# A Collision Risk Assessment Method for Aircraft on the Apron Based on Petri Nets
## 3 Quantitative Evaluation Method for Aircraft Collision Risk
### Calculation of Aircraft Potential Movement Space  
As an object moves from the starting point to the endpoint, its potential position at any given moment can be represented by a circle. The radius of this circle is the product of time and the maximum speed of the object, and this radius gradually increases over time. Assuming the object takes 80 min to travel from the starting point to the endpoint, a diagram illustrating the reachable areas from the starting point and the endpoint can be constructed in Figure 3 [24]:  
\[S(t)=|x-x_{i}|\leq(t-t_{i})V_{m} \tag{2}\]  
\[D(t)=|x-x_{j}|\leq(t_{j}-t)V_{m} \tag{3}\]  
In this context, \(S(t)\) and \(D(t)\) represent the circles that define the reachable areas from the starting point and the endpoint, respectively. \(x_{i}\) and \(x_{j}\) denote the coordinates of the starting point and the endpoint, respectively. \(t_{i}\) represents the time at which the object departs, \(t_{j}\) represents the expected arrival time, and \(V_{m}\) represents the maximum speed of the object.  
Assuming the object moves from the starting point to the endpoint at a constant speed, its position at any given moment can be calculated as the product of time and average speed. However, during taxiing, the object is subject to constraints such as location and surface operational conditions, leading to deviations between its actual position and the  
Figure 3: Example of reachable range diagram.  
Figure 2: Example of a Petri net on the apron.  
ideal position. To ensure timely arrival at the designated location, this deviation is confined within a certain range, referred to as the object's potential movement space at the current time. This space represents the range of positions the object is allowed to occupy at that moment. The calculation of this position range is determined by the relationship between the reachable circle from the starting point and the reachable circle from the endpoint.  
When the object first departs from the starting point, the reachable circle from the starting point is small, while the reachable circle from the endpoint is large, with the starting point's circle contained within the endpoint's circle. As time progresses, the starting point's reachable circle gradually expands, and the endpoint's reachable circle gradually shrinks, eventually intersecting each other. Finally, the endpoint's reachable circle becomes contained within the starting point's reachable circle. Since the object (aircraft) is constrained by the boundaries of the taxiway during taxiing, it can be assumed that the deviation only occurs along the \(x\)-axis of the taxiing route. This can be summarized as follows:  
\[\left\{\begin{array}{ll}S(t)\subset D(t)&t\leq\frac{(t_{i}+t_{j}-D_{ij}/V_{m} )}{2}\\ S(t)\cap D(t)&\frac{(t_{i}+t_{j}-D_{ij}/V_{m})}{2}\leq t\leq\frac{(t_{i}+t_{j} +D_{ij}/V_{m})}{2}\\ S(t)\supset D(t)&\frac{(t_{i}+t_{j}+D_{ij}/V_{m})}{2}\leq t\end{array}\right. \tag{4}\]  
when at time \(S(t)\subset D(t)\), the upper and lower bounds of the corresponding deviation interval are calculated as follows:  
\[L_{x}(t)=U_{x}(t)-2V_{m}(t-t_{i}) \tag{5}\]  
\[U_{x}(t)=(\overline{V}-V_{m})(t_{i}-t) \tag{6}\]  
At time \(S(t)\cap D(t)\), the upper and lower bounds of the corresponding deviation interval are calculated as follows:  
\[L_{x}(t)=(\overline{V}-V_{m})(t_{j}-t) \tag{7}\]  
\[U_{x}(t)=(\overline{V}-V_{m})(t_{i}-t) \tag{8}\]  
At time \(S(t)\supset D(t)\), the upper and lower bounds of the corresponding deviation interval are calculated as follows:  
\[L_{x}(t)=(\overline{V}-V_{m})(t_{j}-t) \tag{9}\]  
\[U_{x}(t)=L_{x}(t)+2V_{m}(t_{j}-t) \tag{10}\]  
where \(L_{x}(t)\) is the lower bound of the aircraft's position at time \(t\), and \(U_{x}(t)\) is the upper bound of the aircraft's position at time \(t\).