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One approach to designing an optimal real-time collision-free trajectory for autonomous underwater vehicles (AUVs) that move in a 3D unknown underwater space presented here. By explicitly considering the kinematic model of AUVs, a class of feasible trajectories is derived in a closed form, and is expressed in terms of two adjustable parameters for the purpose of collision avoidance. Then, a collision avoidance condition is developed to determine a class of collision-free trajectories. Finally, a performance index is established to find an optimal trajectory from the class. All the steps can be implemented in real-time. The advantages of the proposed approach are: (1) The 3D motion planning problem is reduced to a 2D problem. Instead of directly searching in a 3D space, one only needs to determine two parameters in their plane. Therefore computational efforts are greatly reduced, which is suitable for real-time implementation; (2) The vehicle's kinematic model is explicitly considered, and all boundary conditions are met. After the parameters are determined, the trajectory and controls are explicitly solved in closed forms. This method is shown to be effective by computer simulations.