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.