# Hierarchical Synchronization Control Strategy of Active Rear Axle Independent Steering System
## 2 Dynamic Modeling
### Kinematic Model of Rear Wheel Steering  
The front axle steering mechanism of the vehicle is a type of traditional rack and pinion trapezoid steering mechanism, while the rear axle adopts the ARIS system. There is no mechanical connection between the front and rear steering axles, nor between the left and right wheels of the rear axle. In order to ensure the steering performance of the vehicle and reduce the sideslip and the wear of tires, the steering angle of each rear wheels must conform to the kinematic constraint model of the whole vehicle. Firstly, the steering angle scale factor between the front axle and the rear axle is obtained according to the vehicle dynamics model. Then, the steering angles of the left and right wheels of the rear axle are obtained according to the Ackermann steering geometry.  
In order to obtain the ratio of front and rear wheel steering angle at low speed steady driving, the dynamic model of four-wheel steering vehicle is simplified as a two degree of freedom vehicle model, given by [31].  
\[\left\{\begin{array}{l}mv_{x}(\dot{\beta}+\omega)+(k_{af}+k_{ar})\beta+\frac {{}^{\beta}k_{ar}-{}^{\beta}k_{ar}}{v_{x}}\omega=k_{af}\delta_{f}+k_{ar}\delta _{r}\\ I_{z}\dot{\omega}+(ak_{af}-bk_{ar})\beta+\frac{{}^{\alpha}k_{af}+{}^{\beta}k_{ ar}}{v_{x}}\omega=ak_{af}\delta_{f}-bk_{ar}\delta_{r}\end{array}\right. \tag{5}\]  
where \(m\) is the mass of the vehicle body; \(I_{z}\) is the yaw moment of inertia of the vehicle; \(k_{af}\) and \(k_{ar}\) are the cornering stiffness of the front and rear wheels; \(a\) and \(b\) are the distance from the front and rear axes to the vehicle mass center; \(\delta_{f}\) is the front axle wheel angle; \(\delta_{r}\) is the rear axle wheel angle; \(v_{x}\) is the vehicle longitudinal speed; \(\omega\) is the vehicle yaw rate; and \(\beta\) is the vehicle sideslip angle.  
Active rear wheel steering can produce an active sideslip angle, which can reduce the sideslip angle of the vehicle and improve the stability of the vehicle. In a steady state, \(da\omega dt=0\) and \(\beta=0\) are chosen as the objective functions, and the equivalent active steering target angle \(\delta_{r}\) of rear wheel can be obtained by [32]  
\[\delta_{r}=f_{k}(\delta_{f},v_{x})=\delta_{f}\frac{ak_{\alpha f}mv_{x}^{2}-b(a +b)k_{\alpha f}k_{\alpha r}}{bk_{\alpha r}mv_{x}^{2}+a(a+b)k_{\alpha f}k_{ \alpha r}} \tag{6}\]  
Figure 2 shows the kinematic model of four-wheel steering with two degrees of freedom. The dotted box in Figure 2 shows the steering angle kinematic relationship of the ARIS system. Generally, the front axle of the vehicle adopts the rack and pinion steering mechanism. In order to simplify the model, \(N\) is the reduction ratio from the driver's steering wheel angle to the steering angle of the front axle. \(f_{k}\) represents the ratio between front axle steering angle and rear axle steering angle. According to the Ackerman steering geometry principle [33], the ideal kinematic constraint equation of rear wheel steering is established as  
\[\left\{\begin{array}{l}\delta_{f}=\theta_{sw}/N\\ \tan\delta_{r1}=\rho\tan\delta_{r}/(\rho-w/2)\\ \tan\delta_{r2}=\rho\tan\delta_{r}/(\rho+w/2)\\ \rho=l/(\tan\delta_{f}+\tan\delta_{r})\end{array}\right. \tag{7}\]  
where \(\theta_{sw}\) is the driver's steering wheel angle; \(l=a+b\) is the wheel base; \(w\) is the wheel track; \(\rho\) is the turning radius; \(\delta_{r1}\) is the left rear wheel steering angle and \(\delta_{r2}\) is the right rear wheel steering angle.  
Substituting (6) into (7), the constraint equation can be obtained as  
\[\left\{\begin{array}{l}\delta_{r1}=\arctan(\rho\tan\delta_{r}/(\rho-w/2))\\ \delta_{r2}=\arctan(\rho\tan\delta_{r}/(\rho+w/2))\end{array}\right. \tag{8}\]  
where \(\delta_{r}=\delta_{f}\frac{ak_{\alpha f}mv_{x}^{2}-bk_{\alpha f}k_{\alpha r}}{ bk_{\alpha r}mv_{x}^{2}+ak_{\alpha f}k_{\alpha r}}\).