## 4 Kinematic and Dynamic Models of the Robotic Platform
### Kinematic Model
\right]\cdot\eta\]  
For each wheel, a motion constraint, shown in (26), is obtained from the first row of the equation system of (25) [21]. \[\left[ \begin{array}{cc}-s_{\dot{\varphi}_{i}}&c_{\dot{\varphi}_{i}}&L_{i} \cdot c_{\dot{\xi}_{i}}\\ \end{array} \right]\cdot R_{w}^{\mathbb{R}}(\theta)\cdot\dot{\phi}+r_{R_{i}} \cdot\dot{\varphi}_{R_{i}}\cdot c_{\gamma_{i}}=0 \tag{26}\]  
Thus, the motion constraints equation of the entire robot is shown in (27), where (a) \(J_{1}\) is composed by the motion constraint rows of each wheel, as described in (26), and where (b) \(J_{2}\) is defined by the wheel design. The results are the Equations (28) and (29), respectively. \[J_{1}\cdot R_{w}^{R}(\theta)\cdot\dot{\phi}+J_{2}\cdot\dot{\varphi}_{i}=0 \tag{27}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Wheel \(i\)** & \(L_{i}\) & \(\alpha_{i}\) & \(\beta_{i}\) & \(\gamma_{i}\) & \(r_{R_{i}}\) \\ \hline
1 & L & \(360-\alpha\) & \(\frac{\pi}{2}-\alpha_{1}\) & \(\frac{\pi}{4}\) & r \\
2 & L & \(\alpha\) & \(\frac{\pi}{2}-\alpha_{2}\) & \(-\frac{\pi}{4}\) & r \\
3 & L & \(180-\alpha\) & \(\frac{\pi}{2}-\alpha_{3}\) & \(\frac{\pi}{4}\) & r \\
4 & L & \(180+\alpha\) & \(\frac{\pi}{2}-\alpha_{4}\) & \(-\frac{\pi}{4}\) & r \\ \hline \hline  & L & & & 274.06 mm & \\ \hline  & \(\alpha\) & & & 29.5\({}^{\circ}\) & \\ \hline  & r & & & 76.2 mm & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Kinematic parameters of our robot. Figure 8 shows each variable in the robot frame. Figure 8: Wheel parameters relative to the robot. \[J_{1}=\left[ \begin{array}{ccc}-s_{\phi_{1}}&c_{\phi_{1}}&L_{1}\cdot c_{\xi_{1}} \\ -s_{\phi_{2}}&c_{\phi_{2}}&L_{2}\cdot c_{\xi_{2}}\\ -s_{\phi_{3}}&c_{\phi_{3}}&L_{3}\cdot c_{\xi_{3}}\\ -s_{\phi_{4}}&c_{\phi_{4}}&L_{4}\cdot c_{\xi_{4}}\end{array} \right] \tag{28}\]  
\[J_{2}=\left[ \begin{array}{ccc}r_{R_{1}}.c_{\gamma_{1}}&0&0&0\\ 0&r_{R_{2}}.c_{\gamma_{2}}&0&0\\ 0&0&r_{R_{3}}.c_{\gamma_{3}}&0\\ 0&0&0&r_{R_{4}}.c_{\gamma_{4}}\end{array} \right] \tag{29}\]  
The kinematic configuration model for that robot is given by (30) [21], where \(S(q)\) is decomposed in the high and low part, relating to the kinematic posture model and the wheels velocities, respectively. \[\dot{q}=\left[ \begin{array}{c}\dot{\xi}\\ \dot{\varphi}_{1}\\ \dot{\varphi}_{2}\\ \dot{\varphi}_{3}\\ \dot{\varphi}_{4}\end{array} \right]=\left[ \begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{\varphi}\\ \dot{\varphi}_{1}\\ \dot{\varphi}_{2}\\ \dot{\varphi}_{3}\\ \dot{\varphi}_{4}\end{array} \right]=S(q)\cdot\eta=\left[ \begin{array}{c}R_{ w}^{R}(\theta)\\ E\end{array} \right]\cdot\left[ \begin{array}{c}\dot{x}^{R}\\ \dot{y}^{R}\\ \dot{\theta}^{R}\end{array} \right] \tag{30}\]  
The high part of the previous equation is determined by the kinematic posture model for an omnidirectional robot, which is given by (31). The low part of the equation is given by the isolation of \(\dot{\varphi}_{i}\) from (27), in such a way that for all the wheels we obtained (32), which is related to \(E\) from (30). \[\dot{\xi}=R_{R}^{w}(\theta)\cdot\eta \tag{31}\]  
\[\dot{\varphi}_{i}=-J_{2}^{-1}\cdot J_{1}\cdot R_{w}^{R}(\theta).\dot{\xi}=-J_ {2}^{-1}\cdot J_{1}\cdot\eta \tag{32}\]  
Then, substituting the values of Table 1, we obtain our kinematic model, which is presented in (33). \[\dot{q}=\left[ \begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{\theta}\\ \dot{\varphi}_{1}\\ \dot{\varphi}_{2}\\ \dot{\varphi}_{3}\\ \dot{\varphi}_{4}\end{array} \right]=\left[ \begin{array}{ccc}c_{\theta}&-s_{ \theta}&0\\ s_{\theta}&c_{\theta}&0\\ 0&0&1\\ \frac{1}{r}&\frac{1}{r}&\frac{(l_{1}+l_{2})}{(l_{1}+l_{2})}\\ \frac{1}{r}&-\frac{1}{r}&-\frac{(l_{1}+l_{2})}{r}\\ \frac{1}{r}&\frac{1}{r}&-\frac{(l_{1}+l_{2})}{r}\\ \frac{1}{r}&-\frac{1}{r}&\frac{(l_{1}+l_{2})}{r}\end{array} \right]\cdot\left[ \begin{array}{c}\dot{x}^{\text{R}}\\ \dot{y}^{\text{R}}\\ \dot{\theta}^{\text{R}}\end{array} \right] \tag{33}\]