# Hybrid Strategy Improved Beetle Antennae Search Algorithm and Application
## 5 Application to Mechanical Module Design
### Physical and Mathematical Modeling of the Height Compensation Module  
The altitude compensation module is obtained using the synergistic design of the linkage mechanism, constant force coil spring, grooved roller, spring set, and symmetric support. Specifically, as shown in Figure 5, the T-rail on the load platform cooperates with the roller mounted at point A to form a recess. The connecting rods \(AOB\) and \(A^{\prime}O^{\prime}B^{\prime}\) take points \(O\) and \(O^{\prime}\) as the rotation centers, respectively; \(B\), \(B^{\prime}\), \(C\) and \(C^{\prime}\) are the hinge points. The bottom of the load platform is equipped with a parallel structure of adjustable stiffnessas a spring set, with the springs of different stiffnesses arranged in parallel to realize the adjustability of the overall stiffness. Considering that the altitude compensation module needs to compensate for the gravity of solar wing dynamically and cannot provide constant friction, the structural parameters need to be optimized using the intelligent optimization algorithm to make the output of the compensation module stable at about 500 N load. Because of the symmetrical design of the height compensation module, the mechanical analysis of the rod combination \(AOB\) is carried out first to reduce the calculation steps. The torque-equilibrium equation for the rotation center \(O\) can be expressed as follows,  
\[Gl_{AO}\cos(\beta-\sigma)=F_{CB}l_{BO}\cos(\sigma-\gamma), \tag{16}\]  
where \(G\) is the altitude compensation module load on the platform; \(l_{AO}\) and \(l_{BO}\) are the lengths of both sides of the connecting rod, respectively; \(\beta\) is the outer angle of the connecting rod \(AOB\); \(\sigma\) is the arbitrary angle of \(AOB\) rotating around \(O\); \(F_{CB}\) is the force of the connecting rod \(BC\) on the connecting rod \(AOB\); and \(\gamma\) is the angle of the connecting rod \(BC\) and the plumbling, when the altitude compensation module is in the initial working state, \(\gamma\) is 0. The expression of the bearing capacity can be obtained after substituting the analysis of the elastic force and the vertical force. \[G=\frac{1}{2}\frac{kl_{BO}}{l_{AO}}(X+X_{0})\frac{\cos(\sigma-\gamma)}{\cos \gamma\cos(\beta-\sigma)}, \tag{17}\]  
where \(k\) is the stiffness of the compression spring U; \(X\) is the new compression of the compression spring after rotating the connecting rod \(AOB\) by the angle \(\sigma\); and \(X_{0}\) is the preload value of the compression spring in the initial operating state. The expressions for \(\gamma\) and \(X\) can be obtained by geometrically analyzing the model of the height compensation module. \[\left\{ \begin{array}{l}\gamma=\arcsin(\frac{l_{BO}-l_{BO}\cos\sigma}{l_{BC} })\\ X=l_{BC}+l_{BO}\sin\sigma-l_{BC}\cos\gamma\end{array} \right., \tag{18}\]  
where \(l_{BC}\) is the length of the connecting rod \(BC\). Figure 5: The two-dimensional simplified model of the altitude compensation module structure. (**a**) Physical model; (**b**) mathematical model. Taking the gravity of the rod into account in the load and coupling the above three equations,  
\[\left\{ \begin{array}{l}(G+G_{0}+G_{AO})l_{AO}\cos(\beta-\sigma)=2F_{CB}l_{BO} \cos(\sigma-\gamma)+G_{BO}l_{BO}\cos\sigma\\ F_{CB}\cos\gamma=\frac{1}{2}(k(X+X_{0})+G_{CC^{\prime}})-G_{BC}\end{array}  \right., \tag{19}\]  
where \(G_{0}\) is the self-weight of the top loading platform; \(G_{AO}\) and \(G_{BO}\) are the self-weights of the \(AO\) and \(BO\) sections of the connecting rod \(AOB\); and \(G_{CC^{\prime}}\) and \(G_{BC}\) are the self-weights of the spring-loaded support \(CC^{\prime}\) and the rod \(BC\), respectively.