## 4 Geometric Parameters Optimization and Performance Evaluation  
### Geometric Parameters Optimization  
To ensure the excellent performances of the 2-DOF CPM, it is necessary to optimize its geometric parameters. The optimum goal is to guarantee the excellent decoupling performance of the CPM. Because the symmetric structure of the CPM already possesses a total output decoupling character, the minimization of the input coupling will be selected as the optimization objective, i.e., min (_ICD_). The CPM is fabricated from a single block with a specific thickness of 15 mm, and the independent geometric parameters to be optimized are \(L\), \(t_{1}\), \(t_{2}\), and \(d\). The other performances of the CPM, including the workspace, output rotation stiffness, and natural frequency, are guaranteed by the constraint conditions described as follows. (1)  
Workspace: \(\delta_{x}\times\delta_{y}\geq 47.5\) um \(\times\) 47.5 um. According to the analytical models, the workspace of the CPM is influenced by the stiffness of the actuator and the mechanism, and is also limited by the maximum stress of the materials. The nominal motion stroke of the piezoelectric actuator  
Figure 15: Determination of the shape function. adopted in this CPM is 60 \(\upmu\)m, and its static axial stiffness is about 15 N/\(\upmu\)m. Additionally, alloy Al-7075 with a yield stress of 525 MPa is chosen to fabricate the CPM. According to the given workspace, Equations (24) and (26) can be utilized to determine the satisfied ranges of the geometric parameters. 2. Output rotation stiffness: \(\mathbf{K}_{\alpha}\geq 3.5\times 10^{4}\) Nm/rad. The translational deformations of the terminal platform caused by external loads along the \(x\)- and \(y\)-axes can be compensated through the closed-loop control. However, the rotational deformation about the \(z\)-axis cannot be compensated, and thus will affect the motion precision of the CPM. Therefore, the output rotation stiffness is expected to be large enough. 3. Natural frequency: \(f\geq 800\) Hz. The first two natural frequencies of the 2-DOF CPM are identical and correspond to the translational vibration modes along the two axes. Therefore, they should be as high as possible to guarantee the CPM with a rapid dynamic response and robustness against the disturbances. The optimization process of the CPM's geometric parameters is performed with the \"fmincon\" function. It is an optimization function in MATLAB used to find a constrained minimum of a scalar function of several variables and its grammar is given as  
\[[\mathbf{x},\ \text{fval}]=\text{fmincon}(\text{fun},\ \mathbf{x0},\ \mathbf{A},\ \mathbf{b},\ \mathbf{Aeq},\ \mathbf{beq},\ \mathbf{lb},\ \mathbf{ub},\ \text{nonlcon}) \tag{38}\]  
where \(\mathbf{x}\) is the optimized solution, fval is the value of the objective function at the solution \(\mathbf{x}\), fun is the objective function, \(\mathbf{x0}\) is the initial point for \(\mathbf{x}\), \(\mathbf{A}\) is the matrix for linear inequality constraints, \(\mathbf{b}\) is the vector for linear inequality constraints, \(\mathbf{Aeq}\) is the matrix for linear equality constraints, \(\mathbf{beq}\) is the vector for linear equality constraints, \(\mathbf{lb}\) is the vector of lower bounds, \(\mathbf{ub}\) is the vector of upper bounds, and nonlcon is the nonlinear constraint function. The minimization of the ICD is adopted as the objective function and the requirement of the workspace, output rotation stiffness, and the natural frequency of the CPM are used as the nonlinear constraint conditions in the \"fmincon\" function. There are no linear inequality and linear equality constraints in the geometric parameters optimization, and therefore set \(\mathbf{A}=[]\), \(\mathbf{b}=[]\), \(\mathbf{Aeq}=[]\), \(\mathbf{beq}=[]\).