# The Kinematic Analysis of a Wind Turbine Climbing Robot Mechanism
## 3 Kinematic Modeling  
The entire structure of the scaled-down prototype climbing robot is composed of a body frame, mecanum wheels for locomotion, and a stepper motor coupled to a twin pulley and Bowden cable for adhesion. This structure facilitates kinematic modeling and focuses on the robot's motion through the mecanum wheel movements. When the mecanum wheels are energized, the angled peripheral rollers translate a portion of the force in the rotational direction of the wheel to a force normal for the wheels' direction. Depending on each individual wheel direction and velocity, the resulting combination of all these forces produce a total force vector in any desired direction. As a result, the robot can freely move in the direction of the corresponding force vector without the need to change the wheel orientation [12; 13; 14; 15; 16]. For a mecanum wheel to achieve multidirectional movements the wheels are arranged on the robot wheel configuration by mounting four wheels on the robot body frame on each side with a specified angle, as shown in Figure 5. To introduce this new transformative design of climbing robot, the position of the robot to the wind turbine tower and coordinate system needs to be defined. Figure 5: Coordinate System of WT Climbing Robot. Let the radius of the meanum wheel and roller be \(\mathrm{r_{w}}\) and \(\mathrm{r_{r}}\), respectively; the angular velocity of four wheels to be \(\omega_{1}\), \(\omega_{2}\), \(\omega_{3}\)  and \(\omega_{4}\)  the speeds of rollers on each wheel to be \(\mathrm{v_{1r}}\), \(\mathrm{v_{2r}}\), \(\mathrm{v_{3r}}\)  and \(\mathrm{v_{4r}}\) and the velocity of robot in \(Z\)-direction, Y-direction, and angular velocity in the X-axis around the center mass point O' to be \(\mathrm{v_{Z}}\), \(\mathrm{v_{Y}}\) and \(\mathrm{\omega_{0}}\). The coordinate origins at O, the center of the robot, and each wheel have an origin of O\({}_{1}\), O\({}_{2}\), O\({}_{3}\) and O\({}_{4}\). The distance between the wheel 1 and 2 center point of the robot wheels is while the distance between the wheel 1 and 3 center point of robot wheels is \(\mathrm{L_{Y}}\), the wheels' axis of rolling \(\varphi_{i}\) is the angle of the rollers which is \(\pm 45^{\circ}\), in this case shown in Figure 3, the layout of the rollers. The \(\mathrm{\sigma_{i}}\) is the angle between \(\mathrm{P_{i}}\) to \(\mathrm{Z_{R}}\). The C\({}_{i}\) is the distance from the center point of the caster wheel to the center point of the robot wheels. The \(\mathrm{\lambda}\) is the angle between C\({}_{i}\) and the distance from the center of the body frame to the center point of the caster wheel. The \(\mathrm{v_{Si}}\) and \(\mathrm{v_{Qi}}\) represent the velocity of the wheel \(i\) in the S\({}_{i}\) O\({}_{i}\) Q\({}_{i}\) coordinate system. According to Figure 5 the velocity of the wheel \(i\) and the layout of the rollers in contact to the mockup wind turbine tower without slipping can be derived by:  
\[\mathrm{v_{Si}}=\mathrm{v_{ir}}(\sin\varphi) \tag{1}\]  
\[\mathrm{v_{Qi}}=\mathrm{\omega_{i}}\left(\mathrm{r_{i}}\right)-\mathrm{v_{ir }}(\cos\ \varphi) \tag{2}\]  
The velocity of the wheel's center translated to the robot can be expressed \(\mathrm{Z_{R}O^{\prime}Y_{R}}\) as a coordinate system can be achieved by the Equations (3) and (4). \[\mathrm{v_{iZ_{R}}}=\mathrm{v_{Z}}+\left(\mathrm{C_{i}}\sin\lambda\right)\ \mathrm{\omega_{o}} \tag{3}\]  
\[\mathrm{v_{iY_{R}}}=\mathrm{v_{Y}}+\left(\mathrm{C_{i}}\cos\lambda\right)\ \mathrm{\omega_{o}} \tag{4}\]  
The moving velocity of the wheel \(i\) in the robot coordinate system can be obtained from the Equations (5) and (6).