# Multicriteria Autonomous Vehicle Control at Non-Signalized Intersections
## 4 Simulation Results  
### Verification of the MPC Method  
In order to analyze the effectiveness and time-optimality of the proposed MPC method, it has been compared to the result of a simulation-based optimization algorithm in CarSim environment. Note, that latter optimization technique can only be used for offline analysis and validation due to its big computational time. For the sake of comparison, a four-directional double lane intersection has been chosen with four vehicles having initial conditions and turning intentions shown in Figure 3. As illustrated, Vehicle 1 and Vehicle are heading straight, while Vehicle 2 and Vehicle 4 are turning left with a 28 km/h speed constraint for entering the intersection. The simulation-based optimization algorithm is searching for the optimal acceleration values for each vehicles, by which they can safely pass the intersection without the danger of collision or skidding, while the total traveling time is minimized. The optimization algorithm running iteratively in MATLAB environment works as follows:  
* Initialization: the upper and lower bounds for the acceleration values are calculated based on the intersection scenario. These values are used as bound constraints for the solution of the optimization. The starting point for the optimization is chosen to be zero acceleration for each vehicles. * The constrained optimization algorithm runs the CarSim simulation with initial conditions and turning intentions depicted in Figure 3. The scheme of the CarSim multi-vehicle simulation is shown in Figure 4. Note, that the acceleration values for which the optimization algorithm is searching for are noted with \(a_{i}\in[1...4]\), and are used as inputs for the simulated vehicles. The minimization function for the optimization algorithm is the total traveling time, which is measured by the CarSim simulation. The simulation ends when the last vehicle leaves the intersection conflict zone. * To guarantee collision free passing of the vehicles, the positions of each vehicles are measured, and a inter-vehicle distant constraint is built in the optimization procedure for avoiding them to get closer to each other more than 3 m. In this case, a large number is added to the traveling time measured in CarSim, thus the optimization algorithm discards the corresponding acceleration values. * The constrained optimization runs the CarSim simulation iteratively until it bounds acceleration values for the autonomous vehicles by which a local minimum for the traveling time is reached. Figure 3: Initial conditions for the simulation. Next, the proposed time-optimal MPC intersection control has been evaluated with the same initial conditions depicted in Figure 3. The scheme of the real-time MPC control implemented in CarSim environment is depicted in Figure 5.  
Figure 4: Optimization scheme in CarSim environment. Figure 5: Real-time MPC control in CarSim environment. The result of the vehicle ordering and the traveling times spent in the intersection are shown in Figure 6.  
In the proposed time-optimal intersection control, Vehicle 3 crosses the intersection first, followed by Vehicle 1 and Vehicle 2, while Vehicle 4 finally leaves the intersection at 10.45 s.  
With the solution of the simulation-based optimization algorithm, the ordering of the vehicles are the same to that suggested by the proposed MPC controller. Here, the total traveling time of the vehicles are slightly smaller, with the last vehicle leaving the intersection at 10.31 s.  
Although the ordering of the vehicles are the same and the total traveling time is similar, the velocity profiles are different for the proposed MPC and the simulation-based optimization method, as depicted in Figure 7,b. Note, that while the velocity of Vehicle 4 is similar for both cases, the other vehicles choose different acceleration values due to the nature of the underlying algorithm. With the solution of the simulation-based optimization, Vehicle 1, Vehicle 2 and Vehicle 3 selects very small acceleration values compared to the proposed MPC method.