Ai = Rz(9i) Trz(di) Trx(ai) Rx(ai). A useful decomposition of matrix Ai is


 Ai = Ai Bi (2.7)


with the definitions


 Ai = Rz(ei) (2.8)

and

 Bi = Trz(di) Trx(ai) Rx(ai). (2.9)


Explicitly, matrix Bi is



 1 0 0 ai

 0 Ti -ai 0 G ki
 Bi (2.10)
 0 ai Ti di 0 0 0 1

 0 0 0 1


where Gi is the upper left 3 x 3 in Bi and ki is the upper right 3 x 1 vector of Bi. The upper left 3 x 3 matrix in Ai is the rotation matrix Ri necessary to align the unit vectors of Fi with their counterparts in Fi_1, while vector



 aiCi

 i = aiSi

 di


positions the origin of Fi with respect to Fi_.1

..