Departmental Papers (ESE)


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.

Document Type

Journal Article

Subject Area

GRASP, Kodlab

Date of this Version

January 2003


Copyright 2003 IEEE. Reprinted from IEEE Transactions on Automatic Control, Volume 48, Issue 1, January 2003, pages 42-56.

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NOTE: At the time of publication, author Daniel Koditschek was affiliated with the University of Michigan. Currently (August 2005), he is a faculty member in the Department of Electrical and Systems Engineering at the University of Pennsylvania.


Bipeds, hybrid systems, Poincaré sections, underactuated system, zero dynamics



Date Posted: 04 August 2005

This document has been peer reviewed.