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.