Eric Westervelt

Hybrid zero dynamics of planar biped walkers

Planar, underactuated, biped walkers form an important domain of applications for hybrid dynamical systems. This paper presents the design of exponentially stable walking controllers for general planar bipedal systems that have one degree-of-freedom greater than the number of available actuators. The within-step control action creates an attracting invariant set - a two-dimensional zero dynamics submanifold of the full hybrid model

Stable walking of a 7-DOF biped robot

whose restriction dynamics admits a scalar linear time-invariant return map. Exponentially stable periodic orbits of the zero dynamics correspond to exponentially stabilizable orbits of the full model. A convenient parameterization of the hybrid zero dynamics is imposed through the choice of a class of output functions. Parameter optimization is used to tune the hybrid zero dynamics in order to achieve closed-loop, exponentially stable walking with low energy consumption, while meeting natural kinematic and dynamic constraints. The general theory developed in the paper is illustrated on a five link walker, consisting of a torso and two legs with knees.

Switching and PI control of walking motions of planar biped walkers

The primary goal of this paper is to demonstrate a means to prove asymptotically stable walking in an underactuated, planar, five-link biped robot model. The analysis assumes a rigid contact model when the swing leg impacts the ground and an instantaneous double support phase. The specific robot model analyzed corresponds to a prototype under development by the Centre National de la Recherche Scientifique (CNRS), Paris, France. A secondary goal of the paper is to establish the viability of the theoretically motivated control law. This is explored in a number of ways. First, it is shown how known time trajectories, such as those determined on the basis of walking with minimal energy consumption, can be incorporated into the proposed controller structure. Secondly, various perturbations to the walking motion are introduced to verify disturbance rejection capability. Finally, the controller is demonstrated on a detailed simulator for the prototype which includes torque limits and a compliant model of the walking surface, and thus a noninstantaneous double support phase.

Design of asymptotically stable walking for a 5-link planar biped walker via optimization

A companion paper has addressed the problem of designing controllers that induce exponentially stable, periodic walking motions at a fixed walking rate for a planar, biped robot with one degree of underactuation. This note provides two additional control features: 1) the ability to compose such controllers to obtain walking at several discrete walking rates with guaranteed stability during the transitions; and 2) the ability to regulate the average walking rate to a continuum of values.

Zero dynamics of underactuated planar biped walkers

Closed-loop, asymptotically stable walking motions are designed for a 5-link, planar bipedal robot model with one degree of underactuation. Parameter optimization is applied to the hybrid zero dynamics, a 1-DOF invariant subdynamics of the full robot model, in order to create asymptotically stable orbits. Tuning the dynamics of this 1-DOF subsystem via optimization is interesting because asymptotically stable orbits of the zero dynamics correspond to asymptotically stabilizable orbits of the full hybrid model of the walker. The optimization process uses a sequential quadratic programming (SQP) algorithm and is able to satisfy kinematic and dynamic constraints while approximately minimizing energy consumption and ensuring stability. This is in contrast with traditional approaches to the design of walking controllers where approximately optimal walking (time-) trajectories are derived and then enforced on the robot using a trajectory tracking controller.

Controlled Periodic Motion in a Nonlinear System with Impulse Effects: Walking of a Biped Robot

Abstract The zero dynamics of a hybrid model of bipedal walking are introduced and studied for a class of N-link, planar robots with one degree of underactuation and outputs that depend only on the configuration variables. Asymptotically stable solutions of the zero dynamics correspond to asymptotically stabilizable orbits of the full hybrid model of the walker. The Poincare map of the zero dynamics is computed and proven to be diffeomorphic to a scalar, linear, time-invariant system, thereby rendering transparent the existence and stability properties of periodic orbits.

Feedback Control of Dynamic Bipedal Robot Locomotion

Abstract The goal is to demonstrate a means to prove asymptotically stable walking in a planar, under actuated, five-link biped robot model. The analysis assumes a rigid contact model when the swing leg impacts the ground and an instantaneous double-support phase: under theses hypotheses, the robot is modeled by a dynamic nonlinear system and an impulse model. The controller induces finite-time stabilization of four of the robots five degrees of freedom, resulting in a reduced Poincare stability analysis that can be carried out by computing a one dimensional map.