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Stable Walking for Bipedal Locomotion under Foot-Slip via Virtual Nonholonomic Constraints

Leonardo Colombo, Álvaro Rodríguez Abella, Alexandre Anahory Simoes, Anthony Bloch

Abstract

Foot slip is a major source of instability in bipedal locomotion on low-friction or uncertain terrain. Standard control approaches typically assume no-slip contact and therefore degrade when slip occurs. We propose a control framework that explicitly incorporates slip into the locomotion model through virtual nonholonomic constraints, which regulate the tangential stance-foot velocity while remaining compatible with the virtual holonomic constraints used to generate the walking gait. The resulting closed-loop system is formulated as a hybrid dynamical system with continuous swing dynamics and discrete impact events. A nonlinear feedback law enforces both classes of constraints and yields a slip-compatible hybrid zero dynamics manifold for the reduced-order locomotion dynamics. Stability of periodic walking gaits is characterized through the associated Poincaré map, and numerical results illustrate stabilization under slip conditions.

Stable Walking for Bipedal Locomotion under Foot-Slip via Virtual Nonholonomic Constraints

Abstract

Foot slip is a major source of instability in bipedal locomotion on low-friction or uncertain terrain. Standard control approaches typically assume no-slip contact and therefore degrade when slip occurs. We propose a control framework that explicitly incorporates slip into the locomotion model through virtual nonholonomic constraints, which regulate the tangential stance-foot velocity while remaining compatible with the virtual holonomic constraints used to generate the walking gait. The resulting closed-loop system is formulated as a hybrid dynamical system with continuous swing dynamics and discrete impact events. A nonlinear feedback law enforces both classes of constraints and yields a slip-compatible hybrid zero dynamics manifold for the reduced-order locomotion dynamics. Stability of periodic walking gaits is characterized through the associated Poincaré map, and numerical results illustrate stabilization under slip conditions.

Paper Structure

This paper contains 11 sections, 4 theorems, 25 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Hybrid Locomotion Framework
  3. Hybrid locomotion model for walking with foot slip
  4. Virtual holonomic constraints and hybrid zero dynamics
  5. Output dynamics, gait parameterization and stability
  6. Virtual Nonholonomic Constraints for Slip Regulation and Stable Walking
  7. Virtual affine nonholonomic constraints
  8. Control Design
  9. Stability of Periodic Walking Gaits
  10. Numerical Results
  11. Conclusions and Future Work

Key Result

Proposition 1

Assume that $A(\cdot)$ has constant rank $1$ on the stance phase domain, and that $M_s(q)$ in eq:Ms has full row rank $1$ for every $q$ in that domain. Then, locally on the stance phase domain, there exists a smooth feedback law rendering the constraint eq:affine_constraint_sec3 invariant under the

Figures (2)

  • Figure 3: Step-to-step response under variable slip conditions. Top: prescribed slip level at each step. Middle: pre-impact forward velocity for the open-loop and controlled cases. Bottom: slip-output magnitude under the controlled case.
  • Figure 4: Hip trajectory in the sagittal plane for the open-loop and controlled cases under variable slip conditions.

Theorems & Definitions (13)

  • Definition 1
  • Proposition 1
  • proof
  • Remark 1
  • Remark 2
  • Proposition 2
  • proof
  • Definition 2
  • Remark 3
  • Theorem 1
  • ...and 3 more