A compact and useful expression for Ai is
(2.11)
Rotation matrices are orthogonal, so Ri-l= RT, where the superscript T denotes the transpose operation, and the inverse of matrix Ai can be expressed as
Ci
Sci
-s0
0
Si
CiT i
-Cii
-c0 a
0
-ai
-oidi
-ridi
1
RiT (-RiTli)
= (2.12)
0 0 0 1
Problem Definition
If the orientation of the end-effector is specified by the rotation matrix R, necessary to align the unit vectors of the end-effector frame Fn with the corresponding vectors of base frame F0, and the position of the origin of the endeffector frame is given as a vector p with respect to the base frame F0, then the end-effector pose is adequately described by the 4 x 4 matrix
nx bx tx n b ty nz b tz 0 0 0
n b t p R p
0 0 0 1 0 0 0 1
Ai-1
(2.13)
..