# Adaptive Cruise Control Based on Model Predictive Control with Constraints Softening
## 5 Case Studies  
The effectiveness of ADAS is usually assessed through simulation studies. Traditional vehicle models are mostly based on Matlab/Simulink, which need to set up complex simulation working conditions and driver models to complete the closed-loop operating mode simulation. Instead, this paper utilizes Matlab/Simulink and CarSim to verify the proposed ACC strategy in a more real transportation circumstance. CarSim has such advantages as a high precision and stable operation and it can set up a complete driver model. For comparison, the proposed multi-objective ACC strategy with constraint softening is represented by ACC-Soften. The ACC strategy proposed in [24] is represented by a multi-objective ACC algorithm (MO-ACC), which is based on the MPC algorithm, considering the car-following and economy performance. At the same time, the constant time headway was utilized to analyze the mutual longitudinal kinematics model. The results indicate their superiority as compared with the linear quadratic regulator (LQR)-based ACC algorithm (LQR-ACC). The performances of car following, safety, comfort, and economy were tested to validate the performance of the proposed ACC algorithm. The working condition of the preceding vehicle was set according to Figure 2.  
Figure 1(a) shows the running trajectory of the preceding vehicle in the Federal Highway Administration (FHWA) Alt 3 Road working condition, which can be set up in CarSim. To better compare the algorithm in [24], we set the simulation parameters to the same values. At the same time, Alt 3 Road FHWA in the vehicle simulation software CarSim was selected to simulate the straight and curve of an ideal dry road. The parameters were set by the system itself without modification. The adhesion coefficient of the road was set to the value of 0.85, which simulates a dry road surface and is close to a real road surface. The radius of curvature of the roads was set to the values of 155 m, 150 m, and 125 m, respectively, which can effectively simulate the tracking situation of an actual vehicle under the conditions of a curve of a road. In this condition, the host vehicle needs to accelerate and decelerate frequently. The time for the simulation was set to 50 s.  
Figure 2: The working condition of the preceding vehicle. Figure 2b shows the location of the preceding vehicle in the Alt 3 Road FHWA condition. Before the simulation, the preceding vehicle is located 45 meters from the starting point. The host vehicle is located at the start point from the beginning. The velocity of the preceding vehicle is shown in Figure 2c. In order to verify the effectiveness of the proposed ACC algorithm, the velocity of the proceeding vehicle changes every 12 seconds. It changes dramatically from 30.6 m/s to 19.5 m/s during the first 40 seconds. Within the last 10 seconds, the preceding vehicle begins to brake increasingly. As a result, its velocity changes from 30.6 m/s to zero. The comparison results for ACC-Soften and MO-ACC are shown in Figure 3.  
Figure 3: Comparison results for ACC-Soften and multi-objective ACC algorithm (MO-ACC). Figure 3a shows the response curve of the inter-distance. It can be seen that both ACC-Soften and MO-ACC can guarantee safety during car following effectively, keeping an inter-distance larger than the length of the vehicle body. This paper adopted the variable time headway strategy to analyze the mutual longitudinal kinematics model. Substituting the traditional constant time headway with the variable time headway can ensure driving safety and improve traffic flows in complex and changing traffic environments. Figure 3b-d shows the velocity of the host vehicle, the relative velocity, and the acceleration of the host vehicle, respectively. Both ACC-Soften and MO-ACC can adjust the velocity to adapt to changes in the preceding vehicle, which helps to ensure the car-following performance.