# Design and Development of an Omnidirectional Three-Wheeled Industrial Mobile Robot Platform
## 4 Mathematical Modeling
### Kinematic Modeling
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}  
From the reference system of the omnidirectional wheel (Figure 9), we obtain the mathematical expressions (2)  
\[ \begin{cases}x_{i}=r\cdot\varphi_{i}\\ y_{i}=-R\cdot\phi_{ix}\end{cases}  \tag{2}\]  
The negative direction is for rotating clockwise. Equating the Equations (1) in (2), and isolating the expression (\(r\cdot\varphi_{i}\)), we obtain the following. First, it concentrates on the displacement of the platform on the abscissa axis of the system \(X_{C}\):  
\[ \begin{split}& X_{c}=(r\cdot\varphi_{i}+L)\cdot\cos(\alpha_{i})-y_{ i}\cdot\sin(\alpha_{i})\\ &(r\cdot\varphi_{i}+L)\cdot\cos(\alpha_{i})=X_{c}+y_{i}\cdot \sin(\alpha_{i})\\ & r\cdot\varphi_{i}=\frac{X_{c}}{\cos(\alpha_{i})}+\frac{\sin( \alpha_{i})}{\cos(\alpha_{i})}\cdot y_{i}-L\\ & r\cdot\varphi_{i}=\frac{1}{\cos(\alpha_{i})}\cdot X_{c}-\frac{ \sin(\alpha_{i})}{\cos(\alpha_{i})}\cdot R\cdot\phi_{ix}-L\end{split} \] (3a) Now the displacement of the platform on the ordinate axis of the system \[ \begin{split}& Y_{c}=(r\cdot\varphi_{i}+L)\cdot\sin(\alpha_{i})+y_{ i}\cdot\cos(\alpha_{i})\\ &(r\cdot\varphi_{i}+L)\cdot\sin(\alpha_{i})=Y_{c}-y_{i}\cdot \cos(\alpha_{i})\\ & r\cdot\varphi_{i}=\frac{Y_{c}}{\sin(\alpha_{i})}-\frac{y_{i} \cdot\cos(\alpha_{i})}{\sin(\alpha_{i})}-L\\ & r\cdot\varphi_{i}=\frac{1}{\sin(\alpha_{i})}\cdot Y_{c}+\frac{ \cos(\alpha_{i})}{\sin(\alpha_{i})}\cdot R\cdot\phi_{ix}-L\end{split}  \tag{3b}\]  
Equations (3a) and (3b) are equated to obtain  
\[\phi_{ix}=\frac{sin(\alpha_{i})}{R}X_{c}-\frac{cos(\alpha_{i})}{R}Y_{c} \tag{4}\]  
Replacing the values of \(\alpha_{i}\) in Table 2, the wheel translation expression (5) is expressed as follows:  
\[ \begin{cases}\phi_{1x}=\frac{1}{R}Y_{c}\\ \phi_{2x}=\frac{\sqrt{3}}{2R}X_{c}-\frac{1}{2R}Y_{c}\\ \phi_{3x}=-\frac{\sqrt{3}}{2R}X_{c}-\frac{1}{2R}Y_{c}\end{cases}  \tag{5}\]  
With respect to its rotation, all variables are known to be equal as illustrated in Figure 10, where \(\hat{S}\) is the tangential velocity, \(L\) is the turning radius, and \(\theta_{c}\) is the angular velocity of the robot. Figure 9: Omnidirectional wheel reference system. We know that the formula for the perimeter traveled for the rotational motion of the prototype is expressed by the Equation (6)  
\[y_{i} =L\cdot\theta_{c}\] \[-R\cdot\theta_{ix} =L\cdot\theta_{c} \tag{6}\]  
We obtain the expressions of rotation (7) in each wheel  
\[ \begin{cases}\theta_{1x}=-\frac{L}{R}\theta_{c}\\ \theta_{2x}=-\frac{L}{R}\theta_{c}\\ \theta_{3x}=-\frac{L}{R}\theta_{c}\end{cases}  \tag{7}\]  
By the superposition theorem, translation and rotation are combined (8):  
\[\theta_{ix}=\phi_{ix}+\phi_{ix} \tag{8}\]  
Finally, the angular equations of each wheel (9) are expressed in the matrix below:  
\[ \begin{pmatrix}\theta_{1x}\\ \theta_{2x}\\ \theta_{3x}\end{pmatrix} = \begin{pmatrix}0&\frac{1}{R}&-\frac{L}{R}\\ \frac{\sqrt{3}}{2R}&-\frac{1}{2R}&-\frac{L}{R}\\ -\frac{\sqrt{3}}{2R}&-\frac{1}{2R}&-\frac{L}{R}\\ \end{pmatrix} \cdot \begin{pmatrix}X_{c}\\ Y_{c}\\ \theta_{c}\end{pmatrix}  \tag{9}\]  
The matrix is derived to express the wheel variables in velocities (10)  
\[ \begin{pmatrix}\omega_{1x}\\ \omega_{2x}\\ \omega_{3x}\end{pmatrix} = \begin{pmatrix}0&\frac{1}{R}&-\frac{L}{R}\\ \frac{\sqrt{3}}{2R}&-\frac{1}{2R}&-\frac{L}{R}\\ -\frac{\sqrt{3}}{2R}&-\frac{1}{2R}&-\frac{L}{R}\\ \end{pmatrix} \cdot \begin{pmatrix}v_{Cx}\\ v_{Cy}\\ \omega_{C}\end{pmatrix} =J\cdot \begin{pmatrix}v_{Cx}\\ v_{Cy}\\ \omega_{C}\end{pmatrix}  \tag{10}\]  
where \(J\) is known as the Jacobian matrix, which is used to calculate the actuation required on the wheels for the robot to reach a certain velocity state. The velocities of the body (11) are expressed as a function of those of the wheel, by means of the inverse Jacobian \(J^{-1}\) (obtained by the inversion of the \(J\) matrix):  
\[