## 3 Development of a Dynamic Model of the Robotino with Bond Graphs
### Coordinate Systems  
We start by analyzing the Robotino system to define the basic coordinate systems that will be used to model it. Figure 18 shows a scheme of Robotino's chassis--the base (see also Figure 9). There are several rigidly attached coordinate frames. Three of them are local body-fixed frames \(C_{i}x_{i}y_{i}z_{i}\) (\(i=1,\,2,\,3\)), whose axes \(x_{i}\) are fixed and originate at the corresponding wheel centers \(C_{i}\) (\(i=1,\,2,\,3\)). They are drawn through the base's center \(C\); the \(z_{i}\)-axes are orthogonal to the plane of motion and are directed upwards. There is another _Cxyz_ coordinate frame whose \(x\)-axis is directed in the direction of the camera and thus shows the trajectory of the Robotino. Hence, due to the parallelism of the \(z\)-axes, the angular velocity expressed in the world and the local coordinate frames are identical:  
\[\omega^{w}=\omega=\dot{\varphi}, \tag{1}\]  
where the angle \(\varphi\) is the rotation angle of the chassis. In this study, the vectors represented in the global coordinate frame are denoted by \(w\) in the exponent. For vector components, the capital letters \(X\) and \(Y\) in the indices indicate that the components refer to the corresponding global coordinate frame. In the case of a radius vector of a point, projections onto the axis of the global coordinate frame are denoted by the capital letters \(X\) or \(Y\), with the corresponding point indicated in the index. The translational part of the base motion with respect to the world coordinates is described by the components of the center \(C\) velocity:  
\[\mathbf{v}_{C}^{w}= \begin{pmatrix}v_{CX}\\ v_{CY}\\ 0\end{pmatrix} . \tag{2}\]  
The orientation of the local coordinate frames with respect to the world one is defined by rotation matrices:  
\[\mathbf{R}_{i}= \begin{pmatrix}\cos(\varphi_{i})&-\sin(\varphi_{i})&0\\ \sin(\varphi_{i})&\cos(\varphi_{i})&0\\ 0&0&1\end{pmatrix} ,\;(i=1,2,3), \tag{3}\]  
Figure 18: [PERSON]’s base with the coordinate frames _Cxyz_ and \(C_{i}x_{i}y_{i}z_{i}\) (\(i=1,\,2,\,3\)). where \(\varphi_{i}\) is the rotation angle determined by (see Figure 18):  
\[\varphi_{1}=\varphi+\pi/3,\ \varphi_{2}=\varphi+\pi,\ \varphi_{3}=\varphi-\pi/3. \tag{4}\]  
Thus, if the velocity vector \(\mathbf{v}_{i}\) is defined in the local system, its components in the world system are defined by the transform:  
\[\mathbf{v}_{i}^{w}=\mathbf{R}_{i}\mathbf{v}_{i},(i=1,2,3). \tag{5}\]  
Similarly, the components of a vector, e.g., force \(\mathbf{F}_{i}^{w}\) defined in the world frame, can be expressed in the local coordinate frame by the inverse transformation matrix. Because the matrices (3) are orthogonal, their inverse matrices are simply their transpose, \(\mathbf{R}_{i}^{-1}=\mathbf{R}_{i}^{T}\). Therefore, it follows that:  
\[\mathbf{F}_{i}=\mathbf{R}_{i}^{T}\mathbf{F}_{i}^{w}. \tag{6}\]  
The power defined by the scalar product of these quantities in the world system can be evaluated using the above expressions:  
\[(\mathbf{F}_{i}^{w})^{T}\mathbf{v}_{i}^{w}=(\mathbf{R}_{i}\mathbf{F}_{i})^{T} \mathbf{R}_{i}\mathbf{v}_{i}=\mathbf{F}_{i}^{T}\mathbf{R}_{i}\mathbf{v}_{i}= \mathbf{F}_{i}^{T}\mathbf{v}_{i}, \tag{7}\]  
Thus, it is equal to the power evaluated by the local representation of these variables. We may describe this transform using the bond graph component **Transform** in Figure 19a. By opening the component, a document appears (Figure 19b), which has the same title as the component, in this case the **Transform**. This serves to define the internal structure of the component model. The bond graph model describing the transform in Figure 19b describes transforms corresponding to Equations (5) and (6). Thus, going from the lower to the upper ports represents the direct transform of velocities from the local to the world system in (5). The **0**-junctions represent the summation of the transformed quantities.