Page 102 - Handbook of Biomechatronics

98 Naser Mehrabi and John McPhee

where x represents the model coordinates (e.g., positions and velocities) and
T and F represent net joint torques and the external loads acting on the sys-
tem. Eq. (1b) represents the kinematic constraints that restrict the move-
ments of two rigid bodies relative to each other (e.g., joints). Since x and
F have been measured beforehand in the laboratory, the joint torques (T)
can be computed by simply substituting the measured kinematics and exter-
nal loads into Eq. (1) and evaluating T at each time step. The torque and
force requirement of a task are important to know when designing a system
controller and its actuator capacity. For example, the maximum ankle torque
during normal walking can be used to select the stiffness or the actuator
power of an ankle-foot orthosis (AFO), which is a wearable assistive device
that supports and corrects ankle motion.
If muscle-level information is required, a static optimization can be per-
formed to resolve the muscle indeterminacy problem and compute the share
of each muscle contributing to the resultant joint torque. The muscle inde-
terminacy problem results from the number of muscles crossing a joint
exceeding the degrees of freedom of that joint; it is difficult to identify indi-
vidual muscle forces because different combinations of forces can produce
the same net joint torque. To resolve this problem, the static optimization
is subjected to the torque equilibrium equation:

n
X
m
T j ¼ (2)
r F i
i, j
i¼1
m
where r i,j represents the moment arm of the muscle force F i about the joint j,
and index i refers to the individual muscles crossing the joint of interest.
A unique muscle activation pattern similar to that of humans can be
achieved by minimizing a physiological cost function during static optimi-
zation, such as

n
X p
J ¼ a (3)
i
i¼1
where a is the muscle activation level at the current time step, n is the num-
ber of muscles crossing the joint, and p is an exponent (usually, p ¼2). The
inverse dynamics can only provide insight into a task whose kinematics and
kinetics have already been measured in the laboratory, and it cannot predict
the dynamics of a new task based on previously measured data.