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Gabriel Baud-bovy
12-05-1997, 11:53 PM
Dear all,

I am writting a program that simulate the motion of an INERT
human arm under the influence of the gravity.

My question is about how to model correctly the friction
forces as well as the range of biomechanically possible motion
(see the specifics below).

Thank you for your help.

Gabriel Baud-Bovy


1) The arm is modeled by an open chain made of 2 cylindric
segments with a 3DF shoulder joint (elbow azimuth, elbow
elevation, axial rotation around the humerus) and a 1DF
elbow joint (elbow flexion).

2) The configuration space equation can be written as:

q = M(theta)ddtheta + B(theta)[dtheta dtheta]
+ C(theta) [dtheta^2] + G(theta)
where theta, dtheta and ddtheta correspond to joint angles,
joint velocities and joint accelerations,
q is correspond to the torques
M(theta) is the inertia or mass matrix
B(theta) is the matrix of Coriolis coefficients
C(theta) is the matrix of centrifugal coefficients
G(theta) is the gravity matrix

3) The friction forces can be modeled by adding a term
F(theta,dtheta) to the configuration space equation.

example: F(theta,dtheta) = k dtheta, for viscous friction

4) In the case of an INERT human arm, friction forces can have
at least two origins:

a) in the valid range of joint angles value, some viscous
or Coulomb friction forces may be present because of the necessity
to stretch some muscles and ligaments when the arm moves
even if no muscular activity is present.

b) when the joint angles are reaching their limits, then
the friction forces becomes very large because of the resistance
due to the contact between bones and to the resistance of
the ligaments that maintain the articulation in place. In fact,
these forces may be best described as contact forces and may be
modelled, in principle, by adding unilateral constraints on the
joint angles to the system.

5) Questions:

a) For EACH ONE of degrees of freedom, what is the
corrsponding viscous friction constant in the normal
range of motion?

b) Does anybody have an ANALYTICAL definition of the range
of biomechanically valid joint angles? This is easy for the
elbow flexion, for example:

elbow flexion minimum = 0 (stetched arm)
elbow flexion maximum = approximately 150 (bend arm)

For the shoulder, I have seen many graphical (e.g. globographic)
representations but I have not found corresponding analytical
expressions (which could be obtained by fitting methods). In
principle, the minimum and maximum could be implicitely expressed
by constraints such as:

gmin(axial rotation, elbow azimumth, elbow elevation) = 0
gmax(axial rotation, elbow azimumth, elbow elevation) = 0

Of course, elbow azimuth and elevation minimum and maximum were
independent from axial rotation, it would be nice to have a more
explicit expression such as:

gmin(elbow azimumth, elbow elevation) = 0
gmax(elbow azimumth, elbow elevation) = 0
axial rotation minimum = fmin(elbow azimumth, elbow elevation)
axial rotation maximum = fmax(elbow azimumth, elbow elevation)

Gabriel Baud-Bovy baudbovy@fpshp1.unige.ch
Université de Genève, FAPSE tel. +41 22 705 97 67
9, route de Drize fax +41 22 300 14 82
1227 Carouge, Switzerland home tel. +41 22 320 21 38