## 4 Kinematic and Dynamic Models of the Robotic Platform
### Kinematic Model  
Firstly, to describe the robot posture in the ground plane, the reference systems shown in Figure 7 have been created, where we find reference systems of: (a) the world, \(\sum_{W}\), (b) the robot, \(\sum_{R}\), (c) the wheel \(i\) with the same orientation that the robot one, \(\sum_{Ri}\), and (d) the wheel \(i\) with the orientation of the mecanum wheel rollers, \(\sum_{ri}\). In that picture, we describe: (a) the distance between \(\sum_{R}\) and \(\sum_{Ri}\), \(L_{i}\), (b) the angle between the axis \(R_{x}\) and the contact point of the wheel \(i\) rollers with the floor, \(\alpha_{i}\), (c) the angle between the axis \(r_{ix}\) and the axis \(R_{iy}\), \(\beta_{i}\), and (d) the angle between the axis \(R_{iy}\) and the axis \(r_{iy}\), \(\gamma_{i}\). All of them are well known: (a) \(L_{i}\), (b) \(\alpha_{i}\) and (c) \(\beta_{i}\) are determined by the dimensions of the robot design, and (d) \(\gamma_{i}\) is determined by the wheel design (usually 45\({}^{\circ}\)). Moreover, we describe (e) \(\phi_{i}\) as a sum of \(\alpha_{i}\), \(\beta_{i}\) and \(\gamma_{i}\), and (f) \(\delta_{i}\) as a sum of \(\beta_{i}\) and \(\gamma_{i}\). Figure 6: Robot configuration while crossing the doors. Like the vast majority of wheeled robots, the position of the robot is described by 3 d.o.f., this is, position and orientation \(x\), \(y\) and \(\theta\). Thus, the vector \(\xi\) which defines the posture of the robot is presented in (21). Furthermore, we define (a) \(\hat{\xi}\) as the velocity vector of the robot in \(\sum_{W}\), and (b) \(\eta\) as the velocity vector of the robot in \(\sum_{R}\), calculated from \(\hat{\xi}\) in (22), where \(R_{w}^{R}(\theta)\) is the rotation matrix between \(\sum_{R}\) and \(\sum_{W}\). \[\xi=\left[ \begin{array}{c}x\\ y\\ \theta\end{array} \right]\in\Re^{3} \tag{21}\]  
\[\eta=R_{w}^{R}(\theta)\cdot\hat{\xi}=\left[ \begin{array}{ccc}c_{\theta}&s_{ \theta}&0\\ -s_{\theta}&c_{\theta}&0\\ 0&0&1\end{array} \right]\cdot\left[ \begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{\theta}\end{array} \right] \tag{22}\]  
The required velocity in each wheel \(i\), \(\mathcal{O}_{Ri}\), to move the robot at a given velocity \(\eta\) is described in (23). In addition, in \(\eta\) the velocities are referred to \(\sum_{R}\), so in (24) we displace them to \(\sum_{ri}\) through the rotation matrix \(R_{R}^{R_{i}}(\phi_{i})\), which does so. \[\left[ \begin{array}{c}\dot{\mathcal{O}}_{R_{1}}\\ \dot{\mathcal{O}}_{R_{2}}\\ 0\end{array} \right]^{R}=\left[ \begin{array}{c}\dot{x}^{R}-L_{i}\cdot s_{x_{ i}}\dot{\theta}\\ \dot{y}^{R}-L_{i}c_{x_{i}}\dot{\theta}\\ 0\end{array} \right] \tag{23}\]  
\[\left[ \begin{array}{c}\dot{\mathcal{O}}_{R_{1}}\\ \dot{\mathcal{O}}_{R_{2}}\\ 0\end{array} \right]^{R_{i}}=R_{R}^{R_{i}}(\phi_{i})\cdot\left[ \begin{array}{c }\dot{\mathcal{O}}_{R_{1}}\\ \dot{\mathcal{O}}_{R_{2}}\\ 0\end{array} \right]^{R}= \tag{24}\]  
Figure 7: Overall of the robot reference systems (\(\sum_{W}\), \(\sum_{R}\), \(\sum_{Ri}\) and \(\sum_{ri}\)). According to [21], we define: (a) the wheel speed, \(\dot{\varphi}_{R_{i}}\), (b) the wheel radius, \(r_{R_{i}}\), (c) the roller speed, \(\dot{\varphi}_{r_{i}}\), and (d) the roller radius \(r_{r_{i}}\). In our case, the robot values are shown in Table 1.  
For omnidirectional robots with mecanum wheels, the delivered speed by the wheel is \(r_{R_{i}}\cdot\varphi_{R_{i}}\) in the \(R_{ix}\) direction, whereas the roller lineal velocity is \(r_{r_{1}}\cdot\varphi_{r_{i}}\) in the \(r_{ix}\) direction. Thus, making use of (24), we transform the delivered speed by the wheel to \(\sum_{r_{i}}\) in (25). \[\left[ \begin{array}{cc}r_{R_{i}}\cdot\dot{\varphi}_{R_{i}}\cdot c_{\gamma_{i} }\\ r_{r_{i}}\cdot\dot{\varphi}_{r_{i}}+r_{R_{i}}\cdot\dot{\varphi}_{R_{i}}\cdot \varsigma_{\gamma_{i}}\\ 0\end{array} \right]= \tag{25}\]  
\[\left[ \begin{array}{ccc}s_{\dot{\varphi}_{i}}&-c_{\dot{\varphi}_{i}}&L_{i} \cdot c_{\dot{\xi}_{i}}\\ c_{\dot{\varphi}_{i}}&s_{\dot{\varphi}_{i}}&L_{i}\cdot c_{\dot{\xi}_{i}}\\ 0&0&0\end{array}