## 5 Validation and Analysis of Kinetostatic Model with Computational Simulations
### Computational Simulations of Spatial Pointing Trajectory  
This section continues the verification of the kinetostatic model through an example of a spatial pointing trajectory. As shown in Figure 14a, the spatial pointing of the mechanism  
Figure 12: (**a**) Total-deformation results plot of the FE-model under the set of forces; (**b**) analytical and FE spiral trajectories. Figure 13: (**a**) Absolute errors of displacements \(\delta_{x}\) and \(\delta_{z}\), during spiral trajectory; (**b**) relative errors in percentage, between analytical and FE-results for both trajectories during the spiral trajectory. is represented by the normal vector \(l_{EC}\) of the mobile platform plane, where point \(C\) is the center of the mobile platform and point \(E\) is the intersection of the normal line of the mobile platform plane and the \(y\)-axis. By defining the included angle between the projection of vector \(l_{EC}\) on the \(O^{\prime}xz\) plane and the \(z\)-axis as the azimuth \(\alpha\), and the included angle between vector \(l_{EC}\) and \(y\)-axis as the pitch \(\Psi\), then the spatial pointing of the mechanism can be expressed as (\(\alpha\), \(\Psi\)) [28]. Given in the example is a set of spatial pointing trajectories, where the azimuth \(\alpha\) takes 72 points at equal intervals within the range of \([0,360^{\circ}]\), and the pitch angle \(\Psi\) is \(0.025^{\circ}\) (in the calculation process, the angle is in radian system). First, we convert the spatial pointing (the azimuth \(\alpha\) and the pitch angle \(\Psi\)) into RPY (the Roll-Pitch-Yaw representation method of rotation around the \(x\)-, \(y\)-, and \(z\)-axes of the fixed coordinate frame) angles \(\theta_{x}\) and \(\theta_{z}\), which rotates around the fixed \(x\)-axis and \(z\)-axis, respectively. A method of converting the spatially pointing RPY angle is provided in Appendix A.3. Then the angular displacements \(\theta_{x}\) and \(\theta_{z}\) of the 72 points were successively substituted in Equation (26) to obtained the input forces \(f_{1,y}\) and \(f_{2,y}\), the curves of which are shown in Figure 14b. \[ \begin{bmatrix}\theta_{x}\\ \theta_{z}\end{bmatrix} = \begin{bmatrix}C_{\theta_{x},f_{1,y}}&C_{\theta_{x},f_{ 2,y}}\\ C_{\theta_{z},f_{1,y}}&C_{\theta_{z},f_{2,y}}\end{bmatrix}\begin{bmatrix}f_{1,y }\\ \left\lceil{f_{2,y}}\right\rceil\\ \end{bmatrix} = \begin{bmatrix}\left(\mathbf{C}_{TOF_{1}}\right)_{[rows1,3,col5]}& \left(\mathbf{C}_{TOF_{2}}\right)_{[rows1,3,col5]}\end{bmatrix}\begin{bmatrix} f_{1,y}\\ \left\lceil{f_{2,y}}\right\rceil\end{bmatrix}  \tag{26}\]  
We define the matrix consisting of \(C_{\theta_{x},f_{1,y}}C_{\theta_{x},f_{2,y}}C_{\theta_{z},f_{1,y}}\) and \(C_{\theta_{x},f_{2,y}}\) as the rotation mapping matrix, in which \(C_{\theta_{x},f_{1,y}}\) represents the angular displacement of the mobile platform around the \(x\)-direction caused by a force along the \(y\)-direction at \(F_{1}\). By employing the obtained input forces \(f_{1,y}\) and \(f_{2,y}\) in the Finite Element Analysis, the angular displacement of the mobile platform can then be obtained. It should be noted that the angle obtained from the finite element analysis is the RPY angle rotating around the fixed coordinate frame, and the RPY angle also needs to be converted into the azimuth \(\alpha\) and pitch \(\Psi\). A conversion method is given in Appendix A.4. As shown in Figure 15a, the inner cone and outer cone respectively represent the FE-results and analytical results of the spatial pointing trajectory. The pitch is enlarged 1650 times for easier observation, and each generatrix on the cone represents a set of spatial points. To quantify the deviation between the spatial pointing azimuth \(\alpha\) and pitch \(\Psi\) between the theoretical calculation results and the finite element analysis, Figure 15b is given, where the polar diameter represents the pitch of spatial pointing, and the polar angle represents the azimuth angle. The absolute error and relative error of the azimuth angle are shown in Figure 15c,d, respectively.