## 5 Validation and Analysis of Kinetostatic Model with Computational Simulations
### Computational Simulations of Spatial Pointing Trajectory
The relative error of pitch is also given in Figure 15e. Figure 14: (**a**) Spatial pointing of mechanism; (**b**) curves of input forces. As shown in Figure 15, the analytical results of spatial pointing show good consistency with the FE result. The maximum absolute errors of the azimuth and pitch are 2.56 mrad and 3.74 mrad, respectively, and the relative errors of pitch fluctuate between 0.845% and 0.86%, which verifies the accuracy of the kinetostatic model. It can be inferred that for the spatial pointing trajectory, the relative error of pitch is within 1% when the pitch angle \(\Psi\) does not exceed 0.025\({}^{\circ}\). Meanwhile, it is observed that the errors caused by the kinetostatic model have a certain regularity with the change in the azimuth angle. Thus, the errors can be compensated by the control at the mobile platform to achieve the desired spatial pointing. It can be seen from Figure 16a,c,e that \(C_{\delta_{x},f_{1},g}\) and \(C_{\delta_{x},f_{1},g}\) are less affected by the structure parameters of the flexure hinge, while the absolute values of \(C_{\delta_{x},f_{1},g}\) and \(C_{\delta_{x},f_{2},g}\) increase as \(r\) increases and decrease as \(w\) and \(t_{0}\) decrease. It is indicated that when only the forces \(f_{1,g}\) or \(f_{2,g}\) are applied, increasing \(r\) will cause a larger linear displacement of the mobile platform center, while increasing \(w\) or \(t_{0}\) will yield a smaller linear displacement. As shown in Figure 16b,d,f, \(C_{\delta_{x},f_{1},g}\) and \(C_{\delta_{x},f_{2},g}\) are less affected by the structure parameters of the flexure hinge, and the absolute values of \(C_{\delta_{x},f_{2},g}\) and \(C_{\delta_{x},f_{1},g}\) increase as \(r\) increases and decreases as \(w\) or \(t_{0}\) increase. It is revealed that when only forces \(f_{1,g}\) or \(f_{2,g}\) are applied, increasing \(r\) will result in a larger angular displacement of the mobile platform center, and increasing \(w\) or \(t_{0}\) will cause a smaller angular displacement. Figure 17 shows the variation of the translation mapping matrix \(\mathbf{C}_{T}\) and the rotation mapping matrix \(\mathbf{C}_{R}\) in terms of the branches number \(n\). As can be seen from Figure 17a, with the increase in the branches number \(n\), the absolute values of \(C_{\delta_{x},f_{1},g^{\prime}}\), \(C_{\delta_{x},f_{1},g}\), and \(C_{\delta_{x},f_{2},g}\) decrease, while the absolute values of \(C_{\delta_{x},f_{2},g}\) decrease to 0 first and then gradually increase. Figure 16: Variation of mapping matrix \(\mathbf{C}_{T}\) and \(\mathbf{C}_{R}\) in terms of flexure hinge structure parameters: (**a**) variation of \(\mathbf{C}_{T}\) in terms of the parameter \(r\); (**b**) variation of \(\mathbf{C}_{R}\) in terms of the parameter \(r\); (**c**) variation of \(\mathbf{C}_{T}\) in terms of the parameter \(w\); (**d**) variation of \(\mathbf{C}_{R}\) in terms of the parameter \(w\); (**e**) variation of \(\mathbf{C}_{T}\) in terms of the parameter \(t_{0}\); (**f**) variation of \(\mathbf{C}_{R}\) in terms of the parameter \(t_{0}\). Another interesting phenomenon is that when the branch number \(n\) increased from 4 to 5, the value of \(C_{\delta_{x},f_{2y}}\) changed from negative to positive. It indicates that when the number of the mechanism's branches is less than or equal to 4, only applying a force \(f_{2y}\) at \(F_{2}\) along the positive direction of the \(y\)-axis will cause a displacement of the mobile platform center along the negative direction of the \(x\)-axis, while the opposite conclusion is drawn when the branch number is more than or equal to 5. As can be seen from Figure 17b, with the increase in the branch number \(n\), the absolute values of \(C_{\delta_{x},f_{1y}}\), \(C_{\delta_{x},f_{2y}}\) and \(C_{\delta_{x},f_{1y}}\) decrease, while the absolute values of \(C_{\delta_{x},f_{2y}}\) decrease to 0 first and then gradually increases.