# Tracked Wall-Climbing Robot for Calibration of Large Vertical Metal Tanks
## 2 Design of Tracked Wall-climbing Robot (TWCR)
### Design of Permanent Magnetic Adhesion and Tracked Locomotion Mechanism  
The TWCR uses permanent magnets to adhere to the metal surface of the tanks and moves forward and backward by the tracked locomotion mechanism, which provides adhesion stability, and the adhesion mechanism remains working in the event of unexpected power failures. The adhesion mechanism, shown in Figure 3, is designed based on the theoretical analysis and a number of previous studies [23]. The material of the magnet is sintered NdFeB 52 SH. A single magnet unit comprises two pieces of permanent magnet and a yoke. The tracks use two aluminum plates to fix the permanent magnet units and protect them from potential damage by the rust, which could be dislodged and block the track. Figure 3: Structure of adhesion mechanism: (**a**) Left view; (**b**) front view; (**c**) bottom view. Figure 2: Mechanical structure of the tracked wall-climbing robot (TWCR): (**a**) Right view (**b**) top view (1—baseboard, 2—driving wheel, 3—scale, 4—tracks, 5—driven wheel, 6—permanent magnetic units, 7—driving, transmission, and steering mechanism, 8—scale moving, and collapsing mechanism, 9—microcontroller). Previous studies [7, 24] have pointed out that the main risks encountered by the TWCR during operation are sliding along the wall and rolling around the lowest contact point the track and the wall. A force analysis model is built to determine the necessary adhering force. In Figure 4, O is the center of mass, P is the lowest contact point, \(h\) is the distance between the center of mass and the wall, \(l\) is the distance of the highest and lowest contact point, \(F_{f}\) is the friction force between the track and the wall, \(G\) is the total weight of the TWCR and the cables, \(F_{m}\) is the adhesion force of each permanent magnetic unit, \(n\) is the number of the permanent magnetic unit in contact with the wall and \(N_{i}\) is the supporting force of the wall to each permanent magnetic unit. Every magnetic unit on the track bears the same adhering force, but the supporting force differs, due to the torque of the gravity. The necessary friction force that the robot needs to avoid sliding along the wall is given by:  
\[F_{f}\geq G \tag{1}\]  
We have,  
\[F_{f}=\sum_{i=1}^{n}\mu N_{i}=\mu\sum_{i=1}^{n}N_{i}, \tag{2}\]  
where \(\mu\) is the friction coefficient. We also have,  
\[nF_{m}=\sum_{i=1}^{n}N_{i}. \tag{3}\]  
According to Equations (1-3), we have,  
\[F_{m}\geq\frac{G}{n\mu}. \tag{4}\]  
The adhesion force also has to be great enough to avoid overturning around point P. As the torque at point P cannot be transmitted by the flexible chain of the track, the first upper magnetic unit will be lifted if the overturning happens at point P, the torque balance equation is  
\[\frac{1}{2}Gh=(F_{m}-N_{1})l \tag{5}\]  
Figure 4: Static force analysis of the TWCR. where 1/2\(G\)_th_ is the 1/2 torque (as there are two tracks share the load) of the gravity, \((F_{m1}-N_{1})\cdot l\) is the torque of the first upper magnetic unit, and we have,  
\[N_{1}>0, \tag{6}\]  
if the first upper magnetic unit is still able to adhere to the wall, then, according to Equations (5) and (6), we have,  
\[F_{m}\geq\frac{G\textit{th}}{2l} \tag{7}\]  
According to Equations (4) and (7), we have the necessary condition for the stable adhesion of the TWCR:  
Figure 2:0  
The weight of the TWCR is about 10 kg according to the evaluation of the 3-D modeling tool (SolidWorks Version 2015, Dassault Systemes SolidWorks Corp., Waltham, MA, USA), and the weight of the suspension cable (about 30 m) is 3 kg, \(G=(10+3)\times 9.8=127.4\) N. The number of the permanent magnetic unit in contact with the wall (both side) \(n\) is 26, the friction coefficient \(\mu\) is 0.1-0.2, the height of the center of the mass \(h\) is 31.2 mm according to the evaluation of the 3-D modeling tool, the distance of the highest and lowest contact point \(l\) is 212 mm.