## 4 Geometric Parameters Optimization and Performance Evaluation
### Performance Evaluation with Finite Element Analysis  
The optimized mechanism's performance will be evaluated through FEA, performed with the ANSYS WorkBench software (Workbench 14.5, ANSYS Inc., Cannonburg, PA, USA). The physical and mechanical parameters of the adopted material alloy Al-7075 are: [PERSON]'s modulus = 71.7 GPa, Poisson's ratio = 0.33, Density = 2770 kg/m\({}^{3}\), and yield stress = 525 MPa. A good finite element model with a reasonably high number of DOFs is first required to guarantee the accuracy of FEA results. The finite element model of the CPM is presented in Figure 16, wherein the mesh is generated by using the hex dominant method. The maximum element size is set as 2 mm and the numbers of the nodes and elements are 139,471 and 32,016, respectively. The input axial stiffness of the CPM is calculated by applying a force at the actuation point \(A_{1}\), as shown in Figure 17a. The relationship between the applied force and the deformation at point \(A_{1}\) is obtained and is given in Figure 17b. The force-displacement curve fits into a line very well, which indicates the linearity of the CPM. The input axial stiffness is obtained by calculating the slope of the fitted line and resorting to the least squares method, which is 3.36 N/\(\upmu\)m. In actual CPM, preloading and actuation forces will cause initial displacements along each axis, which may influence the CPM's stiffness. Taking a small initial displacement (less than 100 \(\upmu\)m) along one axis into consideration, both the theoretical and finite element models have demonstrated that the CPM's stiffness remains unchanged. Similarly, the output rotation stiffness of the CPM can be analyzed through applying bending moment at the terminal platform, as shown in Figure 18a. The relationship between the rotation angle and applied moment is given in Figure 18b, and the output rotation stiffness can be calculated as 3.33 \(\times\) 10\({}^{4}\) Nm/rad. It can be observed that the input axial stiffness and the output rotation stiffness of the CPM calculated through FEA are 8.7% and 4.9% smaller than the theoretical values. The reason is because the actuation platform and terminal platform are considered to be rigid parts in the analytical model, but they also suffer deformation under the applied loads. This is evident from the stress distribution diagrams shown in Figure 19. Therefore, the compliance of the platforms reduces the stiffness of the CPM. Figure 16: Finite element model of the CPM. Figure 17: Finite element model and results of input axial stiffness. (**a**) Finite element model. (**b**) Force-displacement curve. The finite element model shown in Figure 17a can be utilized to calculate the ICD of the CPM. When the driving force 100 N is applied at the actuation point \(A_{1}\) along the \(x\)-axis, the axial deformation at point \(A_{1}\) and the bending deformation at point \(A_{2}\) are 29.8 \(\upmu\)m and 8.66 \(\times\) 10\({}^{-2}\)\(\upmu\)m, respectively. The ICD can be calculated as 0.291%, which is close to the theoretical value. This not only verifies the validity of the analytical model but also demonstrates the excellent input decoupling characteristic of the CPM. Furthermore, the resonant frequencies and corresponding vibration modes of the CPM are also simulated. The first six natural frequencies are 808 Hz, 808 Hz, 1802 Hz, 3153 Hz, 3153 Hz, and 3430 Hz, and the first three vibration modes are shown in Figure 20. The simulation results show that the first two natural frequencies are identical, and their vibration modes exist along the \(x\)- and \(y\)-axes, which correspond to the structural characteristic of the CPM. The third vibration mode is out-of-plane motion and its frequency is more than twice as large as the first two natural frequencies. Additionally, the rotational vibration mode corresponds to the eighth vibration mode with a frequency of 4636 Hz. Therefore, the vibration modes with higher frequencies have less of an effect on the translation motion of the CPM.