## 3 Development of a Dynamic Model of the Robotino with Bond Graphs
### Dynamics of the Chassis  
The bond graph model of the chassis is shown in Figure 25. The model follows the structure of the real device, which contains three drive units represented by the components **Drive 1** to **Drive 3**. The drivers develop angular velocities \(\omega_{i}\) and torques \(\tau_{i}\), which are applied to the wheels (Figure 24). The component **Dynamics** in Figure 25 models the kinematics and dynamics of the chassis motion. We start from the relation between the velocity of the wheel centers in the local coordinates and the world velocity of the chassis center:  
\[\mathbf{v}_{\mathrm{C}}^{\mathrm{m}}=\left(\dot{X}_{\mathrm{C}}\ \dot{Y}_{ \mathrm{C}}\ 0\right)^{T}, \tag{20}\]  
where \(X_{\mathrm{C}}\) and \(Y_{\mathrm{C}}\) are its coordinates in the world frame. This relation reads as:  
\[\mathbf{v}_{i}+\mathbf{v}_{\mathrm{C}/Ci}=\mathbf{v}_{\mathrm{C}}^{i}= \mathbf{R}_{i}^{T}\mathbf{v}_{\mathrm{C}^{\prime}}^{\mathrm{m}},\ (i=1,2,3), \tag{21}\]  
or  
\[ \begin{pmatrix}v_{ix}\\ v_{iy}\\ 0\end{pmatrix} + \begin{pmatrix}0\\ 0\\ \dot{\varphi}\end{pmatrix} \times \begin{pmatrix}-R\\ 0\\ 0\end{pmatrix} = \begin{pmatrix}\cos(\varphi_{i})&\sin(\varphi_{i})&0\\ -\sin(\varphi_{i})&\cos(\varphi_{i})&0\\ 0&0&1\end{pmatrix}\begin{pmatrix}\dot{X}_{\mathrm{C}}\\ Y_{\mathrm{C}}\\ 0\end{pmatrix} ,\ (i=1,2,3), \tag{22}\]  
Figure 24: Basic bond graph model of the wheels. Figure 25: Structure of the **Chassis** model. where \(R\) is the radius of the circle, which contains the wheel centers (Figure 18). Expanding the last equation yields:  
\[ \begin{split} v_{ix}&=\dot{X}_{C}\cos(\varphi_{i})+ \dot{Y}_{C}\sin(\varphi_{i})\quad\quad(a)\\ v_{iy}&-R\dot{\varphi}=-\dot{X}_{C}\sin(\varphi_{i} )+\dot{Y}_{C}\cos(\varphi_{i})\ \ (b),\ (i=1,2,3).\end{split}  \tag{23}\]  
The bond graph model of the base dynamics is shown in Figure 26a. It plays a central role in Equation (23). The three transformers **TF** describe rotation-to-linear transformation (similar to Figure 21). They also generate the moments of the y-axis forces of the wheels acting on the chassis about their \(Z\)-axis. The **Transform** describes the right side of (23b) (Figure 26b). It also transforms the y-axis forces of the wheels acting on the chassis to the world frame and applies them to the center of mass of the assembly (see **Translation** in Figure 26a). The **Side rolling** in Figure 26a describes the motion of the wheels due to the rotation of the rollers. The velocities of this motion are described by the bond graph representation in Figure 27a. It generates the \(x\) component of the wheel center, as defined by (23a). The roles of the transforms serve not only to transform the velocities of the chassis' centers from world to local frames, but also to transform to the world frame the x-axis components of the forces acting as the rollers come into contact with the ground (Figure 20), which are transferred by the wheels to the chasses. The sum of these forces is transferred through the bottom port into **Translation**'s upper port (Figure 27b). The component **Translation** describes the translational dynamics of the **Chassis**, including the wheels (Figure 27b). The inertial element **I** describes the translational dynamics of the chassis. We now return to the model of **Chassis** in Figure 25. There are three **Driver** components, one for every wheel, which model the driver units of the Robotino. Figure 26: Dynamics of the **Chassis**: (a) structure of the **Dynamics** model; (b) model of **Transform**. Figure 28 shows the structure of the drivers. The drivers consist of DC motors [86] and reducers (belt drives and planetary reducers). The DC motors are modeled using gyrators (Damic and Montgomery [76]) and lossless reducers by simple transformers. In this model, the control of the drivers is provided by common PID controllers.