Object-oriented Modeling of Mechatronics Systems in Modelica Using Wrapped Bond Graphs

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Object-oriented Modeling of Mechatronics Systems
in ModelicaUsing Wrapped Bond Graphs
François E. Cellier and Dirk Zimmer ETH Zürich
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Outline

• Motivation
• Graphical Modeling
• Modeling Software Requirements
• Bond Graphs
• Electronic Circuits
• Multi-bond Graphs
• 3D Mechanics

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Motivation

• In todays engineering practice, mathematical
  models of physical processes are frequently
  produced that are composed of thousands of
  equations. These models are difficult to create
  and even more difficult to maintain.
• A typical example of systems leading to highly
  complex models are mechanical multi-body systems.
• Tools are needed that enable us to keep the
  complexity of individual component models within
  limits.
• Model wrapping presents itself as a tool suitable
  for such purpose.

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Graphical Modeling

• Graphical modeling is generally more suitable for
  the creation of models of complex systems than
  equation-based modeling.
• This is true because graphical models are
  naturally two-dimensional. Errors in
  hierarchically structured and topologically
  interconnected graphical models are usually
  discovered more easily and rapidly than errors in
  corresponding equation-based models.
• Evidently, the graphical models must be replaced
  by equation-based models at the lowest-possible
  level in the modeling hierarchy.

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Example 3D Mechanics
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Example 3D Mechanics II
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Example 3D Mechanics III
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Example 3D Mechanics IV
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Software Requirements

• The semantic distance from the lowest graphical
  layer to the equation layer should be kept as
  small as possible. In this way, as much as
  possible can be modeled graphically.
• Mechanical multi-body component models are too
  complex to be used conveniently as building
  blocks of the lower-most graphical modeling
  layer.
• To this end, multi-bond graphs are considerably
  more suitable, as we shall demonstrate.

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Bond Graphs Example
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Bond Graphs Example II
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Bond Graphs Example II
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Causal Bond Graphs
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Advantages of Bond Graphs

• Bond graphs represent a generally usable approach
  to modeling physical systems of arbitrary types.
  They offer a suitable balance between general
  usability and domain orientation.
• The concepts of energy and power flows define a
  suitable semantic framework for bond graphs of
  all physical systems.
• The semantic meaning of each bond graph component
  model is sufficiently simple to afford easy
  maintainability of the equation layer below.

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The BondLib Library of Dymola

• Bond graphs can be drawn graphically on the
  computer.
• The resulting model can be simulated immediately.
• The library affords application specific
  solutions, such as a sub-library for electrical
  circuits.

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Multiple Modeling Interfaces in BondLib
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Electronic Circuit Modeling in BondLib
The Bipolar Junction Transistor
Icon window
Diagram window
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Electronic Circuit Modeling in BondLib II
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Electronic Circuit Modeling in BondLib III
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Electronic Circuit Modeling in BondLib IV
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Electronic Circuit Modeling in BondLib V
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Electronic Circuit Modeling in BondLib VI
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Electronic Circuit Modeling in BondLib VII
Inverter Circuit
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Electronic Circuit Modeling in BondLib VIII
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Electronic Circuit Modeling in BondLib IX
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Bond Graphs For Mechanical Systems

• Mechanical systems are three-dimensional. Every
  mechanical body that can move freely has six
  degrees of freedom. For this reason, the
  d?Alembert principle must be formulated six times
  for each mechanical body.
• Mechanical bond graphs have a tendency of quickly
  becoming very large.
• Holonomic constraints cannot be formulated
  directly in the bond graph.

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Example A Planar Pendulum
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Example A Planar Pendulum II
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Mechanical Bond Graphs
It has been possible to describe the motion of
the planar pendulum by a bond graph enhanced by
activated bonds for the description of the
holonomic constraint. Unfortunately, the bond
graph doesnt tell us much that we didnt know
already.

• We shouldnt have to derive the equations first
  in order to be able to derive the bond graph from
  them.
• The resulting bond graph didnt preserve the
  topological properties of the system in any
  recognizable form.

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Multi-bond Graphs

• Multi-bond graphs are a vectorial extension of
  the regular bond graphs.
• A multi-bond contains a freely selectable number
  of regular bonds of identical or similar domains.
• All bond graph component models are adjusted in a
  suitable fashion.

Composition of a multi-bond for planar mechanics
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The MultiBondLib Library

• A Dymola library for modeling systems by means of
  multi-bond graphs has been developed.
• The library has been designed with an interface
  that looks as much as possible like that of the
  original BondLib library.
• Just like the original library, also the new
  multi-bond graph library contains sub-libraries
  supporting modelers in modeling systems from
  particular application domains, especially from
  mechanics.

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Example A Planar Pendulum III

• Multi-bond graph of a planar pendulum

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Multi-bond Graphs 2nd Example

• Model of a crane crab

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Multi-bond Graphs 2nd Example II
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Multi-bond Graphs 2nd Example II
Mass 1
Mass 2
Wall
Prismatic Joint
Revolute Joint
Rod
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Multi-bond Graphs 2nd Example III
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Model Wrapping

• Model wrapping offers the best properties of two
  worlds

• On the upper mechanical layer, an intuitive and
  simple to use interface is being offered.
• The lower multi-bond graph layer offers a
  graphical interpretation that makes it possible
  to decompose even complex mechanical component
  models graphically into much simpler subcomponent
  models.

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3D Mechanics Example
Multi-bond graph model of an uncontrolled bicycle
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3D Mechanics Example II
Multi-body diagram of an uncontrolled bicycle
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Multi-bond Graphs for 3D Mechanics

• Multi-bond graphs offer too low an interface to
  be used for modeling multi-body systems of 3D
  mechanics directly.
• The basic multi-bond graph component models are
  not at the right modeling level to carry
  meaningful multi-body system semantics.
• Consequently, multi-bond graphs of even fairly
  simple multi-body systems become quickly
  unreadable and therefore also poorly
  maintainable.

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Multi-body Diagrams for 3D Mechanics

• Multi-body diagrams are easily interpretable, as
  their component models carry semantics that can
  be mapped one-to-one to those of the underlying
  physical system to be modeled.
• The standard Dymola library offers a multi-body
  library that is user-friendly and therefore
  widely used. However, the component models of
  that library have been implemented using
  matrix-vector equations directly. These models
  are therefore difficult to understand and
  maintain.
• The multi-bond graph library of Dymola offers a
  sub-library for 3D mechanics that re-implements
  the standard multi-body library. Yet, each of
  its component models has been internally realized
  as a multi-bond graph.

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3D Mechanics Example IV

• State variables
• FrontRevolute.phi
• RearWheel.phi1
• RearWheel.phi2
• RearWheel.phi3
• RearWheel.phi_d1
• RearWheel.phi_d2
• RearWheel.phi_d3
• RearWheel.xA
• RearWheel.xB
• Steering.phi
• 2 systems of 3 and
• 15 linear equations, resp.
• 1 non-linear equation
• Simulation
• 20 sec, 2500 output points
• 213 integration steps

Plot window Lean Angle
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3D Mechanics Example III

• State variables
• FrontRevolute.phi
• RearWheel.phi1
• RearWheel.phi2
• RearWheel.phi3
• RearWheel.phi_d1
• RearWheel.phi_d2
• RearWheel.phi_d3
• RearWheel.xA
• RearWheel.xB
• Steering.phi
• 2 systems of 3 and
• 15 linear equations, resp.
• 1 non-linear equation
• Simulation
• 20 sec, 2500 output points
• 213 integration steps

Animation Window
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Animation

• Dymola offers means for animating models of
  mechanical systems.
• This is another reason, why multi-body diagrams
  are important. It is not meaningful to try to
  animate a multi-bond graph. Multi-bonds dont
  carry suitable semantics for connecting them with
  an animation model.
• In contrast, the basic building blocks of an
  animation model are exactly identical to the
  multi-body component models. Therefore,
  animation models can be easily associated with
  multi-body component models, and this is the
  approach that Dymola took.
• Animation models have also been associated with
  the component models of the multi-body
  sub-library of the multi-bond graph library.

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Simulation Run-time Efficiency
The run-time efficiency of the generated
simulation code of a multi-body system model
depends strongly on the selection of suitable
state variables.
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Simulation Run-time Efficiency II

• The run-time efficiency of the generated
  simulation code using the standard multi-body and
  the 3D mechanics sub-library of the multi-bond
  graph library is essentially the same.
• For simple models, the generated equations are
  identical.
• In more complex cases, the equations may differ
  slightly, because the connectors of the two
  libraries are not identical. Whereas the
  standard multi-body library carries for
  rotational dynamics only angles and torques in
  its connector, the multi-bond graph library
  carries angles, angular velocities and torques.
• This occasionally leads to slightly different
  constraint equations that will reflect upon the
  final set of generated simulation equations.

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Multiple Interfaces in MultiBondLib
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Conclusions

• Dymola offers a consequent and clean
  implementation of the principles of
  object-oriented modeling of physical systems.
  Dymola supports graphical encapsulation of
  models, topological interconnection of component
  models, and hierarchical decomposition of models.
• Model wrapping is essentially nothing new. It
  provides simply a systematic interpretation of
  the object-oriented modeling paradigm.
• Whereas object-oriented modeling provides no
  guidance as to how models should be encapsulated,
  the model wrapping paradigm, through the wrapper
  models, provides clean and consistent connectors
  at each layer of the model hierarchy.

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Conclusions II

• Bond graphs (regular bond graphs, multi-bond
  graphs, thermo-bond graphs) offer the lowest
  graphical modeling framework that still carries
  physical meaning.
• The semantic distance from the bond graph down to
  the equation layer below is sufficiently small,
  so that the bond graph libraries are easily
  maintainable.
• The bond graph models can then be wrapped to
  carry the semantics of the component models up to
  a suitable level that specialists of the
  application domain are familiar with.
• All of the wrapping is done graphically, i.e.,
  there is no equation modeling beyond the level of
  the basic bond graph component models.

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The End