# Design and Analysis of Independently Adjustable Large In-Pipe Robot for Long-Distance Pipeline
## 2 Overview of Robot
### Analysis of the Three-Crawlers Differential Rotation Model  
When a large in-pipe robot is operating inside a pipe, the pipe is filled with tiny obstacles caused by corrosion and foreign matter. The robot has a certain route in straight pipeline, so robot path planning deviation exists [20]. Hence, actively adjusting the attitude to restore the robot to a stable attitude is necessary. Since the hardness of the obstacle cannot be detected, the strategy of the robot in the straight tube differs depending on the shape of the obstacle. To realize the control of differential speed rotation, we ought to establish its mechanical model base on force and pressure analysis. Walking in a straight pipe for a long distance, the upper crawler should fit the wall, but no pressure (\(F_{a}=0\)). Although more pressure means more driving force, reducing resistance is beneficial for an in-pipe robot to go farther when no other equipment is carried. Where: \(F_{a}\), \(F_{b}\) and \(F_{c}\) denote the supporting force. \(f_{a}\), \(f_{b}\) and \(f_{c}\) denote the friction. As in Figure 6a, when the robot is walking stable in straight pipe, assuming that the robot's weight is concentrated in the center and the constraint force is provided by the pipe wall. The following equation can be listed:  
\[F_{a}+mg=(F_{b}+F_{c})cos60 \tag{3}\]  
Figure 5: (**a**) Simplified model of robot frames with dimensions. (**b**) central body (origin is \(O_{1}\)) and crawler (origin is \(O_{3}\) ) force distribution, \(G^{0}\) is the weight of central body, \(F_{N}\) is the reaction force of \(F_{1}^{0}\) and \(F_{2}^{0}\). However, when the attitude of the device is changed, the robot obtains a rotational movement through the differential speed and overcomes the dynamic friction. Assuming the robot uses a three-crawler differential speed strategy to rotate inside the tube. We define that crawler has the radial velocity component \(v_{1}\) and tangential velocity component \(v_{2}\), which have the following relationship: \(v_{a1}\quad v_{c1}\quad v_{b1}\), \(v_{a2}=v_{b2}=v_{c2}\). A Turning Center is formed between every two crawlers, and there is an imaginary turning plane between every two crawlers. The differential speed determines the position of the turning center on the plane. Based on differential steering working principle, a steering center is formed between two different speed crawlers [21]. Meeting the following requirements:  
\[r=\frac{b}{1\quad\frac{v_{b}}{v_{c}}} \tag{4}\]  
Figure 7b shows radial velocity between B crawler and A crawler, every two crawlers have a similar relationship. In particular, the fastest crawler (A) and the slowest one (B) has an opposite steering tendency with the other two groups (B and C, C and A). Figure 6: (a) Positive posture of the robot in the tube; (b) schematic of the force on the robot during differential motion (blue arrow represents the rotation direction). Figure 7: _Cont._  
By using D' Alembert Theory, the force diagram is shown in Figure 6b. Coordinate system with robot, we established dynamic equation as follow:  
\[\left\{ \begin{array}{l}\overset{!}{F_{a1}}=m_{a}\overset{!}{a_{a1}}+m_{a} \overset{!}{a_{a2}}\\ F_{a1y}+F_{a}+m_{a}gcos\varepsilon=0\\ F_{aix}\quad f_{a}\quad m_{a}gsin\varepsilon=0\end{array} \right. \tag{5}\]  
\[\left\{ \begin{array}{l}\overset{!}{F_{b1}}=m_{b}\overset{!}{a_{b1}}+m_{b} \overset{!}{a_{b2}}\\ F_{b}cos60+F_{b1x}\quad m_{b}gcos\varepsilon+f_{b}cos60=0\\ f_{b}cos60+F_{b}cos30\quad m_{b}gsin\varepsilon+F_{b1y}=0\end{array} \right. \tag{6}\]  
\[\left\{ \begin{array}{l}\overset{!}{F_{c1}}=m_{c}\overset{!}{a_{c1}}+m_{c} \overset{!}{a_{c2}}\\ f_{c}cos30\quad F_{c1y}+F_{c}cos60+m_{c}gcos\varepsilon=0\\ f_{c}cos60\quad F_{c1x}\quad F_{c}sin30\quad m_{c}gsin\varepsilon=0\end{array} \right. \tag{7}\]  
The rotation of the robot is a dynamic process.