Three-Dimensional Reachability Set For a Dubins Car: Reduction of the General Case of Rotation Constraints to the Canonical Case

In mathematical control theory, a Dubins car is a nonlinear motion model described by differential relations, in which the scalar control determines the instantaneous angular rate of rotation. The value of the linear velocity is assumed to be constant. The phase vector of the system is three-dimensional. It includes two coordinates of the geometric position and one coordinate having the meaning of the angle of inclination of the velocity vector. This model is popular and is used in various control tasks related to the motion of an aircraft in a horizontal plane, with a simplified description of the motion of a car, small surface and underwater vehicles, etc. Scalar control can be constrained either by a symmetric constraint (when the minimum rotation radii to the left and right are the same) or asymmetric constraint (when rotation is possible in both directions, but the minimum rotation radii are not the same). Usually, problems with symmetric and asymmetric constraints are considered separately. It is shown that when constructing the reachability set at the moment, the case of an asymmetric constraint can be reduced to a symmetric case.