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Dynamic Modelling of Electro-Mechanical Multibody Systems

John McPhee Marcus Scherrer

Systems Design Engineering, University of Waterloo, Ontario, Canada

1. INTRODUCTION

Over the past two decades, a number of approaches havebeen developed for systematically formulatingthe equationsof motion for multibody systems. Principles of analyticaland vectorial mechanics have been combined with topolog-ical representations, so that the dynamics of a wide rangeof mechanical systems can be automatically and efficientlyanalyzed [1].

Several authors have recently proposed extensions to the

Principle of Virtual Work (and/or Lagranges Equations) sothat electrical components can be included in a model of amechatronic system [2-4]. In these papers, the mecha-tronic system consists of rigid multibody sub-systems andelectrical networks of analog components (resistors, capac-itors, etc). Although linear graph theory is used to gener-ate Kirchoffs laws for the electrical sub-systems, it is mis-perceived as being inefficient [2] and is dismissed as a uni-fied modelling theory.

In fact, linear graph theory provides a natural represen-tation of multi-disciplinary problems and, when combinedwith principles of mechanics, results in efficient models forelectro-mechanical multibody systems. The application ofgraph theory to electrical networks has long been estab-

lished [5] and, more recently, graph theory has been com-bined with principles of vectorial [6] and analytical mechan-ics [7] to obtain systematic formulations for rigid and flex-ible multibody systems. The extension of these methods toelectro-mechanical systems is natural and straight-forward.

y1x1

Y

X V2

V1

2

1

y2

x2

Figure 1. Two-Link Robot Driven by DC Motors

2. SYSTEM MODELLING

To demonstrate this, consider the example in Figure 1 inwhich a two-link robot arm is being driven by two DC-motors powered by voltage sources V

1

and V2

. The topologyof this electro-mechanical system is encapsulated by the lin-ear graph representation shown in Figure 2.

In contrast with other representations, e.g. bond graphs,the linear graph is relatively simple and bears a striking re-semblance to the physical system. This is emphasized byoverlaying the graph with the links and motors in dashedlines. The edges of the graph correspond directly to physicalcomponents: J 1 and J 2 are the two revolute joints, r 3 r 6represent the location of these joints relative to body-fixedreference frames, M 7 and M 8 are the two motors, and V 1and V 2 are the voltage sources.

g

g

r4

r5

r6

M7

V2

r3

G

J1

J2

M8

V1

Figure 2. Linear Graph of Two-Link robot

Note that the electro-mechanical transducers, the DC-motors, are represented by two edges one in the mechan-

ical system and one in the electrical network. The dynamicequations for the two sub-systems are coupled by the con-stitutive equations for the motors:

V

i

= K

i

i

+ R

i

I

i

+ L

i

I

i

(1)

T

i

= B

i

i

+ J

i

i

; C

i

I

i

(2)

where Vi

and Ii

are the voltage across and current through

motor Mi ( i = 7 8 )

,T

i

and

i

are the motor torque andspeed, R

i

and Li

are the armature resistance and inductance,K

i

and Ci

are the voltage and torque constants, and Bi

andJ

i

are the damping coefficient and inertia of the motor shaft.

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Using graph-theoretic topological equations and princi-ples of mechanics, the dynamic equations for the mechan-ical sub-system can be systematically formulated in abso-lute coordinates, joint coordinates, or some combination ofthese and other coordinates [6]. Furthermore, the mechan-ical equations can be expressed in either recursive or non-recursive formats. For this example, the dynamic equationsare automatically generated in terms of the joint coordinates

1

and

2

, using symbolic Maple routines [7] that exploit thetopological equations to reduce the number of variables and

equations, and virtual work to eliminate non-working jointreactions.These Maple routines have been extended to include

models of electrical networks and a number of electro-mechanical transducers. A graph-theoretic approach againallows some freedom in selecting the system variables; theelectrical sub-system equations are automatically formu-lated in currents or voltages, as desired by the user.

Assuming the links to be rigid in the robot example, ourdynamic formulation produces two symbolic second-orderdifferential equations for the multibody sub-system:

M (

1

2

)

1

2

=

Q

1

(

1

2

1

2

I

7

)

Q

2

(

1

2

1

2

I

8

)

(3)

Had the links been modelled as elastic beams, additionalequations would be generated for the elastic coordinates,which would also appear in the mass matrix

M

and gen-eralized forces f Q g . Selecting currents as the variables forthis problem, two first-order differential equations are ob-tained for the electrical sub-network:

L

7

0

0 L

8

I

7

I

8

=

V

1

( t ) ; R

7

I

7

; K

7

1

V

2

( t ) ; R

8

I

8

; K

8

2

(4)

Thus, a minimal number of system equations (3-4) is auto-matically generated by the graph-theoretic formulation andsymbolic implementation. Although the electrical networksin this example are relatively trivial, networks of any com-

plexity can be efficiently treated using graph theory.With the equations expressed in symbolic form, it is of-

ten possible to find closed-form solutionsfor the generalizedinverse dynamics problem. In this case, given desired jointtrajectories of

1

=

2

= t = 8 , i.e. both links rotate through90 degrees in 4 seconds, one can solve (3-4) to get analyticalexpressions for the required motor currents:

I

7

( t ) =

1

6 4 K

T

7

h

8 B

7

; 3

2

m

2

l

5

( l

3

+ l

4

) s i n

8

t

+ 6 4

m

2

g ( l

3

+ l

4

) c o s

8

t

+ m

2

g l

5

c o s

4

t

+ m

1

g l

3

c o s

8

t

i

(5)

I

8

( t ) =

1

6 4 K

T

8

h

8 B

8

+

2

m

2

l

5

( l

3

+ l

4

) s i n

8

t

+ 6 4 m

2

g l

5

c o s

4

t

i

(6)

This solution can be verified by a forward dynamic sim-ulation in which the motor input voltages are regulated bya PD-controller to respond to errors in the joint trajectories.For this case, a numerical integration of equations (3-4) re-sults in the motor currents shown in solid line in Figure 3.As expected, they oscillate about the analytical solutions (5-6) shown in dotted lines.

0 0.5 1 1.5 2 2.5 3 3.5 440

20

0

20

40

60

80

100

120

I7

I8

Time(s)

Current(A)

0 0.5 1 1.5 2 2.5 3 3.5 440

20

0

20

40

60

80

100

120

I7

I8

Time(s)

Current(A)

0 0.5 1 1.5 2 2.5 3 3.5 440

20

0

20

40

60

80

100

120

I7

I8

Time(s)

Current(A)

Figure 3. Motor Currents Required for Given Joint Trajectories

3. CONCLUSIONS

In summary, a unified and efficient modelling methodol-ogy for electro-mechanical multibody systems has been ob-tained by combining linear graph theory with principles of

mechanics. From a single graph representation, a relativelysmall number of system equations is generated in a methodi-cal manner that is well-suited for computer implementation.It is also worth noting that a graph-theoretic approach is notrestricted to analog components, in contrast to approachesbased solely on virtual work [2-4]. Thus, components of adiscrete-time nature can be readily included in a model ofa mechatronic system containing digital controllers; this ap-pears to be a promising area for future research.

4. ACKNOWLEDGEMENT

The financial support of this research by the Natural Sci-ences and Engineering Research Council of Canada is grate-fully acknowledged.

REFERENCES

[1] W. Schiehlen, Multibody System Dynamics: Rootsand Perspectives, Multi. Sys. Dyn., 1, 149-188 (1997).

[2] V. Hadwich and F. Pfeiffer, The Principle of VirtualWork in Mechanical and Electromechanical Systems,Arch. Appl. Mech., 65, 390-400 (1995).

[3] P. Maisser, O. Enge, H. Freudenberg, and G. Kielau,Electromechanical Interactions in Multibody Sys-tems Containing Electromechanical Drives, Multi.Sys. Dyn., 1, 281-302 (1997).

[4] K. Schlacher, A. Kugi, R. Scheidl, Tensor AnalysisBased Symbolic Computation for Mechatronic Sys-

tems, Math. Comp. Simul., 46, 517-525 (1998).[5] H. Koenig, Y. Tokad, and H. Kesavan: Analysis of Dis-

crete Physical Systems, McGraw-Hill, 1967.

[6] J. McPhee, Automatic Generation of Motion Equa-tions for Planar Mechanical Systems Using the NewSet of Branch Coordinates , Mech. Mach. Theory,33, 805-823 (1998).

[7] P. Shi and J. McPhee, Dynamics of Flexible Multi-body Systems using Virtual Work and Linear GraphTheory, Multi. Sys. Dyn., 4, 355-381 (2000).