## 5 Validation and Analysis of Kinetostatic Model with Computational Simulations
### Computational Simulation of Spiral Trajectory  
As shown in Figure 11a, let the center of the mobile platform move along a given spiral trajectory in the \(x\)-direction and \(z\)-direction, and the equation of the trajectory is as follows:  
\[ \begin{cases}\delta_{x}=R_{c}\cos(10000R_{c}\tau)\\ \delta_{z}=R_{c}\sin(10000R_{c}\tau)\end{cases} ,\ 0\leq R_{c}\leq 5\times 10^{-4} \ \text{m} \tag{24}\]  
where \(\delta_{x}\) and \(\delta_{z}\) denote the displacement of the mobile platform center in the \(x\)-direction and \(z\)-direction, respectively, and \(R_{c}\) is the radius of the spiral trajectory. Suppose the forces\(f_{1,y}\) and \(f_{2,y}\) along the \(y\)-direction are applied at loading position of forces \(\mathbf{F}_{1}\) and \(\mathbf{F}_{2}\) on branch 1 and branch 2, respectively. According to Equation (14), the mapping relationship between force \(f_{1,y}\),\(f_{2,y}\) and the displacement \(\delta_{x}\), \(\delta_{z}\) of the mobile platform center can be obtained:  
\[ \begin{bmatrix}\delta_{x}\\ \delta_{z}\end{bmatrix} = \begin{bmatrix}C_{\delta_{x},f_{1,y}}&C_{\delta_{x},f_ {2,y}}\\ C_{\delta_{x},f_{1,y}}&C_{\delta_{x},f_{2,y}}\end{bmatrix}\begin{bmatrix}f_{1, y}\\ f_{2,y}\end{bmatrix} =\begin{bmatrix}\left(\mathbf{C}_{TOF_{1}}\right)_{[rows4,6,col 5]}&\left(\mathbf{C}_{TOF_{2}}\right)_{[rows4,6,col5]}&\left] \begin{bmatrix}f_{1, y}\\ f_{2,y}\end{bmatrix} \end{array} \tag{25}\]  
where \(\left(\mathbf{C}_{TOF_{1}}\right)_{[rows4,6,col5]}\) denotes the elements of the fifth column of the fourth and sixth rows of the mapping matrix \(\mathbf{C}_{TOF_{1}}\), and \(\left(\mathbf{C}_{TOF_{2}}\right)_{[rows4,6,col5]}\) denotes the elements of the fifth column of the fourth and sixth rows of the mapping matrix \(\mathbf{C}_{TOF_{2}}\). We define the matrix consisting of \(C_{\delta_{x},f_{1,y},\delta_{x},f_{2,y}}C_{\delta_{x},f_{1,y}}\) and \(C_{\delta_{x},f_{2,y}}\) as the translation mapping matrix. A total of 90 points, obtained by equally spacing the domain \(0<R_{c}<5\times 10^{-4}\ \text{m}\), were used to form the spiral trajectory. The displacements \(\delta_{x}\) and \(\delta_{z}\) of the 90 points were substituted in Equation (25) successively to obtain the corresponding input forces \(f_{1,y}\) and \(f_{2,y}\). In the process, the structure parameters of the mechanism and the loading position of force \(\mathbf{F}_{1}\) are listed in Table 2, and the loading position of force \(\mathbf{F}_{2}\) can be obtained from the loading position of force \(\mathbf{F}_{1}\) by rotation transformation. Curves of the input forces are shown in Figure 11b. Using the input forces for Finite Element Analysis (a computational model of the 4-4R mechanism was constructed with the dimensions listed in Table 2). Figure 11: (a) Analytical spiral trajectory; (b) curves of the input forces \(f_{1,y}\),\(f_{2,y}\). [PERSON]'s modulus of the material of the flexure hinge was set as \(E=206\) GPa and [PERSON]'s ratio was set as \(\
u=0.3\). A tetrahedron mesh with an element size of 2 mm was created for this model, and mesh refinements of 0.3 mm were performed at the flexure hinge. The corresponding trajectory of the center of the mobile platform can thus be obtained, as shown in Figure 12a. A comparison of the analytical and FE-result of spiral trajectory is shown in Figure 12b. As can be seen from Figure 12b, the analytical spiral trajectory agrees well with the simulated one, indicating the correctness of the kinetostatic model. The absolute error and relative error of the analytical and simulated results of the spiral trajectory in the \(x\)-direction and \(z\)-direction are presented in Figure 13a,b, respectively. It can be seen from Figure 13 a that with the increase in the helix radius, the absolute error tends to increase, while the relative error fluctuates between 0.033% and 0.038%, as shown in Figure 13b.