# Precision Analysis and Error Compensation of a Telescope Truss Structure Based on Robotics
## 2 Structural Analysis of the Telescope Manipulator Based on Stiffness Performance Index
### Feasibility Analysis of the Robotic Truss Structure  
The feasibility of the secondary mirror truss based on robotics must be analyzed before studying the adjustment capability and method of the secondary mirror. The proposed parallel mechanism needs to be able to realize the adjustment function of the secondary mirror. First, we theoretically analyzed the kinematics of the parallel mechanism. The parallel structure proposed in this paper is a 3-6R mechanism. We established the relationship between the screw theory and geometric algebra and analyzed the kinematics characteristic of the parallel mechanism. According to the definition of the screw theory and geometric algebra, N-dimensional geometric algebra space can be written as the composition of orthogonal groups \(\{e_{1},e_{2},\ldots e_{n}\}\), screw can be seen as a line vector with a pitch, and the equation of the screw is \(r\times S=S^{0}-hS\), written in Plucker coordinates as \(\left(S;S^{0}-hS\right)\). The screw can then be represented by geometric algebra as:  
\[S=v_{1}e_{1}+v_{2}e_{2}+v_{3}e_{3}+b_{1}e_{4}+b_{2}e_{5}+b_{3}e_{6}, \tag{12}\]  
in which \(b_{1}\) is a scalar. The secondary mirror truss structure can be represented as the model shown in Figure 4. We established a coordinate system at the center of symmetry. The fixed platform is set in the XOY plane in Figure 4 and the radius of the fixed platform is r. The moving platform is set on the above. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Angle of Optical Axis** & **0** & **45\({}^{\circ}\)** & **90\({}^{\circ}\)** \\ \hline \(\Delta x\) (\(mm\)) & \(-0.033\) & 0.0276 & 0.0720 \\ \(\Delta y\) (\(mm\)) & \(-0.018\) & \(-0.0064\) & 0.0090 \\ \(\Delta z\) (\(mm\)) & 0.135 & 0.0716 & 0.0338 \\ RMS of wavefront aberration (\(\lambda\)) & 0.25 & 0.18 & 0.48 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Secondary mirror misalignment and wavefront aberration. Figure 3: Misalignment curve as a function of angle. The moving subspace of the parallel mechanism can be solved after obtaining the moving subspace of each serial. The position vector of each joint is shown in Table 3.  
According to Equation (12), each twist on the first limb can be written as:  
\[ \begin{split} S_{M_{11}}=e_{3}\\ S_{M_{12}}=e_{2}+re_{6}\\ S_{M_{13}}=e_{2}-z_{A_{1}}e_{4}+x_{A_{1}}e_{6}\\ S_{M_{14}}=e_{1}+z_{B_{1}}e_{4}-y_{B_{1}}e_{6}\\ S_{M_{15}}=e_{2}-z_{C_{1}}e_{4}+x_{C_{1}}e_{6}\\ S_{M_{16}}=e_{3}+y_{C_{1}}e_{4}-x_{C_{1}}e_{6}\\ \end{split}  \tag{13}\]  
The motion subspace of the first limb can be regarded as series of independent kinematic pairs on the limb, then the union of the twists is:  
\[S_{M_{1}}=S_{M_{11}}\cup S_{M_{12}}\cup S_{M_{13}}\cup S_{M_{14}}\cup S_{M_{15} }\cup S_{M_{16}}=ae_{1}\bigwedge e_{2}\bigwedge e_{3}\bigwedge e_{4}\bigwedge e _{5}\bigwedge e_{6}, \tag{14}\]  
Similarly, the motion subspace of the second limb is the union of the twists:  
\[S_{M_{2}}=S_{M_{21}}\cup S_{M_{22}}\cup S_{M_{23}}\cup S_{M_{24}}\cup S_{M_{25 }}\cup S_{M_{26}}=be_{1}\bigwedge e_{2}\bigwedge e_{3}\bigwedge e_{4}\bigwedge e _{5}\bigwedge e_{6}, \tag{15}\]  
Figure 4: Schematic diagram of the parallel truss mechanism. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Position Vector** & **Coordinates** & **Twist** \\ \hline \(OM_{1}\) & \([r,\ 0,\ 0]\) & \(S_{M_{11}},S_{M_{12}}\) \\ \(OM_{2}\) & \( \begin{bmatrix}-\frac{1}{2}r,\ \frac{\sqrt{3}}{2}r,\ 0\end{bmatrix} \) & \(S_{M_{21}},S_{M_{22}}\) \\ \(OM_{3}\) &  
The motion subspace of the third limb is the union of the twists:  
\[S_{M_{3}}=S_{M_{31}}\cup S_{M_{32}}\cup S_{M_{33}}\cup S_{M_{34}}\cup S_{M_{35}} \cup S_{M_{36}}=ce_{1}\bigwedge e_{2}\bigwedge e_{3}\bigwedge e_{4}\bigwedge e_{5 }\bigwedge e_{6}, \tag{16}\]  
\(a,b,c\) are scalars in the equations that do not influence the result. It can be obtained from the calculation that the motion subspace of each limb at the end of the kinematic chain is a 6-blade. N-dimensional geometric algebra comprises a 0-n order blade and the blade of 0 order is a scalar.