# Design and Development of an Omnidirectional Three-Wheeled Industrial Mobile Robot Platform
## 4 Mathematical Modeling
### Kinematic Modeling  
The kinematic model forms the basis of the mathematical development necessary to design the controller and simulate the behavior of the vehicle, being a fundamental pillar in the planning and control of movements. This model establishes the relationship between the movements of the wheels and the desired trajectory, making it possible to predict and adjust the robot's displacement accurately. For its construction, certain constraints are considered that simplify the equations and provide a framework for analysis:  
* The robot moves on a flat surface. * There are no flexible elements in the robot structure (including wheels). * The wheels have one or no steering axis so that the latter is always perpendicular to the ground. * No frictions are considered in the moving elements of the vehicle or against the ground. The geometrical configuration of the omnidirectional robot platform, which is the object of the kinematic and dynamic study, is shown in Figure 6.  
As shown in Figure 6, the kinematic configuration of the robot is based on an equilateral triangular structure, where an omnidirectional wheel is located at each of its sides. The origin of the system \(\{C\}\), located at the geometric center of the platform, is at a distance \"L\" [25] from each of the wheels. In addition, the wheels are distributed at specific angles \(\alpha_{i}\) with respect to the axis \"X\" as detailed in Table 2. This angular arrangement allows the precise and uniform control of the omnidirectional motions, enabling both kinematic modeling and system dynamics. Figure 6: Kinematic diagram of the omnidirectional robot platform. In addition, each element is described by the following parameters:  
* \(X_{C}\): Displacement of the platform on the abscissa axis of the system \(\{C\}\). * \(Y_{C}\): Displacement of the platform on the ordinate axis of the system \(\{C\}\). * \(x_{i}\): Displacement of the wheel on its abscissa axis of its respective system \(\{R_{i}\}\). * \(y_{i}\): Displacement of the wheel on its ordinate axis of its respective system \(\{R_{i}\}\). * \(r\): Radius of the rollers. * \(R\): Radius of the wheels. * \(\varphi_{i}\): Angular displacement of each roller, for translation. * \(\phi_{i}\): Angular displacement of each wheel, for translation. * \(\theta_{i}\): Angular displacement of each wheel, for rotation. * \(\theta_{i}\): Angular displacement of each wheel, total. The coordinate system of the body and the \(n\) wheels are shown in the following Figure 7.  
The following generalized equations of \(X_{c}=X_{c}(x_{i},y_{i})\) y \(Y_{c}=Y_{c}(x_{i},y_{i})\) show only the translational motion (1) taken from Figure 7:  
\[ \begin{cases}X_{c}=(x_{i}+L)\cdot\cos(\alpha_{i})-y_{i}\cdot\sin(\alpha_{i})\\ Y_{c}=(x_{i}+L)\cdot\sin(\alpha_{i})+y_{i}\cdot\cos(\alpha_{i})\end{cases}  \tag{1}\]  
The kinematic behavior is established on the principle that the wheels in contact with the ground behave as a planar joint with three degrees of freedom as shown in Figure 8.  
\begin{table} \begin{tabular}{l l l l} \hline  & **Wheel 1** & **Wheel 2** & **Wheel 3** \\ \hline \(\alpha_{i}\) & 180\({}^{\circ}\) & 60\({}^{\circ}\) & \(-60^{\circ}\) \\ \hline \end{tabular}  In addition, each element is described by the following parameters:  
* \(X_{C}\): Displacement of the platform on the abscissa axis of the system \(\{C\}\). * \(Y_{C}\): Displacement of the platform on the ordinate axis of the system \(\{C\}\). * \(x_{i}\): Displacement of the wheel on its abscissa axis of its respective system \(\{R_{i}\}\). * \(y_{i}\): Displacement of the wheel on its ordinate axis of its respective system \(\{R_{i}\}\). * \(r\): Radius of the rollers. * \(R\): Radius of the wheels. * \(\varphi_{i}\): Angular displacement of each roller, for translation. * \(\phi_{i}\): Angular displacement of each wheel, for translation. * \(\theta_{i}\): Angular displacement of each wheel, for rotation. * \(\theta_{i}\): Angular displacement of each wheel, total.