# Research on the Decoupling of the Parallel Vehicle Tilting and Steering Mechanism
## 2 Mechanism, Model, and Decoupling Method
### Mathematical Model  
The wheels on both sides need to be kept tangent to the ground while tilting. As such, the proposed model uses M/C circular section tires [20], as shown in Figure 2, wherein \(O_{\mathrm{w}}\) is the center of the tire section. The radius of the tire section is \(r_{0}\). The tire radius of \(O_{\mathrm{w}}\) is \(r_{\mathrm{w}}\). Taking the left wheel as an example, the mathematical model of the tire tread is established by the application of the circular vector function [21]. When modeling, the tire tread is assumed to be an infinite rigid.  
As shown in Figure 2, a right-handed rectangular coordinate \(S_{1}(O_{1};\mathbf{i}_{1},\mathbf{j}_{1},\mathbf{k}_{1})\) is established on the left wheel. The intersection \(O_{1}\) of the wheel axis and the wheel symmetry plane is the origin of \(S_{1}\). The point \(\mathbf{k}_{1}\) is along the \(O_{2}O_{1}\) direction. The point \(\mathbf{j}_{1}\) is located in the intersection of the horizontal plane and the wheel symmetry plane; \(\mathbf{j}_{1}\) is the same as the forward direction of the wheel. In the \(S_{1}\), the vector radius \(\mathbf{r}\), the unit normal vector \(\mathbf{n}\) of any point on the tire tread, and the unit vector \(\mathbf{d}\) in the forward direction of the wheel can be expressed as:  
\[\left\{\begin{array}{l}(\mathbf{r})_{1}=(r_{\mathrm{w}}+r_{0}\cos\varphi_{ \mathrm{w}})\mathbf{e}_{1}(\theta_{\mathrm{w}})+r_{0}\sin\varphi_{\mathrm{w}}\mathbf{k }_{1}\\ (\mathbf{n})_{1}=\left[\frac{\partial(\mathbf{r}_{1})_{1}}{\partial\theta_{\mathrm{w}} }\times\frac{\partial(\mathbf{r}_{1})}{\partial\varphi_{\mathrm{w}}}\right]/ \left[\frac{\partial(\mathbf{r}_{1})}{\partial\theta_{\mathrm{w}}}\times\frac{ \partial(\mathbf{r}_{1})}{\partial\varphi_{\mathrm{w}}}\right],\\ (\mathbf{d})_{1}=\mathbf{j}_{1}\end{array}\right. \tag{1}\]  
where \(\mathbf{e}_{1}(\theta_{\mathrm{w}})\) is a circular vector function [21] and \(\theta_{\mathrm{w}}\) and \(\varphi_{\mathrm{w}}\) are two variables of the tire tread.  
Figure 2: Diagram of the tire tread.  
A coordinate \(S_{2}\) (\(O_{2}\); \(i_{2}\), \(j_{2}\), and \(k_{2}\)) is able to be established on the kingpin \(HI\), with the steering knuckle \(O_{2}\) as the origin. The point \(i_{2}\) is along the \(O_{2}H\) direction and \(j_{2}\) is parallel to the LSPV and points to the outside of the vehicle. In the \(S_{2}\), \(r\), \(n\) and \(d\) can be expressed as:  
In the \(S_{5}\), \(\mathbf{O}_{5}\mathbf{F}\) can be expressed as:  
\[\left(\mathbf{O}_{5}\mathbf{F}\right)_{5}=|\mathbf{O}_{5}\mathbf{C}|\mathbf{k}_{5}-|\mathbf{C}\mathbf{D}|\mathbf{ i}_{5}-|\mathbf{D}\mathbf{E}|\mathbf{e}_{5}(\delta_{\rm t})+|\mathbf{E}\mathbf{F}|\mathbf{g}_{5}(\delta_{ \rm t}), \tag{8}\]  
where \(\mathbf{e}_{5}(\delta_{\rm t})\) and \(\mathbf{g}_{5}(\delta_{\rm t})\) are circular vector functions [21].  
So, for \(\mathbf{G}\mathbf{F}\):  
\[|\mathbf{G}\mathbf{F}|=|(\mathbf{O}_{5}\mathbf{F})_{5}-(\mathbf{O}_{5}\mathbf{G})_{5}|, \tag{9}\]  
From (9), \(\delta_{2}\) can be expressed as a function of \(\delta_{\rm t}\).  
The intermediate variable \(\varphi_{3}\) in (3) can be expressed as:  
\[\varphi_{3}=\pi-\theta_{\rm t}, \tag{10}\]  
The vehicle is arranged symmetrically from left to right. Similarly, the vector radius \((\mathbf{r}^{\prime})_{5}\), the unit normal vector \((\mathbf{n}^{\prime})_{5}\) of any point on the tire tread of the right wheel, and the unit vector \((\mathbf{d}^{\prime})_{5}\) in the forward direction of the right wheel can also be obtained in the \(S_{5}\).