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Adaptive Non-linear Centroidal MPC with Stability Guarantees for Robust Locomotion of Legged Robots

Mohamed Elobaid, Giulio Turrisi, Lorenzo Rapetti, Giulio Romualdi, Stefano Dafarra, Tomohiro Kawakami, Tomohiro Chaki, Takahide Yoshiike, Claudio Semini, Daniele Pucci

TL;DR

The paper addresses robust locomotion for legged robots under constant disturbances by reformulating online centroidal MPC with an adaptive-control-inspired, Lyapunov-backed design. It introduces a coordinate change and adaptive law that, together with stabilizing MPC constraints and friction-cone feasibility, yields stability guarantees and bounded tracking error. The approach is validated experimentally on the humanoid ergoCub and the quadruped Aliengo, demonstrating improved robustness to payload changes and pushes, with open-source code. This work advances practical, provably stable locomotion for general-purpose robots, enabling more reliable operation in unstructured environments.

Abstract

Nonlinear model predictive locomotion controllers based on the reduced centroidal dynamics are nowadays ubiquitous in legged robots. These schemes, even if they assume an inherent simplification of the robot's dynamics, were shown to endow robots with a step-adjustment capability in reaction to small pushes, and, moreover, in the case of uncertain parameters - as unknown payloads - they were shown to be able to provide some practical, albeit limited, robustness. In this work, we provide rigorous certificates of their closed loop stability via a reformulation of the centroidal MPC controller. This is achieved thanks to a systematic procedure inspired by the machinery of adaptive control, together with ideas coming from Control Lyapunov functions. Our reformulation, in addition, provides robustness for a class of unmeasured constant disturbances. To demonstrate the generality of our approach, we validated our formulation on a new generation of humanoid robots - the 56.7 kg ergoCub, as well as on a commercially available 21 kg quadruped robot, Aliengo.

Adaptive Non-linear Centroidal MPC with Stability Guarantees for Robust Locomotion of Legged Robots

TL;DR

The paper addresses robust locomotion for legged robots under constant disturbances by reformulating online centroidal MPC with an adaptive-control-inspired, Lyapunov-backed design. It introduces a coordinate change and adaptive law that, together with stabilizing MPC constraints and friction-cone feasibility, yields stability guarantees and bounded tracking error. The approach is validated experimentally on the humanoid ergoCub and the quadruped Aliengo, demonstrating improved robustness to payload changes and pushes, with open-source code. This work advances practical, provably stable locomotion for general-purpose robots, enabling more reliable operation in unstructured environments.

Abstract

Nonlinear model predictive locomotion controllers based on the reduced centroidal dynamics are nowadays ubiquitous in legged robots. These schemes, even if they assume an inherent simplification of the robot's dynamics, were shown to endow robots with a step-adjustment capability in reaction to small pushes, and, moreover, in the case of uncertain parameters - as unknown payloads - they were shown to be able to provide some practical, albeit limited, robustness. In this work, we provide rigorous certificates of their closed loop stability via a reformulation of the centroidal MPC controller. This is achieved thanks to a systematic procedure inspired by the machinery of adaptive control, together with ideas coming from Control Lyapunov functions. Our reformulation, in addition, provides robustness for a class of unmeasured constant disturbances. To demonstrate the generality of our approach, we validated our formulation on a new generation of humanoid robots - the 56.7 kg ergoCub, as well as on a commercially available 21 kg quadruped robot, Aliengo.
Paper Structure (15 sections, 2 theorems, 17 equations, 2 figures, 1 table)

This paper contains 15 sections, 2 theorems, 17 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Notation and nomenclature
  4. The centroidal momentum dynamics
  5. Parametric-pure feedback forms
  6. Problem statement
  7. Online centroidal MPC with stability and robustness guarantees
  8. Robust adaptive redesign
  9. Stable Centroidal MPC for robust locomotion
  10. Validation and experiments
  11. Do we need the stabilizing constraints ?
  12. The case of a humanoid robot
  13. The case of a quadruped robot
  14. Comments on repeatability and performance
  15. Conclusion

Key Result

Lemma III.1

Consider the centroidal dynamics (momentumdyn_ct), and let Assumption assumption:1 hold true, then the feedback: with $u_n$ given by (adaptive_feedback) and $\nu$ an additional term solving the inequality together with the coordinates change (coordinates_change) and the adaptation law (adaptation) solve Problem problem_statement.

Figures (2)

  • Figure 1: Left - nominal prediction horizon and controller frequency. Center - shorter horizon and nominal frequency. Right - nominal horizon and lower frequency. In the last two cases, the proposed method succeeds in stabilizing the robot's motion as opposed to the nominal one thanks to the additional constraints (\ref{['stability_cstr']}).
  • Figure 2: Top left: The nominal feedback (5) (scaled by robot the mass for clarity) for the humanoid experiment. Top right: the extra stabilizing feedback term $\nu$. Bottom left: the proposed MPC satisfying the first stabilizing constraint in (17). Bottom right: the nominal MPC violating the stabilizing constraint when the quadruped fails (see accompanying media).

Theorems & Definitions (9)

  • Remark II.1
  • Remark II.2
  • Remark III.1
  • Lemma III.1
  • proof
  • Remark III.2
  • Proposition III.1
  • Remark III.3
  • Remark IV.1