## 2 Six-Hardpoint Positioning Mechanism and Its Mathematical Model
### Error Model of Six-Hardpoint Positioning Mechanism
\begin{array}{cccccccc}\frac{\partial f_{1}}{\partial x }&\frac{\partial f_{1}}{\partial y}&\frac{\partial f_{1}}{\partial x}&\frac{ \partial f_{1}}{\partial x_{Ri}}&\frac{\partial f_{1}}{\partial x}&\frac{ \partial f_{1}}{\partial y}\\ \frac{\partial f_{2}}{\partial x}&\frac{\partial f_{2}}{\partial y}&\frac{ \partial f_{2}}{\partial x}&\frac{\partial f_{2}}{\partial x}&\frac{\partial f _{2}}{\partial x}&\frac{\partial f_{2}}{\partial x}&\frac{\partial f_{2}}{ \partial x}&\frac{\partial f_{2}}{\partial y}\\ \frac{\partial f_{3}}{\partial x}&\frac{\partial f_{3}}{\partial y}&\frac{ \partial f_{3}}{\partial z}&\frac{\partial f_{3}}{\partial x}&\frac{\partial f _{3}}{\partial y}&\frac{\partial f_{3}}{\partial y}&\frac{\partial f_{3}}{ \partial y}\\ \frac{\partial f_{4}}{\partial x}&\frac{\partial f_{4}}{\partial y}&\frac{ \partial f_{4}}{\partial z}&\frac{\partial f_{4}}{\partial z}&\frac{\partial f _{4}}{\partial y}&\frac{\partial f_{4}}{\partial y}&\frac{\partial f_{4}}{ \partial y}\\ \frac{\partial f_{5}}{\partial x}&\frac{\partial f_{5}}{\partial y}&\frac{ \partial f_{5}}{\partial x}&\frac{\partial f_{5}}{\partial x}&\frac{\partial f _{5}}{\partial x}&\frac{\partial f_{5}}{\partial y}\\ \frac{\partial f_{6}}{\partial x}&\frac{\partial f_{6}}{\partial y}&\frac{ \partial f_{6}}{\partial z}&\frac{\partial f_{6}}{\partial z}&\frac{\partial f _{6}}{\partial z}&\frac{\partial f_{6}}{\partial z}&\frac{\partial f_{6}}{ \partial z}&\frac{\partial f_{6}}{\partial z}\\ \end{array} \right] \tag{19}\]  
\[\textbf{J}_{2}=\left[ \begin{array}{cccccccccccc}\frac{\partial f_{1}}{ \partial x_{Ai}}&\frac{\partial f_{1}}{\partial y_{Ai}}&\frac{\partial f_{1}}{ \partial x_{Ai}}&\frac{\partial f_{1}}{\partial x_{Ai}}&\frac{\partial f_{1}}{ \partial h_{1}}&\frac{\partial f_{1}}{\partial x_{Bi}}&\frac{\partial f_{1}}{ \partial y_{Bi}}&\frac{\partial f_{1}}{\partial x_{Bi}}&\frac{\partial f_{1}}{ \partial x_{Bi}}&\frac{\partial f_{1}}{\partial x_{Bi}}&\frac{\partial f_{1}}{ \partial x_{Bi}}&\frac{\partial f_{1}}{\partial x_{Bi}}&\frac{\partial f_{6}}{ \partial z_{Bi}}\\ \end{array} \right] \tag{21}\]  
\[d\textbf{C}=\left[dx_{A1}\quad dy_{A1}\quad dz_{A1}\quad dh_{1}\quad dx_{B1} \quad dy_{B1}\quad dz_{B1}\quad\cdots\quad dx_{A6}\quad dy_{A6}\quad dz_{A6} \quad dh_{6}\quad dx_{B6}\quad dy_{B6}\quad dz_{B6}\quad\right]^{T} \tag{22}\]  
where \(\textbf{J}_{1}\) is 6 \(\times\) 6 matrix and \(\textbf{J}_{2}\) is 6 \(\times\) 42 matrix. \(\textbf{d}\textbf{q}\) represents the pose errors of the primary mirror, which is a 6 \(\times\) 1 vector, and \(\textbf{d}\textbf{c}\) represents the errors of kinematic parameters, which is a 42 \(\times\) 1 vector. Since \(\textbf{J}_{1}\) is a reversible matrix, Equation (19) can be transformed as follows:  
\[d\textbf{q}=\textbf{J}_{1}^{-1}\textbf{J}_{2}d\textbf{C}=\textbf{J}_{b}d \textbf{C} \tag{23}\]  
where \(\textbf{J}_{\text{b}}\) is a 6 \(\times\) 42 matrix, and \(\textbf{J}_{\text{b}}\) is a function of \(q\), **H**, **A**, **B**, with **H**, **A**, **B** expressed as follows:  
\[\textbf{H}= \begin{bmatrix}h_{1}&h_{2}&h_{3}&h_{4}&h_{5}&h_{6}\end{bmatrix} ^{T} \tag{24}\]  
\[\textbf{A}= \begin{bmatrix}\textbf{a}_{1}&\textbf{a}_{2}&\textbf{a}_{3}&\textbf{ a}_{4}&\textbf{a}_{5}&\textbf{a}_{6}\end{bmatrix} ^{T} \tag{25}\]  
\[\textbf{B}= \begin{bmatrix}\textbf{b}_{1}&\textbf{b}_{2}&\textbf{b}_{3}&\textbf{ b}_{4}&\textbf{b}_{5}&\textbf{b}_{6}\end{bmatrix} ^{T} \tag{26}\]  
Equation (23) is the preliminary error model of the six-hardpoint positioning mechanism. The model provides the relationship between the pose error vector \(\textbf{d}\textbf{q}\) of the primary mirror and the kinematic parameter error vector \(\textbf{d}\textbf{C}\) of the six-hardpoint positioning mechanism. However, Equation (23) cannot be solved directly, because \(\textbf{d}\textbf{q}\) contains 6 parameters, whereas \(\textbf{d}\textbf{C}\) contains 42 parameters to be solved. Therefore, at least 7 error equations must be combined to achieve the solution, so at least 7 poses must be measured.