The problem of inverse kinematics is fundamental in robot control. Many traditional inverse kinematics solutions, such as geometry, iteration, and algebraic methods, are inadequate in high-speed solutions and accurate positioning. In recent years, the problem of robot inverse kinematics based on neural networks has received extensive attention, but its precision control is convenient and needs to be improved. This paper studies a particle swarm optimization (PSO) back propagation (BP) neural network algorithm to solve the inverse kinematics problem of a UR3 robot based on six degrees of freedom, overcoming some disadvantages of BP neural networks. The BP neural network improves the convergence precision, convergence speed, and generalization ability. The results show that the position error is solved by the research method with respect to the UR3 robot inverse kinematics with the joint angle less than 0.1 degrees and the output end tool less than 0.1 mm, achieving the required positioning for medical puncture surgery, which demands precise positioning of the robot to less than 1 mm. Aiming at the precise application of the puncturing robot, the preliminary experiment has been conducted and the preliminary results have been obtained, which lays the foundation for the popularization of the robot in the medical field.