Planar, underactuated, biped walkers form an important domain of
application for hybrid dynamical systems.
This dissertation presents the design of controllers that induce
exponentially stable dynamic walking for general planar biped robots
that have one degree of freedom greater than the number of available
actuators during the single support phase. The within-step control
action creates an attracting invariant seta two-dimensional zero
dynamics submanifold of the full hybrid modelwhose 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. Thus, walking
controllers may be designed via the two-dimensional zero dynamics. 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.
Two additional control features are developed: 1) the ability to
compose controllers that induce walking at a fixed average walking
rate to obtain walking at several, discrete average walking rates with
guaranteed stability during the transitions; and 2) the ability to
regulate the average walking rate to a continuum of values. The
general theory developed in the dissertation is experimentally
verified on a five-link prototype walker, RABBIT,
consisting of a torso and two legs with knees.
This set of pages documents the Ph.D. thesis work of Eric
R. Westervelt completed June 2003. The thesis work was done under the
supervision of Jessy
W. Grizzle and Daniel E. Koditschek
and supported in part by NSF grants INT-9980227 and IIS-9988695 and by
the University of Michigan Center for Biomedical Engineering Research
The foundational work of this thesis was begun by Jessy W. Grizzle
during his sabbatical in Strasbourg, from September 1998 through
February 1999. His French partners in this research were, and
continue to be, Franck Plestan and Gabriel Abba.