## 2 Structure Design of the 2-DOF CPM
### Realization of Input Decoupling Performance
Therefore, the influence of each limb's mechanical parameters on the stiffness of Part I is investigated first. The stiffness of the \(i\)th limb can be expressed in the coordinate system [\(A\)-\(x_{1}y_{1}z_{1}\)] as  
\[\mathbf{k}_{i}^{o}=\mathbf{T}_{ii}^{-1}\mathbf{k}_{i}^{l}\mathbf{T}_{i\text{d}}^{-1} \tag{9}\]  
where \(\mathbf{k}_{i}^{l}\) is the stiffness matrix of the \(i\)th limb expressed in its local coordinate frame [\(o\)-\(x_{i}y_{i}z_{i}\)], which takes the form  
\[\mathbf{k}_{i}^{l}=\left[ \begin{array}{cc}k_{ix-Fx}&0&0\\ 0&k_{iy-Fy}&k_{iy-Mz}\\ 0&k_{ix-Fy}&k_{ia-Mz}\end{array} \right] \tag{10}\]  
The axial and bending stiffness of the limb can be obtained from Equation (9) as  
\[k_{ix-Fx}^{o} =k_{ix-Fx}-\sin^{2}\theta_{i}\cdot\left(k_{ix-Fx}-k_{iy-Fy}\right) \tag{11}\] \[k_{iy-Fy}^{o} =k_{ix-Fy}+\sin^{2}\theta_{i}\cdot\left(k_{ix-Fx}-k_{iy-Fy}\right) \tag{12}\]  
Considering the stiffness components of the leaf spring satisfy \(k_{ix-Fx}\gg k_{iy-Fy}\), the value of angle \(\theta_{i}\) should be zero to ensure the large bending stiffness and small axial stiffness of Part I. Then the rotation stiffness of the limb can be written as  
\[k_{ia-Mz}^{o}={r_{iy}}^{2}\cdot k_{ix-Fx}+{r_{ix}}^{2}\cdot k_{iy-Fy}+k_{ia-Mz }+2{r_{ix}}\cdot k_{iy-Mz} \tag{13}\]  
Because each component of the stiffness matrix \(\mathbf{k}_{i}^{J}\) is greater than zero, the increment of \(r_{ix}\) and \(r_{iy}\) will both increase the rotation stiffness. According to the discussion above, in order to meet the stiffness requirement of Part I of the decoupler, each limb should be parallel to the \(x\)-axis and their positions should be far from the centric of the actuation platform if possible. Figure 5: Kinematic structure of Part I of flexure-based decoupler. Next, the influence of the leaf spring's geometric parameters is studied. The stiffness matrix of a single leaf spring can be expressed as  
\[\mathbf{k}_{L}=\left[ \begin{array}{cc}\frac{EDI}{L}&0&0\\ 0&\frac{EDI^{3}}{L^{3}}&\frac{EDI^{3}}{2L^{3}}\\ 0&\frac{EDI^{3}}{2L^{2}}&\frac{EDI^{3}}{3L}\end{array} \right] \tag{14}\]  
where \(E\) is the elastic modulus, and \(L\), \(t\), and \(D\) are the length, thick and height of the leaf spring, respectively. According to the stiffness requirement of Part I and the above analysis, the bending stiffness of the leaf spring should be small, while the axial and rotation stiffness should be large. According to Equation (13), however, it is impossible to reduce the bending stiffness of the single leaf spring while guaranteeing that the axial and rotation stiffness remain unchanged or increased by adjusting its geometric parameters. Therefore, another type of hinge should be proposed to construct a satisfied limb. As opposed to utilizing a single flexure hinge, a compound hinge composed of \(n\) identical parallel leaf springs is chosen, as shown in Figure 6.  
The length, height and total mass of the compound hinge are the same as the single leaf spring, and its stiffness matrix can be expressed as  
\[\mathbf{K}_{n}=\left[ \begin{array}{ccc}k_{1x-Fx}&0&0\\ 0&(1/n)^{2}\cdot k_{1y-Fy}&(1/n)^{2}\cdot k_{1y-Mz}\\ 0&(1/n)^{2}\cdot k_{1\alpha-Fy}&\left((n+1)/(n-1)\cdot\gamma^{2}+(1/n)^{2} \right)\cdot k_{1\alpha-Mz}\end{array} \right] \tag{15}\]  
where \(\gamma=d/2t\), and \(k_{1m-n}\) is the stiffness component of the single leaf spring. Equation (14) shows that the axial stiffness of the compound hinge is equal to the single leaf spring, and the bending stiffness is always smaller and decreases rapidly as number \(n\) grows. The rotation stiffness depends on the number \(n\) and parameter \(\gamma\). When \(\gamma>1\), the rotation stiffness is always larger than that of a single leaf spring. Therefore, the compound hinge will meet that requirement quite well. Because the stiffness requirement of Part II of the decoupler is contrary to Part I, its limbs should be parallel to the \(y\)-axis according to the investigation above.