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A Real-Time Control Barrier Function-Based Safety Filter for Motion Planning with Arbitrary Road Boundary Constraints

Jianye Xu, Chang Che, Bassam Alrifaee

TL;DR

This work addresses the lack of formal safety guarantees for learning-based motion planners navigating arbitrary road boundaries. It introduces a real-time safety filter built on High-order Control Barrier Functions (TTCBF) that operates on polylines by using a pseudo-distance and a circle-based occupancy to tightly approximate the vehicle's footprint, enabling a Quadratic Program that minimally adjusts the nominal control at each step. The approach provides $2n_{ ext{cir}}$ collision-avoidance constraints per circle and demonstrates feasibility and safety in a nonlinear kinematic bicycle model across four complex road scenarios, achieving execution at up to $40$ Hz. The practical impact lies in enabling safe, real-time verification and augmentation of learning-enabled planners without conservative overapproximations, making it suitable for challenging autonomous driving contexts with intricate road geometries.

Abstract

We present a real-time safety filter for motion planning, such as learning-based methods, using Control Barrier Functions (CBFs), which provides formal guarantees for collision avoidance with road boundaries. A key feature of our approach is its ability to directly incorporate road geometries of arbitrary shape without resorting to conservative overapproximations. We formulate the safety filter as a constrained optimization problem in the form of a Quadratic Program (QP). It achieves safety by making minimal, necessary adjustments to the control actions issued by the nominal motion planner. We validate our safety filter through extensive numerical experiments across a variety of traffic scenarios featuring complex roads. The results confirm its reliable safety and high computational efficiency (execution frequency up to 40 Hz). Code & Video Demo: github.com/bassamlab/SigmaRL

A Real-Time Control Barrier Function-Based Safety Filter for Motion Planning with Arbitrary Road Boundary Constraints

TL;DR

This work addresses the lack of formal safety guarantees for learning-based motion planners navigating arbitrary road boundaries. It introduces a real-time safety filter built on High-order Control Barrier Functions (TTCBF) that operates on polylines by using a pseudo-distance and a circle-based occupancy to tightly approximate the vehicle's footprint, enabling a Quadratic Program that minimally adjusts the nominal control at each step. The approach provides collision-avoidance constraints per circle and demonstrates feasibility and safety in a nonlinear kinematic bicycle model across four complex road scenarios, achieving execution at up to Hz. The practical impact lies in enabling safe, real-time verification and augmentation of learning-enabled planners without conservative overapproximations, making it suitable for challenging autonomous driving contexts with intricate road geometries.

Abstract

We present a real-time safety filter for motion planning, such as learning-based methods, using Control Barrier Functions (CBFs), which provides formal guarantees for collision avoidance with road boundaries. A key feature of our approach is its ability to directly incorporate road geometries of arbitrary shape without resorting to conservative overapproximations. We formulate the safety filter as a constrained optimization problem in the form of a Quadratic Program (QP). It achieves safety by making minimal, necessary adjustments to the control actions issued by the nominal motion planner. We validate our safety filter through extensive numerical experiments across a variety of traffic scenarios featuring complex roads. The results confirm its reliable safety and high computational efficiency (execution frequency up to 40 Hz). Code & Video Demo: github.com/bassamlab/SigmaRL
Paper Structure (18 sections, 19 equations, 10 figures, 1 table)

This paper contains 18 sections, 19 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Control Barrier Function Preliminaries
  4. Optimal Control Problem Formulation
  5. cbf-Based Safety Filter
  6. Pseudo-Distance
  7. cbf Construction
  8. Vehicle Geometry Approximation
  9. Case Study with Kinematic Bicycle Model
  10. Numerical Experiments
  11. Simulation Environment and Traffic Scenarios
  12. Simulation Environment
  13. Traffic Scenarios
  14. Simulation Parameters
  15. Computational Efficiency
  16. ...and 3 more sections

Figures (10)

  • Figure 1: Visualization of drivable sets $\mathcal{X}^{\mathrm{left}}_{\mathrm{road}}$ (beneath the left boundary $\mathcal{L}^{\mathrm{left}}_{\mathrm{road}}$), $\mathcal{X}^{\mathrm{right}}_{\mathrm{road}}$ (above the right boundary $\mathcal{L}^{\mathrm{right}}_{\mathrm{road}}$), and $\mathcal{X}_{\mathrm{road}}$ (within the road boundaries, jointly imposed by both boundaries). Vehicle in blue. Circles approximating the vehicle in gray, and cbf $h$ conceptually depicted in black arrows (details in \ref{['sec:vehicle-approximation']}).
  • Figure 2: Pseudo-distance from an arbitrary point $p$ to the line segment $G=(\overline{p_{1}p_{2}},\bm{t}_{1},\bm{t}_{2})$ziegler2014trajectory.
  • Figure 3: Gradient field of the pseudo-distance above an example polyline with three line segments.
  • Figure 4: Overapproximation of a rectangular vehicle with length $\ell$ and width $w$ using three identical, equidistant circles with radius $r_{\mathrm{cir}}$.
  • Figure 5: 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.
  • ...and 5 more figures

Theorems & Definitions (3)

  • Definition 1: Forward invariant set ames2019control
  • Definition 2: Extended class-$\mathcal{K}$ function ames2019control
  • Definition 3: Relative degree xiao2019control