## 2 Structure Design of the 2-DOF CPM
### Realization of Output Decoupling Performance  
The requirement of the output decoupling performance on the structure design of the 2-DOF CPM is studied first. It is expected that translational motion of the terminal platform along the \(x\)- and \(y\)-axes are independent, thereby ensuring the CPM with a total output decoupling characteristic. Therefore, the output stiffness matrix of the CPM should satisfy the form as  
\[\mathbf{K}=\left[ \begin{array}{ccc}K_{x-Fx}&0&0\\ 0&K_{y-Fy}&0\\ 0&0&K_{\alpha-Mz}\end{array} \right] \tag{1}\]  
The output stiffness of the CPM can be calculated through the summation of the stiffness of each limb as [31]  
\[\mathbf{K}=\sum_{i=1}^{n}\mathbf{T}_{i\text{I}}^{-1}\mathbf{K}_{i}^{\prime}\mathbf{T}_{i\text{ Id}}^{-1} \tag{2}\]  
where \(\mathbf{K}_{i}^{\prime}\) is the local stiffness matrix of the \(i\)th limb, and \(\mathbf{T}_{i\text{I}}\) and \(\mathbf{T}_{i\text{d}}\) are the force and displacement transformation matrices, respectively, which take on the form  
\[\mathbf{T}_{i\text{f}}=\mathbf{T}_{i\text{d}}^{\text{T}}=\left[ \begin{array}{ccc} \cos\theta_{i}&-\sin\theta_{i}&0\\ \sin\theta_{i}&\cos\theta_{i}&0\\ r_{iy}&-r_{ix}&1\end{array} \right] \tag{3}\]  
The output stiffness matrix of the CPM should satisfy the form expressed in Equation (1). However, it is difficult to obtain a satisfied structure by solving the equations directly. To simplify this problem, the structure of the CPM is divided into two parts and the local coordinate systems of each subsystem are established, as shown in Figure 2. There are two ways to build the local coordinate systems, i.e., mirror symmetry and rotation symmetry. The stiffness matrices of the two subsystems shown in Figure 2a under their corresponding local coordinate systems are expressed as  
\[\mathbf{K}_{i}^{\text{m}}=\left[ \begin{array}{ccc}K_{i-Fx}^{\text{m}}&K_{ix-Fy}^ {\text{m}}&K_{ix-Mz}^{\text{m}}\\ K_{iy-Fx}^{\text{m}}&K_{1y-Fy}^{\text{m}}&K_{iy-Mz}^{\text{m}}\\ K_{ix-Fx}^{\text{m}}&K_{1x-Fy}^{\text{m}}&K_{ix-Mz}^{\text{m}}\end{array}  \right]\quad(i=1,\ 2) \tag{4}\]  
Then, the stiffness matrix of the CPM under the globe coordinate system can be obtained according to Equation (2) as  
\[\mathbf{K}^{\text{m}}=\sum_{i=1}^{2}\left(\mathbf{T}_{i\text{f}}^{\text{m}}\right)^{-1 }\mathbf{K}_{i}^{\text{m}}\left(\mathbf{T}_{i\text{d}}^{\text{m}}\right)^{-1} \tag{5}\]  
where \(\mathbf{T}_{1\text{f}}^{\text{m}}=\left(\mathbf{T}_{1\text{d}}^{\text{m}}\right)^{ \text{T}}=\text{diag}([1,\ 1,\ 1])\), and \(\mathbf{T}_{2\text{f}}^{\text{m}}=\left(\mathbf{T}_{2\text{d}}^{\text{m}}\right)^{ \text{T}}=\text{diag}([-1,\ -1,\ 1])\). Figure 1: Kinematic structure of a 2-DOF CPM. If the structure of the CPM is mirror symmetry, the local stiffness matrix of the two subsystems satisfies \(\mathbf{K}_{1}^{\text{m}}=\mathbf{K}_{2}^{\text{m}}\). Then Equation (5) can be simplified as  
\[\mathbf{K}^{\text{m}}=\left[ \begin{array}{ccc}2K_{1x-Fx}^{\text{m}}&0&2K_{1x- Mz}^{\text{m}}\\ 0&2K_{1y-Fy}^{\text{m}}&0\\ 2K_{1\alpha-Fx}^{\text{m}}&0&2K_{1\alpha-Mz}^{\text{m}}\end{array} \right] \tag{6}\]  
This shows that the translational motion along the \(y\)-axis will not affect the other two axes, while the translational motion along the \(x\)-axis and the rotational motion about the \(z\)-axis are coupled. Similarly, when the structure of the CPM is rotation symmetry, its stiffness matrix can be obtained as  
\[\mathbf{K}^{\text{s}}=\left[ \begin{array}{ccc}2K_{1x-Fx}^{\text{s}}&2K_{1x- Fy}^{\text{s}}&0\\ 2K_{1y-Fx}^{\text{s}}&2K_{1y-Fy}^{\text{s}}&0\\ 0&0&K_{1\alpha-Mz}^{\text{s}}\end{array} \right] \tag{7}\]  
This indicates that the rotational motion about the \(z\)-axis has no influence on the other two axes, while the translational motions along the \(x\)- and \(y\)-axes are coupled. It is verifiable that, when the structure of the CPM both satisfies the mirror and rotation symmetry, the translational and rotational motions of the terminal platform are independent.