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
..