# Research on the Decoupling of the Parallel Vehicle Tilting and Steering Mechanism
## 2 Mechanism, Model, and Decoupling Method
### The DCPM and the ASC  
The conditions that are required for ensuring that \(H\!I\) and \(H^{\prime}I^{\prime}\) are translated relative to the frame \(AA^{\prime}O_{4}O_{4}{}^{\prime}\) are:  
\[\left\{\begin{array}{l}|\mathbf{A}\mathbf{B}|=|\mathbf{O}_{3}\mathbf{O}_{4}|=|\mathbf{A}^{\prime }\mathbf{B}^{\prime}|=\left|\mathbf{O}^{\prime}{}_{3}\mathbf{O}^{\prime}{}_{3}\right|\\ |\mathbf{B}\mathbf{O}_{3}|=|\mathbf{O}_{4}\mathbf{A}|=\left|\mathbf{B}^{\prime}\mathbf{O}^{\prime}{}_{3 }\right|=\left|\mathbf{O}^{\prime}{}_{3}\mathbf{A}^{\prime}\right|\end{array}\right., \tag{11}\]  
In order to facilitate the study of the vehicle's tilt motion, it is assumed that the vehicle's pitch angle is zero. As such, the ground plane \(\Sigma\) is parallel to \(i_{5}\). Then in the \(S_{5}\), the unit normal vector \(\mathbf{n}_{\Sigma}\) of the \(\Sigma\) can be expressed as:  
\[\mathbf{n}_{\Sigma}=\cos\theta_{\rm v}\mathbf{k}_{5}-\sin\theta_{\rm v}\mathbf{j}_{5}, \tag{12}\]  
where \(\theta_{\rm v}\) is the tilting angle of the vehicle (the angle between \(\mathbf{k}_{5}\) and the \(\Sigma\)).  
The vector radius of the two tangent points between the tire treads and \(\Sigma\) satisfy:  
\[\left\{\begin{array}{l}\left[(\mathbf{r})_{5}-(\mathbf{r}^{\prime})_{5}\right]\cdot \mathbf{n}_{\Sigma}\!\!=0\\ \mathbf{n}_{\Sigma}=(\mathbf{n})_{5}\\ \mathbf{n}_{\Sigma}=(\mathbf{n}^{\prime})_{5}\end{array}\right., \tag{13}\]  
By applying (12) and (13), \(\theta_{\rm v}\) and \(\mathbf{n}_{\Sigma}\) can be expressed as a function of \(\theta_{\rm t}\), \(\theta_{\rm t}\), and \(\delta_{\rm t}\).  
By applying (1)-(5), given \(\theta_{\rm t}\), \(\theta_{\rm t}\), and \(\delta_{\rm t}\), the left wheel steering angle \(\delta\) and the right wheel steering angle \(\delta^{\prime}\) can be obtained.  
\[\left\{\begin{array}{l}\cos\delta=\left(\mathbf{d}\right)_{5}\cdot\mathbf{i}_{5}\\ \cos\delta^{\prime}=\left(\mathbf{d}^{\prime}\right)_{5}\cdot\mathbf{i}_{5}\end{array}\right., \tag{14}\]  
In order to make \(\delta\) and \(\delta^{\prime}\) independent of \(\theta_{\rm v}\), by the application of (1)-(5), the conditions for eliminating \(\theta_{\rm t}\) and \(\theta_{\rm t}\)\({}^{\prime}\) in (14) are:  
\[\left\{\begin{array}{l}\left[(\mathbf{O}_{5}\mathbf{F})_{5}-(\mathbf{O}_{5}\mathbf{G})_{5} \right]\cdot(\mathbf{k}_{3})_{5}=|\mathbf{A}\mathbf{B}|\\ diag\big{(}0,\quad 1,\quad 1\big{)}\big{[}(\mathbf{O}_{5}\mathbf{F})_{5}-(\mathbf{O}_{5}\mathbf{G})_{5} \big{]}\times(\mathbf{k}_{3})_{5}=0\end{array}\right.^{\prime}, \tag{15}\]  
where  
\[(\mathbf{k}_{3})_{5}=R\Big{(}\mathbf{k}_{5},-\frac{\pi}{2}\Big{)}R\Big{(}\mathbf{i}_{5}, \varphi_{\rm i}-\frac{\pi}{2}\Big{)}R\Big{(}\mathbf{j}_{4},\frac{\pi}{2}-\theta_{ \rm t}\Big{)}\mathbf{k}_{3}.\]  
In conclusion, (11) and (15) are the DCPM.  
For the parallel mechanism that is symmetrically arranged on the left and right sides, the DCPM is described as: (a) when the vehicle is tilting, the two kingpins \(H\!I\) and \(H^{\prime}I^{\prime}\)translate relative to the frame and (b) when the projections of \(GF\) and \(G^{\prime}F^{\prime}\) in the TPV are parallel and equal as \(AB\) and \(A^{\prime}B^{\prime}\), respectively.  
In order to make \(\delta^{\prime}\) and \(\delta\) satisfy the ASC [16] when the vehicle steers left:  
\[\frac{L_{\mathrm{y}}}{\tan\delta^{\prime}}-\frac{L_{\mathrm{y}}}{\tan\delta}= L_{\mathrm{x}}, \tag{16}\]  
where \(L_{\mathrm{y}}\) is the track width and \(L_{\mathrm{x}}\) is the wheelbase.  
The steering geometry correction rate (SGCR) of the prototype can be expressed as: [22]:  
\[K=\frac{\delta-\delta^{\prime}}{\delta_{0}-\delta^{\prime}}. \tag{17}\]  
where \(\delta_{0}\) is the steering angle of the left wheel and \(\delta_{0}\) is satisfying the ASC.