Autonomous Robots

, 27:277

A framework for the control of stable aperiodic walking in underactuated planar bipeds

  • T. Yang
  • E. R. Westervelt
  • A. Serrani
  • J. P. Schmiedeler
Article

DOI: 10.1007/s10514-009-9126-y

Cite this article as:
Yang, T., Westervelt, E.R., Serrani, A. et al. Auton Robot (2009) 27: 277. doi:10.1007/s10514-009-9126-y

Abstract

This paper presents a new definition of stable walking for point-footed planar bipedal robots that is not necessarily periodic. The inspiration for the definition is the commonly-held notion of stable walking: the biped does not fall. Somewhat more formally, biped walking is shown to be stable if the trajectory of each step places the robot in a state at the end of the step for which a controller is known to exist that generates a trajectory for the next step with this same property. To make the definition useful, an algorithm is given to verify if a given controller induces stable walking in the given sense. Also given is a framework to synthesize controllers that induce stable walking. The results are illustrated on a 5-link biped ERNIE in simulation and experiment.

Keywords

Bipedal robotics Stable aperiodic walking Nonlinear control theory 

Abbreviations

q

Vector of joint angles

\(x:=(q;\dot{q})\)

State vector of ERNIE’s swing phase dynamics

\(T\mathcal{Q}\)

State space of ERNIE’s swing phase dynamics

D

Mass-inertia matrix

C

Matrix of centripetal and Coriolis terms

G

Gravity vector

Δ

Impact map

Σ

Complete hybrid model of walking

φ

Solution of the swing phase dynamics

TI

The time of next impact

Φ

Feasible trajectory set

ΦSP

Strictly proper feasible trajectory set

S

Switching set

SP

Proper switching set

SSP

Strictly proper switching set

θ

Monotonic function over a step

s

Normalization variable of θ

\(h_{di,\alpha^{i}}\)

Holonomic constraints on the actuated joints

α

Bézier polynomial parameter set

y

Output to define the holonomic constraints

LgLfh

Decoupling matrix from input u to output y

\(\mathcal{Z}_{\alpha}\)

Zero dynamics manifold

Γα

Individual controller

12)

Coordinates for the zero dynamics manifold

δzero2

Impact coefficient

Vzero

Integral of the zero dynamics

ρ

Restricted Poincaré map

SD

Domain of definition of a Poincaré map

SI

Image of domain of definition of a Poincaré map

\((\Upsilon_{q},\Upsilon_{\dot{q}})\)

Mapping from zero dynamics manifold to \(T\mathcal{Q}\)

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • T. Yang
    • 1
  • E. R. Westervelt
    • 2
  • A. Serrani
    • 3
  • J. P. Schmiedeler
    • 4
  1. 1.Digital Technology Laboratory CorporationDavisUSA
  2. 2.General Electric Global Research CenterNiskayunaUSA
  3. 3.Department of Electrical Engineering, 412 Dreese LaboratoryThe Ohio State UniversityColumbusUSA
  4. 4.Department of Aerospace and Mechanical EngineeringUniversity of Notre DameNotre DameUSA

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