Page 159 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

136 BIOMECHANICS OF THE HUMAN BODY

⎛ ⎞
X
⎜ ⎟
L = C Y (6.15)
⎜ ⎟
⎝ ⎠
Z
Since C is not a square matrix (it is 4 × 3), the unknown X, Y, Z coordinates can be solved using the
Moore-Penrose generalized inverse, as follows:
⎛ X⎞
⎜ ⎟ T ) C L (6.16)
T
−1
Y = (CC
⎜ ⎟
⎝ Z⎠
which, in essence, yields a least-squares solution for the X, Y, Z coordinates of the tracking target.
The solution is easily expanded to account for n > 2 cameras.
While only two cameras are necessary to reconstruct the three-dimensional coordinates of a
tracking target in object-space, more than two cameras are recommended to help ensure that a
minimum of two cameras see the target every point in time. The cameras should be positioned so
that each has a unique perspective of the workspace. Ideally the cameras should be placed at an angle
of 90º with respect to one another. In practice this may not be possible, and every effort should be
taken to maintain a minimum separation angle of at least 60º.

6.4 ANALYSIS OF HUMAN MOTION: AN INVERSE
DYNAMICS APPROACH

The inverse dynamics approach is the most commonly used method to solve for unknown joint
reaction forces and moments. The analysis begins with the most distal segment, moving upward
through the kinematic chain, requiring that all external forces acting on the system are known.
A free-body diagram appropriate for a two-dimensional inverse dynamics analysis of the foot and
shank is illustrated in Fig. 6.8. This can be expressed mathematically in a generalized form suitable
for a two- or three-dimensional analysis of n segments.
∑M = I dω /dt (6.17)
i i i
∑F = m dv /dt (6.18)
i i i
where ∑M is the sum of the moments acting on segment i, I is the inertia tensor for segment i about
i
i
its center of mass (COM), and ω is the angular velocity of the segment. Forces acting on segment i,
i
mass and linear velocity of the segment correspond to F , m , and v , respectively.
i
i
i
The three types of measurement tools described in Sec. 6.3 all provide kinematic data of some form.
For example, goniometers provide an estimate of joint angular position, while electromagnetic track-
ing systems output the relative position and orientation of the sensors attached to the segments. In
the case of video-based motion analysis, output data are in the form of target coordinates expressed
in the object-space. It is important to note that output from all of these devices contains some degree
of error. That is, the sampled signal is actually a combination of “true” signal and “noise.” This is an
important consideration because the equations of motion contain linear and angular acceleration
terms, values that are obtained by numerically differentiating the position data. Differentiating the
raw data will have the undesirable effect of magnifying the noise, which can severely compromise
the integrity of the results. An excellent discussion of this topic can be found in Winter (1990). The
point we wish to make is that the raw data should be treated to reduce or eliminate the amount of
contaminating noise before the data are used in subsequent calculations. This process of treating the
raw data is commonly referred to as data smoothing.