Planning for the Unexpected: Explicitly Optimizing Motions for Ground Uncertainty in Running

We propose a method to generate actuation plans for a reduced order, dynamic model of bipedal running. This method explicitly enforces robustness to ground uncertainty. The plan generated is not a fixed body trajectory that is aggressively stabilized: instead, the plan interacts with the passive dynamics of the reduced order model to create emergent robustness. The goal is to create plans for legged robots that will be robust to imperfect perception of the environment, and to work with dynamics that are too complex to optimize in real-time. Working within this dynamic model of legged locomotion, we optimize a set of disturbance cases together with the nominal case, all with linked inputs. The input linking is nontrivial due to the hybrid dynamics of the running model but our solution is effective and has analytical gradients. The optimization procedure proposed is significantly slower than a standard trajectory optimization, but results in robust gaits that reject disturbances extremely effectively without any replanning required.

I. Introduction

Dynamic locomotion such as running and walking has many dimensions beyond position trajectories, which are merely one symptom of the resulting behavior. As such, new approaches are needed to incorporate powerful existing motion planning and control methods with the dynamic behaviors of legged locomotion. Complicating factors include underactuation, nonlinear hybrid dynamics, large system dimensionality and significant uncertainties in ground properties. However, legged locomotion is not so complex as it first appears, because most behaviors can be described by relatively simple reduced-order models, showing some promise for planning within this dynamic space. Many reduced-order models consist of a point mass body and a massless leg that can apply forces from a contact point toward the point mass, where body motion is only influenced by gravity and the forces applied by the leg. Examples of this type of model include the inverted pendulum (IP) model, the linear inverted pendulum (LIP) model, the spring loaded inverted pendulum (SLIP) model, and the actuated spring loaded inverted pendulum (ASLIP) model. The differentiating factor between these models is the calculation of the applied leg force.