doi:10.1016/j.humov.2007.08.003      

Copyright © 2007 Elsevier B.V. All rights reserved.

The ‘extrapolated center of mass’ concept suggests a simple control of balance in walking

At L. Hof

Laboratory of Human Movement Analysis, Center for Rehabilitation, University Medical Center Groningen, The Netherlands

Available online 1 November 2007. 

Abstract

Next to position x and velocity v of the whole body center of mass (CoM) the ‘extrapolated center of mass’ (XcoM) can be introduced: ξ = x + v/ω₀, where ω₀ is a constant related to stature. Based on the inverted pendulum model of balance, the XcoM enables to formulate the requirements for stable walking in a relatively simple form. In a very simple walking model, with the effects of foot roll-over neglected, the trajectory of the XcoM is a succession of straight lines, directed in the line from center of pressure (CoP) to the XcoM at the time of foot contact. The CoM follows the XcoM in a more sinusoidal trajectory. A simple rule is sufficient for stable walking: at foot placement the CoP should be placed at a certain distance behind and outward of the XcoM at the time of foot contact. In practice this means that a disturbance which results in a CoM velocity change Δv can be compensated by a change in foot position (CoP) equal to Δv/ω₀ in the same direction. Similar simple rules could be formulated for starting and stopping and for making a turn.

Keywords: Stability; Control; Gait; Locomotion; Robotics

Psychological classification codes: 2330; 4140

Nomenclature

bx

offset in forward direction

bz

offset in lateral direction

CoM

center of mass, coordinates (x, y, z)

CoP

center of pressure, coordinates (ux, 0, uz)

g

acceleration of gravity = 9.81 m s⁻¹

k₁

gain of proportional control lateral movement

l

effective pendulum length

s_c

desired step length

sn

actual step length of nth step

t

time

T_c

desired step duration

Tn

duration of nth step

ux

forward CoP position

uxn

constant forward CoP position during step n

uz, uzn

lateral CoP position

vx, vz

CoM velocity in forward/lateral direction

vxn, vzn

CoM velocity at the time of foot placement of step n

w

step width

w_c

desired step width

x, z

forward/lateral position of CoM

xn, zn

position of CoM at beginning of step n

XcoM

extrapolated center of mass, coordinates (ξ, 0, ζ)

ΔT

difference between actual step time and T_c

Δvx

difference between actual velocity and steady-state value

σ(·)

standard deviation of noise in (·)

ξ, ζ

forward/lateral position of XcoM

ξn, ζn

forward/lateral position of XcoM at beginning of step n

ω₀

Article Outline

Nomenclature

1. Introduction

2. Theory

3. Possibilities for control

3.1. Forward control

3.2. Lateral control

3.3. Start and stop walking

3.4. Walking in turns

4. Discussion

References