where

 nx b tx Px

n = ny b = by t= t p = p

 nz bz tz Pz


and

 n b t

 R = n b ty

 nz bz tz



 The inverse kinematics problem for a n-degree-offreedom manipulator consists of finding a set of joint variables values, called a solution set, that will satisfy the equation


 A1 A2 A3 A4 A5.An = P. (2.14)


 This matrix equation gives rise to a system of nonlinear equations whose complexity depends on the manipulator geometry, as described by the DH-parameters.

 At least six degrees of freedom are required to arbitrarily position and orient a rigid body in space. Therefore, when n is larger than six, the manipulator is redundant and the system of equations implied by Eq. (2.14) is underconstrained. If n is less than 6," the system becomes overconstrained and when n is equal to 6, the inverse kinematic problem is exactly specified. In this

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