# Strategy and Evaluation of Vehicle Collision Avoidance Control via Hardware-in-the-Loop Platform
## 3 Proposed System  
The timing of slamming on the brakes is according to the relative velocity and distance of the two vehicles. The rear-end collision system is usually evaluated according to the maximum distance. If the distance achieves the warning zone, the system will only initiate an alarm to the driver. Conversely, if it achieves the minimum limit, the system actively controls the vehicle to brake to avoid the collision. Figure 2 shows the detection of the front and rear cars. Figure 1: Relationship between all technologies of vehicle stability assistance. The key to one-dimensional AEB is the relative velocity \(v_{r}\), the deceleration \(a^{\prime}\) of the front car, and the maximum braking acceleration \(a_{r}\) of the rear car. Note that \(v_{r0}\) is the triggering speed, \(v_{f}\) is the ground speed of the front car, \(s\) is the relative distance of the two vehicles. Therefore, the related motion equations are as follows:  
\[{v_{r}}^{2}={v_{r0}}^{2}+2a_{r}s \tag{1}\]  
\[a^{\prime}=\frac{dv_{f}}{dt} \tag{2}\]  
Note that \(|a_{r}|>|a^{\prime}|\) and the target is \(v_{r}\approx 0\). In addition, the braking deceleration cannot take place beyond the maximum deceleration of the AEB. Otherwise, the traction force saturation will occur because of the nonlinear behavior of the tire's dynamics. Safety is the first issue. The distance \(s^{\prime}\), which is the collision avoidance zone, is set as  
\[s^{\prime}=-\frac{{v_{r0}}^{2}}{2a_{r}} \tag{3}\]  
Then, according to the distance \(s^{\prime}\), the system can invasively initiate emergency braking. However, the one-dimensional automatic brake causes different saturated and slippery tire conditions against the different velocities and distances of the front vehicle; therefore, it is immediately difficult to measure and compensate for. Table 1 gives the proposed fuzzy limits. Under this table, the rear vehicle can have a performance that is more comfortable while the AEB is activated. This rule is based on a series of engineering tests and adjustments. Safety and comfort are the dilemmas of the one-dimensional automatic brake. Therefore, steering to avoid collision from two-dimensional motion control is then proposed. It improves the driving safety and energy efficiency. Figure 3 shows the vehicle changing lanes to avoid a collision. When steering to avoid a two-dimensional collision begins, the vehicle requires one or two instances of steering. Therefore, it will cause significant yaw moment. The vertical and horizontal tractive forces of the tires rapidly decline due to nonlinear saturation in case the vehicle is out of control and in danger of losing its tractive force when steering. Therefore, control is the key to avoiding two-dimensional collisions. Figure 4 gives the model of the vehicle engaged in two-dimensional motion, and the vehicle of the motion equation is shown the following formula:  
\[Ma_{y}=MV_{x}\left(\gamma+\hat{\beta}\right)=2C_{f}\left(\delta_{f}+\frac{l_{ f}}{V_{x}}\gamma-\beta\right)-2C_{r}\left(\frac{l_{r}}{V_{x}}\gamma+\beta\right) \tag{4}\]  
\[N_{in}=I_{n}\dot{\gamma}=2l_{f}C_{f}\left(\delta_{f}+\frac{l_{f}}{V_{x}}\gamma -\beta\right)-2l_{r}C_{r}\left(\frac{l_{r}}{V_{x}}\gamma+\beta\right)+N_{z}+ N_{d} \tag{5}\]  
where \(\beta\) is the side slip angle, \(\gamma\) is the yaw-rate of the center of the vehicle's mass, \(M\) is its mass, \(a_{y}\) is the lateral acceleration, \(\delta_{f}\) is the steering angle of the front wheels, \(V_{x}\) is the longitudinal velocity of the vehicle, \(I_{n}\) is the nominal moment of inertia. \(C_{f}\) and \(C_{r}\) are the cornering stiffness of the front and rear wheels. \(l_{f}\) and \(l_{r}\) are the wheelbase of the center of the front and rear wheels. \(N_{z}\) is the active compensation torque. \(N_{d}\) is the torque from environmental disturbances, such as crosswind.