# The Kinematic Analysis of a Wind Turbine Climbing Robot Mechanism
## 3 Kinematic Modeling
\begin{array}{rrrr}1&1&1&1\\ 1&-1&-1&1\\ -(E+F)&(E+F)&-(E+F)&(E+F)\end{array} \right] \tag{18}\]In this model the forward kinematic equation of the system is obtained:  
\[\left[ \begin{array}{c}\
u_{Z}\\ \
u_{Y}\\ \omega_{o}\end{array} \right]=\frac{r}{4}\left[ \begin{array}{cccc}1&1&1&1\\ 1&-1&-1&1\\ -(E+F)&(E+F)&-(E+F)&(E+F)\end{array} \right]\left[ \begin{array}{c}\omega_{1} \\ \omega_{2}\\ \omega_{3}\\ \omega_{4}\end{array} \right] \tag{19}\]  
For the up-down motion the Z-axis velocity is not zero, while the Y-axis and the angular velocity are zero, then Equation (18) is obtained. \[\
u_{Z}(t)= \begin{array}{c}\tau\\ \tau\\ \lambda\end{array} \left(\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4}\right) \tag{20}\]  
For the rotational motion the Y-axis is not zero, while the Z-axis and angular velocity are zero, then Equation (19) is obtained. \[\
u_{Y}(t)= \begin{array}{c}\tau\\ \lambda\end{array} \left(\omega_{1}-\omega_{2}-\omega_{3}+\omega_{4}\right) \tag{21}\]  
For the spiral motion both Z-axis and Y-axis is not zero, while angular velocity is zero it can be concluded that:  
\[\
u_{ZY}(t)= \begin{array}{c}\tau\\ \lambda\end{array} \left(\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4}\right)+ \frac{r}{4}(\omega_{1}-\omega_{2}-\omega_{3}+\omega_{4}) \tag{22}\]  
\[\
u_{ZY}(t)= \begin{array}{c}\tau\\ \lambda\end{array} \left(\omega_{1}+\omega_{4}\right) \tag{23}\]  
Then the angular velocity is not zero, Equation (20) can be obtained:  
\[\omega_{O}(t)=\frac{r}{4}[-(E+F)\omega_{1}+(E+F)\omega_{2}-(E+F)\omega_{3}+( E+F)\omega_{4}] \tag{24}\]  
The Equations (18)-(20) show the relationship between the angular velocity of the wheels and the movement of the robot in its different movement states. The presence of the four added DC motors is considered an essential kinematic equation of motion because an equation that relates velocity has a direct control of the wheel velocity. The robot parameters as shown in Table 2 with the wheels are mounted in a fixed angle position and orientation with respect to the robot body frame provided that each wheel touches the wind turbine tower surface as it travels. The simulation of the robot's projected movements was implemented in MATLAB. The rotational speeds of the wheels were calculated using the climbing robot parameters and the kinematic equations. Since there are three movement states (up-down, rotatation, and spiral), the distance traveled by the robot during these states are also calculated. For the up-down movement, the distance used was the tower's height, which is 1200 mm. For the rotation of the robot at a point in the tower, the circumference is used, which is 1184 mm. This is taken at a point on the tower where the diameter is 400 mm. Then for the spiral movement, the length of the spiral was calculated using an Archimedean spiral \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(i\) & **Wheels** & \(\sigma_{i}\) & \(\varphi_{i}\) & \(\lambda\) \\ \hline
1 & 1 & 90\({}^{\circ}\) & 45\({}^{\circ}\) & 24.44\({}^{\circ}\) \\
2 & 2 & 90\({}^{\circ}\) & \(-\)45\({}^{\circ}\) & 24.44\({}^{\circ}\) \\
3 & 3 & 90\({}^{\circ}\) & \(-\)45\({}^{\circ}\) & 24.44\({}^{\circ}\) \\
4 & 4 & 90\({}^{\circ}\) & 45\({}^{\circ}\) & 24.44\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Climbing Robot Parameters. along a circular truncated cone. The formula is presented in Equation (25), reaching a value of 8885 mm. \[\int\limits_{a}^{b}\sqrt{r^{2}+\left(\frac{dr}{d\theta}\right)^{2}}d\theta \tag{25}\]  
The velocities \(\
u_{Z}\) and \(\
u_{Y}\) were assumed, such that they reflect the ideal movement scenarios for our robot. For instance, if the robot climbs up the tower at a constant speed, \(\
u_{Z}\) is an arbitrary positive value, while \(\
u_{Y}\) is zero since it is not rotating around the tower. On the other hand, if the robot goes down the tower, \(\
u_{Z}\) is an arbitrary negative value, while \(\
u_{Y}\) is zero.