## 2 System Design and Optimization
### Design and Analysis of the Adsorption of Bionic Structures
\[h_{s}=\sqrt{(r_{w}^{2}+h_{1}^{2}+h_{2}^{2}-2\cdot r_{w}\cdot\sqrt{(h_{1}^{2}+h _{2}^{2})cos(\sigma_{0}+\sigma_{L})})} \tag{4}\]  
\[tan(\sigma_{0})=\frac{h_{1}}{h_{2}} \tag{5}\]  
Figure 4: Adsorption process of the robot (LAWCDR) compared with the gecko (rotor lift keeps the wheels fixed on the wall). (**a**) The wheels start to contact the wall. (**b**) The wheels are compressed. (**c**) The robot uses the lift force generated by the propeller to attach the wheels to the wall. (**d**) The suction cup and support bars are driven towards the wall. (**e**) The suction cup makes the robot adsorb to the wall. (**f**) The correspondence between geckos and robots. It must be ensured that the propeller cannot exceed the plane formed by the two front wheels and the suction cup when the robot is in contact with the wall. The coordinate of the suction cup in the world coordinate system is expressed as follows:  
\[p_{sc}=\left[ \begin{array}{c}x_{sc}\\ y_{sc}\\ z_{sc}\end{array} \right]=\left[ \begin{array}{c}x_{b}\\ y_{b}\\ z_{b}\end{array} \right]+R_{b}^{w}\left[ \begin{array}{c}-d_{2}sin\eta_{1}-d_{ 1}sin(\eta_{1}+\eta_{2})\\ 0\\ h+d_{2}cos\eta_{1}+d_{1}cos(\eta_{1}+\eta_{2})\end{array} \right] \tag{6}\]  
The coordinate of the contact point between the wheels and the wall is expressed as follows:  
\[p_{wheel}=\left[ \begin{array}{c}x_{wheel}\\ y_{wheel}\\ z_{wheel}\end{array} \right]=\left[ \begin{array}{c}x_{b}\\ y_{b}\\ z_{b}\end{array} \right]+R_{b}^{w}\left[ \begin{array}{c}-l-r_{q}sin\sigma_{L} \\ -l_{3}\\ -r_{q}cos\sigma_{L}\end{array} \right]+\left[ \begin{array}{c}-rcos\psi\\ rsin\psi\\ 0\end{array} \right] \tag{7}\]  
The coordinate of the point on the propeller closest to the wall is expressed as follows:  
\[p_{prop}=\left[ \begin{array}{c}x_{prop}\\ y_{prop}\\ z_{prop}\end{array} \right]=\left[ \begin{array}{c}x_{b}\\ y_{b}\\ z_{b}\end{array} \right]+R_{b}^{w}\left[ \begin{array}{c}-(l+\text{R})\\ \pm l\\ h_{p}\end{array} \right] \tag{8}\]  
In order to protect the propeller, when the suction cup and the wheels are in contact with the wall, the propellers are prevented from obtaining contact with the wall. In order to simplify the model, when the wheel is at the maximum compression, \(p_{prop}\) and \(p_{b}\) are simultaneously on the same side of the line formed by the suction cup and the wheel. In the \(x\)-\(z\) plane, the equation of the line passing through \(p_{wheel}\) and \(p_{prop}\) is shown in the following equation:  
\[f(x,z)=[(z-z_{wheel})(x_{sc}-x_{wheel})-(x-x_{wheel})(z_{sc}-z_{wheel})] \tag{9}\]  
Therefore,  
\[f(x_{prop},z_{prop})\bullet f(x_{b},z_{b})>0 \tag{10}\]  
The constraint of \(\eta_{1}\) and \(\eta_{2}\) can be calculated by Equations (6)-(8) and (10). Figure 5: Diagram of the spring mechanism installed on the front wheels. The red circles represent rotating shaft, orange device is the shock absorber. The force of the robot is shown in Figure 6a. If the wheel is in a free state, the friction between the wall and the front wheels belongs to rolling friction, which is much smaller than the sliding friction (ignoring the rolling friction). \(f_{1}\) denotes the friction between the suction cup and the wall, and \(f_{2}\) denotes the friction between the support bars and the wall. \(N_{1}\) and \(N_{2}\) are the support forces of the wall for the suction cup and the support bars, respectively. \(T\) and \(G\) denote the total lift and gravity force, respectively. As shown in Figure 6b, after adsorption succeeds, the robot loses the lift force \(T\) generated by the propeller. Therefore, we need the frictional forces \(f_{1}\) and \(f_{2}\) provided by the wall to balance with the gravity force \(G\). At this time, the two servos can be in a free state to avoid the heat generated by the long working time. In order to increase the weight capacity of the adsorption module, the structure of the suction cup and the support bars is shown in Figure 7. The angle between the carbon fiber tube which is connected to the suction cup and the support bars is \(\theta_{s}\).