## 2 Problem Statement and Input Parameters  
The International Civil Aviation Organization (ICAO) has its own mathematical model for the offfracking calculation of the main undercarriage center of the aircraft [11]. Based on this mathematical model, fillets at taxiways turns and intersections are designed. The same model applies to movement of an aircraft leaving its parking position on an apron or maneuvering on a holding bay. Basic terms and symbols related to aircraft kinematic model, taxiway and fillet design are displayed in Figures 1 and 2. In Figure 2b the path of the aircraft main undercarriage center (\(U\)), while the aircraft cockpit negotiates circular curve, is shown. Actually, the shortest distance between the circular curve denoting aircraft's cockpit guideline and the path of the main undercarriage center represents offfracking. The position of aircraft while its datum point, or cockpit, (\(S\)) follows an arc of a circular radii \(R\) can be determined based on the arc length expressed by its central angle \(\theta_{S}\) and the steering angle \(\beta\). According to ICAO mathematical model [11], the steering angle \(\beta\) between the tangent on the arc in the datum point (\(S\)) and the aircraft longitudinal axis is calculated as follows:  
\[tan\left(\frac{\beta}{2}\right)=\frac{1-e^{\mathrm{K}\Theta_{S}}}{X-K-X\cdot e ^{\mathrm{K}\Theta_{S}}-K\cdot e^{\mathrm{K}\Theta_{S}}} \tag{1}\]  
where the angle \(\theta_{S}\) should be entered in radians. As the result of the calculation, Equation (1) returns the value of the steering angle \(\beta\) also expressed in radians. It should be noted that Equation (1) was derived assuming that \(R>d\), where \(d\) is the aircraft datum length in meters (Figure 1b). In addition to the taxiway center line, which is followed by the airplane cockpit during turning maneuver, the example of two fillets, each composing of an arc of a circle and two tangents, can be seen in Figure 2c. The parameters \(X\) and \(K\) used in Equation (1), respectively, refer to the following:  
\[X=\frac{R}{d} \tag{2}\]  
\[K=\sqrt{X^{2}-1} \tag{3}\]  
The maximum steering angle \(\beta_{max}\) is achieved when the aircraft datum point (\(S\)) reaches the end of a circular arc (Figure 3). However, according to the authors of mathematical models for transient offtracking calculation [7; 27], the maximum transient offtracking of the main undercarriage center is not reached at the same point as the angle \(\beta_{max}\) (at the end of arc), but some distance beyond the end of the arc (Figure 4). Thus, as will be demonstrated later, the ICAO's assumption that the aircraft main undercarriage center reaches the maximum transient offtracking in the same position in which the \(\beta_{max}\) is achieved causes a significant error. Figure 1: Terms and symbols related to the aircraft kinematic model (adapted from [11]). Figure 2: Terms and symbols related to taxiway and fillet design (adapted from [11]). While the datum point (\(S\)) follows a straight guideline coinciding with the arc exit tangent, the steering angle \(\beta\) progressively decreases and the aircraft main undercarriage center (\(U\)) follows a tractrix (Figure 3). As defined in the ICAO mathematical model [11], the steering angle \(\beta\) is calculated as a function of the distance \(F\) which the datum point (\(S\)) has covered along the straight exit tangent:  
\[ln\left(tan\left(\frac{\beta}{2}\right)\right)=ln\left(tan\left(\frac{\beta_{ max}}{2}\right)\right)-\frac{F}{d} \tag{4}\]  
where \(\beta_{max}\) represents, again, the maximum steering angle at the end of an arc, and \(d\) is the aircraft datum length. In the last 60 years the ICAO mathematical model has been used for fillet designs at international airports by civil aviation authorities all over the world, and the accuracy of Equations (1) and (4) is unquestionable.