Hybrid Zero Dynamics of Planar Biped Walkers
General Robotics, Automation, Sensing and Perception Laboratory
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—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.
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Copyright 2003 IEEE. Reprinted from IEEE Transactions on Automatic Control, Volume 48, Issue 1, January 2003, pages 42-56. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to email@example.com. By choosing to view this document, you agree to all provisions of the copyright laws protecting it. NOTE: At the time of publication, author Daniel Koditschek was affiliated with the University of Michigan. Currently (June 2005), he is a faculty member in the Department of Electrical and Systems Engineering at the University of Pennsylvania.