## 3 Development of a Dynamic Model of the Robotino with Bond Graphs
### Dynamics of Wheels  
Figure 20 shows a side view of a wheel. Every wheel is driven by a DC motor over a geared belt and a planetary reducer. We describe the motion of the wheel with respect to the corresponding local coordinate frame \(C_{i}x_{ij}z_{i}\). The angular velocity of the rotation of the wheel is \(\omega_{i}\), and it is directed along the \(x_{i}\)-axis (this is conventional--the real direction depends on the design of drivers' chains and will be discussed later in relation to the wheels' drivers). In addition to rolling around the \(x_{i}\)-axis, the wheels may execute a side motion, i.e., in the direction of the \(x_{i}\)-axis, due to the rollers (Figure 20, see also Section 2.4). The roller that is in contact with the ground may freely rotate in its holder about its longitudinal axis, allowing for the side motion of the wheel. This motion is an essential characteristic of Robotino, because it allows it to move in any direction, irrespective of the current orientations of the wheels' axes, i.e., they are omnidirectional. As already mentioned in the **Introduction**, omnidirectional wheels are of significant interest and are the subject of many studies [69; 70; 71; 72]. Figure 20: Side view of a wheel. Assuming that the rolling occurs without slipping, i.e., the velocity of the temporary contact point \(P_{i}\) is zero, the velocity of the wheel's center is determined by:  
\[\mathbf{v}_{i}=\mathbf{v}_{Pi}+\mathbf{v}_{Ci/Pi}= \begin{pmatrix}0\\ 0\\ 0\end{pmatrix} + \begin{pmatrix}\omega_{i}\\ 0\\ 0\end{pmatrix} \times \begin{pmatrix}0\\ 0\\ R_{w}\end{pmatrix} = \begin{pmatrix}0\\ -R_{w}\cdot\omega_{i}\\ 0\end{pmatrix} , \tag{8}\]  
where \(R_{w}\) is the radius of the wheel. Figure 20 shows the components of the reaction forces at \(P_{i}\). Component \(F_{iy}\) is the reaction force opposing the slipping of the wheel on the ground during its rotation around the \(x_{i}\)-axis. Similarly, \(F_{ix}\) is the reaction of the ground to the slipping of the current roller during its rotation due to the sideways movement of the assembly along the \(x_{i}\)-axis. The component \(F_{iz}\) is the reaction that occurs due to the weight of the complete wheel assembly. We reduce the \(F_{iy}\) component to the center \(C_{i}\) of the wheel (Figure 20). Hence, the respective moment with respect to point \(C_{i}\) is determined by:  
\[\mathbf{M}_{i}= \begin{pmatrix}0\\ 0\\ R_{w}\end{pmatrix} \times \begin{pmatrix}0\\ F_{y}\\ 0\end{pmatrix} = \begin{pmatrix}-R_{w}\cdot F_{y}\\ 0\\ 0\end{pmatrix} . \tag{9}\]  
Hence, the rotation-to-linear-motion transform determined by (8) and (9) can be simply modeled by the **RotLin** component shown in Figure 21a, consisting of a single transformer **TF** in Figure 21b. It uses the parameter \(-R_{w}\) as the transformer arm. We also introduce the wheel body-fixed frame \(x_{\text{ib}}y_{\text{ib}}z_{\text{ib}}\) (\(i=1,\,2,\,3\)), which overlaps with the \(x_{i}y_{i}z_{i}\) (\(i=1,\,2,\,3\)) frame when the current roller touches the ground at its greatest diameter (Figure 20). Thus, the current body-fixed wheel frame is defined with respect to the current roller with which the wheel touches the ground. As the roller changes, the body-fixed frame changes as well. We study the rotation of the wheel with respect to the world frame coordinates \(XYZ\). However, we may also study it in the current wheel body-fixed frame. In this frame, the temporary contact point \(P_{i}\) apparently moves in the opposite direction. When the wheel rotates around the \(x_{i}\)-axis by \(\theta_{i}\) radian, the \(z_{ib}\)-axis rotates by an angle of \(\theta_{ib}\). We calculate a helper angle according to the expression:  
\[\theta_{temp}=\text{mod}(\theta_{i},\pi/3). \tag{10}\]  
Using a mod operator definition, this angle lies in the range \([0,\,\pi/3]\). However, as \(\theta_{0}=\pi/3\), when this angle is greater than \(\theta_{0}/2\), the contact point switches to the next  
Figure 21: Rotation-to-linear-motion transform: (**a**) word model; (**b**) its internal model.