Walking gait optimization for accommodation of unknown terrain height variations

Abstract

We investigate the design of periodic gaits that will also function well in the presence of modestly uneven terrain. We use parameter optimization and, inspired by recent work of Dai and Tedrake, augment a cost function with terms that account for perturbations arising from a finite set of terrain height changes. Trajectory and control deviations are related to a nominal periodic orbit via a mechanical phase variable, which is more natural than comparing solutions on the basis of time. The mechanical phase variable is also used to penalize more heavily deviations that persist "late" into the gait. The method is illustrated both in simulation and in experiments on a planar bipedal robot.

Virtual Constraints and Hybrid Zero Dynamics for Realizing Underactuated Bipedal Locomotion

“…The optimization criterion can be selected to account for uncertainty in the hybrid model. Terrain variation is addressed in [18]. Nonholonomic virtual constraints have just been introduced in [17]; they allow swing foot placement to be planned as a function of velocity.…”

“…In the case of systems with more than one degree of underactuation, systematic methods have been developed which modify the virtual constraints "at the boundary" and allow the conditions to be met [30]. Very straightforward implementations of the result are presented in a robotics context in [12] and [18]. Remark: When low-dimensional pendulum models are used for gait design, they approximate the swing phase dynamics and the impact map is ignored.…”

“…References [18], [13] show how to include common gait perturbations, such as terrain variations, into the optimization. The cost function (77) on the periodic orbit is augmented with terms that account for additional solutions of the model responding to perturbations.…”

DIRTREL: Robust Trajectory Optimization with Ellipsoidal Disturbances and LQR Feedback

“…Griffin and Grizzle [8] and Dai and Tedrake [3] augment direct trajectory optimization methods with cost functionals that weight the tracking performance of linear feedback controllers. However, there are several important differences from our approach.…”

“…However, there are several important differences from our approach. In [8], a fixed-gain proportional-integral controller is assumed, placing potentially significant limits on closed-loop performance. In [3], the elements of the timevarying cost-to-go matrix associated with an LQR tracking controller are added as decision variables to the optimization problem, and disturbances are handled through a sampling scheme that scales poorly with the dimensionality of the disturbance vector.…”