In order to understand the UR5 robot realtime trajectory changes, according to the Denavit-Hartenberg parameter method, by using the Link function in MATLAB Robotics Toolbox, a threedimensional mathematical model of UR robot was built. The positive and inverse kinematics of UR5 robot was analyzed. The relation between the angles of each joint was deduced. Threepolynomial interpolation method was used to analyze the theoretical space trajectory planning between the points. Combined with the MATLAB Robotics toolbox, the space trajectory planning and simulation of the robot model were done and the continuous varied trajectory curve was obtained. The results verify the correctness and rationality of the robot kinematics model. It provides a theoretical basis for further research and application of UR series robots