Graphics Reference
In-Depth Information
skeleton. Going forward, we'll denote the kinematic model parameters by a vector
,
and the observed skeleton points by a vector
r
. Let's compactly denote the forward
kinematic relationship in Equation (
7.8
)by
θ
r
=
f
(
θ
)
(7.13)
We'd like to determine
from a set of measured values of
r
; that is, to invert
Equation (
7.13
). Therefore, such problems are termed
inverse kinematics
. Unfor-
tunately, this inversion is problematic for several reasons. First, the relationship in
Equation (
7.13
) is highly nonlinear, involving products of trigonometric functions of
the parameters. Second, in some applications, the relationship in Equation (
7.13
)is
many to one; that is, there are more values of
θ
than of
r
. In such cases, the inverse
kinematics problem is underdetermined and has many feasible solutions.
9
In this section, we describe several basic methods for solving the inverse kine-
matics problem. Algorithms for inverse kinematics were developed in the robotics
community many years before their application to motion capture and animation;
for example, see Chiaverini et al. [
92
]. The techniques based on dynamical systems
that we discuss in Section
7.7.1
can also be viewed as methods for inverse kinematics.
There's anoffset between themarkers and the skeleton thatwe can't ignore (see the
left side of Figure
7.8
). While some markers can be placed on a performer's suit/skin
fairly close to a joint, at places where the motion capture technician can easily feel
a bone, other markers are further from the underlying kinematic joint (e.g., those
on the shoulders and spine). This relationship can still be taken into account by
Equation (
7.13
); however, we need amodel for the relationship between the markers'
position on the surface of the skin and the underlying bones. We discuss this issue
more in Section
7.4.4
.
θ
7.4.1
Inverse Differential Kinematics
We'll generalize Equation (
7.13
) to be a function of time, since in motion capture
we're interested in recovering the kinematic model at every instant of a performer's
continuous motion:
r
(
t
)
=
f
(
θ
(
t
))
(7.14)
First, we discuss a general approach based on
inverse differential kinematics
,
which is based on differentiating Equation (
7.14
) with respect to time:
d
r
(
t
)
=
∂
f
(
θ
)
∂
θ
d
dt
(
(
t
)
t
)
dt
(7.15)
d
dt
(
=
J
(
t
)
t
)
where in Equation (
7.15
) we introduced the
Jacobian
)
=
∂
f
(
θ
)
∂
θ
J
(
t
(
t
)
(7.16)
9
For example, if the wrist and shoulder positions are fixed, there is still one degree of freedom for
the elbow's position — it can rotate in a circle.
