This article presents a non-collision trajectory planning algorithm in three-dimensional space based on velocity potential field for robotic manipulators, which can be applied to collision avoidance among serial industrial robots and obstacles, and path optimization in multi-robot collaborative operation. The algorithm is achieved by planning joint velocities of manipulators based on attractive, repulsive, and tangential velocity of velocity potential field. To avoid oscillating at goal point, a saturated function is suggested to the attractive velocity potential field that slows down to the goal progressively. In repulsive velocity potential field, a spring damping system is designed to eliminate the chattering phenomenon near obstacles. Moreover, a fuzzy logic approach is used to optimize the spring damping coefficients for different velocities of manipulators. Different from the usual tangential velocity perpendicular to the repulsive velocity vector for avoiding the local minima problem, an innovative tangential velocity potential field is introduced that is considering the relative position and moving direction of obstacles for minimum avoidance path in three-dimensional space. In addition, a path priority strategy of collision avoidance is taken into account for better performance and higher efficiency when multi-robots cooperation is scheduled. The improvements for local minima and oscillation are verified by simulations in MATLAB. The adaptabilities of the algorithm in different velocities and priority strategies are demonstrated by simulations of two ABB robots in Robot Studio. The method is further implemented in an experimental platform with a SCARA and an ABB robot cooperation around a stationary obstacle and a moving object, and the result shows real time and effectiveness of the algorithms.