## 3 Compliance Model of _n_-4R Compliant Parallel Pointing Mechanism
### Validation of Compliance Model with Computational Simulations  
In this section, the 4-4R compliant parallel pointing mechanism is taken as an example for the verification of the compliance model by FEA simulation assuming small displacements in the commercial software ANSYS2019[8]. The structure parameters of the 4-4R compliant parallel pointing mechanism are listed in Table 2. [PERSON]'s modulus of the material of the flexure hinge was set as \(E=206\) GPa and [PERSON]'s ratio was set as \(\
u=0.3\). By substituting the structural parameters into Equation (6), the overall compliance matrix of the 4-4R compliant parallel pointing mechanism was obtained: \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Adjoint Transformation Matrix** & \(\mathbf{x}\) & \(\mathbf{y}\) & \(\mathbf{z}\) & \(\mathbf{\alpha}\) & \(\mathbf{\beta}\) & \(\mathbf{\gamma}\) \\ \hline \(Ad_{01}^{O}\) & \(-r\cos\frac{\varphi}{2}\) & \(r\sin\frac{\varphi}{2}-L\) & \(R\) & \(0\) & \(0\) & \(\pi-\frac{\varphi}{2}\) \\ \(Ad_{02}^{O}\) & \(r\sin\frac{\varphi}{2}-l\) & \(l\tan\frac{\varphi}{2}+r\cos\frac{\varphi}{2}-L\) & \(0\) & \(-\frac{\pi}{2}\) & \((\frac{\varphi-\pi}{2})\) & \(0\) \\ \(Ad_{03}^{O}\) & \(-r\sin\frac{\varphi}{2}-l\) & \(r\cos\frac{\varphi}{2}-l\tan\frac{\varphi}{2}\) & \(0\) & \(-\frac{\pi}{2}\) & \(-\frac{\varphi}{2}\) & \(0\) \\ \(Ad_{04}^{O}\) & \(r\cos\frac{\varphi}{2}\) & \(r\sin\frac{\varphi}{2}\) & \(-R\) & \(0\) & \(0\) & \(\frac{\varphi}{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of translation transformation and rotation transformation. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & **Values (m)** & **Parameters** & **Values (m)** & **Parameters** & **Values (m)** \\ \hline \(L\) & \(0.1\) & \(t_{0}\) & \(0.0005\) & \(d_{1}\) & \(0.041\) \\ \(R\) & \(0.066\) & \(r\) & \(0.00375\) & \(d_{2}\) & \(0.088\) \\ \(l\) & \(0.0666\) & \(w\) & \(0.005\) & \(d_{3}\) & \(0.059\) \\ \(\varphi\) & \(60^{\circ}\) & & & & \\ \hline \The element in row \(i\) and column \(j\) of the compliance matrix represents the displacement of the mobile platform center in the \(i\)-direction under the action of the unit load in the \(j\)-direction. The validation of analytical results in the compliance model of the 4-4R compliant parallel pointing mechanism is provided by commercial software ANSYS 19.2. A tetrahedron mesh with an element size of 2 mm was created for the links, mobile platform, and fixed platform, and mesh refinements of 0.3 mm were performed at the right-circular flexure hinges. First, we fix the fixed platform of the mechanism, and apply the unit load to the mobile platform center. Therefore, the displacement of the mobile platform center is the compliance of the mechanism in this direction. We repeat this process to obtain the compliance matrix of the mechanism. The compliance matrix obtained by the finite element method is as follows:  
\[\mathbf{c}_{4-4R}^{\text{FE}}=\left[ \begin{array}{ccccc}7.24\times 10^{-2}&0&0&1. 52\times 10^{-5}&0&3.62\times 10^{-3}\\ 0&3.20\times 10^{-3}&0&0&9.86\times 10^{-5}&0\\ 0&0&7.24\times 10^{-2}&-3.62\times 10^{-3}&0&1.52\times 10^{-5}\\ 1.52\times 10^{-5}&0&-3.62\times 10^{-3}&2.12\times 10^{-4}&0&0\\ 0&9.86\times 10^{-5}&0&0&5.50\times 10^{-5}&0\\ 3.62\times 10^{-3}&0&1.52\times 10^{-5}&0&0&2.12\times 10^{-4}\\ \end{array} \right]\]  
The comparison of analytical results and the FE-results of compliance is listed in Table 3. It can be seen from Table 3 that the compliance of the main functional direction (rotation direction around the \(x\)-axis and \(z\)-axis) in the compliance matrix is the same and much larger than those of the nonfunctional direction (rotation direction around the \(y\)-axis), indicating that the design of the mechanism and the selection of structure parameters of the flexure hinge are reasonable. In the compliance matrix, the relative errors of the six compliance on the main diagonal are less than 7%, and the relative errors of the functional direction are less than 1%.