Kinematic Linkages - Computer Animation: Algorithms and Techniques - page 168

T 0 (global position and orientation)

Data for Link 0 (the root)

T 1 (transformation of Link 1 relative to Link 0 )

Data for Link 1

T 1.1 (transformation of Link 1.1 relative to Link 1 )

Data for Link 1.1

FIGURE 5.8
Example of a tree structure.

Link 1.1

1.1

Link 1

1

T 0

FIGURE 5.9
Variable rotations at the joints.

To locate a vertex of Link 1 in world space, one must first transform it via the joint rotation matrix.
Once that is complete, then the rest of the transformations up the hierarchy are applied (see Eq. 5.4 ).

0 1

V ¼ T 0 T 1 R 1 ðy 1 ÞV 1 (5.4)

A vertex of Link 1.1 is transformed similarly by compositing all of the transformations up the hier-
archy to the root, as in Equation 5.5 .

0 1 : 1

V ¼ T 0 T 1 R 1 ðy 1 ÞT 1 : 1 R 1 : 1 ðy 1 : 1 ÞV 1 : 1 (5.5)