Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Robust Multi-Agent Safety via Tube-Based Tightened Exponential Barrier Functions

Armel Koulong, Ali Pakniyat

TL;DR

The paper addresses safety for nonlinear multi-agent systems with disturbances, focusing on high relative-degree dynamics modeled in Brunovsky form. It couples a robust state-based ancillary feedback that confines tracking errors to robust positively invariant tubes with nominal trajectory planning, enabling formal safety via tightened exponential control barrier functions. By deriving explicit RPI tube bounds and their support-function-based tightening terms, the authors guarantee that any nominal trajectory satisfying the tightened eCBF constraints yields a provably safe true trajectory under disturbances. The framework is implemented as a distributed Tube-MPC that preserves affine eCBF structure, enabling scalable real-time safety in leader-follower formations with obstacles. The results demonstrate robust, collision-free navigation and formation maintenance, illustrating practical applicability to safety-critical multi-agent coordination tasks.

Abstract

This paper presents a constructive framework for synthesizing provably safe controllers for nonlinear multi-agent systems subject to bounded disturbances. The methodology applies to systems representable in Brunovsky canonical form, accommodating arbitrary-order dynamics in multi-dimensional spaces. The central contribution is a method of constraint tightening that formally couples robust error feedback with nominal trajectory planning. The key insight is that the design of an ancillary feedback law, which confines state errors to a robust positively invariant (RPI) tube, simultaneously provides the exact information needed to ensure the safety of the nominal plan. Specifically, the geometry of the resulting RPI tube is leveraged via its support function to derive state-dependent safety margins. These margins are then used to systematically tighten the high relative-degree exponential control barrier function (eCBF) constraints imposed on the nominal planner. This integrated synthesis guarantees that any nominal trajectory satisfying the tightened constraints corresponds to a provably safe trajectory for the true, disturbed system. We demonstrate the practical utility of this formal synthesis method by implementing the planner within a distributed Model Predictive Control (MPC) scheme, which optimizes performance while inheriting the robust safety guarantees.

Robust Multi-Agent Safety via Tube-Based Tightened Exponential Barrier Functions

TL;DR

The paper addresses safety for nonlinear multi-agent systems with disturbances, focusing on high relative-degree dynamics modeled in Brunovsky form. It couples a robust state-based ancillary feedback that confines tracking errors to robust positively invariant tubes with nominal trajectory planning, enabling formal safety via tightened exponential control barrier functions. By deriving explicit RPI tube bounds and their support-function-based tightening terms, the authors guarantee that any nominal trajectory satisfying the tightened eCBF constraints yields a provably safe true trajectory under disturbances. The framework is implemented as a distributed Tube-MPC that preserves affine eCBF structure, enabling scalable real-time safety in leader-follower formations with obstacles. The results demonstrate robust, collision-free navigation and formation maintenance, illustrating practical applicability to safety-critical multi-agent coordination tasks.

Abstract

This paper presents a constructive framework for synthesizing provably safe controllers for nonlinear multi-agent systems subject to bounded disturbances. The methodology applies to systems representable in Brunovsky canonical form, accommodating arbitrary-order dynamics in multi-dimensional spaces. The central contribution is a method of constraint tightening that formally couples robust error feedback with nominal trajectory planning. The key insight is that the design of an ancillary feedback law, which confines state errors to a robust positively invariant (RPI) tube, simultaneously provides the exact information needed to ensure the safety of the nominal plan. Specifically, the geometry of the resulting RPI tube is leveraged via its support function to derive state-dependent safety margins. These margins are then used to systematically tighten the high relative-degree exponential control barrier function (eCBF) constraints imposed on the nominal planner. This integrated synthesis guarantees that any nominal trajectory satisfying the tightened constraints corresponds to a provably safe trajectory for the true, disturbed system. We demonstrate the practical utility of this formal synthesis method by implementing the planner within a distributed Model Predictive Control (MPC) scheme, which optimizes performance while inheriting the robust safety guarantees.
Paper Structure (10 sections, 4 theorems, 65 equations, 3 figures)

This paper contains 10 sections, 4 theorems, 65 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation and Methodology
  3. State Based Ancillary Feedback:
  4. Robust Positive Invariant (RPI) Tube Construction:
  5. Tightened Safety and Support Functions
  6. Main Result
  7. Proposed Tightened eCBF Functions
  8. Distributed Tube-MPC eCBF Formation Framework
  9. Numerical Example
  10. Conclusion

Key Result

Lemma 2.1

Let $\|w^{\,i}(t)\|\le \bar{w}_i$ for all $t \geq t_0$ and that $Q_i$ and the corresponding $P_i$ in eq:Lyapunov_eq_1 are such that, $\frac{\lambda_{\min}(Q_{i})}{2\lambda_{\max}(P_{i})} > L_{f^i}$. Then the per-agent ellipsoidal tube is robust positively invariant for all $\rho_i$ satisfying $\square$

Figures (3)

  • Figure 1: The considered communication topology ${\mathcal{G}}$ and the augmented graph $\bar{\mathcal{G}}$ for the example in Section \ref{['NUMERICALEXAMPLE']}.
  • Figure 2: The corresponding evolution of the positional states $x^i_1 \in \mathbb{R}^2$, for the leader $i=0$, and follower agents $i \in \{1,2,\cdots,5\}$ with respect to obstacles
  • Figure 3: Relative $[x^i_{1,1},x^i_{1,2}]$ state Error between Agents and Leader in y-direction

Theorems & Definitions (10)

  • Definition 2.1: Robust Positive Invariance (RPI)
  • Lemma 2.1: RPI ellipsoid
  • Definition 2.2: Relative degree $r$
  • Definition 2.3
  • Definition 2.4: Tightened Safety Function
  • Definition 2.5: Support Function of RPI Tubes
  • Remark 2.1
  • Lemma 2.2
  • Lemma 2.3: Support Function of Tubes
  • Theorem 3.1: Forward Invariance of Set