## 3 Dynamic Model and Integrated Controller of FWIDSV
### Tire Force Allocator Layer  
To facilitate the two-dimensional dynamic modeling and analysis of the vehicle, the following rationalization assumption is made: the road surface is flat, and the vertical motion of the vehicle is ignored. The schematic diagram of FWIDSV is shown Figure 8. The meanings of the symbols are as shown in Table 2. \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbols** & **Value** \\ \hline \(\alpha\) & 0.072 \\ \(\beta\) & 1 \\ \(q\) & 13 \\ \(p\) & 15 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of controller. According to [PERSON]'s second law, the balanced force equations of the vehicle along the x-axis, y-axis, and z-axis are obtained, as shown in (21). \[\mathbf{V}=\left[ \begin{array}{c}ma_{xd}-my\dot{\varphi}\\ ma_{yd}+mx\dot{\varphi}\\ l_{z}\dot{\varphi}\end{array} \right]=\mathbf{N}\mathbf{F} \tag{21}\]  
where  
\[\mathbf{F}=\left[ \begin{array}{cccccccc}F_{x11}&F_{x12}&F_{x21}&F_{x22}&F_{y11}&F_ {y12}&F_{y21}&F_{y22}\end{array} \right]\]  
\[\mathbf{N}=\left[ \begin{array}{cccccccc}a_{12}&a_{12}&a_{21}&a_{22}&-b_{11}&-b_{1 2}&-b_{21}&-b_{22}\\ b_{11}&b_{12}&b_{21}&b_{22}&a_{11}&a_{12}&a_{21}&a_{22}\\ c_{11}&c_{12}&c_{21}&c_{22}&d_{11}&d_{21}&d_{22}\end{array} \right]\]  
\[ \begin{array}{cccccccc}a_{ij}=\cos\delta_{ij};&&c_{ij}=(-1)^{j}d\cos\delta_{ ij}+(-1)^{i+1}l_{i}\cos\delta_{ij};\\ c_{ij}=\sin\delta_{ij};&&d_{ij}=(-1)^{j+1}d\sin\delta_{ij}+(-1)^{i+1}l_{i}\cos \delta_{ij}\end{array} \]  
As a redundant system, the dynamics model of FWIDSV has three degrees of freedom but it has eight controllable variables, including the lateral and longitudinal forces of each wheel in FWIDSV. The forces and moments output in the following control layer are distributed to four wheels for specific optimization purposes. LQP and pseudo-inverse are used to solve the math calculation in the most studies of the tire \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbols** & **Meaning** & **Symbols** & **Meaning** \\ \hline \(\delta_{ij}\) & Steering angles of roadwheels & \(a_{yd}\) & Desired lateral acceleration \\ \(a_{ij}\) & Tire side angles & \(\ddot{\varphi}\) & Yaw angle acceleration \\ \(\dot{x}\) & Longitudinal speed of vehicle & \(m\) & Mass \\ \(\dot{y}\) & Lateral speed of vehicle & \(l_{1}\) & Front wheelbase \\ \(\varphi\) & Yaw of the vehicle & \(l_{2}\) & Rear wheelbase \\ \(\dot{\varphi}\) & Yaw rate of the vehicle & \(d\) & Half of the tread \\ \(F\) & The tire forces & \(\mu\) & Ground adhesion coefficient \\ \(a_{xd}\) & Desired longitudinal acceleration & \(l_{z}\) & Yaw inertia \\  & in vehicle coordinate system & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The meaning of the symbols. Figure 8: The schematic of FWIDSV. force distribution [11; 12]. In this paper, LQP is introduced to distribute the tire force, as it is easier to solve optimization problems and add constraints. The objective function is shown in (22). \[\min\lvert\lvert\mathbf{N}\mathbf{F}-\mathbf{V}\rvert^{2} \tag{22}\]  
To solve the tire force through LQP, the optimization function needs to be converted into a standard form, as shown in (23). \[\min\left(\frac{1}{2}\tilde{x}^{T}G\tilde{x}+\tilde{x}^{T}c\right) \tag{23}\]  
Let matrix G and matrix \(\widetilde{\mathbf{x}}\) in linear quadratic programming be  
\[ \begin{array}{l}G=\left[\begin{array}{cc}2N^{T}N&-2N\\ -2N&2I\end{array}\right]\\ \widetilde{x}=\left[\begin{array}{cc}F&V\end{array}\right]^{T}\end{array}  \tag{24}\]  
Proof:  
\[ \begin{array}{l}J&=\frac{1}{2}\left[\begin{array}{cc}F^{T}&V^{T}\end{array} \right]\left[\begin{array}{cc}2N^{T}N&-2N^{T}\\ -2N^{T}&2I\end{array}\right]\left[\begin{array}{cc}F\\ V\end{array}\right]+\left[\begin{array}{cc}F^{T}&V^{T}\end{array}\right] \left[\begin{array}{cc}0\\ 0\end{array}\right]\\ &=\left[\begin{array}{cc}F^{T}N^{T}N-N^{T}N&-F^{T}N^{T}+V^{T}\end{array} \right]\left[\begin{array}{c}F\\ V\end{array}\right]\\ &=F^{T}N^{T}NF-V^{T}NF-F^{T}N^{T}V+V^{T}V\\ &=(NF-V)^{T}(NF-V)\\ &=|NF-V|^{2}\end{array}