CHAPTER 7
SOLVING SIX-DOF MANIPULATORS
Reduction to.a 4-DOF Problem
At least six degrees of freedom are required for a robot manipulator to be able to arbitrarily position and orient its end-effector within its workspace. Equation
(2.14), with n equal to six, yields
A1 A2 A3 A4 A5 A6 = P. (7.1)
If both sides of this equality are multiplied by (A1 A2)-, the equation becomes
A3 A4 A5 A6 = Q (7.2)
with
Q a2-1 A1- P. (7.3)
When 81 and 82 are known, matrix Q is fully determined and can be viewed as a pose matrix for a 4-DOF arm whose structure is described by the left hand side of Eq. (7.2) which merely expresses a 4-DOF problem. In Chapter 5, we have seen that a 4-DOF problem can always be solved in closed-form, hence the remaining joint variables can be computed from Eq. (7.2).
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