Stability Principle Inherent in Wheel Gait of Planar X-Shaped Walker Generated Using Constant Torque Drive and Mechanical Stoppers

Abstract

Since the late 19th century when the first walking toys were developed, it has been known that mechanical stoppers at the hip joint are crucial for generating stable passive dynamic walking. Recent research on passive-dynamic and limit-cycle walkers has also confirmed that mechanical stoppers at the hip and knee joints are effective for generating stable walking motion, but theoretical research on how this mechanical constraint enhances the overall gait stability remains insufficient. This paper introduces a planar X-shaped walker equipped with mechanical stoppers at the hip joint, and investigates the effect of the mechanical constraint on the stability of the wheel gait generated by constant torque drive on a downslope. By simply falling forward while using the stoppers to constrain itself to the target impact posture, the robot can generate a highly stable wheel gait. We divide the motion of one step into four phases, derive approximate analytical solutions for the state error transition function matrix in each phase using a linearized model, and analyze the increase or decrease in the state error norm using metrics such as its maximum singular value. Numerical simulations demonstrate that while two phases are unstable in terms of the increase in the state error norm, the remaining two phases are stable, resulting in overall asymptotic stability.