# Inspection Robot and Wall Surface Detection Method for Coal Mine Wind Shaft
## 3 Overall Design of Tooth Rail Wind Shaft Inspection Robot
### Design of Mobile Module
#### 3.3.2 Installation Design Analysis of Rack Rail  
The stability of rack-and-rail installation directly affects the safety of the inspection system. According to the analysis of the working conditions of the wind shaft and the structural characteristics of the toothed rail, the installation methods of the toothed rail are divided into three types: cross-type, side-by-side-type and side-by-side-beam-type, and the installation schematic diagram is shown in Figure 11.  
By comparing and analyzing the advantages and disadvantages of the above three installation methods, combined with the functional requirements of the inspection system, it can be found that the side-by-side beam installation method is more reliable, which not only has better bearing performance, but also can ensure the stability of the track. It will not affect the ventilation performance of the wind shaft, so the final installation method adopts two I-beams symmetrically installed at the bottom of the rack rail to fix the rack rail. However, the installation spacing of I-beams and the included angle of the camera affect the diameter of the rack and the reliability of the whole system. Therefore, in order to obtain a suitable rack diameter, it is necessary to design suitable installation spacing. Specifically, as shown in Figure 12, Point O represents the origin of the coordinate system, points C, D and E are the wellbore wall, point B is the position of the detection device, and point A is the projection of OB on the central axis of the wellbore, the green circle represents the detection module, the black dashed line represents the designed gear track, the black solid line represents the I-beam and the blue solid line represents the effective inspection area.  
In order to find out the diameter of the rack rail and the appropriate installation spacing, a rectangular coordinate system, as shown in red in the above figure, is established, and the coordinate of point B (unilateral detection device) is (\(s\), \(t\)). According to the fact  
Figure 11: Installation mode of gear rail: (**a**) cross-type; (**b**) side-by-side-type; and (**c**) side-by-side-beam-type.  
Figure 12: Installation schematic of gear rail.  
that the diameter of the rack rail is smaller than the diameter of the shaft, the shooting area should meet the principle of complete circumferential inspection so the relationship can be established.  
\[\begin{cases}4\leq s^{2}+t^{2}\leq 9&(7\text{-a})\\ \sqrt{\left(t-3\right)^{2}+s^{2}}\leq\frac{2\sqrt{3}s}{\sqrt{s^{2}+t^{2}}}&(7 \text{-b})\end{cases} \tag{7}\]  
According to the above conclusions, it can be seen that increasing the spacing of I-beams is beneficial to enhance the bearing capacity of the whole structure. From Figure 13, The blue curve and the green curve together enclose the upper and lower bounds of the interval of Formula (7-a), and the red curve is the boundary formed according to Formula (7-b). it can be seen that the theoretical optimal value of the solution is (0.837, 1.816), the theoretical design diameter of the tooth rail is about 4 m and the appropriate installation spacing of the tooth rail is 1.674 m. The design parameters of the rack are shown in Table 3.