing) while maintaining comparable computation times, enabling more efficient maneuvers in dense environments.

Abstract

We propose a learning-based Control Barrier Function (CBF) to reduce conservatism in collision avoidance for car-like robots. Traditional CBFs often use the Euclidean distance between robots' centers as a safety margin, which neglects their headings and approximates their geometries as circles. Although this simplification meets the smoothness and differentiability requirements of CBFs, it may result in overly conservative behavior in dense environments. We address this by designing a safety margin that considers both the robot's heading and actual shape, thereby enabling a more precise estimation of safe regions. Because this safety margin is non-differentiable, we approximate it with a neural network to ensure differentiability. In addition, we propose a notion of relative dynamics that makes the learning process tractable. In a case study, we establish the theoretical foundation for applying this notion to a nonlinear kinematic bicycle model. Numerical experiments in overtaking and bypassing scenarios show that our approach reduces conservatism (e.g., requiring 33.5% less lateral space for bypassing) without incurring significant extra computation time. Code: https://github.com/bassamlab/sigmarl

A Learning-Based Control Barrier Function for Car-Like Robots: Toward Less Conservative Collision Avoidance

TL;DR

The work tackles conservatism in collision avoidance for car-like robots by replacing circle-based safety margins with an MTV-based margin that accounts for actual geometry and heading. A relative-dynamics framework enables learning a differentiable neural network surrogate, , to approximate the non-differentiable MTV margin, yielding a CBF with relative degree . The authors provide a theoretical foundation for applying this margin to the nonlinear kinematic bicycle model, train the margin on a large dataset (≈80k points) with an ~2.8% width error, and implement a CBF-QP that minimally adjusts a nominal RL controller to enforce safety. Case studies on overtaking and bypassing demonstrate that the MTV-based margin reduces conservatism (e.g., 33.5% less lateral space for bypassing) while maintaining comparable computation times, enabling more efficient maneuvers in dense environments.

Abstract

We propose a learning-based Control Barrier Function (CBF) to reduce conservatism in collision avoidance for car-like robots. Traditional CBFs often use the Euclidean distance between robots' centers as a safety margin, which neglects their headings and approximates their geometries as circles. Although this simplification meets the smoothness and differentiability requirements of CBFs, it may result in overly conservative behavior in dense environments. We address this by designing a safety margin that considers both the robot's heading and actual shape, thereby enabling a more precise estimation of safe regions. Because this safety margin is non-differentiable, we approximate it with a neural network to ensure differentiability. In addition, we propose a notion of relative dynamics that makes the learning process tractable. In a case study, we establish the theoretical foundation for applying this notion to a nonlinear kinematic bicycle model. Numerical experiments in overtaking and bypassing scenarios show that our approach reduces conservatism (e.g., requiring 33.5% less lateral space for bypassing) without incurring significant extra computation time. Code: https://github.com/bassamlab/sigmarl

Paper Structure

This paper contains 21 sections, 3 theorems, 24 equations, 8 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Paper Contributions
  4. Notation
  5. Paper Structure
  6. Preliminaries
  7. Kinematic Bicycle Model
  8. Control Barrier Functions
  9. MTV-Based Safety Margin as cbf
  10. MTV-Based Safety Margin
  11. Learning-Based Safety-Margin Approximator
  12. Construction of the Control Barrier Function
  13. Case Study
  14. Relative Dynamics of the Kinematic Bicycle Model
  15. Optimal Control Problem Formulation
  16. ...and 6 more sections

Key Result

Theorem 1

Given a candidate hocbf $h(\bm{x})$ for system eq:general-dynamics as in def:hocbf, if the initial state satisfies $\bm{x}(t_0) \in \bigcap_{i=1}^{r} C_i$, then any Lipschitz continuous controller that satisfies eq:hocbf-constraint renders the set $\bigcap_{i=1}^{r} C_i$ forward invariant for all $t

Figures (8)

  • Figure 1: An example of conservatism caused by the circle approximation. Vehicle $i$ is prevented from overtaking $j$.
  • Figure 2: The kinematic bicycle model. $C$: center of gravity; $x, y$: $x$- and $y$-coordinates; $v$: velocity; $\beta$: slip angle; $\psi$: yaw angle; $\delta$: steering angle; $\ell_{wb}$: wheelbase; $\ell_{r}$: rear wheelbase.
  • Figure 3: An example illustrating \ref{['alg:mtv']} for the case where $g_{x_A^i} < 0$, $g_{y_A^i} < 0$, $g_{x_A^j} > 0$, and $g_{y_A^j} < 0$.
  • Figure 4: Flow diagram of the cbf-qp formulation with the MTV-based safety margin.
  • Figure 5: Overtaking scenario with c2c-based safety margin. The lowest safety margin occurs for $t \ge 5.4s$.
  • ...and 3 more figures

Theorems & Definitions (12)

  • Definition 1: Class $\mathcal{K}$ Function
  • Definition 2: cbf ames2014controlames2017control
  • Definition 3: Forward Invariant
  • Definition 4: Relative Degree xiao2019control
  • Definition 5: High-Order cbf xiao2019control
  • Theorem 1: Thm. 4 in xiao2019control
  • Theorem 2
  • proof
  • Remark 1
  • Theorem 3
  • ...and 2 more