Kinematic Linkages - Computer Animation: Algorithms and Techniques - page 166

Arc i

Arc i contains
￿ a constant transformation of Link i to
its neutral position relative to Link i 1
￿ a variable transformation responsible
for articulating Link i

Node i contains
￿ a transformation to be applied to
object data to position it so its
point of rotation is at the
origin (optional)
￿

Node i

object data

FIGURE 5.6
Arc and node definition.

Because all of the other parts of the hierarchy will be defined relative to this part, this transformation
affects the entire assemblage of parts and thus will transform the position and orientation of the entire
structure. This transformation can be changed over time in order to animate the position and orientation
of the rigid structure. Link 1 is defined relative to the untransformed root object by transformation T 1 .
Similarly, Link 1.1 is defined relative to the untransformed Link 1 by transformation T 1.1 . These relation-
ships can be represented in a tree structure by associating the links with nodes and the transformations
with arcs. In the example shown in Figure 5.8 , the articulation transformations are not yet included in
the model.
An arc in the tree representation contains a transformation that applies to the object represented by
the node to which the arc immediately connects. This transformation is also applied to the rest of the
linkage farther down the hierarchy. The vertices of a particular object can be transformed to their
final positions by concatenating the transformations higher up the tree and applying the composite
transformation matrix to the vertices. A vertex, V 0 , of the root object, Link 0 , is located in the world
coordinate system, V 0 0 by applying the rigid transformation that affects the entire structure; see

Equation 5.1 .

0 0

V ¼ T 0 V 0 (5.1)

A vertex of the Link 1 object is located in the world coordinate system by transforming it first to its
location relative to Link 0 and then relocating it (conceptually along with Link 0 ) to world space by
Equation 5.2 .

0 1

V ¼ T 0 T 1 V 1 (5.2)

Similarly, a vertex of the Link 1.1 object is located in world space by Equation 5.3 .

0 1 : 1

¼ T 0 T 1 T 1 : 1 V 1 : 1 (5.3)
Notice that as the tree is traversed farther down one of its branches, a newly encountered arc trans-
formation is concatenated with the transformations previously encountered.
As previously discussed, when one constructs the static position of the assembly, each arc of the tree
has an associated transformation that rotates and translates the link associated with the child node

V