## 2 Six-Hardpoint Positioning Mechanism and Its Mathematical Model
### Kinematic Model of Six-Hardpoint Positioning Mechanisms
#### 2.1.2 Forward Kinematic Model  
The forward kinematic model is used to solve the pose of the primary mirror through the given lengths of the six hardpoints, based on the Gauss-Newton iteration method. According to the inverse kinematic Equation (1), the length \(h_{i}\) of the hardpoint is:  
\[ \begin{array}{l}h_{i}=|I_{i}|\\ =\sqrt{I_{i}{}^{T}I_{i}}\\ =\sqrt{\left(t+R\mathbf{a}_{i}-\mathbf{b}_{i}\right)^{T}\left(t+R\mathbf{a}_{i}-\mathbf{b}_{i} \right)}\end{array}  \tag{3}\]  
where \(i\) = 1, 2,..., 6. Equation (3) is the forward kinematic equation, which is classified as a nonlinear equation and is difficult to solve directly. Therefore, according to the Gauss-Newton iteration method, the following function \(F(\mathbf{q})\) is constructed:  
\[F(\mathbf{q})=\sum_{i=1}^{6}\biggl{[}h_{i}-\sqrt{\left(t+R\mathbf{a}_{i}-\mathbf{b}_{i} \right)^{T}\left(t+R\mathbf{a}_{i}-\mathbf{b}_{i}\right)}\biggr{]}^{2} \tag{4}\]  
In this way, the problem of solving the nonlinear equation is transformed into the problem of finding the pose \(\mathbf{q}\) of the primary mirror that minimizes \(F(\mathbf{q})\). It can be solved by the least squares method. For the convenience of calculation, make:  
\[f_{i}(\mathbf{q})=h_{i}-\sqrt{\left(t+R\mathbf{a}_{i}-\mathbf{b}_{i}\right)^{T}\left(t+R \mathbf{a}_{i}-\mathbf{b}_{i}\right)} \tag{5}\]  
\[\mathbf{f}(\mathbf{q})=\left[f_{1}(\mathbf{q}),f_{2}(\mathbf{q}),f_{3}(\mathbf{q}),f_{4}(\mathbf{q}), f_{5}(\mathbf{q}),f_{6}(\mathbf{q})\right]^{T} \tag{6}\]  
Then, substituting Equation (6) into Equation (4) yields:  
\[F(\mathbf{q})=\mathbf{f}(\mathbf{q})^{T}\mathbf{f}(\mathbf{q}) \tag{7}\]  
The iteration formula according to the Gauss-Newton iteration method is:  
\[\mathbf{q}_{k+1}=\mathbf{q}_{k}-\left[(\mathbf{J}_{k})^{T}\mathbf{J}_{k}\right]^{-1}(\mathbf{J}_{ k})^{T}\mathbf{r}_{k} \tag{8}\]  
Figure 2: Structural diagram of the six-hardpoint positioning mechanism. (**a**) The mechanism in the initial pose; (**b**) The mechanism in the state after a pose adjustment. where:  
\[\mathbf{J}_{k}= \begin{bmatrix}\frac{\partial f_{1}(\mathbf{q})}{\partial x}&\frac{\partial f _{1}(\mathbf{q})}{\partial y}&\frac{\partial f_{1}(\mathbf{q})}{\partial z}&\frac{ \partial f_{1}(\mathbf{q})}{\partial x}&\frac{\partial f_{1}(\mathbf{q})}{\partial\beta }&\frac{\partial f_{1}(\mathbf{q})}{\partial y}\\ \frac{\partial f_{2}(\mathbf{q})}{\partial x}&\frac{\partial f_{2}(\mathbf{q})}{ \partial y}&\frac{\partial f_{2}(\mathbf{q})}{\partial z}&\frac{\partial f_{2}( \mathbf{q})}{\partial x}&\frac{\partial f_{2}(\mathbf{q})}{\partial\beta}&\frac{ \partial f_{2}(\mathbf{q})}{\partial\gamma}\\ \frac{\partial f_{2}(\mathbf{q})}{\partial x}&\frac{\partial f_{2}(\mathbf{q})}{ \partial y}&\frac{\partial f_{2}(\mathbf{q})}{\partial z}&\frac{\partial f_{2}( \mathbf{q})}{\partial x}&\frac{\partial f_{2}(\mathbf{q})}{\partial\beta}&\frac{ \partial f_{2}(\mathbf{q})}{\partial\gamma}\\ \frac{\partial f_{2}(\mathbf{q})}{\partial x}&\frac{\partial f_{2}(\mathbf{q})}{ \partial y}&\frac{\partial f_{2}(\mathbf{q})}{\partial z}&\frac{\partial f_{2}( \mathbf{q})}{\partial x}&\frac{\partial f_{2}(\mathbf{q})}{\partial\beta}&\frac{ \partial f_{2}(\mathbf{q})}{\partial\gamma}\\ \frac{\partial f_{2}(\mathbf{q})}{\partial x}&\frac{\partial f_{2}(\mathbf{q})}{ \partial x}&\frac{\partial f_{2}(\mathbf{q})}{\partial x}&\frac{\partial f_{2}( \mathbf{q})}{\partial x}&\frac{\partial f_{2}(\mathbf{q})}{\partial\beta}&\frac{ \partial f_{2}(\mathbf{q})}{\partial\gamma}\\ \frac{\partial f_{2}(\mathbf{q})}{\partial x}&\frac{\partial f_{2}(\mathbf{q})}{ \partial y}&\frac{\partial f_{2}(\mathbf{q})}{\partial z}&\frac{\partial f_{2}( \mathbf{q})}{\partial x}&\frac{\partial f_{2}(\mathbf{q})}{\partial\beta}&\frac{ \partial f_{2}(\mathbf{q})}{\partial\gamma}\\ \end{bmatrix}