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Unifying Complementarity Constraints and Control Barrier Functions for Safe Whole-Body Robot Control

Rafael I. Cabral Muchacho, Riddhiman Laha, Florian T. Pokorny, Luis F. C. Figueredo, Nilanjan Chakraborty

TL;DR

This work addresses safety-critical, whole-body robot control by linking two dominant approaches: complementarity-based methods and control barrier functions (CBFs). It proves their equivalence for sampled-data, first-order systems, first in the single-constraint setting and then in the general multi-constraint case, using mapping and KKT arguments to show that LCQP and CBF-QP yield the same optimal velocity $\mathbf{u}^*$. The contributions include a formal equivalence result, a geometric interpretation, and numerical validation on a 3-DOF planar robot demonstrating identical trajectories and quantifying near-machine-precision agreement. The findings enable transferring robustness guarantees and solver advances between frameworks, offering a unified perspective and practical benefits for safe, reactive robotic control, with future work extending to higher-order dynamics and broader constraint classes.

Abstract

Safety-critical whole-body robot control demands reactive methods that ensure collision avoidance in real-time. Complementarity constraints and control barrier functions (CBF) have emerged as core tools for ensuring such safety constraints, and each represents a well-developed field. Despite addressing similar problems, their connection remains largely unexplored. This paper bridges this gap by formally proving the equivalence between these two methodologies for sampled-data, first-order systems, considering both single and multiple constraint scenarios. By demonstrating this equivalence, we provide a unified perspective on these techniques. This unification has theoretical and practical implications, facilitating the cross-application of robustness guarantees and algorithmic improvements between complementarity and CBF frameworks. We discuss these synergistic benefits and motivate future work in the comparison of the methods in more general cases.

Unifying Complementarity Constraints and Control Barrier Functions for Safe Whole-Body Robot Control

TL;DR

This work addresses safety-critical, whole-body robot control by linking two dominant approaches: complementarity-based methods and control barrier functions (CBFs). It proves their equivalence for sampled-data, first-order systems, first in the single-constraint setting and then in the general multi-constraint case, using mapping and KKT arguments to show that LCQP and CBF-QP yield the same optimal velocity . The contributions include a formal equivalence result, a geometric interpretation, and numerical validation on a 3-DOF planar robot demonstrating identical trajectories and quantifying near-machine-precision agreement. The findings enable transferring robustness guarantees and solver advances between frameworks, offering a unified perspective and practical benefits for safe, reactive robotic control, with future work extending to higher-order dynamics and broader constraint classes.

Abstract

Safety-critical whole-body robot control demands reactive methods that ensure collision avoidance in real-time. Complementarity constraints and control barrier functions (CBF) have emerged as core tools for ensuring such safety constraints, and each represents a well-developed field. Despite addressing similar problems, their connection remains largely unexplored. This paper bridges this gap by formally proving the equivalence between these two methodologies for sampled-data, first-order systems, considering both single and multiple constraint scenarios. By demonstrating this equivalence, we provide a unified perspective on these techniques. This unification has theoretical and practical implications, facilitating the cross-application of robustness guarantees and algorithmic improvements between complementarity and CBF frameworks. We discuss these synergistic benefits and motivate future work in the comparison of the methods in more general cases.

Paper Structure

This paper contains 16 sections, 2 theorems, 34 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Constraints for Safe Whole-Body Robot Control
  5. Safe Control Approaches
  6. Complementarity-based Approach
  7. CBF-based Approach
  8. Equivalence of Solutions
  9. Single Constraint Case
  10. General Case
  11. Geometric Lens on the Equivalence
  12. Discussion
  13. Numerical Validation
  14. Setup
  15. Results
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\mathbf{b} \in \mathbb{R}^m$, and $\mathbf{A} \in \mathbb{R}^{m \times n}$ with rows $\mathbf{a}_i$ assuming $\lVert \mathbf{a}_i\rVert\neq 0$. Using $\mathbf{G}$ as in eq:g-operator, define where $\mathbf{H} = \mathbf{G}(\mathbf{A})$. Then the problems have the same optimal solutions.

Figures (2)

  • Figure 1: A 3-DoF planar robot is guided from an initial configuration (left) to reach a goal with the end effector, depicted here through the star. The robot follows the complementarity and the QP-CBF policies, leading to identical paths. The illustration in the center shows an example configuration along the path, and the one on the right shows the robot configuration when reaching the goal. The notation described in the right figure applies equally to the left and center figures.
  • Figure 2: The safety constraint $h'\geq0$ (top) and the solution error $e$ (bottom) are plotted at each time step of the simulation.

Theorems & Definitions (3)

  • Remark 1
  • Theorem 1: Equivalence of Solutions
  • Corollary 1: Convexity of Safe Control