Exponentially Stabilizing Controllers for Multi-Contact 3D Bipedal Locomotion

Abstract

Models of bipedal walking are hybrid with continuous-time phases representing the Lagrangian stance dynamics and discrete-time transitions representing the impact of the swing leg with the walking surface. The design of continuous-time feedback controllers that exponentially stabilize periodic gaits for hybrid models of underactuated 3D bipedal walking is a significant challenge. We recently introduced a method based on an iterative sequence of optimization problems involving bilinear matrix inequalities (BMIs) to systematically design stabilizing continuous-time controllers for single domain hybrid models of underactuated bipedal robots with point feet. This paper addresses the exponential stabilization problem for multi-contact walking gaits with nontrivial feet. A family of parameterized continuous-time controllers is proposed for different phases of the walking cycle. The BMI algorithm is extended to the multi-domain hybrid models of anthropomorphic 3D walking locomotion to look for stabilizing controller parameters. The Poincaré map is addressed and a new set of sufficient conditions is presented that guarantees the convergence of the BMI algorithm to a stabilizing set of controller parameters at a finite number of iterations. The power of the algorithm is ultimately demonstrated through the design of stabilizing virtual constraint controllers for dynamic walking of a 3D humanoid model with 28 state variables and 275 controller parameters.

On the Existence and Uniqueness of Poincaré Maps for Systems With Impulse Effects

“…This map executes the trajectory of the system from a point on the guard to its next The Poincaré map for systems with impulse effects have been introduced in [19] (see also [36]). It was mainly employed in the search of periodic gaits (limit cycles) of bipedal robots, together with the use of several methods such as geometric abelian Routh reduction, hybrid zero dynamics and virtual constraints, hybrid Hamiltonian systems, symmetries, etc [5], [10], [11], [14], [25], [26], [20], [21], [22], [23], [32], [33], [34]. These methods permit one to gain extra information regarding the behavior of the system, which provide advantages for the construction of the Poincaré map.…”

Hierarchical and Safe Motion Control for Cooperative Locomotion of Robotic Guide Dogs and Humans: A Hybrid Systems Approach

“…Then it translates the exponential stabilization problem into a recursive optimization problem that is set up based on linear and bilinear matrix inequalities. Sufficient conditions for the convergence of the algorithm to a set of stabilizing parameters have been addressed in [28], [29].…”