# Wheeled Permanent Magnet Climbing Robot for Weld Defect Detection on Hydraulic Steel Gates
## 2 Mobile Robot System
### Robot Kinematic and Dynamic Model Analysis  
The kinematic model of the robot is developed based on the following assumptions:  
1. The local radius of curvature of the tracked weld at any given point is significantly larger than the minimum turning radius of the robot;  
Figure 4: Control cabinet. Figure 3: Climbing robot. 2. The front and rear wheels on the same side of the robot maintain identical linear velocities; 3. The center of mass of the robot coincides with its center of rotation. To further investigate the kinematic characteristics of the climbing robot, it is first necessary to establish an appropriate coordinate system. As illustrated in Figure 5, x-o-y denotes the world coordinate system, while x-o-y-r represents the robot's local coordinate system. In the robot coordinate system, O\({}^{\prime}\) corresponds to the geometric center of the climbing robot, b denotes the distance between the left and right wheels, and D is the diameter of the magnetic wheels, v\({}_{\text{L}}\), v\({}_{\text{R}}\), and v represent the linear velocities of the left wheel, right wheel, and the overall forward velocity of the robot, respectively. \(\omega\) denotes the angular velocity of the climbing robot about its geometric center. The variables and their corresponding meanings are shown in Table 1.  
The calculation methods for \(v\) and \(\omega\) of the climbing robot in its local coordinate system are presented in Equations (1) and (2):  
\[v=\frac{v_{L}+v_{R}}{2}, \tag{1}\]  
\[\omega=\frac{v_{L}-v_{R}}{b}, \tag{2}\]  
The position and posture of the climbing robot are represented by [x, y, \(\theta\)]T, where x and y denote the coordinates of the center of the robot, and \(\theta\) denotes the heading angle of the robot. The kinematic model of the climbing robot can be formulated as follows:  
\[ \begin{bmatrix}\dot{x}\\ \dot{y}\\ \dot{\theta}\end{bmatrix} = \begin{bmatrix}cos\theta&0\\ sin\theta&0\\ 0&1\end{bmatrix}\begin{bmatrix}v\\ \omega\end{bmatrix}  \tag{3}\]  
Combining Equations (1)-(3), we can get Equation (4) as shown below:  
\[ \begin{bmatrix}\dot{x}\\ \dot{y}\\ \dot{\theta}\end{bmatrix} = \begin{bmatrix}(v_{L}+v_{R})/2\times cos\theta\\ (v_{L}+v_{R})/2\times sin\theta\\ (v_{L}+v_{R})/b\end{bmatrix}  \tag{4}\]  
Next, we will conduct some dynamic analyses. When the robot is adsorbed onto a vertical steel surface and moves upward in a straight line, let the driving torque of the motor be denoted by M. Considering power losses in the transmission process, and assuming a transmission efficiency of \(\eta\), the following equation holds:  
\[M_{t}=\eta M=\eta P/\omega \tag{5}\]  
Figure 5: Coordinate system of robot. In the above equation, P denotes the power of the driving motor, \(\omega\) represents the motor's angular velocity, and Mt is the driving torque transmitted to the wheels. Considering the robot as a whole for force analysis, when it climbs vertically upward, the force balance satisfies the following equation:  
\[ma=3(\tau_{R}+\tau_{L})-F_{f}-G \tag{6}\]  
\[F_{f}=f_{m}(\;F-N) \tag{7}\]  
In the above equation, \(\tau_{R}\) and \(\tau_{L}\) represent the driving forces of the right and left wheels, respectively; m is the mass of the robot; a is the acceleration in the vertical upward direction; F\({}_{\text{f}}\) denotes the dynamic friction force acting on the robot; F is the total magnetic adhesion force; f\({}_{\text{m}}\) denotes the coefficient of kinetic friction and N is the support force acting on the robot in the horizontal direction. Accordingly, the following equation can be established:  
\[ma=3(\tau_{R}+\tau_{L})-f_{m}(\sum_{j=1}^{6}F_{j}-\sum_{i=1}^{6}N_{i})-G \tag{8}\]  
Here, \(\tau_{R}=\tau_{L}=\text{F}_{\text{f}}/6\), where F\({}_{\text{f}}\) is the traction force required for upward motion. The resistance force F\({}_{x}\) represents the resultant rolling friction and the gravitational force component in the vertical direction. By combining the above two equations, the following relationship is obtained: