The problem of planning the motion path of an unmanned vehicle is presented in the article. The results of development and the software implementation, and the research of the algorithm for constructing quasi- optimal trajectory of an unmanned vehicle in a known environment are shown. The RRT standard algorithm as the basis for the path construction between two points is used in the article. To improve the efficiency, the basic algorithm of the following modifications such as the orientation to the finish point, the removal of intermediate vertices are introduced. The orientation to the finish point allows to check the possibility of the direct connection to the last point which could be found by the RRT algorithm. The orientation also reduces the trajectory searching, because the basic RRT algorithm searches the point until a randomly generated point appears in the vicinity of the finish line. The deleting process of the intermediate vertices is carried out for such route sections where the trajectory could be straighten by the intermediate vertices’ removing without crossing the obstacles. The consideration of the kinematic constraints on the minimum turning radius of the vehicle, which is based on the Dubins curves is implemented in the article. As a result of all these algorithm modifications, its performance has been increased about 30% according to the computer simulation results.