Hybrid zero dynamics of planar biped walkers

• Published in: IEEE transactions on automatic control Vol. 48; no. 1; pp. 42 - 56
• Main Authors: Westervelt, E.R., Grizzle, J.W., Koditschek, D.E.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01-01-2003; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Actuators; Applied sciences; Computer science; control theory; systems; Constraint optimization; Control system analysis; Control systems; Control theory. Systems; Dynamical systems; Dynamics; Energy consumption; Exact sciences and technology; Invariants; Kinematics; Knee; Leg; Legged locomotion; Low energy; Mathematical models; Orbits; Parametrization; Studies; System theory; Torso; Walking; Legged locomotion; Energy consumption; Invariant; Zero order; Stability; Periodic orbit; Linear time; Two dimensional model; Discrete system; Dynamical system; Modeling; Optimization; Hybrid system; Walking; Poincaré section; Kinematics; Hybrid model; Bipedal walking; Dynamic model; Actuator
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2002.806653

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.