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Safety-Critical Stabilization of Force-Controlled Nonholonomic Mobile Robots

Tianyu Han, Bo Wang

TL;DR

The paper tackles safety-critical stabilization of force-controlled nonholonomic mobile robots, addressing nonholonomic constraints by designing a nominal GAS controller in polar coordinates for the error dynamics and enforcing safety with a Cartesian CBF. The main method combines a strict Lyapunov function, derived from a polar-coordinate nominal controller, with a reciprocal CBF construction for cascaded kinematic–kinetic subsystems via integrator backstepping, all unified through a gamma_m-quadratic program. Key contributions include a Lyapunov function that remains valid on any compact set, a backstepping-based CBF framework for cascaded systems, and a feasible QP that guarantees forward invariance of the safety set while achieving stabilization. The results demonstrate practical safety-critical control suitable for autonomous parking and inter-vehicle collision avoidance, with a path toward handling input saturation and multi-agent formations in future work.

Abstract

We present a safety-critical controller for the problem of stabilization for force-controlled nonholonomic mobile robots. The proposed control law is based on the constructions of control Lyapunov functions (CLFs) and control barrier functions (CBFs) for cascaded systems. To address nonholonomicity, we design the nominal controller that guarantees global asymptotic stability and local exponential stability for the closed-loop system in polar coordinates and construct a strict Lyapunov function valid on any compact sets. Furthermore, we present a procedure for constructing CBFs for cascaded systems, utilizing the CBF of the kinematic model through integrator backstepping. Quadratic programming is employed to combine CLFs and CBFs to integrate both stability and safety in the closed loop. The proposed control law is time-invariant, continuous along trajectories, and easy to implement. Our main results guarantee both safety and local asymptotic stability for the closed-loop system.

Safety-Critical Stabilization of Force-Controlled Nonholonomic Mobile Robots

TL;DR

The paper tackles safety-critical stabilization of force-controlled nonholonomic mobile robots, addressing nonholonomic constraints by designing a nominal GAS controller in polar coordinates for the error dynamics and enforcing safety with a Cartesian CBF. The main method combines a strict Lyapunov function, derived from a polar-coordinate nominal controller, with a reciprocal CBF construction for cascaded kinematic–kinetic subsystems via integrator backstepping, all unified through a gamma_m-quadratic program. Key contributions include a Lyapunov function that remains valid on any compact set, a backstepping-based CBF framework for cascaded systems, and a feasible QP that guarantees forward invariance of the safety set while achieving stabilization. The results demonstrate practical safety-critical control suitable for autonomous parking and inter-vehicle collision avoidance, with a path toward handling input saturation and multi-agent formations in future work.

Abstract

We present a safety-critical controller for the problem of stabilization for force-controlled nonholonomic mobile robots. The proposed control law is based on the constructions of control Lyapunov functions (CLFs) and control barrier functions (CBFs) for cascaded systems. To address nonholonomicity, we design the nominal controller that guarantees global asymptotic stability and local exponential stability for the closed-loop system in polar coordinates and construct a strict Lyapunov function valid on any compact sets. Furthermore, we present a procedure for constructing CBFs for cascaded systems, utilizing the CBF of the kinematic model through integrator backstepping. Quadratic programming is employed to combine CLFs and CBFs to integrate both stability and safety in the closed loop. The proposed control law is time-invariant, continuous along trajectories, and easy to implement. Our main results guarantee both safety and local asymptotic stability for the closed-loop system.
Paper Structure (13 sections, 6 theorems, 40 equations, 4 figures)

This paper contains 13 sections, 6 theorems, 40 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Constructions of CBFs
  3. Control Barrier Functions and Safety-Critical Control
  4. Constructions of CBFs for Cascaded Systems
  5. Problem Formulation and Main Results
  6. Stabilization and Constructions of CLFs
  7. Control of the error kinematics
  8. Velocity tracking
  9. Constructions of strict Lyapunov functions
  10. Construction of the CBF
  11. Safety-Critical Control Design
  12. Simulation Results
  13. Conclusion

Key Result

Theorem 1

Consider the system eq:cascade and the admission set $\mathcal{C}\subset\mathbb{R}^n$. Suppose that we know a CBF $B_1:\operatorname{int}\mathcal{C}\to \mathbb{R}_{>0}$ for system eq:nls and a continuously differentiable "virtual" controller $x_2^*:\mathbb{R}^n\to\mathbb{R}^m$ such that for some $\alpha_B\in\mathcal{K}$ and for all $x_1\in\operatorname{int}\mathcal{C}$. Then the function $B:\math

Figures (4)

  • Figure 1: Illustration of the mobile robot paths in stabilization (Example 1).
  • Figure 2: Convergence of the configuration variables of the mobile robot in polar coordinates (Example 1).
  • Figure 3: Illustration of the mobile robot paths in stabilization (Example 2).
  • Figure 4: Convergence of the configuration variables of the mobile robot in polar coordinates (Example 2).

Theorems & Definitions (15)

  • Definition 1: Safety ames2019control
  • Definition 2: Control Lyapunov Function (CLF)
  • Definition 3: Control Barrier Function (CBF)
  • Theorem 1: Integrator backstepping
  • proof
  • Proposition 1: Nominal controller
  • proof
  • Remark 1
  • Proposition 2
  • proof
  • ...and 5 more