Unified aggregate component modelling of dynamic mechatronic systems using Bond Graphs

International Journal of Electrical Engineering Education, Oct 2004 by Dickinson, Stephen J

Abstract

Traditional dynamic modelling of multidisciplinary systems involves separate sets of analysis skills for each domain. Bond Graphs embrace commonality, providing a concise unified means of system representation and a systematic analysis method. This paper introduces Bond Graphs, benefits of acausal component models, and software for automated equation generation and simulation.

Keywords Bond Graphs; component modelling; dynamic systems; mechatronics

From the smallest nano-robot to the largest hydro-electric power plant, assessment of dynamic performance is vital in order to optimise and verify designs, and to reduce the probability of costly mistakes. A most challenging task in this area is the creation of dynamic system models: mathematical representations of systems which are simple enough to understand, compute and analyse, and yet at the same time are detailed enough to capture all essential dynamic aspects within particular operational envelopes. Bond Graph notation and methodology address three key areas which are fundamental to the understanding, analysis, simulation and hence teaching of physical dynamic mechatronic systems. First, that of a highly concise unified graphical notation that seamlessly crosses boundaries between engineering disciplines. Secondly, the ability to represent complete non-causal sub-models which greatly simplifies the construction of aggregate models; and thirdly, simple analytical methods which allow relationships and system equations to be systematically deduced at a glance. Bond Graphs also lend themselves well to automated transformation to and from other notations such as circuit schematics or block diagrams, and also the fully automated generation of system equations.

Ideally, the creation of a dynamic system model is best treated as a process consisting of two distinct phases. The first, and most challenging, phase involves making important decisions on the high level model structure: which physical elements to include, which to leave out, and which may be combined. A single-discipline example of this would be the creation of a circuit schematic where decisions such as whether or not to include parasitic elements, such as capacitance between PCB tracks, would be made depending on their significance within the envelope of operation, mainly frequency range. The development of a dynamic model is generally an iterative process where basic assumptions may be changed many times; the use of a rapid prototype-type environment where high level changes are instantly reflected in system equations can therefore yield great benefits.

The problem of incompatibility between sub-models due to conflicting assignment of causality is addressed well by Bond Graphs,' as will be seen later in this paper. When dealing with mixed discipline mechatronic systems the ability to connect sub-models from different disciplines is obviously essential, however the possibility of merging schematics from such multiple disciplines can be problematic, especially when attempting to automate the generation of composite system equations. An approach commonly adopted is to generate independent sets of equations for each sub-system, and then merge them to produce the final aggregate system model. This however can only be done if the causality of each sub-model is suitably assigned, and it therefore becomes necessary to manually fix causalities for each sub-system interconnect prior to the generation of sub-system equations. The problem is compounded where component libraries are employed, as alternative sets of equations must be maintained for each sub-model.

This paper introduces Bond Graph notation and methodology mainly through examples and by comparison with other well-known techniques. Important issues regarding the assignment of causality will be discussed, particularly within the context of developing component model libraries. Finally, a teaching lab session example will be described where students use Bond Graph software to simulate systems directly from the Bond Graph representation.

Bond Graph concepts and notation

Before proceeding to Bond Graphs, it is useful first to consider how a set of firstorder differential equations may be obtained for the system shown in Fig. 1 using 'traditional' techniques.

There are several ways of going about this task;2 ways which may on the surface seem dissimilar and vary widely in rigour. However, whether done formally or informally, each method must at some point assume a particular set of causal assignments which will dictate an equation structure such as that shown graphically by the block diagram3 of Fig. 2.

The issue of causality can be developed from Fig. 2. If we first consider the voltage source, by definition an ideal voltage source will determine (or cause) the voltage on whatever it is connected to, which in this case is the rest of the RC circuit, as shown by the v^sub 1^ signal line arrow pointing away from the voltage source in Fig. 2. Conversely, it follows that any current flowing in the voltage source, in this case i^sub 1^, is determined by the rest of the circuit. This is shown by the i^sub 1^ arrow pointing into the voltage source. In the case of a capacitor, the decision as to causal direction is not quite so cut and dry, as a capacitor can equally well be treated as a differentiator of voltage, or an integrator of current. In Bond Graph 'language' these two causal options are referred to as 'differential causality' and 'integral causality' respectively. It is however preferable to avoid any differentiating elements within the final model, as the resulting equations will generally be problematic. Integral causality has therefore been assumed for both capacitors in the example, and this is shown by the current lines pointing into the capacitor node and the voltage lines pointing out, and of course, the node indicating the appropriate integral function. There is no preference to resistor causality except that the assignment must be compatible with the adjoining system. In this example it has been assumed that the system is causing the voltage on both resistors, and conversely the resistors are therefore causing the current. Another important point to note regarding the block diagram is that voltage lines and current lines have been arranged in pairs. This is by no means accidental, as the instantaneous product of each pair yields the instantaneous power. Each pair therefore represents a single path for power to flow, a power flow which is positive when the product of the two is positive. An important aspect that is not however apparent from this diagram is the direction of positive power, an issue which, as will be seen, is addressed by Bond Graph notation.