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.

A common approach in designing legged robots is to build fully actuated machines and control the machine dynamics entirely in software, carefully avoiding impacts and expending a lot of energy. However, these machines are outperformed by their human and animal counterparts. Animals achieve their impressive agility, efficiency, and robustness through a close integration of passive dynamics, implemented through mechanical components, and neural control. Robots can benefit from this same integrated approach, but a strong theoretical framework is required to design the passive dynamics of a machine and exploit them for control. For this framework, we use a bipedal spring-mass model, which has been shown to approximate the dynamics of human locomotion. This paper reports the first implementation of spring-mass walking on a bipedal robot. We present the use of template dynamics as a control objective exploiting the engineered passive spring-mass dynamics of the ATRIAS robot. The results highlight the benefits of combining passive dynamics with dynamics-based control and open up a library of spring-mass model-based control strategies for dynamic gait control of robots.

This paper presents a methodology for achieving efficient multi-domain underactuated bipedal walking on compliant robots by formally emulating gaits produced by the Spring Loaded Inverted Pendulum (SLIP). With the goal of achieving locomotion that displays phases of double and single support, a hybrid system model is formulated that faithfully represents the full-order dynamics of a compliant walking robot. The SLIP model is used as a bases for constructing human-inspired controllers that yield a dimension reduction through the use of hybrid zero dynamics. This allows for the formulation of an optimization problem that produces hybrid zero dynamics that best represents a SLIP model walking gait, while simultaneously ensuring the proper reduction in dimensionality that can be utilized to produce stable periodic orbits, i.e., walking gaits. The end result is stable robotic walking in simulation and, when implemented on the compliant robot ATRIAS, experimentally realized dynamic multi-domain locomotion.

Legged robots enjoy kilohertz control rates but are still making incremental gains towards becoming as nimble as animals. In contrast, bipedal animals are amazingly robust runners despite lagged state feedback from protracted neuromechanical delays. Based on evidence from biological experiments, we posit that much of disturbance rejection can be offloaded from feedback control and encoded into feed-forward pre-reflexive behaviors called preflexes. We present a framework for the offline numerical generation of preflex behaviors to optimally stabilize legged locomotion tasks in the presence of response delays. By coupling directly collocated trajectory optimizations, we optimize the preflexive motion of a simple bipedal running model to recover from uncertain terrain geometry using minimal actuator work. In simulation, the optimized preflex maneuver showed 30-77% economy improvements over a level-ground strategy when responding to terrain deviating just 2-4cm from the nominal condition. We claim this “preflex-and-replan” framework for designing efficient and robust gaits is amenable to a variety of robots and extensible to arbitrary locomotion tasks.