Graphics Reference
In-Depth Information
Y
Z
Coordinate system
after rigid motion
X
v
Y
Z
Figure 7.10.
Any rigid motion can be expressed as
a rotation around some axis followed by a transla-
tion along the same axis.
X
ρ
Z
screw axis
X
Y
Origin of world
coordinate system
The notation exp in Equations (
7.11
)-(
7.12
) denotes the exponential map of a matrix,
the required exponential map in Equation (
7.12
) is given by the Rodrigues formula
in Equation (
6.58
) and is simply the matrix rotation corresponding to the axis-angle
parameters. Twistswere introduced to the animationcommunitybyBregler et al. [
67
],
and are commonly used in robotics applications [
342
].
Conversion between quaternions, twists, axis-angle representations, and rotation
matrices is straightforward (see [
526
], and Problems
6.21
,
7.3
, and
7.5
). Therefore, we
canassume that a kinematicmodel for the humanbody is generally representedusing
six degrees of freedom for the body's root, and some number of “angles” (parame-
terized by either quaternions or twists) to represent the joints. In total, the number
of degrees of freedom in typical parameterizations of the human kinematic model
ranges from thirty to fifty.
In the next section, we address the critical problem of the change of coordinates
from the Euclidean domain of 3D marker locations to the hierarchy of joints of a
kinematic model. This coordinate transformation is more difficult than the forward
transformations given earlier; instead wemust solve an
inverse kinematics
problem.
7.4
INVERSE KINEMATICS
In motion capture, we're faced with the problem of determining the underlying
parameters of a kinematic model from observations of points on (or near) the
8
Hence, Equation (
7.8
) can be interpreted as a product of exponential maps.
