Hybrid zero dynamics of planar biped walkers

Abstract

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. KeywordsBipeds, hybrid systems, Poincaré sections, underactuated system, zero dynamics 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 pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it. Abstract-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.

Complete Stability Analysis of a Control Law for Walking Robots with Non-permanent Contacts

“…Since the control law (6) has been designed by potential shaping, the energy of the system appears as a natural candidate for its stability analysis. For the same reason, the choice of a control law trying to compensate completely the system dynamics such as a computed torque [11,13], appears to be not very judicious because this energy can't be used any longer for the stability analysis. Finally, it is not possible to compensate completely the external forces in the case of walking machines because of the underactuation (2) what explains the difference with the control law proposed in [3].…”

“…The only stability analyses of control laws that have been proposed so far for walking robots have been distinctly focusing on each contact phases, with the strong assumption that these phases are never perturbated [11,13]. In this study we aim at analysing the stability of a regulation of the position of a walking robot without any assumption on the state of these contacts.…”

Achieving Bipedal Running with RABBIT: Six Steps Toward Infinity

“…As a result of enforcing the virtual constraints, in stance phase, the robot behaves as an unactuated 1 DOF system whose properties may be tuned by choosing different constraint parameters. Apart from different boundary conditions on the virtual constraints, this control is identical to the walking controllers developed in [19,21]. The set of parameters a s is detailed next.…”

“…The terms θ − s and θ + s are the values of the function θ s (q) evaluated at the end and the beginning of the stance phase. In [19,21], h d is expressed in terms of Bézier polynomials. Here, a slightly different class of polynomials is used.…”

“…1(a)), has provided a means to test conceptually new approaches to underactuated, active, dynamic walking and running. Since March 2003, RABBIT has been used to experimentally verify a mathematical framework for the systematic design, analysis, and optimization of controllers that induce stable walking gaits in N -link planar bipedal robots with one degree of underactuation [9,21]. Walking controllers designed within this framework act by enforcing virtual constraints, which are holonomic constraints on the robot's configuration that are asymptotically imposed through feedback control.…”