Kinematic Linkages - Computer Animation: Algorithms and Techniques - page 161

CHAPTER

5

Kinematic Linkages

In describing an object's motion, it is often useful to relate it to another object. Consider, for example,
a coordinate system centered at our sun in which the moon's motion must be defined. It is much
easier to describe the motion of the moon relative to the earth and the earth's motion relative
to the sun than it is to come up with a description of the moon's motion directly in a sun-centric
coordinate system. Such sequences of relative motion are found not only in astronomy but also
in robotics, amusement park rides, internal combustion engines, human figure animation, and pogo
sticks.
This chapter is concerned with animating objects whose motion is relative to another object, espe-
cially when there is a sequence of objects where each object's motion can easily be described relative
to the previous one. Such an object sequence forms a motion hierarchy . Often the components of
the hierarchy represent objects that are physically connected and are referred to by the term linked
appendages or, more simply, as linkages . Another common aspect of relative motion is that the
motion is often restricted. The moon's position relative to the earth, for example, can be specified
by a single parameter (in this case, an angle) since, at least for this discussion, it rotates around
the earth in a fixed plane at a fixed distance. The plane and distance are built into the hierarchical
model so the animator is only concerned with specifying one rotational parameter. This is an example
of a model with reduced dimensionality because the hierarchical structure enforces constraints and
requires fewer parameters than would be needed to specify the position of the moon otherwise.
This chapter addresses how to form data structures that support such linkages and how to animate
the linkages by specifying or determining position parameters over time. As such, it is concerned with
kinematics .
Of course, a common use for kinematic linkages is for animating human (or other) figures in
which limbs are defined by a hierarchy of rotational joints connected by rigid links. The two
approaches to positioning such a hierarchy are known as forward kinematics , in which the animator
must specify rotation parameters at joints, and inverse kinematics (IK), in which the animator
specifies the desired position of the hand, for example, and the system solves for the joint angles
that satisfy that desire. Figure 5.1 , demonstrating forward kinematics, shows a sample sequence
of rotating a limb's joints. Figure 5.2 , demonstrating IK, shows a sample sequence of positioning
the hand at the desired location as a procedure automatically solves the required joint angles. These
techniques are the subject of this chapter after discussing the fundamentals of modeling such
hierarchies.