## 2 Problem Statement and Input Parameters
By applying simple transformation, Equation (4) can be written in a more convenient form:  
\[F=d\cdot ln\left[\frac{tan\left(\frac{\beta_{max}}{2}\right)}{tan\left(\frac {\beta}{2}\right)}\right] \tag{5}\]  
Figure 4: Aircraft position where its main undercarriage center (\(U\)) reaches the maximum transient offtracking \(OTmax\). Figure 3: Offtracking of the aircraft main undercarriage center (\(U\)) when its datum point (\(S\)) follows a straight exit guideline (adapted from [11]). It is important to emphasize that all the formulas in the ICAO mathematical model, derived for the calculation of the aircraft undercarriage offtracking, can also be applied to the calculation of offtracking for a single-unit vehicle. In contrast to the aircraft undercarriage, where the datum point was located exactly in the center of the aircraft cockpit, the datum point on the datum length of the single-unit vehicle is usually defined as the most prominent point in the center line of the vehicle front part, i.e., the point in the middle of the vehicle front bumper (Figure 5). Looking at Figure 4, and based on the previously described formulas from the ICAO mathematical model, the calculation of the exact aircraft position where its main undercarriage center (\(U\)) reaches the maximum transient offtracking \(OT_{max}\) arises as a key problem that will be addressed in this paper. This particular aircraft position is determined by two parameters: the distance \(F_{OTmax}\) which the datum point (\(S\)) has covered along the straight exit tangent, and the steering angle \(\beta_{d}\) between the longitudinal aircraft axis and the exit tangent direction. Therefore, in the maximum transient offtracking position, based on the previous Equation (5), a new equation is established:  
\[F_{OTmax}=d\cdot ln\left[\frac{tan\left(\frac{\beta_{max}}{2}\right)}{tan \left(\frac{\beta_{d}}{2}\right)}\right] \tag{6}\]  
Since there are two unknown variables, distance \(F_{OTmax}\) and the angle \(\beta_{d}\), basic mathematical logic requires forming a system composed of two different equations in order to calculate these two variables. As Equation (6) applies to all positions of the aircraft when its datum point follows the exit tangent, for the second equation of the system it is necessary to find another reliable relation between the distance \(F_{OTmax}\) and the angle \(\beta_{d}\) that would be valid only in the position where the maximum transient offtracking is reached. The required additional relation between the distance \(F_{OTmax}\) and the angle \(\beta_{d}\) is obtained from the study [27] focused on the development of a mathematical model for transient offtracking calculation. This study proved that the maximum transient offtracking of a single-unit vehicle occurs when the direction of the motion of the rear axle (vector \(\overrightarrow{m}\)) is parallel to the tangent to the steering path (guideline). The geometry describing this case is shown in \[X=\frac{R}{d} \tag{2}\]0 In other words, only in the maximum transient offtracking position is the direction of the vector \(\overrightarrow{m}\) perpendicular to the line passing through the center of the rear axle (\(U\)) and the circular arc center \(C\). Only in this particular position is the distance of the vehicle datum point (\(S\)) from the beginning of the exit tangent \(F_{d}\) equal to the unknown distance \(F_{OTmax}\):  
\[X=\frac{R}{d} \tag{2}\]1  
\[X=\frac{R}{d} \tag{2}\]2 Terms and symbols related to the kinematic model of a single-unit vehicle. Using simple geometric relations retrieved from Figure 6 and the previous identity in Equation (7), the following equation is established:  
\[X=\frac{R}{d} \tag{2}\]3  
where the required distance \(F_{d}\) can further be expressed as follows:  
\[X=\frac{R}{d} \tag{2}\]4  
Since the required angle \(\beta_{d}\) varies in the range \(0<\beta_{d}<\beta_{max}\), and \(\beta_{max}\) is always \(\beta_{max}<90^{\circ}\), both sides of Equation (9) can be multiplied by \(cos\beta_{d}\).