# Adaptive Cruise Control Based on Model Predictive Control with Constraints Softening
## 3 Mathematical Model  
### Mutual Longitudinal Kinematics Model  
Figure 1 shows the longitudinal kinematics model for the ACC system [17]. Inter-distance control is one of the important evaluation indexes of the ACC system.  
The inter-distance error between the actual and desired inter-distance at time step \(k\) can be expressed as  
\[\delta(k)=\delta_{\rm real}(k)-\delta_{\rm target}(k) \tag{1}\]  
with  
\[\delta_{\rm real}(k)=x_{\rm p}(k)-x(k)\]  
\[\delta_{\rm target}(k)=d_{\rm o}+h_{\rm p}v(k) \tag{2}\]  
Figure 1: Longitudinal kinematics for the adaptive cruise control (ACC) system.  
where \(\delta(k)\) represents the inter-distance deviation at time step \(k\); \(\delta_{\text{real}}(k)\) represents the real inter-distance, and \(\delta_{\text{target}}(k)\) represents the target inter-distance at time step \(k\); \(x(k)\) and \(x_{\text{p}}(k)\) represent the positions of the host vehicle and the preceding vehicle at time step \(k\), respectively; \(d_{\text{o}}\) represents the minimum safety inter-distance between the host vehicle and the preceding vehicle at time step \(k\), where the velocity of the host vehicle is low or zero; \(t_{\text{h}}\) represents the desired time headway, \(v(k)\) represents the velocity of the host vehicle.  
Different from the constant time headway (CTH) strategy, the variable time headway (VTH) strategy is adopted, which considers the effects of the host vehicle velocity and the relative velocity between the host vehicle and the preceding vehicle. In our previous work, we verified the superior performance of VTH [18]. The desired time headway can be expressed as  
\[t_{\text{h}}=\begin{cases}t_{1}+t_{2}v(k)-t_{3}v_{\text{rel}}(k)&v(k)<v_{\text {max}}\\ t_{1}+t_{2}v_{\text{max}}-t_{3}v_{\text{rel}}(k)&v(k)\geq v_{\text{max}}\end{cases} \tag{3}\]  
where \(t_{1}\), \(t_{2}\), and \(t_{3}\) are positive constants, which can be approximated by the reaction time of the driver; \(v_{\text{max}}\) represents the maximum velocity of the host vehicle; and \(v_{\text{rel}}(k)\) represents the relative velocity between the host vehicle and the preceding vehicle at time step \(k\).  
According to the discrete differential theory, the acceleration and jerk of the host vehicle at time step k can be represented as  
\[\begin{array}{l}a(k)=\frac{v(k)-v(k-1)}{T_{\text{s}}}\\ j(k)=\frac{a(k)-a(k-1)}{T_{\text{s}}}\end{array} \tag{4}\]  
where \(a(k)\) and \(j(k)\) represent the acceleration and the jerk of the host vehicle at time step \(k\), respectively, and \(T_{\text{s}}\) is the sampling period of the ACC system.  
Then, the relationship between the actual acceleration and the desired acceleration of the host vehicle satisfies the following conditions [19]:  
\[a(k+1)=(1-\frac{T_{\text{s}}}{\tau})a(k)+\frac{T_{\text{s}}}{\tau}u(k) \tag{5}\]  
where \(\tau\) is the time lag of the ACC system, and \(u(k)\) is the desired acceleration at time step \(k\).  
According to the mutual longitudinal kinematics characteristics between the host vehicle and the preceding vehicle of the ACC system, the discrete kinematics equation can be expressed as  
\[\begin{array}{l}\Delta x(k+1)=\Delta x(k)+v_{\text{rel}}(k)T_{\text{s}}+ \frac{1}{2}a_{\text{p}}(k){T_{\text{s}}}^{2}-\frac{1}{2}a(k){T_{\text{s}}}^{2 }\\ v_{\text{rel}}(k+1)=v_{\text{rel}}(k)+a_{\text{p}}(k)T_{\text{s}}-a(k)T_{ \text{s}}\\ v(k+1)=v(k)+a(k)T_{\text{s}}\\ a(k+1)=(1-\frac{T_{\text{s}}}{\tau})a(k)+\frac{T_{\text{s}}}{\tau}u(k)\\ j(k+1)=-\frac{1}{\tau}a(k)+\frac{1}{\tau}u(k)\end{array} \tag{6}\]  
where \(a_{\text{p}}(k)\) is the acceleration of the preceding vehicle at time step \(k\).