## 3 Methodology
### Geometric-Translation Based Intersection Coordination Algorithm (TICA)  
As we mentioned above, the intersection manager needs to allocate the spatial and temporal resources of the intersection according to real-time traffic flow data, that is, to conduct online scheduling while keeping a balance between solving efficiency and quality. Therefore, the complexity of the optimization algorithm should not be too high in order to pursue higher computing efficiency and shorter computing time. In this section, we will propose a Geometric Translation-based Intersection Coordination Algorithm (TICA) to address the problem. In fact, the collision can be avoided if vehicle B can be delayed for a while to reach the fourth conflict point (as shown in Figure 11a). The expected time difference (as shown in Figure 11b) can be calculated as follows:  
\[\Delta t=\left|TB_{8}^{1}-TB_{7}^{2}\right|+t^{safe} \tag{27}\]where \(TB_{8}^{1}\) and \(TB_{7}^{2}\) are known values, and \(t^{safe}\) represents the buffer time for ensuring safety. Furthermore, we can calculate the ideal time of vehicle B arriving at conflict point 4, that is  
\[TB^{\prime}_{7}=TB_{7}^{2}+\Delta t \tag{28}\]  
where \(TB^{\prime}_{7}^{2}\) represents the arriving time of vehicle B at the fourth conflict point after speed control. After that, we can calculate the modified design speed for vehicle B based on the known ideal time difference. As shown in Figure 12, \(t_{unif}^{def}\) and \(t_{unif}^{\text{mod}}\) represent the time needed to reach the fourth conflict point before and after speed control; \(t_{acel}^{def}\) and \(t_{acel}^{\text{mod}}\) represent the time of acceleration before and after speed control. Therefore, we have:  
\[\Delta t=\left|t_{unif}^{def}-t_{unif}^{\text{mod}}\right| \tag{29}\]  
where \(\Delta t\) represents the expected time difference. Additionally, \(v^{def}\) and \(v^{\text{mod}}\) represent the design speed given by the intersection manager before and after speed control, respectively. Among them, only \(t_{acel}^{\text{mod}}\) and \(v^{\text{mod}}\) are unknown, and the relationship between the two variables is:  
\[v^{\text{mod}}=v^{init}+a_{\text{max}}\times t_{acel}^{\text{mod}} \tag{30}\]  
Assume that the distance between the fourth conflict point and the stop line at the intersection are known, that is  
\[s^{\text{mod}}=len^{enta}+dist_{2,4}^{cp} \tag{31}\]  
where \(dist_{2,4}^{cp}\) represents the distances of vehicle B between the fourth conflict point and the stop line. Besides, the movement distance \(s_{1}\) of vehicle B before speed control and the distance \(s_{2}\) after speed control can be calculated as follows:  
\[s_{1}=\left[\left(v^{init}+v^{def}\right)\times t_{acel}^{def}\div 2\right]+ \left[v^{def}\times\left(t_{unif}^{def}-t_{acel}^{def}\right)\right] \tag{32}\]  
\[s_{2}=\left[\left(v^{init}+v^{\text{mod}}\right)\times t_{acel}^{\text{mod}} \div 2\right]+\left[v^{\text{mod}}\times\left(t_{unif}^{\text{mod}}-t_{acel}^{ \text{mod}}\right)\right] \tag{33}\]  
It is obvious that \(s_{1}\) and \(s_{2}\) are equal to \(s^{\text{mod}}\); therefore we have:  
\[s^{\text{mod}}=s_{1}=s_{2}, \tag{34}\]  
Figure 11: Diagram of speed control. (**a**) Ideal conflict resolution results of vehicles A and B; (**b**) speed control of vehicle B.  
Aggregating all these formulas together, we can calculate \(t_{acel}^{\text{mod}}\) and \(v^{\text{mod}}\), which can guarantee the safety and efficiency of vehicles. The velocity difference of design speed before and after speed control can be calculated as:  
\[\Delta v=\left|v^{def}-v^{\text{mod}}\right| \tag{35}\]  
The procedures of conflict detection and resolution strategy are concluded in Algorithm 1, and the details of TICA are shown in Algorithm 2. The algorithm framework, which has a computation complexity of \(O\left(n^{3}\right)\), adopts an iterative method to generate optimal traffic solution for vehicles with no collisions, which can optimize vehicle sequences to reduce average delays and ensure safety with a linear computational complexity.