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Safe and Dynamically-Feasible Motion Planning using Control Lyapunov and Barrier Functions

Pol Mestres, Carlos Nieto-Granda, Jorge Cortés

TL;DR

This work tackles safe, dynamically feasible motion planning for control-affine systems by marrying RRT-based search with control Lyapunov and barrier function theory. The authors introduce the Compatible-CLF-CBF-RRT (C-CLF-CBF-RRT), which uses rigorous CLF-CBF compatibility verification to guarantee that each planned edge can be tracked by a safe controller, avoiding unreliable open-loop trajectories. They provide tractable, closed-form or QCQP-based compatibility tests for linear dynamics with polytopic or ellipsoidal obstacles, and extend the framework to higher relative degree constraints via HOCBFs. The approach yields probabilistic completeness and superior safety and robustness compared to prior CLF-CBF+RRT methods, with demonstrated efficiency in simulation and hardware. Collectively, this work advances planning under safety and stability guarantees by integrating compatibility verification directly into sampling-based planning, enabling practical deployment on real robots.

Abstract

This paper considers the problem of designing motion planning algorithms for control-affine systems that generate collision-free paths from an initial to a final destination and can be executed using safe and dynamically-feasible controllers. We introduce the C-CLF-CBF-RRT algorithm, which produces paths with such properties and leverages rapidly exploring random trees (RRTs), control Lyapunov functions (CLFs) and control barrier functions (CBFs). We show that C-CLF-CBF-RRT is computationally efficient for linear systems with polytopic and ellipsoidal constraints, and establish its probabilistic completeness. We showcase the performance of C-CLF-CBF-RRT in different simulation and hardware experiments.

Safe and Dynamically-Feasible Motion Planning using Control Lyapunov and Barrier Functions

TL;DR

This work tackles safe, dynamically feasible motion planning for control-affine systems by marrying RRT-based search with control Lyapunov and barrier function theory. The authors introduce the Compatible-CLF-CBF-RRT (C-CLF-CBF-RRT), which uses rigorous CLF-CBF compatibility verification to guarantee that each planned edge can be tracked by a safe controller, avoiding unreliable open-loop trajectories. They provide tractable, closed-form or QCQP-based compatibility tests for linear dynamics with polytopic or ellipsoidal obstacles, and extend the framework to higher relative degree constraints via HOCBFs. The approach yields probabilistic completeness and superior safety and robustness compared to prior CLF-CBF+RRT methods, with demonstrated efficiency in simulation and hardware. Collectively, this work advances planning under safety and stability guarantees by integrating compatibility verification directly into sampling-based planning, enabling practical deployment on real robots.

Abstract

This paper considers the problem of designing motion planning algorithms for control-affine systems that generate collision-free paths from an initial to a final destination and can be executed using safe and dynamically-feasible controllers. We introduce the C-CLF-CBF-RRT algorithm, which produces paths with such properties and leverages rapidly exploring random trees (RRTs), control Lyapunov functions (CLFs) and control barrier functions (CBFs). We show that C-CLF-CBF-RRT is computationally efficient for linear systems with polytopic and ellipsoidal constraints, and establish its probabilistic completeness. We showcase the performance of C-CLF-CBF-RRT in different simulation and hardware experiments.

Paper Structure

This paper contains 15 sections, 10 theorems, 42 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Control Lyapunov and Control Barrier Functions
  4. Rapidly-exploring Random Trees (RRTs)
  5. Problem Statement
  6. CLF and BNCBF Compatibility Verification
  7. Compatibility Verification for General Dynamics and Obstacles
  8. Compatibility Verification for Linear Systems and Polytopic Obstacles
  9. Compatibility Verification for Linear Systems and Ellipsoidal Obstacles
  10. Compatibility Verification for Higher Relative Degree Systems
  11. C-CLF-CBF-RRT
  12. CLF-CBF Compatible Paths
  13. Algorithm Description
  14. The COMPATIBILITY function
  15. Analysis of C-CLF-CBF-RRT

Key Result

Proposition 2.3

(Sufficient Condition for BNCBF): Suppose there is an extended class $\mathcal{K}_{\infty}$ function $\alpha:\mathbb{R}\to\mathbb{R}$ such that, for all $x\in\mathbb{R}^n$, there exists $u\in\mathbb{R}^m$ with for all $i\in\mathcal{I}(x)$. Then, $h$ is a BNCBF of $\mathcal{C}$.

Figures (1)

  • Figure 1: Visual aid for the arguments described in the proof of Lemma .

Theorems & Definitions (48)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Proposition 4.1
  • proof
  • Remark 4.2
  • proof
  • Remark 4.3
  • proof
  • ...and 38 more