First we show that a similar result can be obtained if 85 and 86 are known or if 81 and 06 are known instead of the first two joint variables.

 In the development of the 4-DOF inverse kinematics solution, we have used the simplifying assumption that the last frame had DH-parameters d, a, and a all equal to zero. Although this assumption is obviously correct in the case of Eq. (7.2), we must show that it can be obtained in other cases. As shown in Eq. (2.7), a homogeneous matrix Ai decomposes into Ai = Ai Bi where Ai and Bi are given by Eqs. (2.8) and (2.9) and Ai is a homogeneous matrix for which DHparameters a, d, and a are zero. If the values of 81 and 86 are known, Equation (7.1) now reduces to the 4-DOF problem


 A2 A3 A4 A5 = Q (7.4)

where

 Q = A P- P A6- 1 B5 1 (7.5)


Similarly, If 85 and 86 are known, the inverse kinematic task becomes that of solving the 4-DOF case


 Al A2 A3 A4 = Q/ (7.6)

with

 Q = P A6-1 A5-I B4-I. (7.7)


 In the following section, we will show how a twodimensional iterative technique can be implemented to solve the inverse kinematics problem of six-DOF robot

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