Singularities represent configurations from which certain directions of motion may be unattainable. It is possible to decouple the determination of a singular configurations for those manipulators with a spherical wrist into two simpler problems. The first is to determine the arm singularities, that is, singularities resulting from motion of the arm, which consists of the first three or more links, while the second is to determine the wrist singularities resulting from motion of the spherical wrist. Suppose that n=6, that is, the manipulator consists of a 3-DOF arm with a 3 - DOF spherical wrist. In this case the Jacobian matrix is a 6x6 matrix and a configuration is singular if and only if
if we now partition the Jacobian matrix into 
blocks as
then, since the final three joints are always revolute
Since The wrist axes intersect at a common point O, if we choose the coordinate frames so
that 
, then 
becomes
and the i-th column 
of 
is
if joint i is revolute and
if joint i is prismatic. In this case the Jacobian matrix has the block triangular form
with determinant
where 
and 
are each 3x3 matrices. has i-th column
if joint i is revolute, and 
if joint i is prismatic,
while
Appendix D shows how to caculate the dererminant for RRP:RRR manipulator.
