a5 r/2, a4=7r/2, a3=0, a2=0, and al =7r/2


belongs to the class 11-001 of orthogonal manipulators.

 Since most industrial robot arms are orthogonal, it is worthwhile to consider the inverse kinematics problem with respect to these manipulators. The A-matrices associated with orthogonal arms have one of the two following forms


Ai (a=0) -


-Si Ci

0

0


aiCi siSi di

1


(8.1)


Ai(a==7r/2) =


Si aiCi

-Ci asii 0 di

0 1


 Further computational simplification is obtained in the inverse kinematic equations with orthogonal manipulators since

 RiZ Rilz = Z if ai= 0 and 

 Ri1z y if ai= v/2.


 Doty (1986) has shown that, of the 24 classes of nontrivial orthogonal manipulators, those with 2 non-zero


(8.2)

..