## 2 Omni-Directional Wheeled Mobile Robot Model
### Kinematics Model  
For the OMR, each wheel has two position parameters: the position relative to the center of the platform, and the respective attitude angle. The coordinates of the mecanum wheel are shown in Figure 3. In this figure, _xoy_ is the coordinate frame attached to the center of the platform, \(x^{\prime}o^{\prime}y^{\prime}\) is  
Figure 1: The structure of the mecanum wheel. Figure 2: The omni-directional mobile robot (OMR) structure diagram. a coordinate frame attached to the axis center of one mecanum wheel, \(i\) represents the order of the mecanum wheel, \(\alpha_{i}\) is the deflection angle of each roller, \(\beta_{i}\) is the angle between lines \(\overline{\omega o^{\prime}}\), and \(\overline{\alpha x}\), \(\left(l_{ix},l_{iy},\theta_{i}\right)\) are the wheel's location and pose, respectively, in the coordinate \(xoy\). \(l_{ix}\) and \(l_{iy}\) can be expressed as  
\[l_{ix}=l_{i}\cos\beta_{i},\ \ l_{iy}=l_{i}\sin\beta_{i} \tag{1}\]  
The structure of mecanum wheel is shown in Figure 4. In this figure, \(r\) denotes radius of the mecanum wheel, \(V_{ir}\) denotes velocity vector of the roller center, and \(\omega_{i}\) denotes rotation velocity of the motor. There are three speed vectors. \(\left[v_{x}\ v_{y}\ \omega\right]^{T}\) describes the center speed of the platform relative to the ground, \(\left[v_{ix}\ v_{iy}\ \omega_{i}\right]^{T}\) presents the center speed of the mecanums wheels relative to the coordinate \(xoy\), and \(\left[v^{\prime}{}_{ix}\ v^{\prime}{}_{iy}\ \omega^{\prime}{}_{i}\right]^{T}\) denotes the center speed of the mecanums relative to the coordinate \(x^{\prime}o^{\prime}y^{\prime}\). The kinematic models of the OMR are described as  
\[\left[ \begin{matrix}v^{\prime}{}_{ix}\\ v^{\prime}{}_{iy}\end{matrix} \right]=K_{i1}\left[ \begin{matrix}\omega_{i}\\ v_{ir}\end{matrix} \right],\ K_{i1}=\left[ \begin{matrix}0&\sin\alpha_{i}\\ r&\cos\alpha_{i}\end{matrix} \right] \tag{2}\]  
\[\left[ \begin{matrix}v_{ix}\\ v_{iy}\end{matrix} \right]=K_{i2}\left[ \begin{matrix}v^{\prime}{}_{ix}\\ v^{\prime}{}_{iy}\end{matrix} \right]=K_{i2}K_{i1}\left[ \begin{matrix}\omega_{i} \\ v_{ir}\end{matrix} \right]\ K_{i2}=\left[ \begin{matrix}\cos\theta_{i}&-\sin \theta_{i}\\ \sin\theta_{i}&\cos\theta_{i}\end{matrix} \right] \tag{3}\]  
\[\left[ \begin{matrix}v_{ix}\\ v_{iy}\end{matrix} \right]=K_{i3}\left[ \begin{matrix}v_{x}\\ v_{x}\\ \omega\end{matrix} \right],\ K_{i3}=\left[ \begin{matrix}1&0&-l_{iy}\\ 0&1&l_{ix}\end{matrix} \right] \tag{4}\]  
Based on Equations (3) and (4), the inverse kinematics equation of the system can be obtained as  
\[K_{i2}K_{i1}\left[ \begin{matrix}\omega_{i}\\ v_{ir}\end{matrix} \right]=K_{i3}\left[ \begin{matrix}v_{x}\\ v_{x}\\ \omega\end{matrix} \right],i=1,2,3, \tag{5}\]  
and \(\det\left(K_{i2}K_{i1}\right)\
eq 0\). Figure 3: The wheel location and pose in the platform center. Assuming that  
\[K_{i}=[K_{i1}]^{-1}[K_{i2}]^{-1}*K_{i3},\,i=1,2,3,\,\,\gamma_{i}=\theta_{i}-\alpha_ {i}\]  
then, the inverse kinematics equation of each mecanum wheel is derived as  
\[ \begin{bmatrix}\omega_{i}\\ \varphi_{ir}\end{bmatrix} =[K_{i1}]^{-1}[K_{i2}]^{-1}K_{i3} \begin{bmatrix}v_{x} \\ v_{y}\\ \omega\end{bmatrix} =K_{i} \begin{bmatrix}v_{x}\\ v_{y}\\ \omega\end{bmatrix}  \tag{6}\]  
where  
\[K_{i}=\frac{1}{-r\sin\alpha_{i}} \begin{bmatrix}\cos\left(\gamma_{i}\right)& \sin\left(\gamma_{i}\right)&l_{ix}\sin\left(\gamma_{i}\right)-l_{iy}\cos \left(\gamma_{i}\right)\\ r\cos\theta_{i}&-r\sin\theta_{i}&l_{iy}r\cos\theta_{i}-l_{ix}r\sin\theta_{i} \end{bmatrix} \]  
Assuming that  
\[\mathrm{E}= \begin{bmatrix}\frac{\cos(\gamma_{1})}{\sin\alpha_{1}}&\frac{\sin (\gamma_{1})}{\sin\alpha_{1}}&\frac{l_{1}\sin(\gamma_{1}-\beta_{1})}{\sin \alpha_{1}}\\ \frac{\cos(\gamma_{2})}{\sin\alpha_{2}}&\frac{\sin(\gamma_{2})}{\sin\alpha_{ 2}}&\frac{l_{2}\sin(\gamma_{2}-\beta_{2})}{\sin\alpha_{2}}\\ \frac{\cos(\gamma_{3})}{\sin\alpha_{3}}&\frac{\sin(\gamma_{3})}{\sin\alpha_{ 3}}&\frac{l_{3}\sin(\gamma_{3}-\beta_{3})}{\sin\alpha_{3}}\end{bmatrix}