Hybrid Event Shaping to Stabilize Periodic Hybrid Orbits

Many controllers for legged robotic systems leverage open- or closed-loop control at discrete hybrid events to enhance stability. These controllers appear in several well studied phenomena such as the Raibert stepping controller, paddle juggling, and swing leg retraction. This work introduces hybrid event shaping (HES): a generalized method for analyzing and designing stable hybrid event controllers. HES utilizes the saltation matrix, which gives a closed-form equation for the effect that hybrid events have on stability. We also introduce shape parameters, which are higher order terms that can be tuned completely independently from the system dynamics to promote stability. Optimization methods are used to produce values of these parameters that optimize a stability measure. Hybrid event shaping captures previously developed control methods while also producing new optimally stable trajectories without the need for continuous-domain feedback.

I. Introduction

In general, the walking and running gaits of legged robots are naturally unstable and challenging to control. Hybrid systems such as these are difficult to work with due to the discontinuities in state and dynamics that occur at hybrid events, such as toe touchdown. These discontinuities violate assumptions of standard controllers designed for purely continuous systems, and work is ongoing to adapt these controllers for hybrid systems [1], [2]. One strategy for hybrid control is to cancel out the effects of hybrid events by working with an invariant subsystem [3]–[5]. We propose instead that the effects of hybrid events are valuable due to rich control properties that can be used to stabilize trajectories of a hybrid system.