Autonomous underwater vehicles are vehicles that are entirely or partly independent of human decisions. In order to obtain operational independence, the vehicles have to be equipped with specialized software. The task of the software is to move the vehicle along a trajectory while avoiding collisions. In its role of avoiding obstacles, the vehicle may sometimes encounter situations in which it is very difficult to determine what the next movement should be from an ad hoc perspective. When such a situation occurs, a planning component of the vehicle software should be run with the task of charting a safe trajectory between nearby obstacles. This paper presents a new path planning algorithm for a Biomimetic Autonomous Underwater Vehicle. The main distinguishing feature of the algorithm is its high speed compared with such classic planning algorithms as A*. In addition to presenting the algorithm, this paper also summarizes preliminary experiments intended to assess the effectiveness of the proposed algorithm.