Benjamin Morris

A Restricted Poincaré Map for Determining Exponentially Stable Periodic Orbits in Systems with Impulse Effects: Application to Bipedal Robots

Systems with impulse effects form a special class of hybrid systems that consist of an ordinary, time-invariant differential equation (ODE), a co-dimension one switching surface, and a re-initialization rule. The exponential stability of a periodic orbit in a C1-nonlinear systems with impulse effects can be studied by linearizing the Poincaré return map around a fixed point and evaluating its eigenvalues. However, in feedback design-where one may be employing an iterative technique to shape the periodic orbit subject to it being exponentially stable—recomputing and re-linearizing the Poincaré return map at each iteration can be very cumbersome. For a non- linear system with impulse effects that possesses an invariant hybrid subsystem and the transversal dynamics is sufficiently exponentially fast, it is shown that exponential stability of a periodic orbit can be determined on the basis of the restricted Poincaré map, that is, the Poincaré return map associated with the invariant subsystem. The result is illustrated on a walking gait for an underactuated planar bipedal robot.

Hybrid Invariance in Bipedal Robots with Series Compliant Actuators

Stable walking motions in bipedal robots can be modeled as asymptotically stable periodic orbits in nonlinear systems with impulse effects. The method of hybrid zero dynamics, previously used to analyze planar walking in bipeds with one degree of underactuation, is extended to address the increased degrees of underactuation and the additional impact invariance conditions that arise when actuator dynamics are explicitly modeled. The resultant controller is parameterized and includes a discrete feedback in the parameters that is active only in the instantaneous double support phase. The controller design method is illustrated on a five-link planar walker with series compliant actuation, that is, a robot where a compliant element has been deliberately inserted between each actuated joint and its corresponding motor in order to increase the overall energy efficiency of locomotion

A Policy for Open-Loop Attenuation of Disturbance Effects Caused by Uncertain Ground Properties in Running

Outside of the laboratory, accurate models of ground impact dynamics are either difficult or impossible to obtain. Instead, a rigid ground model is often used in gait and controller design, which simplifies the system model and allows attention to remain focused on other aspects of running. In real-world terrain this simplification may overlook important dynamic effects. Immediately following a foot touchdown event, sensitivity to ground stiffness is at its highest and at the same time the accuracies of state estimates are at their lowest. Even if ground stiffness is known and state estimates are accurate, actuator bandwidth limitations make immediate compensation difficult. Taking inspiration from nature, we propose a novel solution to attenuate the effects of unexpected ground stiffness changes using a unified control system comprised of hardware passive dynamics and open-loop software control policies.