Optimizing robust limit cycles for legged locomotion on unknown terrain

Abstract

Abstract-While legged animals are adept at traversing rough landscapes, it remains a very challenging task for a legged robot to negotiate unknown terrain. Control systems for legged robots are plagued by dynamic constraints from underactuation, actuator power limits, and frictional ground contact; rather than relying purely on disturbance rejection, considerable advantage can be obtained by planning nominal trajectories which are more easily stabilized. In this paper, we present an approach for designing nominal periodic trajectories for legged robots that maximize a measure of robustness against uncertainty in the geometry of the terrain. We propose a direct collocation method which solves simultaneously for a nominal periodic control input, for many possible one-step solution trajectories (using ground profiles drawn from a distribution over terrain), and for the periodic solution to a jump Riccati equation which provides an expected infinite-horizon cost-to-go for each of these samples. We demonstrate that this trajectory optimization scheme can recover the known deadbeat openloop control solution for the Spring Loaded Inverted Pendulum (SLIP) on unknown terrain. Moreover, we demonstrate that it generalizes to other models like the bipedal compass gait walker, resulting in a dramatic increase in the number of steps taken over moderate terrain when compared against a limit cycle optimized for efficiency only.

L<inf>2</inf>-gain optimization for robust bipedal walking on unknown terrain

“…It is well known that the compass gait robot has a very narrow region of attraction and can easily fall down over rough terrain. We compare two limit cycles, the passive one and the robust one [6]. We run simulations of the robot walking over unknown terrain with virtual slope drawn from…”

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.…”

“…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. These design choices substantially increase the size and complexity of the nonlinear program that must be solved, limiting the algorithm's applicability to complex robot planning problems.…”

“…This paper builds on previous work on robust motion planning based on direct trajectory optimization [26,3] and differential dynamic programming (DDP) [27,6,29]. Robust motion planning algorithms often differ in the precise notion of robustness that they seek to optimize.…”

Robust Phase-Space Planning for Agile Legged Locomotion over Various Terrain Topologies

“…Optimal control for legged locomotion over rough terrains is explored in [20,21,22,23,24]. The work in [25] proposed an effective control technique to stabilize non-periodic motions of under-actuated robots, with a focus on walking over uneven terrain.…”