C4 nx

 S4R2 1 R x = R31 R2 R1 ny

 0 n



can be used to compute the last variable e4. This shows

that a 4-DOF inverse kinematic problem will, in general (general in the sense that matrix H is nonsingular), yield a unique solution set. However, for some manipulator

geometries and/or some particular end-effector poses, the problem may have more than one solution.



 Special 4-DOF Manipulator Geometries


 Equation (5.21) is valid only when matrix H is invertible. The determinant of matrix H, computed from Eq. (5.22), is given by


dH = 61 62 al[a2 (a32 wI + a32 w2) + 2 02 03 a3 d2 w3]

 +0 a~ 2 2 +02 d2 22 2
 + C3 a3 12 (a2 '+ o2 2 ) + c2 a1.] w4 (5.27)


where the quantities wI, w2, w3, and w4 are defined, in terms of the components of pose vectors t and p, as


 wI= t x2 + ty2 w2= Px2 + py2

 t
 ~w3.Pxt y tx
 W4 -x tx

..