## 3 Development of a Dynamic Model of the Robotino with Bond Graphs
### Introduction to Bond Graphs
#### 3.1.2 Elementary Components  
The elementary components are models of the elementary processes, i.e., the physical processes in which only one component participates. The bond graph theory defines nine such components, which are listed in Table 1.  
The inertial component is characterized by generalized momentum p. It corresponds to processes governed by momentum, the moment of momentum laws in mechanics, and the Faraday law in electricity. Similar to it is the capacitive element, which is the dual of the inertial element. If we substitute effort for flow, flow for effort, momentum p for charge q, and inertia I for capacitance C, we obtain the capacitive element. Despite this, both elements are retained--perhaps because it is in accordance with the theory of mechanics and electromagnetism. On the other hand, in thermodynamics, there is no counterpart to inertial processes, only to capacitive ones. In these processes, there is an accumulation of  
Figure 14: A component and its document. kinetic energy, i.e., the energy of motion, in the inertial elements, and of the static energy in the capacitive elements. The dissipation of energy is described by the resistive element. The power input to this element is defined according to its constitutive relation in Table 1, determined by:  
\[P=e\cdot f=R\cdot f^{2},\]  
Thus, if the resistance of the element is positive, there is a constant dissipation of the power flowing into the resistive element. The transformer is an element in which there is a linear transformation between port variables. Using the constitutive relations from the table, we may evaluate the power input of the transformer as:  
\[P_{1}=e_{1}\cdot f_{1}=e_{1}\cdot k\cdot f_{2}=e_{2}\cdot f_{2}=P_{2},\]  
Thus, the transformer conserves the transfer of power across the element. The same principle holds for the gyrators. We mostly use the transformer to evaluate the transformation of the component vector over different coordinate systems. We use the gyrators to model the DC motors that drive the Robotino. The sources of efforts and flows imply certain relationships about the effort or flow at their ports, which do not depend on the other variable, i.e., the port effort does not depend on the port flow, and vice versa. Thus, these are the ideal sources. In mechanics, they are often used to imply zero velocity or force components in the specific parts of the system. Finally, we have two junction elements. The effort junction implies that the efforts of the element ports are added, and the sum is equal to zero. The convention is that, if port power is directed into the junction, the corresponding effort gains a plus sign; otherwise, it gains a minus sign (subtracted). On the other hand, the flows of all the junction ports are mutually equal, and we may consider the effort junction to be a flow node. The other junction, the flow junction, behaves in the opposite way to the effort junction; it is its dual. In this element, the junction ports flows are added, and their sum is equal to zero. The sign convention is the same as in the former junction. The junction ports' efforts