where T is the rotation matrix from the SCS to the LCS and O is the translation between coordinate systems.
The motion of rigid segments in space can be fully described by measuring three independent translational degrees-of-freedom (position) and three independent rotational degrees-of-freedom (orientation).
Photogrammetric procedures for obtaining measurements of six degree-of -freedom (dof) segmental motion require that a system of three or more noncolinear points be fixed to each segment. These non-colinear points are used to define orthogonal segment coordinate systems (SCSs) located independently within each of the segments. In addition, an orthogonal laboratory coordinate system (LCS), which is assumed to be stationary, is defined during system calibration.
Measurement of the position and orientation of a local segment coordinate system (SCS) with respect to the laboratory coordinate system (LCS) can be used to completely describe the segment's motion.
A least squares procedure is used by Visual3D to determine the position
and orientation. To understand how this procedure works, consider a point
located on a segment at position A in the SCS. The location of the point
in the LCS (P) is given by:
where T is the rotation matrix from the SCS to the LCS and O is the translation
between coordinate systems.
If the position O and orientation T are defined for some reference position,
then the fixed SCS coordinates of a target A can be determined from measurement
of P at this position
If the segment undergoes motion, the new orientation matrix T and translation
vector O may be computed at any instant, provided that for at least three
noncolinear points A is predetermined and P is measured. The matrix T
and origin vector O are found by minimizing the sum of squares error expression:
under the orthonormal constraint
where m is equal to the number of targets on the segment ( m > 2).
Since the above system of equations represents a constrained maximum
-minimum problem, the method of Lagrangian multipliers can be used to
obtain the solutions. A function
is used to supply the boundary conditions (This solution is adapted from
the solution outlined by Spoor & Veldpaus in the Journal of Biomechanics,
pp. 391- 393, 1980.).
The fact that a least squares fit can be done on an over determined system (m >3) allows the user to employ up to 8 targets to track each segment. Over determination also allows Visual3D to calculate segment positions and orientations if data from one or more targets is lost (However, coordinates must be determined for at least three points on the segment.).
In principle, tracking markers can be placed anywhere on a rigid segment. In practice, the markers should be distributed over the maximum area possible for a segment, they should be placed in areas that exhibit the least soft tissue artifact, and they should be visible from as many cameras as possible throughout the movement.
Depending on your laboratory setup, your subject population, the movement being recorded, most labs need to make custom compromises.
For example, consider tracking the thigh segment for a walking trial relative to the above rules. The markers would cover the maximum volume by being placed on the greater trochanter, medial mid thigh, and lateral knee. The greater trochanter and lateral knee exhibit terrible soft tissue artifact, and the medial thigh marker would probably get knocked off the segment during walking, so none of these 3 locations should probably be used. The compromise is to place markers distributed along the anterior and lateral side of the mid thigh. These markers could be on a rigid cluster (or 3 or 4 markers), or distributed around the leg (eg 8 markers).