## 2 System Design and Optimization
### Design and Analysis of the Adsorption of Bionic Structures
The angle between the carbon fiber tube attached to the suction cup and the horizontal plane is \(\alpha\). Ignoring the mass of the support bars and the carbon fiber tube, the forces in the horizontal and vertical directions are balanced. \[\left\{ \begin{array}{l}N_{1}+N_{2}=F_{suction}\\ f_{1}+f_{2}=mg\end{array} \right. \tag{11}\]  
Figure 6: Force analysis of the robot. (**a**) Force analysis of the approaching state (propellers are working). (**b**) Force analysis of the adsorbing state (propellers are not rotating). Figure 7: Structure of the suction cup and the support bars. Additionally,  
\[\left\{ \begin{array}{l}f_{1}\leq\mu_{1}N_{1}\\ f_{2}\leq\mu_{2}N_{2}\end{array} \right. \tag{12}\]  
\[(F-N_{1})d_{L}[sin\alpha+\tan(\theta_{s}-\alpha)\cos\alpha]=mgd_{1}\cos\alpha \tag{13}\]  
where \(\mu_{1}\) is the coefficient of friction between the suction cup and the wall, and \(\mu_{2}\) is the coefficient of friction between the support bars and the wall. \(d_{L}\) is the distance from the connection point of the support bars to the suction cup. \(m\) is the mass of the robot and \(g\) denotes the acceleration of gravity  
\[N_{2}=\frac{mgd_{1}\cos\alpha}{d_{L}\sin\alpha+d_{L}\cos\alpha\tan(\theta_{s} -\alpha)} \tag{14}\]  
Because suction cup and support bars are not the same material, \(\mu_{1}\) and \(\mu_{2}\) are not consistent. In general, \(\mu_{1}<\mu_{2}\), therefore it is necessary to raise \(N_{2}\) to increase the weight that the robot can hold. The influence of \(N_{2}\) on \(\alpha\) is shown in Figure 8. \(N_{2}\) becomes larger as \(\alpha\) increases. As \(\alpha\) becomes larger, the robot assigns more \(F_{suction}\) to \(N_{2}\). Assuming \(d_{1}=0.35\) m,\(d_{L}=0.2\) m, \(N_{2}\) is approximately equal to \(1.996\times mg\) when \(\alpha=10^{\circ}\) which is \(1.14\) times as large as \(N_{2}=1.75\times mg\) when \(\alpha=0^{\circ}\). The increase of \(\alpha\) may cause the suction cup to be unable to fit against the wall and form a negative pressure environment. Therefore, we increased the contact angle of the suction cup by adding a pivot and a limit block, as shown in Figure 7. The suction cup is connected to the robot by a pivot and is limited by a limit block. The suction cup and the carbon fiber tube are connected by a flexible silicone tube. Therefore, the suction cup can rotate \([-5^{\circ},5^{\circ}]\) on the limit block, and because the double-layer suction cup itself has \([-6.75^{\circ},6.75^{\circ}]\) angle range, the robot's suction cup has \([-11.75^{\circ},11.75^{\circ}]\) angle range. As shown in Figure 9, we tested the variation of \(N_{2}\) for different clamping angles of the support bars \(\theta_{s}\). As the clamping angle \(\theta_{s}\) becomes larger, \(N_{2}\) gradually becomes smaller. However, \(N_{1}\) cannot be lower than a certain threshold, otherwise it will lead to an increase in the leakage rate of the suction cup and further lead to the disengagement of the suction cup. Therefore, we take \(\theta_{s}\) as 45 degrees. When the suction cup works on the surface of the painted wall, the pressure difference is 40 kpa, and the diameter suction cup is 0.03 m. \(F_{suction}\) is about 110 N. The total mass of the robot is 3 kg. Then, the maximum friction force that the robot can provide is about 69.37 N, which is two times greater than the gravity. Figure 8: Influence of angle \(\alpha\) to the support force \(N_{2}\) on absorbing module.