Content uploaded by Vladimir M Zatsiorsky
Author content
All content in this area was uploaded by Vladimir M Zatsiorsky on Nov 11, 2014
Content may be subject to copyright.
Copyright © 2007 by ASME
1
INTRODUCTION
It is often of interest in studies of human movement to quantify
the function of a muscle force or muscular joint torque. Such
information is useful for the identification of the causes of movement
disorders and for predicting the effects of interventions including
surgical procedures, targeted muscle strengthening, focal treatments
for spasticity, and functional electrical stimulation. One useful way to
characterize the actions of muscle forces or muscular joint torques is to
create linked-segment models of the body and analyze these linkages
to determine the joint angular accelerations or end effector forces that
result solely from the application of the muscle force or torque in
question. Such induced acceleration (IA) analyses or induced end
effector force (IEF) analyses have been applied most often to quantify
muscle function during normal and pathological walking [1,2].
Both IA and IEF analyses involve computation of the effects of
individual muscle forces or muscle torques while all other muscle
forces or torques are taken to be zero. Because the other joints in the
linkage bear no moments, the muscle in question will influence forces
and motions at other locations in kinetic chain only through the
propagation of joint reaction forces along the segments. Yet, there are
many occasions when it makes sense to consider joints as having
stiffnesses (angle-dependent resistance) or damping (speed-dependent
resistance), perhaps due to preactivation of antagonist muscle pairs [3],
that will result in the propagation of joint moments as well as joint
reaction forces.
The purpose of this paper is to present examples in which
conventional IEF analysis fails, in our view, to accurately represent the
roles of individual muscle forces or muscle joint torques in producing
end effector forces. We will focus on IEF analysis, but the points
raised apply just as readily to IA analysis.
ILLUSTRATIVE CASES
Analyses of several simple, two-dimensional linkages are
presented which illustrate the need to account for the effects of joint
stiffness under certain circumstances and which show that singular
linkage configurations exist for which typical end effector force
decomposition is not a well-posed problem.
Figure 1. Two-segment linkage with torsional spring. As
the stiffness at the middle joint is increased, the endpoint
force produced at C substantially changes direction.
Case 1: Two-segment linkage with joint stiffness
A computational model of a two-segment linkage with ends
pinned to ground at A and C (Figure 1) was created to illustrate the
role of joint stiffness in modulating end effector force. Segments AB
and B’C were each 0.5 m in length and separated by 135°. Joint B
was modeled as slightly nonconforming, with AB and B’C having
cylindrical articulating surfaces with radii of 10 cm and 11 cm,
respectively. A penalty function was used to apply forces to both
Proceedings of the ASME 2007 Summ
er Bioengineering Conference (SBC2007)
June 20-
24, Keystone Resort & Conference Center, Keystone, Colorado, USA
SBC2007
-
17
6695
ON THE USE OF INDUCED END EFFECTOR FORCE ANALYSIS FOR DETERMINING
MUSCLE ROLES DURING MOVEMENT
Stephen J. Piazza (1,2,3), Vladimir M. Zatsiorsky (1)
(1) Department of Kinesiology
The Pennsylvania State University
University Park, PA
(2) Department of Mechanical Engineering
The Pennsylvania State University
University Park, PA
(3) Department of Orthopaedics & Rehabilitation
The Pennsylvania State University
Hershey, PA
Copyright © 2007 by ASME
2
segments at B and B’ that were proportional to penetration between
the articulating surfaces. The nonconformity of joint BB’ permitted
small changes in joint angle at this joint, and equal and opposite
torques were applied to AB and B’C that represented the action of a
bidirectional torsional spring with spring constant KT. With KT = 0, a
10 N m torque was applied to AB, the equations of motion for the
system were integrated forward in time until static equilibrium was
attained. This was repeated several times as KT was increased to 1000
N m rad-1, representing increasing levels of co-contraction in the
muscles crossing the joint.
When KT = 0, the reaction force at C was directed along B’C but
as KT was increased, the direction of this force changed until it was
nearly perpendicular to AC when the joint was most stiff (Figure 1).
The stiffness of joint BB’ is thus shown to determine the direction of
the end effector force.
Case 2: Role of hip extensor torque during fast walking
A second model was created to examine the ground reaction
forces produced by a hip extension torque during early stance in fast
walking. Three segments (trunk, thigh, and shank) were pin-jointed at
the hip and knee, and pin-jointed to the ground at the ankle (Figure 2).
The segments were assigned realistic inertial properties but the
contralateral leg was not modeled. When the only applied torque was
a hip extensor torque of 25 N m, the resulting ground reaction force
was 206 N, directed along the shank in a nearly upward direction.
When this hip extensor torque was applied with a knee flexor torque of
15 N m, the ground reaction force was 34 N, directed to the right.
Figure 2. A three-segment linkage. Hip extensor moment
alone induces a nearly vertical (support) GRF; adding knee
flexor moment gives a nearly horizontal (propulsive) GRF.
We can use the results of this analysis to draw different
conclusions about the role of a hip extensor muscle such as gluteus
maximus (GM). If we consider the ground reaction force induced by
GM alone, we would conclude that GM contributes mainly to support
of the body. If, however, we consider the possibility that GM is
indirectly responsible for the knee flexor torque because it further
stretches the knee flexor tendons as it extends the thigh, then GM may
be considered to contribute mostly to forward propulsion. It is the
latter of these two possibilities that is better supported by experimental
evidence: The early stance-phase activity of GM has been shown to
increase dramatically with walking speed in fast walking [4,5].
Case 3: Singular Configurations
Consider a two-segment linkage in static equilibrium with joint
torques TP and TD applied at the “proximal” and “distal” joints (Figure
3). If the segments are each of length L, then the end effector force
induced by TD alone is found to be TD / L, directed down at E but the
end effector force induced by T
P is zero because the joint reaction
force induced at the distal joint by TP is normal to the distal segment.
This IEF analysis implies that T
D contributes to end effector force
while TP does not, but application of static equilibrium conditions tells
us that TD depends directly on TP (TD = ½ TP), suggesting that there is
an end effector force induced by TP, equal to TP / 2L.
Figure 3. A singular configuration in which the end effector
force may be described as being simultaneously induced
wholly by eitherTP or by TD.
Another special configuration for which IEF analysis yields
paradoxical results is that of three-segment, pin-jointed linkage ABCD
with massless segments BC and CD forming a right angle (Figure 4).
A torque applied to segment AB will produce reaction forces at B and
C, but these forces will not propagate to the ground at D. If the joints
of this linkage transmit no moments, but only forces, then the
conclusion from IEF analysis is that no end effector force is induced
by this torque, and that a muscle producing this torque thus has no role
at all in support, braking, or propulsion of the body.
Figure 4. A three-segment linkage configured such that TAB
induces no reaction force at D, thus not contributing to
support, braking, or propulsion.
SUMMARY
While the models presented in this paper demonstrate some
inconsistencies associated with IEF analysis, it is important to note
that IEF analysis is still a highly valuable tool for understanding
human movement. Chen [6] described IA analysis as a fundamentally
ill-posed problem, but it is our hope that a more general approach will
be developed that incorporates IEF and IA concepts while also
accounting for the effects of joint stiffness and singular chain
configurations.
REFERENCES
1. Neptune R.R. et al., 2004, Gait & Posture, 19, pp. 194-205.
2. Kimmel, S.A., Schwartz, M.H., Gait & Posture, 23, pp. 211-221.
3. Loeb, G.E. et al., 1999, Exp Brain Res, 126, pp. 1-18.
4. Lieberman, D.E. et al., 2006, J Exp Biol, 209, pp. 2143-2155.
5. Murray, M.P. et al., 1983, Amer J Sports Med, 11, pp. 68-74.
6. Chen, G., 2006, Gait & Posture, 23, pp. 37-44.
