  The paper presents the Inverse Kinematics (IK) close form derivation steps
using combination of analytical and geometric techniques for the UR robot. The
innovative application of this work is used in the precise positioning of
puncture robotics system. The end effector is a puncture needle guide tube,
which needs precise positioning over the puncture insertion point. The IK
closed form solutions bring out maximum 8 solutions represents 8 different
robot joints configurations. These multiple solutions are helpful in the
puncture robotics system, it allow doctors to choose the most suitable
configuration during the operation. Therefore the workspace becomes more
adequate for the coexistence of human and robot. Moreover IK closed form
solutions are more precise in positioning for medical puncture surgery compared
to other numerical methods. We include a performance evaluation for both of the
IK obtained by the closed form solution and by a numerical method.
