When joint i is revolute, parameter ei is the joint variable and, if joint i is prismatic, the joint variable is di. When applicable, di measures the translation along axis zi_.
Homogeneous Matrices
If a vector iu = [iux, iuy, iuz]T is expressed in frame Fi, its expression with respect to frame Fi_, i-lu, satisfies
i-lu Ci -Siri Sici aiCi iux
-x i.11 1 i ii Ux
= (2 .1)
i-1a 1
i z 0 i i di
z i z
1 0 0 0 1 1
or in a compact notation,
i-1u u
= Ai (2.2)
1 1
where Ti=cos(ai), ci=sin(ai), Ci=cos(8i), and Si=sin(ei) and Ai is the homogeneous frame-transform matrix (Paul 1981). The leading superscript indicates the frame of expression.
The homogeneous matrix transform merely expresses the fact that frame Fi can be obtained from frame Fi-1 by the following sequence of basic transforms:
1. Rotation about zi_1 of angle ei whose homogeneous matrix is
..