## 3 Compliance Model of _n_-4R Compliant Parallel Pointing Mechanism
### Compliance of the Branch  
As shown in Figure 4, branch 1 consists of four similar flexure hinges denoted by \(R_{1}\), \(R_{2}\), \(R_{3}\), \(R_{4}\) and three links in series. Compared with the flexure hinge, the stiffness of the link is large enough so that the deformation of the link can be ignored and only the deformation of the flexure hinge is considered.  
Figure 3: Structure parameters and coordinate frame setting of the right-circular flexure hinge.  
Coordinate frame setting of the branch 1 is shown in Figure 4. Flexure elements, \(R_{1}\), \(R_{2}\), \(R_{3}\), and \(R_{4}\), have their own local coordinate frames, \(O_{1}x_{1}y_{1}z_{1}\), \(O_{2}x_{2}y_{2}z_{2}\), \(O_{3}x_{3}y_{3}z_{3}\), and \(O_{4}x_{4}y_{4}z_{4}\). The global coordinate frame \(Oxyz\) is fixed at the initial location of the mobile platform center. Since the compliance of the series compliant mechanism is the sum of the compliance of each flexure element in the global coordinate frame [26], the compliance of branch 1 can be then defined as:  
\[\mathbf{C}_{B1}^{O}=\sum_{i=1}^{4}\mathbf{C}_{Ri}^{O} \tag{2}\]  
where \(i=1\),..., 4 is the number of flexure elements in the branch 1, while \(\mathbf{C}_{B1}^{O}\) and \(\mathbf{C}_{Ri}^{O}\) represent the compliance matrix of branch 1 and \(R_{i}\) in the global coordinate frame, respectively.  
Assuming that the compliance matrix of flexure hinges \(R_{i}\) in local coordinate frame \(O_{i}x_{i}y_{i}z_{i}\) is \(\mathbf{C}_{Ri}^{O_{i}}\), its compliance matrix \(\mathbf{C}_{Ri}^{O}\) in the global coordinate frame \(Oxyz\) can be obtained by the following coordinate transformation:  
\[\mathbf{C}_{Ri}^{O}=\mathbf{A}\mathbf{d}_{Oi}^{O}\cdot\mathbf{C}_{Ri}^{O_{i}}\cdot\left(\mathbf{A }\mathbf{d}_{Oi}^{O}\right)^{\mathrm{T}} \tag{3}\]  
where \(i=1,\ldots\)\(\mathbf{A}\) is the number of flexure elements in the branch 1, and \(\mathbf{A}\mathbf{d}_{Oi}^{O}\) is the so-called 6 \(\times\) 6 adjoint transformation matrix, indicating the transformation from local coordinate frame \(O_{i}x_{i}y_{i}z_{i}\) to global coordinate frame \(Oxyz\). Assuming that the rotation transformation matrix of the local coordinate frame \(O_{i}x_{i}y_{i}z_{i}\) to the global coordinate frame \(Oxyz\) is \(\mathbf{R}_{Oi}^{O}\) and the translation vector is \(\mathbf{t}_{Oi}^{O}=(x,y,z)^{\mathrm{T}}\), the adjoint transformation matrix \(\mathbf{A}\mathbf{d}_{Oi}^{O}\) is then defined as:  
\[\mathbf{A}\mathbf{d}_{Oi}^{O} = \begin{bmatrix}\mathbf{R}_{Oi}^{O}&0\\ \mathbf{T}_{Oi}^{O}\cdot\mathbf{R}_{Oi}^{O}&\mathbf{R}_{Oi}^{O}\end{bmatrix}\!,\ \text{where}\ \mathbf{T}_{Oi}^{O}=\begin{bmatrix}0&-z&y\\ z&0&-x\\ -y&x&0\end{bmatrix},\ \mathbf{R}_{Oi}^{O}=\begin{bmatrix}\mathrm{c}\beta\mathrm{c} \gamma&-\mathrm{c}\beta\mathrm{s}\gamma&\mathrm{s}\beta\\ \mathrm{c}\alpha\mathrm{s}\gamma+\mathrm{c}\gamma\mathrm{s}\mathrm{s}\alpha \mathrm{s}\beta\mathrm{s}\gamma&-\mathrm{c}\beta\mathrm{s}\alpha\\ \mathrm{s}\alpha\mathrm{s}\gamma-\mathrm{c}\alpha\mathrm{c}\gamma\mathrm{s} \beta&\mathrm{c}\gamma\mathrm{s}\alpha+\mathrm{c}\alpha\mathrm{s}\beta\mathrm{s }\gamma&\mathrm{c}\alpha\mathrm{c}\beta\end{bmatrix} \tag{4}\]  
where \(\mathbf{T}_{Oi}^{O}\) is the antisymmetric matrix defined by the translation vector \(\mathbf{t}_{Oi}^{O}\): \(\alpha\), \(\beta\) and \(\gamma\) are the angle of rotation around \(x\)-, \(y\)-, and \(z\)-axis, respectively. s and c represent sin and cos, respectively. The parameters for constructing the translation vector \(\mathbf{t}_{Oi}^{O}\) and rotation matrix \(\mathbf{R}_{Oi}^{O}\) are listed in Table 1.  
Since the structure parameters of all flexure hinges in the \(n\)-4R compliant parallel pointing mechanism are exactly the same, therefore:  
\[\mathbf{C}_{R1}^{O1}=\mathbf{C}_{R2}^{O2}=\mathbf{C}_{R3}^{O3}=\mathbf{C}_{R4}^{04}=\mathbf{C} \tag{5}\]  
Figure 4: Structure and coordinate frame setting of the branch 1.  
where \(\mathbf{C}\) is the compliance matrix of the right-circular flexure hinge, which can be computed in Appendix A.1.