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

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

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

3. Motivation
• In today‘s 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.

4. 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.

5. Example: 3D Mechanics

6. Example: 3D Mechanics II

7. Example: 3D Mechanics III

8. Example: 3D Mechanics IV

9. 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.

10. Bond Graphs: Example

11. Bond Graphs: Example II
Se
U0
0
I L
1
R
R1
0
C
C
R
R2
i1
i1 i1
iL uL uC
iC
u0
i0
u1
u2
i2
v2
v1

12. Bond Graphs: Example II
Se
U0
0
I L
1
R
R1
0
C
C
R
R2
i1
i1 i1
iL uL uC
iC
u0
i0
u1
u2
i2
v2
v1

13. Causal Bond Graphs
Se
U0
0
I L
1
R
R1
0
C
C
R
R2
i1
i1 i1
iL uL uC
iC
u0
i0
u1
u2
i2
v2
v1
u0 = f(t)
i0 = i1 + iL
uL = u0
diL/dt = uL / L
v1 = u0
u1 = v1 – v2
v2 = uC
iC = i1 – i2
duC/dt = iC / C
u2 = uC
i2 = u2 / R2
i1 = u1 / R1

14. 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.

15. 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.

16. Multiple Modeling Interfaces in BondLib

17. Electronic Circuit Modeling in BondLib
The Bipolar Junction Transistor
Diagram window
Icon window

18. Electronic Circuit Modeling in BondLib II

19. Electronic Circuit Modeling in BondLib III

20. Electronic Circuit Modeling in BondLib IV

21. Electronic Circuit Modeling in BondLib V

22. Electronic Circuit Modeling in BondLib VI

23. Electronic Circuit Modeling in BondLib VII
Inverter Circuit

24. Electronic Circuit Modeling in BondLib VIII
Initial number of equations
Final number of equations
Simulation Time

25. Electronic Circuit Modeling in BondLib IX

26. 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.

27. Example: A Planar Pendulum
Holonomic
Constraint

28. Causality
Conflict
Example: A Planar Pendulum II
dvx
dt
m = -F sin(φ)
dvy
dt
m = -F cos(φ) + mg
x = ℓ sin(φ)
vx = ℓ cos(φ) φ
.
y = ℓ cos(φ)
vy = -ℓ sin(φ) φ
.
1
I:m
vx
vx
F sin(φ)
Fx
1
I:m
Se:mg
vy
vy
vy
F cos(φ)
Fy
mg
0
F sin(φ)
ℓ cos(φ) φ
. 0
F cos(φ)
ℓ sin(φ) φ
.
vx
0
φ = asin( x / ℓ )
Dq
x
TF
φ

29.  We shouldn’t have to derive the equations first in order
to be able to derive the bond graph from them.
 The resulting bond graph didn’t preserve the topological
properties of the system in any recognizable form.
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 doesn’t tell us much that we didn’t know
already.

30. • 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.
Multi-bond Graphs
} f
3
v
t

fy
vy
fx
vx Composition of a
multi-bond for
planar mechanics

31. 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.

32. Example: A Planar Pendulum III
Multi-bond graph of a planar pendulum

33. Multi-bond Graphs: 2nd Example
Model of a crane crab
y
prismatic joint
revolute
joint
mass 1
mass 2
rod
x
Prismatic Joint
Revolute
Joint
Rod
Mass 2
Mass 1

34. Multi-bond Graphs: 2nd Example II

35. Multi-bond Graphs: 2nd Example II
Wall Prismatic Joint Mass 1 Revolute Joint Rod Mass 2

36. Multi-bond Graphs: 2nd Example III
Wrapper models
Mechanical connectors

37. Model Wrapping
 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.
• Model wrapping offers the best properties of two
worlds:

38. 3D Mechanics: Example
Multi-bond graph model of an uncontrolled bicycle

39. 3D Mechanics: Example II
Multi-body diagram of an uncontrolled bicycle

40. 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.

41. 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.

42. 3D Mechanics: Example IV
State variables:
• FrontRevolute.phi
• RearWheel.phi[1]
• RearWheel.phi[2]
• RearWheel.phi[3]
• RearWheel.phi_d[1]
• RearWheel.phi_d[2]
• RearWheel.phi_d[3]
• 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
0.7s CPU-time
Plot window: Lean Angle
0 10 20
-0.2
-0.1
0.0
0.1
0.2
0.3
[rad]
RearWheel.phi[2]

43. 3D Mechanics: Example III
State variables:
• FrontRevolute.phi
• RearWheel.phi[1]
• RearWheel.phi[2]
• RearWheel.phi[3]
• RearWheel.phi_d[1]
• RearWheel.phi_d[2]
• RearWheel.phi_d[3]
• 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
0.7s CPU-time
Animation Window:

44. 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 don’t 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.

45. 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.

46. 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.

47. Multiple Interfaces in MultiBondLib

48. 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.

49. 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.

50. The End