THREE-DIMENSIONAL KINEMATICS AND KINETICS - Biomechanics and Motor Control of Human Movement - page 182

are recorded and averaged over the calibration time. The long axis of the
leg segment ( y axis) would then be defined as the line joining the midpoint
between the lateral and medial malleolii ( m T 2 and m C 1 ) and the midpoint
between the head of the fibula and the medial epicondyle of the tibia ( m T 3
and m C 2 ). These midpoints are the ankle and knee joints, respectively. The
leg y axis and the line from m C 1 to m T 2 define a plane normal to which is
the leg x axis. The direction of the leg z axis would be defined as a line
normal to the leg x - y plane such that the leg x - y - z is a dextral system.
The anatomical axes of the leg are now defined relative to the three tracking
markers. The location of the center of mass of the leg would be a known
distance along the y axis of the leg from the ankle joint; thus, the vector, c ,
from m , the origin of the tracking marker axis system, is also known. The
two calibration markers are now removed and are no longer needed because
the orientation of the axis system of the three tracking markers is now known
and assumed to be fixed relative to the newly defined anatomical axes.
In clinical gait laboratories, it is impossible for many patients, such as
cerebral palsy or stroke patients, to assume the anatomical position for even
a short period of time. Thus, the clinical gait teams have developed a consis-
tent marker arrangement combined with a number of specific anthropometric
measures. These include such measures as ankle and knee diameters which,
when combined with generic X-ray anthropometric measures, allow the team
to input an algorithm so that offset displacements from the tracking markers
to the joint centers are known. Patients are then asked to assume a static
standing position with a number of temporary calibration markers similar
to what has been described previously. In effect, the major single differ-
ence in the clinical laboratory is that calibration is performed on the patient
in a comfortable standing position rather than the anatomical position. For
the complete detailed steps in arriving at the joint kinetics in an operational
clinical laboratory, the reader is referred to Davis et al. (1991) and Ounpuu
et al. (1996).
In Figure 7.2, we see two matrix rotations. [G to M] is a 3 × 3 rotation
matrix that rotates from the GRS to the tracking marker axes, x m − z m .

This is a time-varying matrix because the tracking marker axes will be con-
tinuously changing relative to the GRS. [M to A] is a 3 × 3 matrix that rotates
from the tracking marker axes to the anatomical axes. This matrix is assumed
to be constant and results from the calibration protocol. The combination of
these two rotation matrices gives us the [G to A] rotation matrix, which,
when solved for a selected angle sequence, yields the three time-varying
rotation angles, θ 1 , θ 2 , θ 3 . With this final matrix, we can get the orientation
of the anatomical axes directly from the tracking marker coordinates that are
collected in the GRS.
However, from Figure 7.2, we are not yet finished. We also have to find a
translational transformation to track the 3D coordinates of the segment COM,
c , over time. The location of c is defined by the vector, R c , which is a vector

y m −