The inside structure model of mechatronic devices

Mechatronics, Vol. 3, No. 2, pp. 127-138, 1993

0957~.158/93 $6.00+0.00 Pergamon Press Ltd

Printed in Great Britain

THE INSIDE STRUCTURE M O D E L OF MECHATRONIC DEVICES

O. PETRIK-G. SZASZ TechnicalUniversityof Budapest, Deparmaentof PrecisionMechanics& Applied Optics H-1521 Budapest,Egri J. u. 1. I11.1., Hungary Abstract - A determinant phase of the systems analysis is the structure and characteristic modelling. Parts of the mechatronic systems have different physical nature. The model describes the inner stleam of data, and the energy conversions of subsystems, and further the inside interfaces of the device. The external function of a mechatronical unit is based on the harmonical co-operation of the subsystems. The combined application of structure analysis and synthesis are useful for optimal performance approximation. The use of system simulation is presented. The experimental performance of a mechatronic unit and the workshop approbation results are shown versus the theoretical - during the design validation-- requirements. The paper illuslrates the method presented by a practical example.

I. INTRODUCTION

The mechatronic unit is generally an integrated complex system. Out of primer m e c h a n i c a l performance (actuation) it is able to s e n s e the actual position, c o m p a r e it with the prescribed one, then, depending on the difference, it can evaluate the situation and m a k e a d e c i s i o n for further action. After decision making the unit e x e c u t e s the necessary performance, that is checked by the sensing part of the system, etc. It seems on the basis of the above short performance description, that a mechatronic unit consists of some subsystems that have different physical nature. The precise co-operation of these subsystems enables the perfect functioning of the unit (Fig. 1). EMSS

"

,o

vl

i~,~

, .,__,_

°

.~[=...

CPU 1

~ I....

L_

1

I

Xo!

Fig. 1. A sketch o f a mechatronic device. EMSS : electromechanical subsystem, TG : tachogenerator, M: motor, GB : gear box, T: table, U." digital converter INF1 f input voltage ' CB.. code bar of position transducer , X 78.•position. variable. CPU" .computer. A/D" analog . . INF2 : information subsystem N ° 1 and N ° 2

127

128

O. PETRIK and G. SZASZ

When an engineer is confronted with a complex dynamic-system problem, he must first recognize, that the problem is one to which he can apply the principles developed in the system-techniques (inclusive linear, concentrate-parametric theory of technical systems). The practicing engineer, however, is faced with many types of situation. His education has hopefully given him the perspective with which to judge the problem at hand and categorize it properly. There are two jobs for systems handling: the analysis and the synthesis, i.e.: the investigations of the existing system, and the design of a new one. The two procedures are sometimes combined with each other creating a good working system that satisfies numerous requirements. In this paper we are dealing firstly with the analysing methods, and next some of synthesis ones.

2. THE STEPS OF SYSTEM ANALYSIS At the start there is the real system, as a whole entity. One has to define the boundary of the investigating system, so we can separate the environment from the system itself. The influence from the outside environment is defined as an input, and the response of the system as an output. To investigate the dynamic performance of the system one has to set up an adequate p h y s i c a 1 model. This model contains all the essential features and attributes of the real system, connected to our scope and tasks. The next step is the definition of the m a t h e m a t i c a 1 syste~ .~odelon the basis of the physical one. The final form of the system model is (are) the system equation(s), but before that you have to investigate the system elements and the connections between them, i.e. the system structure. The mathematical statement of the governing relationship between system variables are called the formulation problem. Connecting elements impose constraints on the variation of system variables, and the convenient way of specifying these constraints is by a mathematical statement of the way in which the various through-variables are related and the way in which the various across-variables are related. The elemental equations then relate the through- and across-variables for each individual element. This package of equations is a complete mathematical description of the system. Summarising, the steps of the modelling are as follows: - analysing the real system as a whole, recognising the modelling point of view (i.e. the task), - refining the system-boundaries, and the subsystems, identification of system elements, - making clear the connections and the dependency between the elements i.e.: mapping the systemstructure (by graphs), setting up a mathematical model, - solving the mathematical model for the desired input, - checking the validity of the model and of the solution by refining (correcting) the model (if there are intolerable differences between the measured and the calculated results). -

-

-

There are numerous practical processes and methods for setting up a mathematical model. The Bond graphs, the signal-flow graphs and others are well known [see the references]. Below we are showing the method of the structure graphs, as a useful tool for system modelling. This method gives a very good insight into the inner structure of complex mechatronic devices. One of the system elements incorporates the appropriate attributes of the system parts, e.g.: inertia, elasaticity, damping, capacity, inductivity etc. Table 1. shows a set of "ideal" system elements including energy storers, dissipator, sources and transformer. These elements are of course abstract ones, repre-

The inside structure model of mechatronic devices

129

T a b l e 1.

Attribute

Blockdiagr.

Graph.

Elemental equation

F = m dv2 dt

Inertia 1/1

2

I

½

F

k

Energy (ideal)

1

2

Em = ~ mv~

1'

v2 = m ! F dt + vo

F t

Elasticity

F:kS(v2-vl)dt+Fo 1/2 V2

Vl

1 F2

e~-2k

0

Vl

b

F = b(1/: - ~l)

Dissipation 1/2

1/I

(1/2-1/1) :-F

Dissipative power p = b(v2 _ Vl )2

b

I/2

1"2 Trails

VI

m

1/4

1/2 V 4 -- V3 _ - - - n

-

V 2 -- V 1

formation

1/3

1/I

V3

Fo : - l Fa

No energy lost

n

V

(F) Sonrce

(across variable)

Source (through variable)

v = v(t) F=

dependent from system

Infinite (theoretically)

(v) m

dependent from system F =F(t)

Infinite (theoretically)

O. PETRIK and G. S Z A S Z

130

senting the attribute that we want to model with them. A real-life part may consist more separated pure elements connected in a defined structure. The combination of ideal elements that is intended to represent the behaviour of a real system is called the model of the system. The degree to which the behaviour of the ideal model corresponds to the behaviour of the real system, represented by the model, is a function of experience, skill and engineeringjudgement of the modeller. Usually, the model must represent a compromise between its complexity and the degree of accuracy required in the predicted behaviour of the real system. The procedure described above is very important. The engineer must make some simplifying assumptions about a real world situation to make it mathematically tractable and yield a usable solution with a reasonable effort invested. The usefulness of the solution is directly dependent on how closely the model represents the behaviour of the real system.

3. THE MODEL OF A COMPLEX ELECTROMECHANICAL SYSTEM For the illustration of above mentioned modelling techniques we demonstrate an electromechanical subsystem model. This device consists of a direct current servo, a gearbox, a tachogenerator connected to the shaft of the servo, and a screw-driven table with a slide bar for straight line guide. The sketch of the system (see Fig. 1) and its structure graph are shown in the Fig. 2 (without the tachogenerator).

'

I

i jjjjjj/ 0 electrical

rotation transformer (G/ B)

transducer

i

g mechanical i translation transducer

'I Fig. 2.

The system equations based on the structure graph are as follows: The electrical part of the motor

L

di

+Ri+

dt

where: Ut input voltage, R, L resistance and inductivity of the motor, O.)4g motor angular velocity, n~l transducting coefficient, i current value in the motor.

(D 4 g

?'lvl

=Uu,

(U

The inside structure m/)del o f mechatronic devices

131

The equation of electro-mechanical transducer: (04g = n v l U 3 o ,

M1 =

(l/a)

i nvl

For the rotation transformer (gear box): (.05g = glt(..O4g , M 3 =

]

(I/b)

M2 nt

From rotation to translation transducer equation: W7g = gtv20)6g,

(I/c)

M4 f ~ _ ~ nv2

The equation of the mechanical part of the moving table

= K ntO94g where: M, K nt to4g W7g

(2)

nv2 J

torque at the spring, spring constant (rotating), gear box ratio, gear box output, table (support) velocity.

The equations of the interacting subsystems:

mx where: m x w7g

d-""7-= Fly2

(3)

nv 2

mass of the table, velocity of the table,

M~ nv2

torque in the rotating spring, screw driver transmission ratio, B3B 4 dissipation coefficient of the screw spindle and bearing, bx dissipation coefficient of the slide guide.

(_~+ntO21~tg+[B]+nt(B k nt

2 +B3)](04g+M

~ =

i ntn v

+ B 3 WTg, gl~ 2

(4)

where: 01, 02 inertia of the servorotor and screw spindle, tOag angular velocity of the servo, B1, B2 dissipation coefficient of the bearings in the servo and in the gear box The state of the system will be defined by the state vector, through the state variables, which are dependent on the instantaneous loading of energy of the storers. In this system the state variables are as follows:

132

O. PETRIK and G. S Z A S Z

I W7g

.

The STATE SPACE MODEL of this subsystem on basis of eq.-s (1), (2), (3) and (4) is: r

•

__21

~1

-LB1 +n, ~B2 + B3/j

0.)4g

n, ('04g + nv2

01 + Ill2 02

nt

B~ -01 +- g/t202

.

W2g --

1

01 +#02 Mk + . 1(01 +#02) i' 1 2 (B3+B4)+bx n~ 2 mx K

WTg =

" W7g q-

n~B3 nv2mx

-b

1 "

rlv2mx

Mk,

(5)

l(,t = gn,~04e - n~---2w7e, di dt =

1 -

Ilvl'--L " ('04g

--

R 1 T i + -~U/ .

The transfer function of the subsystem, if the output is the displacement of the guided table (XTg) is: A p ( T a s + l)

Y, (s) = # s S + r ~ s 4 + ~ s 3 + # s 2 + r l s .

(6)

The expressions for a p , T a , T a , T ~ , 7"33 , T44 , 7"55 can be found in Appendix 1. If we consider the coupling to be unyielding (i.e. rigid) so K --* ,,~, in this case: Ap

~(s).

r3s3 + r~s2 + ~s"

The expressions for 4 , ~3 can be found in Appendix 2. We can consider a further assumption i.e.: L ---) 0; in case the final simplified model of the electromechanical subsystem is as follows: Ap

rs(s)- ~s 2 + ~s"

(7)

4. DESIGN OF THE APPROPRIATE CONTROLLER The next step is the design of the controller. First we consider an analogue control system, using a controller Y~(s):

Ks Yc(s) =

s+l r~ +1

;

(8)

The inside structure model o f mechatronic devices

133

1

where: Ks = ~-~-. The open loop transfer function is the following:

Y(s)

-

(9)

( zs + 1)T1. s '

f'mally the closed loop transfer function:

W(s) =

If we consider it to be:

r~ ~ = ~;

than it will be:

T2 =

and:

z =

KsAp zTxs 2 + Tls + ApK s Tqr~=

so

1 T2s 2 + 2~T~ + 1

(10)

1 K~ Ap ;

T~2 2K;2 Ap2 ;

T1 2ApK, "

The natural frequency of system will be: O) 0 = - -

TI

The necessary controller finally: 2 "l"2 S + ~'1

L

_

(11)

T12

- - s + T

K s 2Ap

which is a Lead-Lag filter with frequencies: 1

fz = 2rc~

and

1

L_KsAp

4.1. Adjustment (tuning) of controller With data from Appendix 3 they are:

z2 = 0.148 [s];

where: K~ = Ks m a x

Y~(s)

zs+l

. o 0r ).

°r'T? T1 2 AK~

1 = 9.45 [s]; 2 • 5.3" 10-2

134

O. PETRIK and G. SZ,~SZ

finally:

Yc(s) = 3.205-

s +28.9 .

10 -2 S + 0 . 1 0 6 '

(12)

and the frequencies: f~ = 4.67; fp = 0.0168. The realization might be in an analogue or digital way. The up to date digital circuits give the possibility to produce a precise and modem digital control. 4.11. The realisation of digital control. The appropriate digital controller on the basis of sampling control system theory:

using the data of 4.1. we can get:

D(z)=3.205.10 -2 1 - 0 . 8 5 5 3 8 3 z -1 1 - 0.9994701z -l ' with T O = 5" 10 .3 [S] s a m p l i n g time.

Better conditions may be taken by ~T0 = 1 0 - 2 [s] 1 (the result achieved by the iteration method). The final resulting "z" transfer function is:

D(z) = 0.03205.

1 - 0.7111531z -1 1 - 0.9989406 z -1 "

(14)

4.12. The modified Bang-bang control. The digital control enables the achievement of finite settling time. (In the analogue control this is theoretically indefinite.) The Bang-bang method, which means a few altemate actuations to control the moving part of the device, shown in Fig. 3, gives a very good control effect. X, ot 10

-lO Fig. 3.

It is important to find an appropriate value for t 1 and t 2 . The calculations are as follows: t1 =mT0;

X~t(z)=

t 2 =n.T

10z (z 0 _ z z-1

O,

~+z-n).

(15)

The inside structure model of mechatronic devices

135

T h e digital transfer f u n c t i o n c a n be calculate o n b a s i s o f Ys (z):

z

a(z)

(16)

T h e i n p u t (unit step): Xa

=

z

z-1 T h e output:

X , = Y~(z)X.o~. T h e a c t u a t i n g effect:

Xact = D ( z ) ( X a - X s); Xact =

D(Z)[ z--Z_-~- Ys(z)Xb 1.

(17)

T h e t r a n s f e r f u n c t i o n b e l o n g i n g to the B a n g - b a n g control:

D(z)

=

z

z- 1

Sac t (2') B(z) X~t (z);

(18)

A(z)

finally, u s i n g o u r data: 10(1- 2z -~ + z-" ) (19)

D(z) =

C h o o s i n g the "m" a n d E.g. t h e result:

"n" v a l u e s

m i g h t o p t i m i z e the positioning. tl = 27T0

and

t 2 = 2 8 T o.

T h e s e t i m e v a l u e s are g i v i n g the f o l l o w i n g t r a n s f e r function:

D(z)

I0(1 - z 1 + 5.48667.10 io . z-2 _ 2z-27 + 3z 2s _ 1.097334.10 3 . z 29 + 5.4866.10 -l° • z 3o) 1 - 1.036623. z x + 0.0364434 • z -2 + 0.073246 • z 28 _ 0.033031 • z -29 _ 1. 796 • 10 3 . z-3°

F r o m e q u a t i o n (20) w e c a n o m i t the e x t r e m e little v a l u e s in c a s e o f t h e calculation, so the r e m a i n i n g t e r m s are as follows:

D(z)

=

10(1 - z -1 + 5 . 4 8 6 6 7 . 1 0 -l° • z -2 ) -2 •

(21)

1 - 1. 0 3 6 6 2 3 . z -1 + 0 . 0 3 6 4 4 3 4 • z

5. T H E S I M U L A T I O N

AS A TOOL OF SYSTEM DESIGN & ANALYSIS

A v e r y effective a n d useful tool is s i m u l a t i o n . T h e r e are m a n y suitable s i m u l a t i o n l a n g u a g e s for the c h e c k i n g a n d e v a l u a t i n g o f a n y kind o f s y s t e m . In o u r e x a m p l e o n e h a d u s e d the T U T S I M l a n g u a g e for the illustration o f the results. In Fig. 4 the d i a g r a m s are the following:

136 N ° 1: N ° 2: N ° 3: N ° 4:

O. PETRIK and G. SZ,/~SZ Xa: input variable (unit step); X s ~ : output variable in the case of analogue control;

Xs DIX: output variable in the case of direct digital control; X,, DoC: actuating signal for executing the digital control.

PLOT1: Xa -8.OOOOO Z.O000O PLOTZ: Xsa.al -5.00000 4.00000 PLOT3: XsDDC -4.00000 5.00000 PLOT4: XmDDC -I.OOOOOE+I ?.O00OOE+I

T

T

I

r

PDT1 T

digital T

r

realisation T

T

4.

9.. J

f

f

~

j

8.809000

TlnE

1.90000£+Z

Fig. 4.

As the curves show, the process of positioning is very exact, the position (Xvg) will be obtained with great accuracy, but the translation is slow. The settling time is about one minute at the digital control, and it is longer at the analogue one. The next step has to be the parameter changing to improve the system characteristics, e.g.: shorter settling time at the same accuracy. The simulation provides the possibility to model the computer itself. With reference to Fig. 1 we can see, that using a computer in the device we can design an intelligent control to improve the performance of the device. Furthermore, one can consider an adaptive and/or a self learning system too. These continually developing possibilities assure an excellent future for mechatronic devices. Summarizing the statements of the paper we can recommend the structural method for handling mechatronical systems, consisting of subsystems of different physical nature. The built in computer gives excellent features for the complex system, making it intelligent. For investigation and design simulation is a good tool, but you have to take into account sometimes the possibility of numerical lability. In certain cases it is preferable using the analytical methods for simulation.

The inside structure model of mechatronic devices

137

APPENDIX I.

The relation between the transfer function Ys (s) coefficients and system element characteristics:

~, = ~[~ Ta = - ~

,nvl 2

+.~/,~ +,4 [s];

[m~v 1~

~v~x)] + 1

T1 : 1

[s];

L B3 + B4 + n~2bx

B1

+

+ nv2bx

T2- _

+

g{n~le[B1 +/lt2 (n2

+an+n2v2bx)]+l}

Rn21{(B3 + B4 +n22bx )(B 1 + ntB2 )+(B 4 +n22bx )nt2B3 + K[ 01 + nt2( 02 +n22mx )]}

+

K{n21R[B1 + nz2(B2 + B4 +n2v2bx)] + 1}

Is2];

mxn~2 + Ln~l{[Ba + n~2(BE +B 3 )](B4 + nv22bl)+ B3(B1 +n~2B2)+nv23mxK}

K{nv21g[Ol +/~2 (B 2 +B 4 +n22bx)]+1}

+

T44_ L{mxn22 [ B1 + r~2("2 + B3 )]+(01 + nt202 )(B3 + B4 +n22bx)}+

[$3];

Rn2v2mx(O1+ ~202)

K{n~IR[B 1 + n~2(B2 + B4 +nZ2bx)]+ 1} Lmxn~ln22(O1 + # 0 2 ) T55 =K{n2vlR[Bl+nt2(B2+B4+n2v2bx)]+l }

[sS]"

APPENDIX 2.

Time constants 2 T2 --

lim

,.o

z 22 =

3 ,t.3 =

n~l{L[~I + n,~tB2+ ~4+ nv~x )]+ R[01+ n~,102+n:2m~)]} [

n~1R 01 + n

°@1+n,~("~+"4 2

2

++~1]+'

nvltnv2mx n21g[nl +n2(B2 + B 4 +n22bx)]+l

lim r 3 = 0.

L--)0

,~/02 + nv2mx 2 )]

is4];

138

O. P E T R I K a n d G. S Z A S Z A P P E N D I X 3.

Practical data o f the electromechanical s u b s y s t e m , which is a positioner of a straight line guided table driven by a direct current servo motor. m x = 1 kg

02 = 10 -2 k g m 2

bx = l N s / m

B2 = 0 . 0 1 N m s

nt = 0 . 1

K

R

B3 = 0 . 1 N m s

=5 D

=100Nm

81 = 2 . 1 0 - 6 k g m 2 B1 = 10 -2 N m s nvl = 0.1

h n~2 = 2 x

5 . 1 0 -2 m = 7 . 9 6 . 1 0 -3 m 2~ rad.

REFERENCES [1] J. L. Shearer-A. T. Murphy-H. H. Richardson: Introductionto System Dynamics.Addison-WesleyPubl. CompanyReading Massachusetts, London 1967. [2] Roll Isermann:Digital Control Systems Springer-Verlag.Berlin, Heidelberg, New York 1981. [3] O. Petrik: Mechatronic solutionsof microactuatorsand positioners. Mechatronics. Vol. 1, No. 4. pp. 417425. [4] O. Petrik: Krafte (Momen0-Riickkoppelungssystemyon mmechatronischenElementen. Per. Poly. Mech. Eng. Vol. 31. No. 2-3. [5] O. Petrik: ,,Mechatronik" - eine neuartige Beziehungzwischen Feinmechanik, Optik und Elektronik. Per. Poly. Mech. Eng. 31/2-3. [6] O. Petrik: Systemtechnisch modellierte Prezisionsgetriebe. XI. Intern. Tagung ,,Wissenschaftliche Fortschritte der Elektronik Technologie und Feingeratetechnik Dresden". Proceeding p. 35.