# Multicriteria Autonomous Vehicle Control at Non-Signalized Intersections
## 3 Model Predictive Intersection Control Design
### Time-Optimal Intersection Control Design  
One of the design goal is to minimize the crossing time of autonomous vehicles \(T_{total}\), therefore reduce the risk of a congestion emerging. This goal is achieved by ordering and accelerating vehicles to cross the intersection at the highest possible speed. Hence, the control algorithm defines constant acceleration \(a_{i}\)\(i\in[1...n]\) for the vehicles entering the intersection, by which the maximum speeds \(v_{i,max}\) on the given track available can be reached. First, the following computation is processed for vehicles entering the intersection:  
\[a_{i}=\frac{{v_{i,max}}^{2}-{v_{i,0}}^{2}}{2{s_{i,ent}}} \tag{1}\]  
where \(v_{i,0}\) is the initial speed, \(s_{i,ent}\) is the initial distance of all vehicles from the intersection origin. Corresponding to the specified conditions for acceleration, \(a_{i}=\{a_{max};a_{min}\}\) replaces the calculated accelerations when the thresholds are reached. In this case, the maximum entry speed in the conflict zone shall be changed as follows:  
\[v_{i,max}=\sqrt{{v_{i,0}}^{2}+2a_{i}{s_{i,ent}}} \tag{2}\]  
The essence of the coordination is to compare the travel time of autonomous vehicles at the intersection to prevent collisions during their crossing. In the process of minimizing travel time, conflict situations between autonomous vehicles are analyzed to control accelerations. Hence, altering the acceleration followed by the vehicles before entering the conflict zone of the intersection is integrated into the proposed algorithm. In case vehicles are in a conflict situation, the priority for passing through the intersection should be given to the vehicle with the least travel time. The rest of the vehicles reduce their acceleration to give priority for the first vehicle, then give priority to the vehicle leaving second in the same way. In the event ofmultiple vehicle conflicts, the algorithm takes into account the stopping of vehicles at the end of the entry zone and determines the waiting time for entering the conflict zone with a predetermined acceleration. For the proposed method, the entry and exit times of each vehicle in the conflict zone must be calculated. At the intersection, the entry time of vehicles can be determined by solving the following second-order equation:  
\[\frac{1}{2}{a_{i0}}{t_{i,ent}}^{2}+v_{i,0}{t_{i,ent}}-s_{i,ent}=0 \tag{3}\]  
where \(t_{i,ent}\geq 0\)\(i\in[1...n]\) is the entrance time. Assuming constant accelerations in the entering zone, (3) is simplified as:  
\[t_{i,ent}=\frac{s_{i,ent}}{(v_{i,max}+v_{i,0})/2} \tag{4}\]  
The time spent by the vehicles at the intersection shall be determined by assuming that the vehicle is traveling at a constant velocity when it gets to the conflict zone. Selecting \(a_{i}=0\) in the conflict zone the time spent is given as follows:  
\[t_{i,con}=s_{i,int}/v_{i,max} \tag{5}\]  
where \(s_{i,int}\) is the trajectory length corresponding with the turning intention. If the autonomous vehicle has to stop to give priority to other vehicle (or vehicles), a waiting time \(t_{i,wait}\) is also computed. In this case, the value \(v_{i,max}=0\) is set in order for the vehicle to stop in front of the conflict zone. The waiting time is defined as the difference between the final time of the previous vehicle and the entry time of the subject vehicle.