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Synthesizing Robust Walking Gaits via Discrete-Time Barrier Functions with Application to Multi-Contact Exoskeleton Locomotion

Maegan Tucker, Kejun Li, Aaron D. Ames

TL;DR

This paper introduces a robustness metric for bipedal locomotion based on the estimated size of the forward-invariant set of the discrete step-to-step dynamics, certified with discrete-time barrier functions applied to a reduced-order Poincaré-map model. By projecting full-order dynamics to a low-dimensional manifold and solving a barrier-function-based condition, it yields a maximal robust radius $r^*$ that quantifies how perturbations affect gait stability. The authors implement a simulation-in-the-loop framework to synthesize robust nominal gaits and validate them experimentally on the Atalante exoskeleton for both flat-foot and multi-contact walking, showing that $r^*$-driven gaits outperform those optimized solely for stability in real hardware. The approach is compatible with various control frameworks and provides a practical, scalable robustness certificate for high-dimensional legged systems.

Abstract

Successfully achieving bipedal locomotion remains challenging due to real-world factors such as model uncertainty, random disturbances, and imperfect state estimation. In this work, we propose a novel metric for locomotive robustness -- the estimated size of the hybrid forward invariant set associated with the step-to-step dynamics. Here, the forward invariant set can be loosely interpreted as the region of attraction for the discrete-time dynamics. We illustrate the use of this metric towards synthesizing nominal walking gaits using a simulation-in-the-loop learning approach. Further, we leverage discrete-time barrier functions and a sampling-based approach to approximate sets that are maximally forward invariant. Lastly, we experimentally demonstrate that this approach results in successful locomotion for both flat-foot walking and multi-contact walking on the Atalante lower-body exoskeleton.

Synthesizing Robust Walking Gaits via Discrete-Time Barrier Functions with Application to Multi-Contact Exoskeleton Locomotion

TL;DR

This paper introduces a robustness metric for bipedal locomotion based on the estimated size of the forward-invariant set of the discrete step-to-step dynamics, certified with discrete-time barrier functions applied to a reduced-order Poincaré-map model. By projecting full-order dynamics to a low-dimensional manifold and solving a barrier-function-based condition, it yields a maximal robust radius that quantifies how perturbations affect gait stability. The authors implement a simulation-in-the-loop framework to synthesize robust nominal gaits and validate them experimentally on the Atalante exoskeleton for both flat-foot and multi-contact walking, showing that -driven gaits outperform those optimized solely for stability in real hardware. The approach is compatible with various control frameworks and provides a practical, scalable robustness certificate for high-dimensional legged systems.

Abstract

Successfully achieving bipedal locomotion remains challenging due to real-world factors such as model uncertainty, random disturbances, and imperfect state estimation. In this work, we propose a novel metric for locomotive robustness -- the estimated size of the hybrid forward invariant set associated with the step-to-step dynamics. Here, the forward invariant set can be loosely interpreted as the region of attraction for the discrete-time dynamics. We illustrate the use of this metric towards synthesizing nominal walking gaits using a simulation-in-the-loop learning approach. Further, we leverage discrete-time barrier functions and a sampling-based approach to approximate sets that are maximally forward invariant. Lastly, we experimentally demonstrate that this approach results in successful locomotion for both flat-foot walking and multi-contact walking on the Atalante lower-body exoskeleton.
Paper Structure (9 sections, 1 theorem, 17 equations, 7 figures)

This paper contains 9 sections, 1 theorem, 17 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on Locomotion
  3. Robust Walking via Hybrid Forward Invariance
  4. Results
  5. Implementation Details
  6. Flat-Foot Walking
  7. Multi-Contact Walking
  8. Comparison to Optimizing for Stability
  9. Conclusion

Key Result

Theorem 1

Let $\mathbf{x}_{k+1} = P_{\mathbf{X}}(\mathbf{x}_k)$ be the Poincaré map restricted to the set $\mathbf{X} \cap \mathcal{S}$. If there exists a discrete-time barrier function for the set $\mathcal{I}$, then the set $\mathcal{I}$ is forward invariant and exponentially stable. If $\dim(\mathbf{X} \ca

Figures (7)

  • Figure 1: The framework developed in this paper optimizes locomotive robustness using forward invariance, certified via discrete-time barrier functions, as a metric for robustness.
  • Figure 2: Directed graphs describing the hybrid system domain structure for the a) flat-foot and b) multi-contact walking.
  • Figure 3: Model representations. a) The full system model is denoted by the generalized coordinates $x = (q_e^{\top},\dot q_e^{\top})^{\top}$ with $q_e := (p_b^{\top},\phi_b^{\top},q^{\top})^{\top} \in \mathbb{R}^3 \times SO(3) \times \mathcal{Q}$. Here $p_b \in \mathbb{R}^3$ and $\phi_b$ respectively denote the euclidean position and orientation of the global base frame $R_b$ relative to the world frame $R_w$. b) Here, the reduced-order representation of the model is illustrated, defined as the angular velocities of the global frame relative to the world frame, i.e., $\mathbf{x} := (\dot \phi_x, \dot \phi_y, \dot \phi_z)^{\top}$.
  • Figure 4: Diagram of sim-in-the-loop approach towards optimizing robustness.
  • Figure 5: Invariant sets identified in simulation compared to the values seen on hardware during the experiments.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Definition 1
  • Definition 2
  • Theorem 1