Asymptotically Stable Gait Generation and Instantaneous Walkability Determination for Planar Almost Linear Biped with Knees

TL;DR

This paper introduces a linearized model for planar biped robots with knees, enabling instant walkability assessment and stable gait generation through simplified equations of motion and control strategies.

Contribution

It extends existing linearization methods to include knee joints, allowing rapid computation of gait stability and walkability for complex bipedal robots.

Findings

Linearized model accurately predicts gait stability.

Instantaneous walkability determination is feasible.

Control inputs successfully generate stable gait in simulations.

Abstract

A class of planar bipedal robots with unique mechanical properties has been proposed, where all links are balanced around the hip joint, preventing natural swinging motion due to gravity. A common property of their equations of motion is that the inertia matrix is a constant matrix, there are no nonlinear velocity terms, and the gravity term contains simple nonlinear terms. By performing a Taylor expansion of the gravity term and making a linear approximation, it is easy to derive a linearized model, and calculations for future states or walkability determination can be performed instantaneously without the need for numerical integration. This paper extends the method to a planar biped robot model with knees. First, we derive the equations of motion, constraint conditions, and inelastic collisions for a planar 6-DOF biped robot, design its control system, and numerically generate a stable bipedal gait on a horizontal plane. Next, we reduce the equations of motion to a 3-DOF model, and derive a linearized model by approximating the gravity term as linear around the expansion point for the thigh frame angle. Through numerical simulations, we demonstrate that calculations for future states and walkability determination can be completed in negligible time. By applying control inputs to the obtained model, performing state-space realization, and then discretizing it, instantaneous walkability determination through iterative calculation becomes possible. Through detailed gait analysis, we discuss how the knee joint flexion angle and the expansion point affect the accuracy of the linear approximation, and the issues that arise when descending a small step.