## 2 Configuration of the Remote Center Compliance (RCC) Mechanism
### Compliance Modeling  
The parallelism alignment between the template and the substrate in the nanoimprint machines is accomplished by the rotational motions around the remote center of the RCC mechanism when a constant contact force between the template and the substrate is applied. Because the parallelism alignment is a quasi-static process, the compliance characteristics that describe the load-displacement relationship of an RCC mechanism are the key input specifications for the design of an RCC mechanism--especially in such static applications. The static analytical model of the RCC mechanism herein is formulated based on the stiffness matrix method [20]. By integrating the compliance matrix of the single beam flexure together with the geometric and material parameters of the RCC mechanism, close-form equations are generated to describe the mechanism's response, and enable further performance analysis and optimization. The RCC mechanism consists of four individual round beam flexures connected in parallel. In order to obtain the compliance model of the entire RCC mechanism, the compliance matrix of a single beam flexure should be obtained first. The compliance and stiffness matrices for a round beam flexure with the coordinate frame shown in Figure 4 can be described as [17; 21]  
\[\mathbf{C}^{0} =\left[ \begin{array}{cccccc}\frac{4I}{E\pi D^{2}}&0&0&0&0&0\\ 0&\frac{l^{3}}{3 EI}&0&0&0&\frac{l^{2}}{2 EI}\\ 0&0&\frac{l^{3}}{3 EI}&0&-\frac{l^{2}}{2 EI}&0\\ 0&0&0&\frac{l}{2 CI}&0&0\\ 0&0&-\frac{l^{2}}{2 EI}&0&\frac{l}{EI}&0\\ 0&\frac{l^{2}}{2 EI}&0&0&0&\frac{l}{EI}\end{array} \right] \tag{1}\] \[\mathbf{K}^{0} =(\mathbf{C}^{0})^{-1}\]  
where \(l\) is the length of the beam flexure, and \(D\) is the diameter of the round cross-section. \(I=\pi D^{4}/64\) denotes the second moment of the round cross section. \(E\) is the elastic modulus, and \(G\) denotes the shear modulus of the material. \(\mathbf{C}^{0}\) describes the compliance characteristics of free end \(O_{i}\) with respect to the other fixed end \(O\), as shown in Figure 4.  
The coordinate frames of the RCC mechanism are assigned in Figure 5. Because of the small deformation of the flexure mechanism, the original dimensions of the mechanism are used to make a reasonable model of the mechanism. The small deformation assumption is determined by the requirement of the calculation error. For ease of modeling, the origin of the global coordinate system \(O-XYZ\) lies in the centerline of the end-effector, and in the same plane with the ends of the four beam flexures. When an external force is applied to the end-effector of the mechanism, the displacement vector of Point \(O\) can be described as \(\Delta\mathbf{q}=\left[ \begin{array}{cccccc}\Delta x&\Delta y&\Delta z& \Delta\theta_{x}&\Delta\theta_{y}&\Delta\theta_{z}\end{array} \right]^{\text{T}}\). Transforming the displacement vector from \(O-XYZ\) to \(O_{1}-XYZ\) yields  
\[\Delta\mathbf{q}_{1}=\mathbf{J}_{1}^{\text{T}}\Delta\mathbf{q} \tag{2}\]  
Figure 4: Model of a round beam flexure with specified coordinates. (2)  
where  
\[ \begin{split}\mathbf{J}_{1}&=\left[\begin{array}{cc} \mathbf{I}&\mathbf{0}\\ C\left(\overrightarrow{OO_{1}}\right)&\mathbf{I}\end{array}\right]\left[ \begin{array}{cc}\mathbf{R}_{y}\left(\angle O/O_{1}\right)&\mathbf{0}\\ \mathbf{0}&\mathbf{R}_{y}\left(\angle O/O_{1}\right)\end{array}\right]\\ &=\left[\begin{array}{cc}\mathbf{R}_{y}\left(\angle O/O_{1}\right)&\mathbf{0} \\ C\left(\overrightarrow{OO_{1}}\right)\mathbf{R}_{y}\left(\angle O/O_{1}\right)& \mathbf{R}_{y}\left(\angle O/O_{1}\right)\end{array}\right]\end{split}  \tag{3}\]  
\(\mathbf{R}_{y}\left(\angle O/O_{1}\right)\) is the rotation matrix characterized by the rotating angle \(-\theta\)\((0^{\circ}\leq\theta\leq 90^{\circ})\) of the rotated \(O_{1}-XYZ\) with respect to \(O-XYZ\) around the \(Y\) axis, and has the form  
\[\mathbf{R}_{y}\left(\theta\right)=\left[ \begin{array}{ccc}\cos\left(- \theta\right)&0&\sin\left(-\theta\right)\\ 0&1&0\\ -\sin\left(-\theta\right)&0&\cos\left(-\theta\right)\end{array}