## 3 Methodology
### Speed Control-Based Intersection Coordination Model (SICM)
#### 3.3.1 Calculation of Spatio-Temporal Trajectory
\begin{table}
\begin{tabular}{c c} \hline
**Parameters and Variables** & **Description** \\ \hline \(num\) & The total number of vehicles in the simulation period \\ \(ID=\left[1,2,\cdots,num\right]\) & The ID of vehicles \\ \(len\in\) & Serial numbers of lanes \\ \(\left[\text{E1,E2,E3,S1,S2,S3,W1,W2,W3,N1,N2,N3}\right]\) & Serial numbers of lanes \\ \(lane_{ID}=\left[lane_{1},lane_{2},\cdots,lane_{num}\right]\) & Serial numbers of lanes of the vehicles \\ \(t_{ID}^{mix}=\left[t_{id}^{mix},t_{id}^{mix},\cdots,t_{min}^{mix}\right]\) & Time of vehicles arrival in the approach road \\ \(v_{ID}^{init}=\left[v_{id}^{init},t_{id}^{mix},\cdots,v_{num}^{init}\right]\) & Initial speeds of vehicles \\ \(L_{ID}=\left[l_{1},L_{2},\cdots,L_{num}\right]\) & Virtual lengths of vehicles \\ \(\alpha_{i}^{ul}=\left[\alpha_{i}^{ul},\alpha_{i}^{ul},\alpha_{i}^{ul},\alpha_ {i}^{ul},\alpha_{i}^{ul}\right]\) (\(i\in ID,i=1,2,3,4\)) & Index of four conflict points through which the \(id\)th vehicle will pass \\ \(TB_{2}^{id}=\left[TB_{2\times v_{i}^{ul}-1}^{id},TB_{2\times v_{i}^{ul}}^{id} \right]\) (\(i=1,2,\cdots,16\)) & Duration of the \(i\)th vehicle occupation at the \(j\)th conflict point \\ \(\text{dist}_{id,i}^{cp}=\left\{\text{dist}_{id,i}^{cp},\text{dist}_{id,2}^{cp},\text{dist}_{id,3}^{cp},\text{dist}_{id,4}^{cp}\right\}\) & Distance between the \(i\)th conflict point and the stop line at the intersection \\ \(t_{\text{min}}\) & Total simulation time \\ \(dt\) & Time step \\ \(t_{id}^{cross}\) & Travel time for each vehicle \\ \(v^{id}=\left[v_{i}^{ul},v_{i}^{ul},\alpha_{i}^{ul},\cdots,v_{i}^{ul}\right]\) & Speed sequence of the \(i\)th vehicle \\ \(w^{max}=4.5\) m & Width of lane \\ \(v^{def}=30\) km/h & Default design speed \\ \(v^{max}=50\) km/h & Limit speed \\ \(v_{id}^{mod}=\left[v_{i}^{mod},v_{2}^{mod},\cdots,v_{max}^{mod}\right]\) & Modified design speed \\ \(len^{entna}=100\) m & Length of the approach road \\ \(len^{inter}=37\) m & Length of the intersection \\ \(len^{dept}=100\) m & Length of the departure road \\ \(\mu_{\max}|=1\) m/s\({}^{2}\) & The maximum value of acceleration \\ \hline \end{tabular}
\end{table} 
Table 2: Notations and descriptions. The vehicle will speed up to the speed limit in the departure road, and then we have  
\[i_{id}^{act2}=\left|v^{\max}-v^{def}\right|\div a_{\max} \tag{9}\]  
\[s_{id}^{act2}=\left[\left(v^{def}\right)^{2}-\left(v^{\max}\right)^{2}\right] \div 2a_{\max} \tag{10}\]  
where \(i_{id}^{act2}\) represents the time for the \(id\)th vehicle to accelerate uniformly and \(s_{id}^{act2}\) represents the distance traveled during acceleration. 4. The vehicle will move at the constant speed through the remaining part of the departure road, and then we have  
\[t_{id}^{unif2}=\left(len^{depart}-s_{id}^{act2}\right)\div v^{\max} \tag{11}\]  
where \(t_{id}^{unif2}\) represents the time of constant motion. Aggregating all these formulas together, we can calculate the spatio-temporal trajectory of vehicles. Additionally, the time bocks in the conflict-duration graph can be calculated and located (as shown in Figure 10), while the \(TB_{2\times cp_{i}^{il}-1}^{id}\) and \(TB_{2\times cp_{i}^{il}}^{id}\) represent the start and end time of occupation at the \(i\)th conflict point.