# A Rapid Verification System for Automatic Emergency Braking Control Algorithm of Passenger Car
## 4 Verification of Passenger Car AEB Control Algorithm
### AEB Control Algorithm  
The warning model of the AEB control algorithm can be divided into the safety distance model and the time-to-collision (TTC) model [26], which can evaluate the braking safety state base on safety distance and safety time respectively. The TTC model judges the braking time by comparing the TTC value of the current state with the braking time threshold. When TTC is less than the braking time threshold, a warning message will be sent or automatic emergency braking will be executed. The enhanced TTC model including vehicle acceleration can be described as:  
\[TTC=\left\{ \begin{array}{ll}\frac{d_{r}}{v_{r}},&v_{r}>0,a_{b}\leq a_{f}\\ \frac{-v_{r}+\sqrt{v_{r}^{2}+2\left(a_{b}-a_{f}\right)_{d}}}{a_{b}-a_{f}},&a_{ b}>a_{f}\\ T_{s},&\text{others}\end{array} \right. \tag{3}\]  
where \(d_{r}\) is the relative distance between VUT and VT; \(v_{r}\) is the relative speed between VUT and VT; \(a_{b}\) is the acceleration of VUT; \(a_{f}\) is the acceleration of VT; \(T_{s}\) is the set TTC value in safe conditions. The TTC model considers the motion state of the vehicle before the collision. When VT is stationary and VUT is braking, TTC can be calculated according to the relative distance and relative speed. The TTC algorithm will be switched by comparing the relative acceleration when VT is moving at a constant speed or decelerating. If the speed and acceleration value of VUT is less than or equal to VT, they will not collide, and the TTC value is safe. The AEB controller judges the braking status based on the TTC value and determines the expected acceleration value of the VUT. When the vehicle is running in good road condition, the maximum achievable braking acceleration is about \(-8\) m/s\({}^{2}\)[27]. Considering the comfortableness of automatic emergency braking, the expected acceleration of the AEB system is divided into three levels based on the principle of finite state machine [28], namely \(-2\) m/s\({}^{2}\), \(-4\) m/s\({}^{2}\), and \(-6\) m/s\({}^{2}\). The AEB system should also be designed with a collision warning function. Considering that a normal driver's reaction time is about 1 s [29], the warning and braking state can be defined in four stages [11]. The AEB system will switch its working status according to the corresponding TTC value. Figure 8: Verification system. 1. When 4 s \(\leq\) TTC < 3 s, the AEB system enters the brake warning status, and an audible and visual warning signal is issued to remind the driver to brake;
2. When 3 s \(\leq\) TTC < 2.25 s, if the driver still fails to brake timely, the AEB system enters the braking status named level I (L I), and the value of braking acceleration is \(-\)2 m/s\({}^{2}\);
3. When 2.25 s \(\leq\) TTC < 1.75 s, the AEB system enters the braking status named level II (L II), and the value of braking acceleration is \(-\)4 m/s\({}^{2}\);
4. When TTC \(\leq\) 1.75 s, the AEB system enters the braking status named level III (L III). It will remain in this status until the vehicle is completely stationary, or the TTC value increases and switches to another braking status. The acceleration value of L III braking status is \(-\)6 m/s\({}^{2}\). The lower controller of the AEB system adopts the PID control strategy to adjust the braking acceleration value, stabilize VUT acceleration at the target acceleration value, and realize error control of AEB acceleration. The PID control algorithm can be expressed as:  
\[\left\{ \begin{aligned} e(t)=a_{tar}(t)-a_{be}(t)\\ u(t)=K_{p}e(t)+K_{i}f_{0}^{t}e(t)dt+K_{d}\frac{de(t)}{dt}\end{aligned} \right. \tag{4}\]  
where \(e(t)\) is the error between the target acceleration value and the actual one at time \(t\); \(a_{tar}(t)\) is the target acceleration value at time \(t\); \(a_{be}(t)\) is the actual acceleration value at time \(t\); \(K_{p}\), \(K_{i}\), and \(K_{d}\) is the proportional coefficient, integral constant and differential constant respectively.