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Analysis results and tools for the control of planar bipedal gaits using hybrid zero dynamics

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Abstract

New analysis and tools are presented that extend the hybrid zero dynamics (HZD) framework for the control of planar bipedal walkers. Results include (i) analysis of walking on a slope, (ii) analysis of dynamic (decoupling matrix) singularities, and (iii) an alternative method for choosing virtual constraints. A key application of the new tools is the design of controllers that render a passive bipedal gait robust to disturbances without the use of full actuation—while still requiring zero control effort at steady-state. The new tools can also be used to design controllers for gaits having an arbitrary steady-state torque profile. Five examples are given that illustrate these and other results.

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Abbreviations

a :

Augmentation function parameter vector

α :

Ground slope

[α]:

Functional dependence on α; emphasized with square brackets

D(q):

Inertial matrix of the full dynamics

\(F{[}\alpha{]}(q,\dot{q})\) :

Coriolis, centrifugal, gravitational terms of the full dynamics

\(h_{d}(s),\ \bar{h}_{d}(\theta)\) :

Desired joint angles of the actuated joints

I(θ,y):

Closed-loop virtual inertia

I y (θ,y):

Virtual inertia of transverse dynamics

I RB (θ,y):

Rigid body inertia

I AB (θ,y):

Inertia associated with the articulated body angular momentum

K p , K d :

PD controller gains

l, l c , m, J, g 0 :

Parameters of the two-link model

L g L f h(q):

Decoupling matrix from input u to output y

\(L_{\bar{g}}L_{\bar{f}}h(q)\) :

Decoupling matrix from input v to output y

q a :

Vector of actuated coordinates

q u :

Unactuated coordinate (scalar)

s(θ):

Normalization function for θ

S(q):

Function used to determine invertibility of decoupling matrix

\(\mathcal{S}\) :

Switching surface; Poincaré section

Σ :

Full dynamics walking model

Σ zero :

HZD model of walking

σ :

Angular momentum about pivot foot

θ(q):

Scalar function that is a surrogate for time; monotonic over a step

θ +,θ − :

Values of θ at the start and end of a gait

θ s :

Value of θ corresponding to a singularity

T s :

Time from step start to singularity

u :

Control input

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

State vector of the full dynamics

y=h(q):

Output defining virtual constraints

z=(θ,σ):

State vector of the restricted (zero) dynamics

\(\mathcal{Z}\) :

Zero dynamics manifold; constraint surface

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Authors and Affiliations

  1. Department of Mechanical Engineering, The Ohio State University, Columbus, OH, 43210, USA

    E. R. Westervelt & K. D. Farrell

  2. Department of Electrical Engineering, University of Michigan, Ann Arbor, MI, 48109, USA

    B. Morris

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  1. E. R. Westervelt
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  2. B. Morris
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  3. K. D. Farrell
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Correspondence to E. R. Westervelt.

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Westervelt, E.R., Morris, B. & Farrell, K.D. Analysis results and tools for the control of planar bipedal gaits using hybrid zero dynamics. Auton Robot 23, 131–145 (2007). https://doi.org/10.1007/s10514-007-9036-9

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  • DOI: https://doi.org/10.1007/s10514-007-9036-9

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