Page 202 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

BIOMECHANICS OF THE MUSCULOSKELETAL SYSTEM 179

origin and insertion sites of each musculotendinous actuator and the relative positions of the body
segments are known at each instant during the movement (Sec. 7.4).
However, Eq. (7.9) cannot be solved for the m actuator forces because m > n (i.e., the matrix
of actuator moment arms is nonsquare). Static optimization theory is usually used to solve this
indeterminate problem (Seireg and Arvikar, 1973; Hardt, 1978; Crowninshield and Brand, 1981).
Here, a cost function is hypothesized, and an optimal set of actuator forces is found subject to
the equality constraints defined by Eq. (7.9) plus additional inequality constraints that bound the
values of the actuator forces. If, for example, actuator stress is to be minimized, then the static
optimization problem can be stated as followss (Seireg and Arvikar, 1973; Crowninshield and
Brand, 1981): Find the set of actuator forces which minimizes the sum of the squares of actuator
stresses:

m 2
J = ( F i MT F ) (7.11)
∑
MT
oi
i=1
subject to the equality constraints
q F
T MT = R() MT (7.12)
and the inequality constraints

0 ≤ F MT ≤ F o MT (7.13)

F oi MT is the peak isometric force developed by the ith musculotendinous actuator, a quantity that is
directly proportional to the physiological cross-sectional area of the ith muscle. Equation (7.12)
expresses the n relationships between the net actuator torques T MT , the matrix of actuator moment
arms Rq() , and the unknown actuator forces F MT . Equation (7.13) is a set of m equations which
constrains the value of each actuator force to remain greater than zero and less than the peak iso-
metric force of the actuator defined by the cross-sectional area of the muscle. Standard nonlinear
programming algorithms can be used to solve this problem (e.g., sequential quadratic programming
(Powell, 1978).

7.6.4 Forward-Dynamics Method
Equations (7.1) and (7.7) can be combined to form a model of the musculoskeletal system in
which the inputs are the muscle activation histories (a) and the outputs are the body motions
(qqq,, ) (Fig. 7.22). Measurements of muscle EMG and body motions can be used to calculate the
time histories of the musculotendinous forces during movement (Hof et al., 1987; Buchanan et al.,
1993). Alternatively, the goal of the motor task can be modeled and used, together with dynamic
optimization theory, to calculate the pattern of muscle activations needed for optimal performance
of the task (Hatze, 1976; Pandy et al., 1990; Raasch et al., 1997; Pandy, 2001). Thus, one reason
why the forward-dynamics method is potentially more powerful for evaluating musculotendinous
forces than the inverse-dynamics method is that the optimization is performed over a complete
cycle of the task, not just at one instant at a time.
If we consider once again the example of minimizing muscle stress (Sec. 7.6.3), an analogous
dynamic optimization problem may be posed as follows: Find the time histories of all actuator forces
which minimize the sum of the squares of actuator stresses:

m
t f
J = ∑ F ( i MT F ) 2 (7.14)
MT
∫
oi
0 i=1