## 3 Compliance Model of _n_-4R Compliant Parallel Pointing Mechanism
### Analysis of Compliance Performance of n-4R Compliant Parallel Pointing Mechanisms  
According to the compliance model established in Section 3.3, the structure parameters of the flexure hinge, the scale parameters of the mechanism, and the number of mechanism branches will affect the compliance of the mechanism. Therefore, it is necessary to analyze the influence of the changes of these parameters on the overall compliance of the mechanism. For convenience, \(\mathbf{C}_{\theta}\) and \(\mathbf{C}_{\delta}\) are defined as the rotation-related and translation-related compliance on the diagonal of the compliance matrix, respectively. We take the 4-4R compliant parallel pointing mechanism as an example to analyze the influence of structure parameters on the compliance of the mechanism. The analysis is carried out from the following three aspects: (1) The influence of the radius \(r\) of flexure hinge and the distance \(L\) between the mobile and fixed platform on \(\mathbf{C}_{\theta}\) and \(\mathbf{C}_{\delta}\); (2) the influence of parameter \(l\) and the minimum thickness \(t_{0}\) of flexure hinge on \(\mathbf{C}_{\theta}\) and \(\mathbf{C}_{\delta}\); (3) the influence of the distribution radius \(R\) of mobile and fixed platforms and the width \(w\) of flexure hinge on \(\mathbf{C}_{\theta}\) and \(\mathbf{C}_{\delta}\). When analyzing the effect of parameter changes on compliance performance, the parameters in Table 4 were selected, while the other parameters remained the same as in Table 2. Figure 5 shows the variations of diagonal elements of the compliance matrix in terms of the compliance hinge structure parameters and the mechanism scale parameters. The following conclusions can be drawn from Figure 5: (1)\(\mathbf{C}_{\theta}\) and \(\mathbf{C}_{\delta}\) are directly correlated with \(L\) and \(r\), while they are inversely correlated with \(l\), \(t_{0}\), \(R\), and \(w\). (2) Compared with the structural parameters (\(L\), \(l\), and \(R\)) of the mechanism, \(\mathbf{C}_{\theta}\) and \(\mathbf{C}_{\delta}\) are more sensitive to the flexure hinge structure parameters (\(r\), \(t_{0}\), and \(w\)). Therefore, when designing the mechanism, the structure parameters of the flexure hinge can be changed preferentially to ensure the mechanism meets the compliance performance requirements. (3) The compliance of the mechanism in the functional direction \(C_{\theta_{x},m_{x}}\) and \(C_{\theta_{x},m_{z}}\) are equal and far greater than those in the non-functional direction \(C_{\theta_{y},m_{y}}\), which is consistent with the conclusion in Section 3.4. The influence of parameter variations near a given flexure hinge structure parameter and mechanism scale parameter on the compliance of the mechanism was investigated. However, in practical applications, the scale of the mechanism required for different application scenarios often varies greatly and may differ by several times or even tens of times. Therefore, it is also of interest to qualify and quantify the mechanism compliance scales with the defining geometric dimensions. Considering two different scaling situations, we define the scale coefficients \(s_{1}\) and \(s_{2}\). The scale coefficient \(s_{1}\) denotes that only dimensional parameters of the mechanism are scaled by \(s_{1}\) times, while the structural parameters of the flexure hinge remain unchanged. The scale coefficient \(s_{2}\) denotes that both dimensional parameters of the mechanism and the structural parameters of the flexure hinge are scaled by \(s_{2}\) times. The analysis results are given in Figures 6 and 7. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & **Min (mm)** & **Max (mm)** & **Parameters** & **Min (mm)** & **Max (mm)** \\ \hline \(L\) & 70 & 130 & \(r\) & 2.5 & 5 \\ \(l\) & 45 & 85 & \(t_{0}\) & 0.35 & 0.65 \\ \(R\) & 45 & 85 & \(w\) & 3.5 & 6.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Variation range of geometric parameters of mechanism.