# Design of a Dynamic Secondary Mirror Truss Adjustment Mechanism for Large Aperture Telescopes
## 2 Kinematic Modeling of the Mechanism
### Positive Solution Model  
The positive positional solution is the positional variables of the moving platform with known inputs of the four-leg drive rod lengths. Due to the coupling between the four legs of the mechanism, the position of the endpoint of the mechanism's moving platform is decided by the inputs of the four-leg drive rods.  
Multiplying both ends of Equation (8) by their respective transpositions gives:  
\[\mathbf{r}^{T}\mathbf{r}-2\big{(}\mathbf{a}_{i}+l_{i}\mathbf{\omega}_{i}- \mathbf{b}_{i}\big{)}^{T}\mathbf{r}+\mathbf{b}_{i}^{T}\mathbf{b}_{i}-2\big{(} \mathbf{a}_{i}+l_{i}\mathbf{\omega}_{i}\big{)}^{T}\mathbf{b}_{i}+\mathbf{a}_{ i}^{T}\mathbf{a}_{i}+2l_{i}\mathbf{a}_{i}^{T}\  
Based on the above kinematic modeling theory, a new dynamic secondary mirror truss adjustment mechanism is designed to replace the conventional Hexapod stage and adjust the secondary mirror's attitude by adjusting the secondary mirror truss. This study involves a dynamic secondary mirror truss adjustment mechanism, including a ring beam, a secondary mirror chamber, eight blades, and four correction units. The ring beam is fixed to the truss supporting the secondary mirror, and its components (the four alignment units) are fixed to the four sides of the ring beam. Each alignment unit is connected to the secondary mirror chamber by two blades. The chamber blades are applied with a preload so that the blades experience some tension regardless of the position of the secondary mirror chamber. The alignment unit consists of a stepper motor, worm gear reducer, eccentric cam, blade spring, moveable block, and absolute angle encoder. Figure 4 shows the overall structure of this SMAM.  
The dynamic secondary mirror truss adjustment mechanism uses the movement of the four correction units along the optical axis to control the position of the secondary mirror. When the four correction units move in synchronization, the secondary mirror moves in the direction of the optical axis. When the four correction units are in different positions, a height difference is created between the two sides of the secondary mirror, thus controlling the deflection of the secondary mirror. The correction unit is a flat-bottomed direct-acting cam mechanism, as shown in Figure 5. The worm gear reducer reduces the speed and increases the torque. A stepped shaft and a flat key transmit the torque through an eccentric  
Figure 4: The overall structure of the SMAM.  
Figure 3: Secondary mirror support structure.  
can, which acts on the removable block. By compressing the spring, the movable block is brought into tight contact with the eccentric cam, and the position of the movable block is controlled by adjusting the angle of rotation of the stepper motor to regulate the angle of rotation of the eccentric cam. The end of the shaft is coupled with an absolute angle encoder, which monitors and feeds back the position of the eccentric cam in real-time. The four removable blocks are adjusted separately so that the secondary mirror chamber can be translated (Tz) in the direction of the optical axis and deflected along the Rx and Ry of the vertical optical axis to achieve three DOFs. The two DOFs of movement along the x and y can be adjusted manually using pre-tightening bolts connecting the blades to the correction units.