Reachable Sets for Simple Models of Car Motion

In 1889, Andrey Andreevich Markov published a paper in “Soobscenija Charkovskogo Matematiceskogo Obscestva” where he considered four mathematical problems related to the design of railways. The simplest among these problems (and the first one in course of the presentation) is described as follows. Find a minimum length curve between two points in the plane provided that the curvature radius of the curve should not be less than a given quantity and the tangent to the curve should have a given direction at the initial point. In 1951, Rufus Philip Isaacs submitted his first Rand Corporation Report on differential game theory where he stated and lined out a solution to the “homicidal chauffeur” problem. In that problem, a “car” with a bounded turning radius and a constant magnitude of the linear velocity tries as soon as possible to approach an avoiding the encounter “pedestrian”. At the initial time, the direction of the car velocity is given. In 1957, in American Journal of Mathematics, Lester Eli Dubins considered a problem in the plane on finding among smooth curves of bounded curvature a minimum length curve connecting two given points provided that the outgoing direction at the first point and incoming direction at the second point are specified. Obviously, if one takes a particular case of Isaacs’ problem with the immovable “pedestrian”, then the “car” will minimize the length of the curve with the bounded turning radius. The arising task coincides with the problem considered by A. A. Markov. The difference from the problem by L. E. Dubins is in the absence of a specified direction at the incoming point. The fixation of incoming and outgoing directions presents in the other three problems by A. A. Markov. However, they contain additional conditions inherent to the railway construction. In such a way the notion of a “car” which moves only forward and has bounded turning radius appeared. In 1990, in Pacific Journal of Mathematics, James Alexander Reeds and Lawrence Alan Shepp considered an optimization problem where the object with bounded turning radius and constant magnitude of the linear velocity can instantaneously change the direction of motion to the opposite one. In a similar way, carts move around storage rooms. Thus, the model of the car that can move forward and backward has appeared. 8