# Dynamic Decoupling and Trajectory Tracking for Automated Vehicles Based on the Inverse System
## 2 Coupling Mechanism of 4 WID Vehicle  
The two-track vehicle dynamic model representing the 4 WID vehicle established in the Cartesian coordinate system to study the dynamic characteristics and coupling mechanism of the vehicle, as shown in Figure 1.  
Assuming the small steering turning, and for simplification, the steering angles of the left and right front tires are equivalent to steering angle \(\delta\)[25]. The planar motion of the vehicle can be expressed as:  
\[m\ddot{x}=m\dot{y}\dot{\varphi}+F_{x1}\cos\delta-F_{y1}\sin\delta+F_{x2}\cos \delta-F_{y2}\sin\delta+F_{x3}+F_{x4}-k_{D}\dot{x}^{2} \tag{1}\]  
\[m\ddot{y}=-m\dot{x}\dot{\varphi}+F_{x1}\sin\delta+F_{y1}\cos\delta+F_{x2}\sin \delta+F_{y2}\cos\delta+F_{y3}+F_{y4}, \tag{2}\]  
\[l_{x}\ddot{\varphi}=l_{f}\big{(}(F_{x1}+F_{x2})\sin\delta+\big{(}F_{y1}+F_{y2} \big{)}\cos\delta\big{)}+d\big{(}(F_{x1}-F_{x2})\cos\delta-\big{(}F_{y1}-F_{y2 }\big{)}\sin\delta+F_{x3}-F_{x4}\big{)}-l_{x}\big{(}F_{y3}+F_{y4}\big{)}, \tag{3}\]  
where the definition of each symbol is showed in Table 1.  
When the tire force differential algorithm is adopted, the longitudinal forces are synthetically considered as the total longitudinal force \(F_{t}\) and the additional yaw moment \(M_{z}\)[26]. \[F_{t}=F_{x1}+F_{x2}+F_{x3}+F_{x4} \tag{4}\]  
\[M_{z}=d\big{(}\Delta F\cos\delta+\Delta F\big{)}, \tag{5}\]  
where the longitudinal force distribution ratio of the front and rear axes \(k_{r}\) is defined as:  
\[k_{r}=\frac{F_{x3}+F_{x4}}{F_{t}}, \tag{6}\]  
The additional yaw moment \(M_{z}\) is caused by the longitudinal force difference \(\Delta F\) between the vehicle's two sides:  
\[\Delta F=F_{x1}-F_{x2}=F_{x3}-F_{x4}, \tag{7}\]  
Then, introducing the small angle hypothesis, the tire lateral force is proportional to its slip angle [27]. \[\alpha_{f}=\delta-\theta_{f}=\delta-\big{(}\dot{y}+l_{f}\dot{\varphi}\big{)}/ \dot{x}, \tag{8}\]  
\[m\ddot{x}=m\dot{y}\dot{\varphi}+F_{x1}\cos\delta-F_{y1}\sin\delta+F_{x2}\cos \delta-F_{y2}\sin\delta+F_{x3}+F_{x4}-k_{D}\dot{x}^{2} \tag{1}\]0  
\[m\ddot{x}=m\dot{y}\dot{\varphi}+F_{x1}\cos\delta-F_{y1}\sin\delta+F_{x2}\cos \delta-F_{y2}\sin\delta+F_{x3}+F_{x4}-k_{D}\dot{x}^{2} \tag{1}\]1 The two-track model with a reference position trajectory. The two-track model is simplified to a 3 DOF model. \[m\ddot{x}=m\dot{y}\dot{\varphi}+F_{x1}\cos\delta-F_{y1}\sin\delta+F_{x2}\cos \delta-F_{y2}\sin\delta+F_{x3}+F_{x4}-k_{D}\dot{x}^{2} \tag{1}\]2  
\[m\ddot{x}=m\dot{y}\dot{\varphi}+F_{x1}\cos\delta-F_{y1}\sin\delta+F_{x2}\cos \delta-F_{y2}\sin\delta+F_{x3}+F_{x4}-k_{D}\dot{x}^{2} \tag{1}\]3  
\[m\ddot{x}=m\dot{y}\dot{\varphi}+F_{x1}\cos\delta-F_{y1}\sin\delta+F_{x2}\cos \delta-F_{y2}\sin\delta+F_{x3}+F_{x4}-k_{D}\dot{x}^{2} \tag{1}\]4  
The dynamics can be rewritten in the state-space form, with the states being \(\mathbf{X}=\left(\dot{x},\dot{y},\dot{\varphi}\right)^{T}\), and the control inputs being \(\mathbf{U}=\left(F_{t},\delta,M_{z}\right)^{T}\),i.e.,  
\[\dot{\mathbf{X}}=\left[ \begin{array}{cccc}\frac{-k_{D}\dot{x}}{m}&\dot{\varphi} &0\\ -\dot{\varphi}&-2\frac{C_{f}+C_{r}}{m\dot{x}}&-2\frac{C_{f}l_{f}-C_{r}l_{r}}{m \dot{x}}\\ 0&-2\frac{C_{f}l_{f}-C_{r}l_{r}}{I_{z}\dot{x}}&-2\frac{C_{f}l_{f}^{2}+C_{r}l_{ r}^{2}}{I_{z}\dot{x}}\end{array} \right]\mathbf{X}+\left[ \begin{array}{cccc} \frac{1}{m}&2C_{f}\frac{\dot{x}}{\dot{x}}+l_{f}\frac{\dot{\varphi}}{\dot{x}} -\delta&0\\ \frac{(1-k_{r})\delta}{m}&\frac{2C_{f}l_{r}}{I_{z}}&0\\ \frac{l_{f}(1-k_{r})\delta}{I_{z}}&\frac{2C_{f}l_{f}}{I_{z}}&\frac{1}{I_{z}} \end{array} \right]\mathbf{U}, \tag{13}\]  
In Equation (13), the input matrix is a non-diagonal matrix, which means each state of the system is affected by multiple inputs. The input matrix is not a constant coefficient matrix, but contains input elements and state elements, which indicates that there are complex interactions among three directions and causes the attendant disturbances when controlling one of the directions, such as the \(-2C_{f}\beta^{2}/m\) is a longitudinal resistance caused by lateral control input.