## 4 Kinematic and Dynamic Models of the Robotic Platform
### Dynamic Model  
The dynamic model for a wheeled omnidirectional robot can be calculated through (34), where (a) the sum of kinetic energies of the robot, \(T\), is calculated in (35), and where (b) the applied torque in the wheel \(i\), \(\varphi\), is figured out in (36) [21]. \[R_{w}^{R}(\theta)[T]_{\xi}+E^{T}[T]_{\varphi}=E^{T}\tau_{\varphi} \tag{34}\]  
\[[T]_{\varphi}=\frac{d}{dt}\left(\frac{\partial T}{\partial\dot{\varphi}} \right)-\frac{\partial T}{\partial\dot{\varphi}} \tag{35}\]  
\[\tau_{\varphi}=\left[ \begin{array}{ccc}\tau_{\varphi 1}&\tau_{\varphi 2}&\tau_{ \varphi 3}&\tau_{\varphi 4}\end{array} \right]^{T} \tag{36}\]We denote (a) \(m_{R}\) as the robot total mass, (b) \(I_{Rz}\) as the robot total moment of inertial, and (c) \(I_{qy}\) a the wheels moment of inertial. For an omnidirectional robot, \(T\) can be expressed as shown in (37), where \(M\), \(I_{\varphi}\) and \(\phi\) have the values shown in (38)-(40). \[T=\dot{\xi}^{T}\Big{(}R_{w}^{R}(\theta)\Big{)}^{T}\cdot M\cdot R_{w}^{R}( \theta)\cdot\dot{\xi}+\varphi^{T}\cdot\dot{I}_{\varphi}\cdot\dot{\varphi} \tag{37}\]  
\[M=\frac{1}{2}diag\{m_{R},m_{R},I_{Rz}\} \tag{38}\]  
\[I_{\varphi}=\frac{1}{2}diag\{I_{qy},I_{qy},I_{qy},I_{qy}\} \tag{39}\]  
\[\phi=\big{[} \begin{array}{cccc}\varphi_{1}&\varphi_{2}&\varphi_{3}&\varphi_ {4}\end{array} \big{]}^{T} \tag{40}\]  
Developing (37), (41) is obtained. \[ \begin{split} T=\frac{m_{R}}{2}\Big{(}\dot{x}^{2}+\dot{y}^{2} \Big{)}+\frac{I_{Rz}}{2}\dot{\theta}^{2}+\\ \frac{I_{qy}}{2}\Big{(}\dot{\varphi}_{1}^{2}+\dot{\varphi}_{2}^{2}+ \dot{\varphi}_{3}^{2}+\dot{\varphi}_{4}^{2}\Big{)}\end{split}  \tag{41}\]  
\(\left[T\right]_{\xi}\) and \(\left[T\right]_{\varphi}\) terms are given using (35), whose results are reflected in (42) and (44). \[\left[T\right]_{\xi}=M_{R}\cdot\dot{\xi} \tag{42}\]  
\[M_{R}=diag\{m_{R},m_{R},I_{Rz}\},\;\xi=\left[ \begin{array}{c}\dot{x}\\ \dot{y}\\ \ddot{\theta}\end{array} \right] \tag{43}\]  
\[\left[T\right]_{\varphi}=M_{\varphi}\cdot\dot{\varphi} \tag{44}\]  
\[M_{\varphi}=diag\{I_{qy}\},\;\phi\left[ \begin{array}{c}\dot{\varphi}_{1}\\ \dot{\varphi}_{2}\\ \dot{\varphi}_{3}\\ \dot{\varphi}_{4}\end{array} \right] \tag{45}\]  
Upgrading (34) with the previous calculations, (46) is obtained. In order to transform the accelerations from the world frame to the robot frame, (47) and (48) are presented. The second one uses and derives the high part of (33). \[R_{w}^{R}(\theta)\cdot M_{R}\cdot\dot{\xi}+E^{T}\cdot M_{\varphi}\cdot\ddot{ \varphi}=E^{T}\tau_{\varphi} \tag{46}\]  
\[\ddot{\xi}=R_{R}^{w}(\theta)\cdot\dot{\eta}+\dot{R}_{R}^{w}(\theta)\cdot\eta \tag{47}\]  
\[\ddot{\varphi}=E\cdot\dot{\eta} \tag{48}\]  
The result is shown in (49), with (50) and (51), where the dynamic model of the omnidirectional robot is presented as the final model related to the operational space in (49)-(51). \[\bar{M}\cdot\dot{\eta}+\bar{C}\cdot\eta=E^{T}\cdot\tau_{\varphi} \tag{49}\]  
\[\bar{M}=R_{w}^{R}(\theta)\cdot M_{R}\cdot R_{R}^{w}(\theta)+E^{T}\cdot M_{ \varphi}\cdot E \tag{50}\]\[\bar{C}=R^{R}_{w}(\theta)\cdot M_{R}\cdot\dot{R}^{w}_{R}(\theta) \tag{51}\]  
Since the compact form described in [22] (specified in (52)) works in the robot space (unlike \(\eta\) that is described in the operational space), it is needed to apply the inverse kinematic transformation calculated in (33) and specified in (53). \[M(q)\cdot\ddot{q}+C(q,\dot{q})\cdot\dot{q}+g(q)=\tau \tag{52}\]  
\[\dot{q}=\left[ \begin{array}{c}\dot{\varphi}_{1}\\ \dot{\varphi}_{2}\\ \dot{\varphi}_{3}\\ \dot{\varphi}_{4}\end{array} \right]=\left[ \begin{array}{ccc}\frac{1}{r}& \frac{1}{r}&\frac{(l_{1}+l_{2})}{r}\\ \frac{1}{r}&-\frac{1}{r}&-\frac{(l_{1}+l_{2})}{r}\\ \frac{1}{r}&\frac{1}{r}&-\frac{(l_{1}+l_{2})}{r}\\ \frac{1}{r}&-\frac{1}{r}&\frac{(l_{1}+l_{2})}{r}\end{array} \right]\cdot\left[ \begin{array}{c}\dot{x}^{R}\\ \dot{y}^{R}\\ \dot{\theta}^{R}\end{array} \right]=K\cdot\eta \tag{53}\]  
Since \(K\) is not quadratic, the transformation should use the pseudo-inverse matrix (54), as shown in (55).