SYNTHESIS OF HUMAN MOVEMENT—FORWARD SOLUTIONS - Biomechanics and Motor Control of Human Movement - page 203

Simple motions have been modeled reasonably successfully. Phillips et al.
(1983) modeled the swinging limbs of a human using the accelerations of
the swing hip along with the moments about the hip. Hemami et al. (1982 a )
modeled the sway of the body in the coronal plane with each knee locked.
With adductor/abductor actuators at the hip and ankle as input, the stability
of the total system was defined.
More recent modeling of three-dimensional (3D) gait has been somewhat
more successful. One of the major problems with previous attempts was the
modeling of initial contact. These earlier models employed springs to rep-
resent the elastic characteristics of the bottom of the feet or shoes, but this
resulted in extremely large accelerations of the foot segment and similarly
large spikes in the ground reaction forces. This was solved with a viscoelas-
tic model of the foot with an array of parallel springs and dampers under
the rigid foot segment (Gilchrist and Winter, 1996). A 3D nine-segment
model of walking that used ADAMS software (discussed in Section 8.2)
demonstrated some success using the inverse dynamics joint moments as
inputs (Gilchrist and Winter, 1997). Nonlinear springs at the knee, ankle,
and metatarsal-phalangeal joints constrained those joints to their anatomical
range. Linear springs were used at the hips and dampers at all joints to ensure
a smooth motion. However, it became apparent that any small errors in the
joint moments after about 500 ms resulted in accumulating kinematic errors
that ultimately became too large, and the model either became unbalanced
or collapsed. The build-up of these errors is an inherent characteristic of any
forward solution; the double integration of the segment accelerations caused
by the input and reaction moments and forces cause displacement errors that
increase over time and that can only be corrected by continuous fine-tuning of
the input joint moments. Thus, we are forced to violate some of the constraints
of forward solution modeling as listed in Section 8.0.1.

8.2 MATHEMATICAL FORMULATION

For the dynamic analysis of connected segment systems, mathematical models
consisting of interconnected mass elements, springs, dampers, and actuators
(motion generators) are often used. The motion of such models may be deter-
mined by defining the time history of the position of individual segments, or
by applications of motor forces, in which case the motion of the segments is
determined by the laws of physics.
Until quite recently, the nonlinear nature of the problem has been an imped-
iment to the solution of the general dynamic model. With the advancement in
computers, researchers are in a position to address the problem of nonlinearity
and utilize the computer in creating models that consist of many elements. In
the following pages, a systematic method for writing the equations of motion
for a general model configuration is explained. The method is suitable for
hand derivation of the equations of motion of simple to moderate model con-
figurations. For complex models, the method can be adapted easily for writing