# Kinematic Analysis and Verification of a New 5-DOF Parallel Mechanism
## 3 Kinematic Analysis  
### Establishment of Inverse Kinematics Equations  
The coordinate systems of the mechanism consists of a moving coordinate system and a fixed coordinate system, as shown in Figure 5. The transformation matrix of the moving coordinate system \(o_{b}-xyz\) relative to the fixed coordinate system \(O_{p}-XYZ\) is:  
Figure 6: Singularity of improved scheme. \[T=\text{Trans}(x,y,z)\cdot\text{Rot}(\alpha,\beta,\gamma) \tag{7}\] \[=\left( \begin{array}{cccc}1&0&0&x\\ 0&1&0&y\\ 0&0&1&z\\ 0&0&0&1\end{array} \right)\left( \begin{array}{cccc}\cos\beta&\sin\beta\text{ sin}\alpha&\sin\beta\text{cos}\alpha&0\\ 0&\cos\alpha&-\sin\alpha&0\\ -\sin\beta&\cos\beta\text{sin}\alpha&\cos\beta\text{cos}\alpha&0\\ 0&0&0&1\end{array} \right)\] \[=\left( \begin{array}{cccc}\cos\beta&\sin\beta\text{sin}\alpha& \sin\beta\text{cos}\alpha&x\\ 0&\cos\alpha&-\sin\alpha&y\\ -\sin\beta&\cos\beta\text{sin}\alpha&\cos\beta\text{cos}\alpha&z\\ 0&0&0&1\end{array} \right)\]  
where \(x\), \(y\), and \(z\) represent the moving distance of the moving coordinate system along the \(X\), \(Y\), and \(Z\) axes of the fixed coordinate system respectively; \(\alpha\), \(\beta\), and \(\gamma\) represent the rotating angles of the coordinate system along the \(X\), \(Y\), and \(Z\) axes of the fixed coordinate system respectively. Any point in the moving coordinate system can be converted into a specific coordinate value in the fixed coordinate system by  
\[\left( \begin{array}{c}S_{i}\\ 1\end{array} \right)=T\left( \begin{array}{c}s_{i}\\ 1\end{array} \right) \tag{8}\]  
where \(S_{i}\) is the coordinate value of five spherical hinges in the fixed coordinate system; \(s_{i}\) is the coordinate value of five spherical hinges in the fixed coordinate system, and \(T\) is the coordinate transformation matrix. The coordinate values of Hook joint \(U_{l}(i=1\)\(\sim\)5) on the fixed platform and spherical hinge \(S_{l}(i=1\)\(\sim\)5) on the moving platform in their respective coordinate systems are obtained through geometric relationship. Then the coordinate values of each point of the moving platform \(S_{l}(i=1\)\(\sim\)5) in the fixed coordinate system can be obtained from Equations (7). During the movement of the moving platform, the link length remains unchanged. Any drive limb has the following closed-loop vector diagram shown in Figure 7:  
Closed loop vector equation is established for Figure 7:  
\[\overline{S_{i}U_{i}}=\overline{O_{p}U_{i}^{\prime}}+\overline{U_{i}^{\prime} U_{i}}-\overline{O_{p}S_{i}},i=1,2,\ldots 5 \tag{9}\]  
Among them, \(O_{p}\) is the origin of fixed coordinate system, \(S_{i}\) is the geometric center of spherical hinge on moving platform, \(U_{i}\) is the geometric center of Hook joint fixed on slider, \(U_{i}^{\prime}\) is the projection of \(U_{i}\) on the plane of fixed coordinate system \(XOY\), and the modulus of  
Figure 7: Closed-loop vector diagram. the vector corresponds to the length from the center of the Hook joint to the center of the spherical hinge on each driving limb of \(L_{1}{\sim}L_{5}\), and this length is a fixed value during the movement of the moving platform. \[L_{i}=\left|\overline{S_{i}U_{i}}\right|=\sqrt{\left(X_{i}^{U}-X_{i}^{S}\right) ^{2}+\left(Y_{i}^{U}-Y_{i}^{S}\right)^{2}+\left(Z_{i}^{U}-Z_{i}^{S}\right)^{2}} \tag{10}\]  
where \(X_{i}^{U}\), \(Y_{i}^{U}\), and \(Z_{i}^{U}\) represent the \(X\), \(Y\), and \(Z\) coordinates of \(U_{i}\); \(X_{i}^{S}\), \(Y_{i}^{S}\), and \(Z_{i}^{S}\) represent the \(X\), \(Y\), and \(Z\) coordinates of \(S_{i}\). The mapping equation for the position of five driving sliders and the posture coordinates of the moving platform is simplified as follows:  
\[Z_{i}^{U}=Z_{i}^{S}\pm\sqrt{L_{i}^{2}-\left(X_{i}^{U}-X_{i}^{S}\right)^{2}+ \left(Y_{i}^{U}-Y_{i}^{S}\right)^{2}} \tag{11}\]  
Specific inverse equations can be obtained by incorporating coordinate values of \(U_{i}\) and \(S_{i}\). Considering that the five sliders must be above the moving platform during the movement, the position inverse solution equation only takes positive signs.