Restricted Discrete Invariance and Self-Synchronization For Stable Walking of Bipedal Robots

TL;DR

This paper introduces a novel approach to achieve stable bipedal walking by identifying invariant low-dimensional manifolds within the hybrid models of locomotion, extending from the 3D LIP model to complex 9-DOF bipeds, and introduces the concept of self-synchronization.

Contribution

It proposes a new method to find invariant submanifolds in hybrid biped models, including the novel concept of self-synchronization, for stable walking gaits.

Findings

Invariant manifolds lead to stable periodic gaits.

Self-synchronization aligns sagittal and frontal plane motions.

Numerical simulations demonstrate asymptotic stability.

Abstract

Models of bipedal locomotion are hybrid, with a continuous component often generated by a Lagrangian plus actuators, and a discrete component where leg transfer takes place. The discrete component typically consists of a locally embedded co-dimension one submanifold in the continuous state space of the robot, called the switching surface, and a reset map that provides a new initial condition when a solution of the continuous component intersects the switching surface. The aim of this paper is to identify a low-dimensional submanifold of the switching surface, which, when it can be rendered invariant by the closed-loop dynamics, leads to asymptotically stable periodic gaits. The paper begins this process by studying the well-known 3D Linear Inverted Pendulum (LIP) model, where analytical results are much easier to obtain. A key contribution here is the notion of \textit{self-synchronization}, which refers to the periods of the pendular motions in the sagittal and frontal planes tending to a common period. The notion of invariance resulting from the study of the 3D LIP model is then extended to a 9-DOF 3D biped. A numerical study is performed to illustrate that asymptotically stable walking may be obtained.