Local Stability of PD Controlled Bipedal Walking Robots

TL;DR

This paper proves stability for PD control laws in hybrid bipedal robots, extending continuous system results to hybrid dynamics with impacts, and validates findings through simulation.

Contribution

It extends stability analysis of PD control to hybrid, underactuated bipedal robots and demonstrates exponential boundedness of their periodic orbits.

Findings

PD control stabilizes bipedal walking with proper gain tuning.

Hybrid zero dynamics assumptions ensure impact robustness.

Simulation confirms theoretical stability results.

Abstract

We establish stability results for PD tracking control laws in bipedal walking robots. Stability of PD control laws for continuous robotic systems is an established result, and we extend this for hybrid robotic systems, an alternating sequence of continuous and discrete events. Bipedal robots have the leg-swing as the continuous event, and the foot-strike as the discrete event. In addition, bipeds largely have underactuations due to the interactions between feet and ground. For each continuous event, we establish that the convergence rate of the tracking error can be regulated via appropriate tuning of the PD gains; and for each discrete event, we establish that this convergence rate sufficiently overcomes the nonlinear impacts by assumptions on the hybrid zero dynamics. The main contributions are 1) Extension of the stability results of PD control laws for underactuated robotic systems, and 2) Exponential ultimate boundedness of hybrid periodic orbits under the assumption of exponential stability of their projections to the hybrid zero dynamics. Towards the end, we will validate these results in a 2-link bipedal walker in simulation.