## 2 Structure Design of the 2-DOF CPM
### Realization of Input Decoupling Performance  
Next, the requirement of the input decoupling performance on the structure design of the 2-DOF CPM is investigated. The effective way of reducing the input coupling of the CPM is to adopt a flexure-based decoupler between an actuator and the terminal platform. There are two types of the decouplers, and their kinematic structures are shown in Figure 3.  
Figure 2: Division of a 2-DOF CPM: (**a**) mirror symmetry coordinate systems; and (**b**) rotation symmetry coordinate systems. The difference between the two decouplers is the actuator's mounting type. The actuator is fixed on the ground in the first decoupler, while the other installs the actuator on the moving part. The actuator can be inserted into a displacement amplifier in the second decoupler, which increases the motion stroke of the CPM and gives the actuator double protection from input coupling. However, this type of decoupler results in a more complex structure. Moreover, the actuator increases the moving mass of the CPM and thus reduces its natural frequency. Therefore, the first type of decoupler is preferred for this paper, and its detailed structure design will be investigated to guarantee the input decoupling characteristic of the CPM. Because of the symmetric structure of the 2-DOF CPM, we can choose the decoupler along the \(y\)-axis as the study object and its stiffness model is shown in Figure 4.  
For the 2-DOF CPM, excellent input decoupling performance requires that the transverse and rotational deformations at one actuation point are small enough that the terminal platform is actuated by the other actuator. According to Figure 4, utilizing a force equilibrium condition of point \(A_{1}\) allows us to obtain the relationship between the displacement of the terminal platform and the deformation at the actuation point as  
\[\mathbf{K}_{1}\delta_{a}+\mathbf{K}_{2}(\delta_{a}-\delta_{o})=0\quad\Rightarrow\quad \delta_{a}=(\mathbf{K}_{1}+\mathbf{K}_{2})^{-1}\mathbf{K}_{2}\delta_{o} \tag{8}\]  
where \(\delta_{o}\) is the is the displacement of the terminal platform, \(\delta_{a}\) is the deformation of actuation point \(A_{1}\), and \(\mathbf{K}_{1}\) and \(\mathbf{K}_{2}\) are the stiffness matrices of Parts I and II of the decoupler expressed in the coordinate frame \(\{A\)-\(x_{1}y_{1}z_{1}\}\), respectively. The bending and rotational deformations at the actuation point caused by the motion of the terminal platform can be obtained through the components of \(\delta_{a}\); they are expected to be as small as possible. Through a qualitative analysis of Equation (8), the stiffness requirement of the decoupler is:  
Figure 4: Stiffness model of flexure-based decoupler. (**a**) Kinematic structure. (**b**) Stiffness model. Figure 3: Kinematic structures of flexure-based decoupler. (**a**) Kinematic Structure 1. (**b**) Kinematic Structure 2.  
the bending stiffness along the \(x\)-axis and the rotation stiffness about the \(z\)-axis of Part II should be much smaller than in Part I. With the exception of protecting the actuator from suffering transverse load, the decoupler also transmits the motion of the actuator to the terminal platform. Therefore, the axial stiffness along the \(y\)-axis of Part I should not be too large, in order to avoid the motion stroke of the actuator losing too much because of its deformation. In addition, the axial stiffness of Part II should be large in order to reduce the motion lost between the actuator and the terminal platform, thereby improving the motion transmission efficiency. The design requirements on the decoupler will be utilized to guide its detailed structure design. The kinematic structure of Part I of the decoupler adopts a parallel type, as shown in Figure 5. In order to obtain a compact structure, the limb is constructed from a single leaf spring due to its advantages of compact size, lightweight, and easy manufacturing [34; 35]. The stiffness of Part I is the summation of the stiffness of each limb.