Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hybrid Lyapunov-based feedback stabilization of bipedal locomotion based on reference spreading

Riccardo Bertollo, Gianni Lunardi, Andrea Del Prete, Luca Zaccarian

TL;DR

This work tackles robust longitudinal trajectory tracking for bipedal locomotion by formulating a hybrid LIPM in which foot switches occur when the CoM reaches the step boundary, with a reference trajectory parametrized by a timer via a reference-spreading mechanism. A nontrivial hybrid error coordinate is developed, yielding linear flow and nonlinear jump dynamics that are bounded in terms of the CoM position error, enabling Lyapunov-based stabilization through a saturated linear feedback gain design solved via LMIs; stability is proved locally with a certified basin of attraction. The approach is validated through simulations on a full humanoid model (Romeo), combining a lateral MPC and TSID for full-body control, and demonstrates advantages over standard MPC in handling asynchronous step timing and achieving capturing behavior. Practically, the method provides a rigorous, convex-optimization-based path to stabilize periodic locomotion and integrates smoothly with full-body planners and controllers for realistic bipedal robots.

Abstract

We propose a hybrid formulation of the linear inverted pendulum model for bipedal locomotion, where the foot switches are triggered based on the center of mass position, removing the need for pre-defined footstep timings. Using a concept similar to reference spreading, we define nontrivial tracking error coordinates induced by our hybrid model. These coordinates enjoy desirable linear flow dynamics and rather elegant jump dynamics perturbed by a suitable extended class ${\mathcal K}_\infty$ function of the position error. We stabilize this hybrid error dynamics using a saturated feedback controller, selecting its gains by solving a convex optimization problem. We prove local asymptotic stability of the tracking error and provide a certified estimate of the basin of attraction, comparing it with a numerical estimate obtained from the integration of the closed-loop dynamics. Simulations on a full-body model of a real robot show the practical applicability of the proposed framework and its advantages with respect to a standard model predictive control formulation.

Hybrid Lyapunov-based feedback stabilization of bipedal locomotion based on reference spreading

TL;DR

This work tackles robust longitudinal trajectory tracking for bipedal locomotion by formulating a hybrid LIPM in which foot switches occur when the CoM reaches the step boundary, with a reference trajectory parametrized by a timer via a reference-spreading mechanism. A nontrivial hybrid error coordinate is developed, yielding linear flow and nonlinear jump dynamics that are bounded in terms of the CoM position error, enabling Lyapunov-based stabilization through a saturated linear feedback gain design solved via LMIs; stability is proved locally with a certified basin of attraction. The approach is validated through simulations on a full humanoid model (Romeo), combining a lateral MPC and TSID for full-body control, and demonstrates advantages over standard MPC in handling asynchronous step timing and achieving capturing behavior. Practically, the method provides a rigorous, convex-optimization-based path to stabilize periodic locomotion and integrates smoothly with full-body planners and controllers for realistic bipedal robots.

Abstract

We propose a hybrid formulation of the linear inverted pendulum model for bipedal locomotion, where the foot switches are triggered based on the center of mass position, removing the need for pre-defined footstep timings. Using a concept similar to reference spreading, we define nontrivial tracking error coordinates induced by our hybrid model. These coordinates enjoy desirable linear flow dynamics and rather elegant jump dynamics perturbed by a suitable extended class function of the position error. We stabilize this hybrid error dynamics using a saturated feedback controller, selecting its gains by solving a convex optimization problem. We prove local asymptotic stability of the tracking error and provide a certified estimate of the basin of attraction, comparing it with a numerical estimate obtained from the integration of the closed-loop dynamics. Simulations on a full-body model of a real robot show the practical applicability of the proposed framework and its advantages with respect to a standard model predictive control formulation.
Paper Structure (17 sections, 5 theorems, 78 equations, 9 figures, 1 table)

This paper contains 17 sections, 5 theorems, 78 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Hybrid model
  3. Biped longitudinal dynamics hybrid formulation
  4. Reference signal
  5. Complete dynamics and error dynamics
  6. Feedback Control Selection
  7. Main stability result
  8. Simulations
  9. Full humanoid robot simulations
  10. Linear Model Predictive Control
  11. Swing Foot Trajectories
  12. Task-Space Inverse Dynamics
  13. Simulation overview
  14. Simulation Results
  15. Conclusions
  16. ...and 2 more sections

Key Result

Proposition 1

Solutions to eq:dyn_modified satisfy eq:eta when $x \in D$. Moreover, $\bar{v}/\omega - \bar{r} > 0$ and the expression of $\tau_\varepsilon$ in eq:eta is an extended class $\mathcal{K}_\infty$ function of $\varepsilon_p$. Equivalently, $\eta(0)=1$, $\eta$ is strictly increasing, $\lim\nolimits\limi

Figures (9)

  • Figure 1: Scheme of the hybrid linear inverted pendulum model.
  • Figure 2: Comparison between the nominal periodic trajectory (blue) and the proposed parametrization $x_r(\tau)$ in \ref{['eq:reference_tau']},\ref{['eq:tau_dynamics']} when the robot is late (red) or early (green) with respect to the reference: timer state (top), position (middle) and velocity (bottom).
  • Figure 3: Comparison among the set $\mathcal{E}$ in \ref{['eq:BoA_estimate']} (green), the set of initial conditions such that $\dot V$ is non increasing (blue) and the set of initial conditions such that the robot state converges to the periodic reference (red).
  • Figure 4: Time evolution of the robot's center of mass position (first plot) and velocity (second plot), compared to the reference trajectory \ref{['eq:reference_tau']},\ref{['eq:tau_dynamics']}. Corresponding input signal (third plot), where the black dotted lines characterize the saturation limits. Evolution of the Lyapunov function $V$ in \ref{['eq:robot_V']} along the simulated response (fourth plot).
  • Figure 5: General control framework for the full-body simulations.
  • ...and 4 more figures

Theorems & Definitions (5)

  • Proposition 1
  • Lemma 1
  • Lemma 2
  • Proposition 2
  • Theorem 1