## 1 Introduction
However, amongst all of the addressed mathematical and numerical models, only a few of them offered a solution for the calculation of the exact vehicle position where the maximum active transient offtracking is reached [7; 27]. This position indicates the exact cross section where the width of the pavement is at its maximum. The distance between the inner edge of the pavement and the steering path, at this very location, is of crucial importance when it comes to some elements of the crossroads [28; 29]. For example, in the most standards and design practices [1; 30], the usual width of the right-turn channel passing behind the triangular isle is 4.5-5.5 m. Though such a channel is slightly tapered (getting narrower towards its end), its net width is constant, as measured between the marking lane on the left and the curb on its right side. This particular net width comes directly from the maximum transient offtracking. Also, when it comes to the spiral ramps in multilevel garages (180\({}^{\circ}\) turns), their width is a direct consequence of the maximum transient offtracking. Nowadays, as demonstrated in [26], their efficiency and accuracy are not enough for road infrastructure design purposes, especially for autonomous vehicle positioning in horizontal curves. For example, the results of the maximum transient offtracking calculation in [PERSON]'s mathematical model for low-speed offtracking [27] are generated by the computer in a form of a graph. Before plotting the graph, equations determining the maximum transient offtracking vehicle position were numerically solved by the computer program. The graph shows the maximum offtracking for turn angles of up to 300\({}^{\circ}\) and various wheelbase/turn radius ratios. The graph is simple and suitable for everyday use, but the mathematical apparatus underlining the graph is not available to the user. In other words, the user cannot check the validity of the maximum transient offtracking graphically read from the diagram. The computation workload of [PERSON]'s method [7]is equivalent to ICAO's [11]. The method relies on the deployment of an \"equivalent base\", acting as a virtual base of an entire multibody vehicle assembly. [PERSON]'s procedure retrieves the maximum transient offfracking value, but with the error of 0.06-0.10 m in lateral terms and more than 1.2 m in longitudinal terms, as proven in [26]. On the other hand, commercially available software tools for vehicle swept path analysis based on CAD platform require from the user to draw guideline first, and then based on its shape and generated swept path width, determine the maximum transient offfracking for a selected design vehicle [31]. The aforementioned deficiencies were the key motivation for the authors of this paper to develop an exact mathematical solution for the maximum transient offfracking calculation in sharp turns. In fact, the primary aim was to develop mathematical formulas which will calculate accurately single-unit vehicle position where the maximum transient offfracking is reached. The new calculation method will eliminate the need to perform the vehicle movement simulations first and draw any of the vehicle movement trajectories, before determining the maximum transient offfracking value. Thus, by avoiding unnecessary procedures, calculation method presented herein is simple enough just to enter the vehicle datum length \(d\), circular radii \(R\) and turn angle, and instantly produces the maximum transient offfracking value, including the vehicle position where this maximum transient offfracking is reached. The sections in the paper are structured in the following order. In Section 2, mathematical definition of the problem and necessary parameters are presented, while in Section 3, mathematical solution of the problem is given. In Section 4 the accuracy of developed calculation method based on adopted regression model was evaluated and discussed. The last, Section 5, contains conclusions and future research plan to further improve application of the presented calculation method.