Braking within Barriers: Constructive Safety-Critical Control for Input-Constrained Vehicles via the Backup Set Method

TL;DR

This work tackles safe braking on asymmetric split-$\mu$ surfaces under friction-limited inputs by developing a backup-set safety framework built on control barrier functions. It introduces a constructive method to generate valid backup set–backup controller pairs using continuous-time Lyapunov equations and feedback linearization, enabling a predictive safety enforcement via a backup-CBF-QP that respects input bounds. The authors apply the method to a four-wheel vehicle model with tire and driver dynamics, providing theoretical conditions and practical guidelines, plus simulation results that show forward-invariant safety sets $\mathcal{S}_{\mathrm{I}}$ under braking constraints. The approach yields formal safety guarantees while balancing stopping distance and lateral stability, offering a viable pathway to safer, input-bounded braking in autonomous and assisted-vehicle systems.

Abstract

This paper presents a safety-critical control framework to maintain bounded lateral motions for vehicles braking on asymmetric surfaces. We synthesize a brake controller that assists drivers and guarantees safety against excessive lateral motions (i.e., prevents the vehicle from spinning out) while minimizing the stopping distance. We address this safety-critical control problem in the presence of input constraints, since braking forces are limited by the available friction on the road. We use backup control barrier functions for safe control design. As this approach requires the construction of a backup set and a backup controller, we propose a novel, systematic method to creating valid backup set-backup controller pairs based on feedback linearization and continuous-time Lyapunov equations. We use simple examples to demonstrate our proposed safety-critical control method. Finally, we implement our approach on a four-wheel vehicle model for braking on asymmetric surfaces and present simulation results.

Figures (10)

• Figure 1: A vehicle braking on an asymmetric (split-$\mu$) surface. The lateral kinematics (side slip angle and yaw rate) are kept within safe bounds with limited braking forces using the proposed safety-critical controller.
• Figure 2: Illustration of the constraint set $\mathcal{S}$ (green), the backup set $\mathcal{S}_\mathrm{b}$ (blue), and the invariant set $\mathcal{S}_{\mathrm{I}}$ (orange) obtained by enlarging the backup set. The backup set, centered at point $\mathbf{x}^*$ (black), is contained inside both the constraint set $\mathcal{S}$ and the set $\mathcal{S}_{\mathrm{ns}}$ (gray) where the backup controller does not saturate.
• Figure 3: Flow chart of the proposed method, summarizing the steps to generate valid backup set-backup controller pairs for use in the backup set method, which ultimately enables input-constrained safety-critical control.
• Figure 4: Illustration of the constraint set $\mathcal{S}$ (green), backup set $\mathcal{S}_\mathrm{b}$ (blue), backup controller $k_{\mathrm{b}}$ (black), and input constraints $\mathcal{S}_{\mathrm{ns}}$ (gray) for Example \ref{['ex:scalar']}.
• Figure 5: Trajectories of the system \ref{['eq:ex1']} under the backup controller (a), and the backup CBF-QP controller (b) with the corresponding control input (c) and (d). The backup controller, given by \ref{['eq:ex1_kb']}, maintains the forward invariance of the backup set $\mathcal{S}_\mathrm{b}$ (blue) from \ref{['eq:ex1_hb']}. The backup CBF-QP, given by \ref{['eq:QP2']}, ensures the forward invariance of the expanded set $\mathcal{S}_{\mathrm{I}}$ (orange) that is within the constraint set $\mathcal{S}$ (green). Both controllers satisfy the input constraints.

Theorems & Definitions (2)

• proof
• proof