Do limit cycles matter in the long run? Stable orbits and sliding-mass dynamics emerge in task-optimal locomotion

We investigate the task-optimality of legged limit cycles and present numerical evidence supporting a simple general locomotion-planning template. Limit cycles have been foundational to the control and analysis of legged systems, but as robots move toward completing real-world tasks, are limit cycles practical in the long run? We address this question both figuratively and literally by solving for optimal strategies for long-horizon tasks spanning as many as 20 running steps. These scenarios were designed to embody practical locomotion tasks, such as evading a pursuer, and were formulated with minimal constraints (complete the task, minimize energy cost, and don't fall). By leveraging large-scale constrained optimization techniques, we numerically solve the trajectory for a reduced-order running model to optimally complete each scenario. We find, in the tested scenarios in flat terrain, that near-limit-cycle behaviors emerge after a transient period of acceleration and deceleration, suggesting limit cycles may be a useful, near-optimal planning target. On rough terrain, enforcing a limit cycle on every step only degrades gait economy by 2-5% compared to optimal 20-step look-ahead planning. When perturbing the scenario with a single “bump” in the road, the model converged in a manner giving the appearance of an exponentially stable orbit, despite not explicitly enforcing exponential stability. Further, we show that the transient periods of acceleration and deceleration may be near-optimally approximated by planning with a simple “sliding mass” template. These results support the notion that limit cycles can be useful approximations of task-optimal behavior, and thus are useful near-term targets for long-term planning.

I. Introduction

Applied robotics is a task-driven enterprise, but our best mathematical formulations rarely conform precisely to the task at hand. So by and large, robot control formulations are developed in constant compromise between what is task-relevant and what is mathematically tractable. Legged locomotion is no exception.