## 2 Design Principle
### Motor Power  
The calculation for the motor power of this proposed robot is based on the vertical mobility of the robot because this is the stage that the robot requires its maximum load capacity. While the robot is in vertical motion, the required driving force (\(F_{req}\)) for the robot is presented in the following equations:  
\[F_{req}=F_{fric}+(G_{robot}+G_{cable})\sin\theta+(m_{robot}+m_{cable})a \tag{13}\]  
\[F_{fric}=\mu_{rolling}(\frac{d_{1}}{l_{1}}F_{mag1}+\frac{d_{2}}{l_{2}}F_{mag3}) \tag{14}\]  
where \(m_{robot}\) and \(m_{cable}\) are the mass of the robot and cable, respectively, \(a\) is their acceleration, and \(\mu_{rolling}\) is the coefficient of rolling friction when the wheel rolls over a ferromagnetic surface. In this study, the viscous frictional force was assumed to be zero as the robot travels linearly along the pipe and its travel velocity is relatively slow, so the viscous frictional force is very small compared to all other forces. The required motor power (\(P_{req}\)) can be obtained from:  
\[P_{req}=F_{req}\
u_{robot} \tag{15}\]  
where \(\
u_{robot}\) is the robot's travel velocity in the vertical direction. The proposed robot is equipped with two driving motors for forward-backward motion. Each motor is assembled in the front and rear modules. In this study, however, the motor power calculation was considered in the case of the total weight of the robot and related device loads on a single motor. Since some energy losses were not included in the calculation, such as the losses from the mechanical and electronic devices, a factor of safety (\(S_{fs}\)) of two was selected. Then, the required motor power was given by:  
\[P_{motor}=P_{req}S_{fs} \tag{16}\]  
For finding the proper gear ratios, the equations of the motor angular velocity (\(\omega_{motor}\)) and the motor torque (\(\tau_{motor}\)) are presented as follows:  
\[\omega_{motor}=rt_{motor}\times rt_{robot}\times\frac{60\
u_{robot}}{2\pi r_{ w}} \tag{17}\]  
\[\tau_{motor}=\frac{p_{motor}}{\omega_{motor}} \tag{18}\]  
where \(rt_{motor}\) and \(rt_{robot}\) are the gear ratios of the robot and the motor, respectively, and \(r_{w}\) is the radius of the robot's wheel.  
To determine the proper power of the motor, the effect of the distance (between the permanent magnet and the curved surface of the pipe) on the magnetic force was investigated in the experimental study presented in Figure 8. First, the gravitational force was set to zero by the tensile tester machine so that it was not added into the magnetic force. The results show that the increment in the distance results in the exponential decay of the magnetic force. To provide sufficient frictional force for the vertical movement of the robot, the proper distance between the magnet and the pipe's surface can be determined based on Equations (11) and (12). As a result, a distance of 4 mm for the front and rear modules provides a magnetic force of 92.4 N and a distance of 2.9 mm for the middle module provides a magnetic force of 71.6 N. Then, the power, speed, and torque requirements of the motor can be calculated from Equations (16)-(18) and the results are presented in Table 2. From these results, Maxon's EC16 brushless DC servo gearmotor with GP16A planetary gearhead was selected to satisfy the requirements of the motor.