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Hybrid Aspects of Modelling Manufacturing Systems Using Modified Petri Nets
Copyright © IFAC Intelligent Manufacturing Systems, Gramado - RS , Brazil, 1998
HYBRID ASPECTS OF MODELLING MANUFACTURING SYSTEMS USING MODIFIED PETRI NETS
Rainer Dratb, Ulricb Engmann, Susan Scbwucbow
Technical University of Ilmenau, Department ofAutomatic Control and Systems Engineering, Postfach 100565,98684 Ilmenau, Germany email: [email protected] [email protected] [email protected]
Abstract: This paper presents a new special view on hybrid modelling of flexible manufacturing systems using a new description method basing on Petri nets (Hybrid Dynamic Nets, HDN). It shows, why hybrid modelling is useful for the design process instead of the currently preferred view based on discrete events. In order to solve handling problems, arising from the system complexity of larger systems, we combine this method with the object oriented paradigm. In the result we get Hybrid Object Nets (HON). Copyright © 1998 IFAC Keywords: Hybrid, Dynamic Modelling, Petri nets, Object Oriented Modelling, Flexible Manufacturing Systems
1. INTRODUCTION AND BACKGROUND
need a model including higher order variables and disturbing effects and we get more complexity in this way. Other samples are transport and handling processes on flexible manufacturing systems, which allow the processing of workpieces during its movement, or gripping processes of interacting robots. As a third example for special production processes with a combination of process industry system parts and manufacturing production system parts, a closed combined examination of continuous and discrete event processes is needed. Such technical processes are e.g. metalhardening, tempering processes or galvanic processes. For the development of automation solutions of these hybrid systems it's necessary to give a hybrid model emphasis.
The most known modelling techniques and description methods for manufacturing systems are based on the product oriented view. The main manufacturing processes are typically considered as groups of assembling, machining, handling, transport and store parts. They seem to be discrete-event-oriented, therefore these processes are modelled in a discrete-eventoriented way. But to solve detail problems, sometimes the continuous dynamic behavior is necessary to be described. Often constant time values are used on the assumption, that they sufficient reflect the time consumption of continuous systems. Nevertheless, not all problems and properties are reflected and the real process would not be sufficient described.
Considering these demands, a hybrid modelling technique would allow to model more complex dynamic structures. Hybrid systems are the combination of continuous dynamic systems and discrete event dynamic systems (DEDS). In this approach, the derivation of events does not only depends on approximations, but on a dynamic model of the continuous processes, which generate them. If
The following manufacturing applications show the necessity of hybrid modelling techniques because of their hybrid nature. As a typical case e.g. is the handling of parts which have different weights. The transportation of a part from point A to B shows different dynamic behavior due to variability of the target ending points and to the part weight. Such systems
145
Alia, David, LeBail, 1991» a function of the marking from the continuous net places.
disturbances are considered too, consequently the events depends on these processes.
Because of the differences between the origin Petri nets and this new approach, it is no longer called Petri net but Hybrid Dynamic Net (HDN) . The proposed solution for describing hybrid processes is based on the HDN and allows the modelling of the dependencies of all system parts with only one description method. The following conventional discrete and new continuous net elements are included:
For modelling hybrid dynamical systems, a useful description method is needed. Therefore an enhanced hybrid net approach (Hybrid Dynamic Nets, HDN), based on Petri nets, is shortly presented here. Because Petri nets are established and often used to describe manufacturing processes (DiCesare, et al. , 1993), the presented approach bases on Petri nets. In order to solve handling problems, arising from the system complexity of larger systems, we combine this method with the object oriented concept. In the result we get Hybrid Object Nets (HON) which are also shortly presented in this paper. The HON approach is finally demonstrated by means of an example.
TD
discrete transition
! p, I
i~ T1 1,:2 1 P2 ·
m2
I, T2
1..=3
.
~1 o
1
2
m2J
2.1.
m2 o
Continuous basic element
Fig. 2 shows a continuous basic element. The firing speed v is the speed of token flow into the place P2. The transition Tl is always active; it can be inactivated only by empty discrete input places. The test arc does not allow token flow. This makes it possible to model subsystems without feedback: the token quantity of P J here is not influenced.
v1
v2
continuous place
.0
traditional Petri net with tokens
rn1
continuous / Petri net
continuous transition
The main idea is to assign the continuous transitions to a firing speed, which is represented by an equation. This equation is a function of token quantities of arbitrary places of the net. The shown arcs below are used for the main normal arc relations between inhibitor the different transitest arc tion and places types.
rn1
- - -;
CS
Continuous transitions are new and not comparable with the discrete ones. Instead of discrete firing, their activity leads to a continuous flow. Continuous places contain tokens, which are interpreted as real values.
The new approach shortly presented here is characterized by modification of the conventional Petri nets in combination with an extended class of continuous Petri Nets (Alia, et aI., 1991 , 1992; Drath, 1998a), to Hybrid Petri Nets (Drath, et aI., 1997; Drath, 1997).
i
discrete place
PC
0
0
I
2. ABOUT HYBRID DYNAMIC NETS (HDN)
I
TC
PD
I
Fig. 1: From conventional Petri net to continuous Petri net
P1
The state space of the upper traditional net in Fig. 1 is shaped however only by three discrete states. The two initial tokens in place PI can distribute themselves due to the times of the transition like represented here. The model below shows the extension with continuous places and transitions (Alia, 1987). Here, a token is not any more an individual, but a real quantity of token fragments . The transition moves with a velocity of flow the token fragments from the place before into the place thereafter. The state space becomes infinitely in this way and this opens the possibility of modelling continuous dynamic.
T1
P2
@-------[]---6 quantity of tokens in PI quantity of tokens in P2 m3(t) ... mj(t): quantity of remaining places of the net v(t): firing speed ml(t): mit):
Fig. 2: Net, including only continuous net elements ofHDN If a single input arc is directed to P2, we get (1).
v = dm 2 (t) dt
The suggestion and the goal of the in the following presented new approach is, that the firing speed can be (in contrast to the approach from (Alla, 1987;
146
= m,(t) .
(1)
continuous net elements allow the modelling of continuous systems using ordinary differential equations. In (Drath, et aI., 1997) we explained in detail how to combine discrete and continuous elements in order to model hybrid systems. Since a time concept is included, it is possible to model real time systems.
For a place Pj with i input arcs we get (2).
m(t) = dm/t) = "v ~ ~ . J
m
This corresponds to the node theorem. Continuous input and output transitions supply their part to increase or to decrease mj.
3. HYBRID DYNAMIC NETS AND OBJECT ORIENTAnON
We can model different basic elements in this way. Even non-linear coupled subsystems can be described. 2.2.
3.1.
Samples
The most mathematical, textual or graphical approaches to describe hybrid systems are currently usable for small examples. Models of complex systems are unwieldy. Therefore a hierarchical concept to structure a model is needed.
First order system: The fIrst order system (Fig. 3) is a basic continuous element consisting of one input and one output place. The fIring speed function (3) is assigned to the transition Tl . Fig. 3 also shows the step response of the system.
In order to solve the mentioned handling problems arising from the system complexity, we applied the object oriented paradigm to HDN, resulting in a new method to describe both continuous and discrete event systems with reduced effort: Hybrid Object Nets (HON). One of the important advantages of the using this concept is the ability to describe a larger system by the decomposition into interacting objects. Because of the properties of objects, the modifIcation of the system model could be easier achieved. The object oriented concept unites the advantages of the modules and hierarchies and adds useful concepts like reuse, encapsulation and information hiding. In this way we get more flexibility.
Fig. 3: Application example, fIrst order dynamic system v(t) = u - 0.1 · Y , this leads to:
(3)
Y = (u - 0.1 . y) dt
(4)
f
Oscillator: With HDN we can describe differential equations of fIrst order. In order to describe differential equations of order n they have to be formed into n differential equations of fIrst order. So the wellknown differential equation for an oscillator can be transformed into the equation system (5).
With Y=
Xl
and x 2 =
Xl
3.2.
General Properties of the HON
HON includes concepts for attributes, methods, interfaces, encapsulation, inheritance, abstraction, data exchange and reusing.
we get (5)
Attributes are represented by places and their contained token quantities. Methods are given in the form of the net structure. Information hiding is realised by encapsulation the detail informations of the net structure, and by publishing selected places using an interface. Abstraction is the step from a concrete net structure to a class: it is realised by fIlling the objects into a class hierarchy. Inheritance is the step from a class to a concrete object, so called instance of the class. If an object will be abstracted from a class, it inherits the whole net structure including the interface. Data exchange is given by the token flow between the objects. Discrete tokens can model method calls and discrete system states; continuous tokens model continuous control variables. ReUSing, the most important quality of object orientation, is given by inheriting or instancing classes. Derived objects can be refIned; places, transitions, arcs and objects can be added: but no inherited element can be deleted.
With D = 0.25, Wo = 1 and Kp=l, the continuous Petri net below (Fig. 4) can model this equation system (5). The step response is also shown in Fig. 4.
Fig. 4: Application example, second order dynamic system 2.3.
Motivation
Hybrid Dynamic Nets (HDN)
The essence of Hybrid Dynamic Nets is the combination of continuous and discrete net elements. The
147
3.3.
Construction of a class
To construct a new modelled using the published, and the Afterwards the net chy. 3.4.
class, a suited subsystem must be HDN. The places, which shall be parent class have to be defmed. can be fitted into a class hierar-
Structure of an object
Fig. 6: Principal scetch of the manufacturing system for hybrid modelling
To generate an object, it has to be instanced from a class. Every object has a hierarchical structure which contains three layers (see Fig. 5) leaned on the two layer concept in (Schwuchow, 1997). In the supreme layer the object frame is presented, which encapsulates the inner net structure of the object, and which allows the communication with the environment.
If a workpiece reaches the range of the robot, the robot should occupy a position above the passing part from its initial position, process the part in passing and then occupy its initial position again. Its speed should be designated as V respectively Vma;c . The movement of the robot is determined by a guide rail, installed in the direction ofthe x-axis.
In the underlying second layer, the net, inherited by the class, is enclosed in an object frame . In this layer, further net elements and objects can be added in order to modify the behaviour of the object and form new subclasses. In the lowest layer, the net, inherited by the parent class, is represented. It can not be modified here but can only be screened.
initial Conditions: No part might be on the conveyor and the robot might be in position of rest. Destination: A model for this system should be constructed, which encapsulates the functionality and the dynamic of the robot and that of the conveyor in each an object. In such a way, both objects should be connected with each other in order to model the dynamic behaviour of the whole system. 4.2.
Modelling Procedure
Discrete model of the discrete event sequence controller of the robot: The function of the robot can be described by a sequence of operations. The Precondition for any activity of the robot is the occupation of its initial position, since calibrating procedures take place there.
Fig. 5: Example of the Layer-Concept for HON
If a workpiece reaches the range of the robot, the robot trolley must look up the position above the workpiece. If it reaches this position and if there is enough time for finishing left, it starts processing the workpiece. Afterwards the robot returns to its initial position (see Fig. 7).
4. APPLICA TlON EXAMPLE The considered application example is a hybrid manufacturing system, which is characterised by the interaction of a controlled robot and a mobile workpiece. The robot carries out continuous positioning procedures, which are influenced by the movement of the passing part while processing.
•
robot initial position occupied workpiece arrived? search for workpiece position workpiece position achieved ?
4.1.
workpiece position achived
Configuration
start of processing
The considered system consists of a robot, which processes passing workpieces, and of a conveyor, which transports workpieces into the range of the robot. In this case, the movement of the parts occurs in positive direction ofthe x-axis of the system (see Fig. 6). The position of the workpiece might be designated as xw, that of the robot as xr. The length of the range of the robot might be designated as xmax.
workpiece is currently processed processing finished start look for robot initial position robot initial position achived ?
Fig. 7: Control sequence for the robot The represented discrete Petri net models this course in its causal order. Since it is a causal model, it is not
148
possible to detennine the time, which is consumed by the individual processing steps. Since the individual operations are characterised by a situation dependent dynamic, it is not possible to estimate the delay of them.
conveyor".
E**;-;·· · · ··· · . ..
Considering these characteristics, dynamic models for the underlying dynamic processes should be created, which describe the temporal behaviour of the continuous processes.
:
"
...
"
v
xW(l,)
xw(t,)
xw
Fig. 10: HON-model for a conveyor
Fig. 8: Principal Scetch of the conveyor necessary parameters
Hybrid model of a comparator: Comparison operations are carried out several times. The current position of the robot trolley must be compared with that of the workpiece on the conveyor. For this reason, an object, which makes this operation available should be constructed.
Hybrid model of the conveyor: The conveyor (s. Fig. 8) can be described by the net shown in Fig. 9. If a workpiece reaches the conveyor, the place entry will be marked (in=l, the value 1 will be interpreted as TRUE). Afterwards the transition Tl get the concession and fires a token into the place workpiece present (prs= 1). The token remains here during the entire stay time of the workpiece.
A
traditional discrete net model :-
--
. --- ._.- --. - .---. ----_.--- --.-.- ._. _. workpiece
The comparator object should compare two continuous control variables x2 and xl and derive three discrete states from it: x2>xl, x2=xl or x2
_.-.-.- .- .-----
'\
T1 present T2 exit lT3 . -: ._.--_._._._.- -.-.~.-.-.-.-.-. 'max- -.~- -~-: entry
P1
.
TB
~---
; transport
f
~ v
position ;
'-- . ~ .speed ~.-. _. _.- - _._. .- ---_._. _._._.- .-- -._ ._._. --'J oontmuous net pan
r
workpiece is processed
Fig. 9: Hybrid net of a conveyor The process of transport of the conveyor is modelled by the continuous transition TB, where the speed function is assigned to. The transport speed is described by the variable v in the place transport speed. The continuous transition TB is activated by the token of the place workpiece present. In the result the current continuous position of the workpiece is described by the token quantity rw of the place position. If the condition rw = xmax is fulfilled, the transition T2 fires and removes the token of the place workpiece present. Further, the token quantity of place position will be removed. In the result the token arrives the place exit (out= 1).
Fig. 11 : Elementary hybrid net of a comparator In the net in Fig. 11, the 3 different states to be distinguished are represented by the places less, greater and equal. The net is marked by one token (e.g. e=l). The concessions of the transitions Tl to T4 are controlled by the arc weights. The token circulates between the 3 places - and models the correct current state in this way. The published places (grey coloured) can be used to influence the token quantities of the places x2 and xl from outside and to observe the discrete states. These places are shown as interface places in the object representation (s. Fig. 12).
This model is universal and can be reused flexibly for different conveyors with individual rates and lengths. The net will be encapsulated, the published interface places are grey coloured can be modified from the environment, using an instance of the class "hybrid
149
The continuous transition TK models the movement of the robot by integration of the robot speed over the time. The current robot position is described by the token quantity in place robot position. Fig. 12: Instance of a comparator with its interfaces
If the object comparator_I discovers the identity of the positions of the robot and of the workpiece, the transition position achieved? will be enabled. After its firing the place position achieved will be marked. The transition start movement even fires and stores a token quantity, corresponding to the transport rate of the conveyor, into the place robot speed.
Complete hybrid model of the robot: The essential components of the model are the before presented models of the discrete sequencer, the continuous places robot speed and robot position (Fig. 13: places in the middle and right, drawn largely) as well as two instances of the class "comparator" (Fig. 13).
The timed transition TK models the time consumption of the processing task and fires after 3 units of time. Afterwards the place workpiece processed will be marked, the processing is ended. Using the object comparatorj , it will be evaluated, if the robot position is behind or before its initial position. It starts moving with maximal rate until the comparator_2 detect, that the initial position is achieved. This enables the transition initial position achieved - the robot is in its initial position. In order to prevent a multiple processing of a workpiece, the place workpiece processed will be marked.
This model evaluates the signals of the conveyor, compares the current or the initial position of the robot with that of the workpiece and controls the robot for properly processing the workpiece. Interfaces which allow communication with the conveyor are defined (see Fig. 14). If the robot is in its initial position (m2= 1) and a workpiece achieved the conveyor (prs= 1), the transition workpiece achieved will fire and mark the places search processing position as well as P7. By the marking of P7, the object comparator_I determines, if the robot is localised before or behind the current workpiece position. If the robot is in a position xr>xw, the place robot speed will be filled with a token quantity corresponding to -vmax.
..
~--~~
4.3.
Model of the whole system
The total model is made of the composition of each an instance of the class robot as well as for the class
B
initial position workpiece ·d ___ - _prs present occuple
--
comparator_1
workpiece position
-------------------£~'~~o~e ~~------~/~--------,
---------
I I
I
T1 .....>----\
I
I
_---1
I
robot position
l-v
-v
__ - i\..J!comparator
P8
ca
maxJmum speed
Fig. 13: Object model for the hybrid manufacturing system
150
position
conveyor. The objects can communicate with each other by adding three fusion arcs. So both objects describe the hybrid dynamic behaviour of the whole system. The place P demonstrates a possibility to bring workpieces into the system (see Fig. 14).
From the special view on manufacturing systems, the proposed modelling method, based on Petri Nets, will reflect the complex system behaviour in a more realistic way. The main goal hereby is to find an intuitive graphical method to describe complex hybrid systems by allowing the verification of the system model using simulation. All of these aspects are considered by Hybrid Object Nets. All nets are designed and simulated by the tool Visual Object Net ++, which is freely available in an evaluation version in (Drath, 1998b).
p
6. ACKNOWLEDGMENT This research is supported by the DFG (Deutsche Forschungsgemeinschaft, (German research association» as a part of the investigation project "Analysis and synthesis of mixed continuous and discrete technical systems" (KONDISK) with the subject "Analysis and synthesis of hybrid subprocesses in flexible manufacturing systems - examinations to an object oriented systems engineering".
Fig. 14: Connections between object for interaction 4.4.
Simulation results
Fig. 15 represents the trajectories for the continuous position of the robot and the values for its rate. It shows the phases for positioning above the workpiece, processing it and return to the initial position of the robot. The system behaviour is context-related and hard to be described using classical continuous or discrete event description methods.
REFERENCES Alia H., David R., (1987). Continuous Petri Nets, Proceedings of the European Workshop on Applications and Theory of Petri Nets, pp. 275-294, Spain,1987. Alia, H., David, R., Bail, J. (1991). Hybrid Petri Nets, ECC European Control Conference, Grenoble, France. Alia, H., David, R., Bail, J. (1992). Asymptotic Continuous Petri-nets: An Efficient Approximation of Discrete Event Systems, International Conference on Robotics and Automation, Nice, France. DiCesare, F., Harhalakis, G., Proth, J.M., Silva, M., Vemadat, F. B. (1993). Practice of Petri Nets in Manufacturing; Chapman & Hall 1993 . Drath, R., Schwuchow, S. (1997): ModeUierung diskret-kontinuierlicher Systeme mit Petri-Netzen. In: Schnieder E. (Hrsg.); EntwurJ komplexer Automatisierungssysteme, 5. Fachtagung, Braunschweig. Drath, R. (1997): Objektorientierte Modellierung hybrider Prozesse - Vorstellung eines neuen Werkzeuges. 42. IWK, TU Ilmenau, 1997; Bd. 3, p. 533-540. Drath, R. (1998a). Hybrid Object Nets: An Object Oriented Concept for Modelling Complex Hybrid Systems. In: Dynamical Hybrid Systems, ADPM98, Reims. Drath, R. (1998b). Tool Visual Object Net ++ , http://www.daimi.au.dklPetriNets/too Is/comp lete_ db.html, section "Visual Object Net ++" Schwuchow, Susan (1997): Petri-Netz-basierte Struktur- und Verhaltensplanung von Automatisierungssystemen der flexiblen Fertigung; 42. IWK, TU Ilmenau, 1997; Bd. 3, p. 520-527.
I1I
Fig. 15 : Simulation results
5. CONCLUSION The integration of different process types in manufacturing systems is rapid increased by the combination of dynamic influences of each process. Therefore the Hybrid Object Nets are proposed. In this way it's possible to get more flexibility compared with the common approximation principles.
151
HYBRID ASPECTS OF MODELLING MANUFACTURING SYSTEMS USING MODIFIED PETRI NETS
Rainer Dratb, Ulricb Engmann, Susan Scbwucbow
Technical University of Ilmenau, Department ofAutomatic Control and Systems Engineering, Postfach 100565,98684 Ilmenau, Germany email: [email protected] [email protected] [email protected]
Abstract: This paper presents a new special view on hybrid modelling of flexible manufacturing systems using a new description method basing on Petri nets (Hybrid Dynamic Nets, HDN). It shows, why hybrid modelling is useful for the design process instead of the currently preferred view based on discrete events. In order to solve handling problems, arising from the system complexity of larger systems, we combine this method with the object oriented paradigm. In the result we get Hybrid Object Nets (HON). Copyright © 1998 IFAC Keywords: Hybrid, Dynamic Modelling, Petri nets, Object Oriented Modelling, Flexible Manufacturing Systems
1. INTRODUCTION AND BACKGROUND
need a model including higher order variables and disturbing effects and we get more complexity in this way. Other samples are transport and handling processes on flexible manufacturing systems, which allow the processing of workpieces during its movement, or gripping processes of interacting robots. As a third example for special production processes with a combination of process industry system parts and manufacturing production system parts, a closed combined examination of continuous and discrete event processes is needed. Such technical processes are e.g. metalhardening, tempering processes or galvanic processes. For the development of automation solutions of these hybrid systems it's necessary to give a hybrid model emphasis.
The most known modelling techniques and description methods for manufacturing systems are based on the product oriented view. The main manufacturing processes are typically considered as groups of assembling, machining, handling, transport and store parts. They seem to be discrete-event-oriented, therefore these processes are modelled in a discrete-eventoriented way. But to solve detail problems, sometimes the continuous dynamic behavior is necessary to be described. Often constant time values are used on the assumption, that they sufficient reflect the time consumption of continuous systems. Nevertheless, not all problems and properties are reflected and the real process would not be sufficient described.
Considering these demands, a hybrid modelling technique would allow to model more complex dynamic structures. Hybrid systems are the combination of continuous dynamic systems and discrete event dynamic systems (DEDS). In this approach, the derivation of events does not only depends on approximations, but on a dynamic model of the continuous processes, which generate them. If
The following manufacturing applications show the necessity of hybrid modelling techniques because of their hybrid nature. As a typical case e.g. is the handling of parts which have different weights. The transportation of a part from point A to B shows different dynamic behavior due to variability of the target ending points and to the part weight. Such systems
145
Alia, David, LeBail, 1991» a function of the marking from the continuous net places.
disturbances are considered too, consequently the events depends on these processes.
Because of the differences between the origin Petri nets and this new approach, it is no longer called Petri net but Hybrid Dynamic Net (HDN) . The proposed solution for describing hybrid processes is based on the HDN and allows the modelling of the dependencies of all system parts with only one description method. The following conventional discrete and new continuous net elements are included:
For modelling hybrid dynamical systems, a useful description method is needed. Therefore an enhanced hybrid net approach (Hybrid Dynamic Nets, HDN), based on Petri nets, is shortly presented here. Because Petri nets are established and often used to describe manufacturing processes (DiCesare, et al. , 1993), the presented approach bases on Petri nets. In order to solve handling problems, arising from the system complexity of larger systems, we combine this method with the object oriented concept. In the result we get Hybrid Object Nets (HON) which are also shortly presented in this paper. The HON approach is finally demonstrated by means of an example.
TD
discrete transition
! p, I
i~ T1 1,:2 1 P2 ·
m2
I, T2
1..=3
.
~1 o
1
2
m2J
2.1.
m2 o
Continuous basic element
Fig. 2 shows a continuous basic element. The firing speed v is the speed of token flow into the place P2. The transition Tl is always active; it can be inactivated only by empty discrete input places. The test arc does not allow token flow. This makes it possible to model subsystems without feedback: the token quantity of P J here is not influenced.
v1
v2
continuous place
.0
traditional Petri net with tokens
rn1
continuous / Petri net
continuous transition
The main idea is to assign the continuous transitions to a firing speed, which is represented by an equation. This equation is a function of token quantities of arbitrary places of the net. The shown arcs below are used for the main normal arc relations between inhibitor the different transitest arc tion and places types.
rn1
- - -;
CS
Continuous transitions are new and not comparable with the discrete ones. Instead of discrete firing, their activity leads to a continuous flow. Continuous places contain tokens, which are interpreted as real values.
The new approach shortly presented here is characterized by modification of the conventional Petri nets in combination with an extended class of continuous Petri Nets (Alia, et aI., 1991 , 1992; Drath, 1998a), to Hybrid Petri Nets (Drath, et aI., 1997; Drath, 1997).
i
discrete place
PC
0
0
I
2. ABOUT HYBRID DYNAMIC NETS (HDN)
I
TC
PD
I
Fig. 1: From conventional Petri net to continuous Petri net
P1
The state space of the upper traditional net in Fig. 1 is shaped however only by three discrete states. The two initial tokens in place PI can distribute themselves due to the times of the transition like represented here. The model below shows the extension with continuous places and transitions (Alia, 1987). Here, a token is not any more an individual, but a real quantity of token fragments . The transition moves with a velocity of flow the token fragments from the place before into the place thereafter. The state space becomes infinitely in this way and this opens the possibility of modelling continuous dynamic.
T1
P2
@-------[]---6 quantity of tokens in PI quantity of tokens in P2 m3(t) ... mj(t): quantity of remaining places of the net v(t): firing speed ml(t): mit):
Fig. 2: Net, including only continuous net elements ofHDN If a single input arc is directed to P2, we get (1).
v = dm 2 (t) dt
The suggestion and the goal of the in the following presented new approach is, that the firing speed can be (in contrast to the approach from (Alla, 1987;
146
= m,(t) .
(1)
continuous net elements allow the modelling of continuous systems using ordinary differential equations. In (Drath, et aI., 1997) we explained in detail how to combine discrete and continuous elements in order to model hybrid systems. Since a time concept is included, it is possible to model real time systems.
For a place Pj with i input arcs we get (2).
m(t) = dm/t) = "v ~ ~ . J
m
This corresponds to the node theorem. Continuous input and output transitions supply their part to increase or to decrease mj.
3. HYBRID DYNAMIC NETS AND OBJECT ORIENTAnON
We can model different basic elements in this way. Even non-linear coupled subsystems can be described. 2.2.
3.1.
Samples
The most mathematical, textual or graphical approaches to describe hybrid systems are currently usable for small examples. Models of complex systems are unwieldy. Therefore a hierarchical concept to structure a model is needed.
First order system: The fIrst order system (Fig. 3) is a basic continuous element consisting of one input and one output place. The fIring speed function (3) is assigned to the transition Tl . Fig. 3 also shows the step response of the system.
In order to solve the mentioned handling problems arising from the system complexity, we applied the object oriented paradigm to HDN, resulting in a new method to describe both continuous and discrete event systems with reduced effort: Hybrid Object Nets (HON). One of the important advantages of the using this concept is the ability to describe a larger system by the decomposition into interacting objects. Because of the properties of objects, the modifIcation of the system model could be easier achieved. The object oriented concept unites the advantages of the modules and hierarchies and adds useful concepts like reuse, encapsulation and information hiding. In this way we get more flexibility.
Fig. 3: Application example, fIrst order dynamic system v(t) = u - 0.1 · Y , this leads to:
(3)
Y = (u - 0.1 . y) dt
(4)
f
Oscillator: With HDN we can describe differential equations of fIrst order. In order to describe differential equations of order n they have to be formed into n differential equations of fIrst order. So the wellknown differential equation for an oscillator can be transformed into the equation system (5).
With Y=
Xl
and x 2 =
Xl
3.2.
General Properties of the HON
HON includes concepts for attributes, methods, interfaces, encapsulation, inheritance, abstraction, data exchange and reusing.
we get (5)
Attributes are represented by places and their contained token quantities. Methods are given in the form of the net structure. Information hiding is realised by encapsulation the detail informations of the net structure, and by publishing selected places using an interface. Abstraction is the step from a concrete net structure to a class: it is realised by fIlling the objects into a class hierarchy. Inheritance is the step from a class to a concrete object, so called instance of the class. If an object will be abstracted from a class, it inherits the whole net structure including the interface. Data exchange is given by the token flow between the objects. Discrete tokens can model method calls and discrete system states; continuous tokens model continuous control variables. ReUSing, the most important quality of object orientation, is given by inheriting or instancing classes. Derived objects can be refIned; places, transitions, arcs and objects can be added: but no inherited element can be deleted.
With D = 0.25, Wo = 1 and Kp=l, the continuous Petri net below (Fig. 4) can model this equation system (5). The step response is also shown in Fig. 4.
Fig. 4: Application example, second order dynamic system 2.3.
Motivation
Hybrid Dynamic Nets (HDN)
The essence of Hybrid Dynamic Nets is the combination of continuous and discrete net elements. The
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3.3.
Construction of a class
To construct a new modelled using the published, and the Afterwards the net chy. 3.4.
class, a suited subsystem must be HDN. The places, which shall be parent class have to be defmed. can be fitted into a class hierar-
Structure of an object
Fig. 6: Principal scetch of the manufacturing system for hybrid modelling
To generate an object, it has to be instanced from a class. Every object has a hierarchical structure which contains three layers (see Fig. 5) leaned on the two layer concept in (Schwuchow, 1997). In the supreme layer the object frame is presented, which encapsulates the inner net structure of the object, and which allows the communication with the environment.
If a workpiece reaches the range of the robot, the robot should occupy a position above the passing part from its initial position, process the part in passing and then occupy its initial position again. Its speed should be designated as V respectively Vma;c . The movement of the robot is determined by a guide rail, installed in the direction ofthe x-axis.
In the underlying second layer, the net, inherited by the class, is enclosed in an object frame . In this layer, further net elements and objects can be added in order to modify the behaviour of the object and form new subclasses. In the lowest layer, the net, inherited by the parent class, is represented. It can not be modified here but can only be screened.
initial Conditions: No part might be on the conveyor and the robot might be in position of rest. Destination: A model for this system should be constructed, which encapsulates the functionality and the dynamic of the robot and that of the conveyor in each an object. In such a way, both objects should be connected with each other in order to model the dynamic behaviour of the whole system. 4.2.
Modelling Procedure
Discrete model of the discrete event sequence controller of the robot: The function of the robot can be described by a sequence of operations. The Precondition for any activity of the robot is the occupation of its initial position, since calibrating procedures take place there.
Fig. 5: Example of the Layer-Concept for HON
If a workpiece reaches the range of the robot, the robot trolley must look up the position above the workpiece. If it reaches this position and if there is enough time for finishing left, it starts processing the workpiece. Afterwards the robot returns to its initial position (see Fig. 7).
4. APPLICA TlON EXAMPLE The considered application example is a hybrid manufacturing system, which is characterised by the interaction of a controlled robot and a mobile workpiece. The robot carries out continuous positioning procedures, which are influenced by the movement of the passing part while processing.
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robot initial position occupied workpiece arrived? search for workpiece position workpiece position achieved ?
4.1.
workpiece position achived
Configuration
start of processing
The considered system consists of a robot, which processes passing workpieces, and of a conveyor, which transports workpieces into the range of the robot. In this case, the movement of the parts occurs in positive direction ofthe x-axis of the system (see Fig. 6). The position of the workpiece might be designated as xw, that of the robot as xr. The length of the range of the robot might be designated as xmax.
workpiece is currently processed processing finished start look for robot initial position robot initial position achived ?
Fig. 7: Control sequence for the robot The represented discrete Petri net models this course in its causal order. Since it is a causal model, it is not
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possible to detennine the time, which is consumed by the individual processing steps. Since the individual operations are characterised by a situation dependent dynamic, it is not possible to estimate the delay of them.
conveyor".
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Considering these characteristics, dynamic models for the underlying dynamic processes should be created, which describe the temporal behaviour of the continuous processes.
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v
xW(l,)
xw(t,)
xw
Fig. 10: HON-model for a conveyor
Fig. 8: Principal Scetch of the conveyor necessary parameters
Hybrid model of a comparator: Comparison operations are carried out several times. The current position of the robot trolley must be compared with that of the workpiece on the conveyor. For this reason, an object, which makes this operation available should be constructed.
Hybrid model of the conveyor: The conveyor (s. Fig. 8) can be described by the net shown in Fig. 9. If a workpiece reaches the conveyor, the place entry will be marked (in=l, the value 1 will be interpreted as TRUE). Afterwards the transition Tl get the concession and fires a token into the place workpiece present (prs= 1). The token remains here during the entire stay time of the workpiece.
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traditional discrete net model :-
--
. --- ._.- --. - .---. ----_.--- --.-.- ._. _. workpiece
The comparator object should compare two continuous control variables x2 and xl and derive three discrete states from it: x2>xl, x2=xl or x2
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T1 present T2 exit lT3 . -: ._.--_._._._.- -.-.~.-.-.-.-.-. 'max- -.~- -~-: entry
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'-- . ~ .speed ~.-. _. _.- - _._. .- ---_._. _._._.- .-- -._ ._._. --'J oontmuous net pan
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workpiece is processed
Fig. 9: Hybrid net of a conveyor The process of transport of the conveyor is modelled by the continuous transition TB, where the speed function is assigned to. The transport speed is described by the variable v in the place transport speed. The continuous transition TB is activated by the token of the place workpiece present. In the result the current continuous position of the workpiece is described by the token quantity rw of the place position. If the condition rw = xmax is fulfilled, the transition T2 fires and removes the token of the place workpiece present. Further, the token quantity of place position will be removed. In the result the token arrives the place exit (out= 1).
Fig. 11 : Elementary hybrid net of a comparator In the net in Fig. 11, the 3 different states to be distinguished are represented by the places less, greater and equal. The net is marked by one token (e.g. e=l). The concessions of the transitions Tl to T4 are controlled by the arc weights. The token circulates between the 3 places - and models the correct current state in this way. The published places (grey coloured) can be used to influence the token quantities of the places x2 and xl from outside and to observe the discrete states. These places are shown as interface places in the object representation (s. Fig. 12).
This model is universal and can be reused flexibly for different conveyors with individual rates and lengths. The net will be encapsulated, the published interface places are grey coloured can be modified from the environment, using an instance of the class "hybrid
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The continuous transition TK models the movement of the robot by integration of the robot speed over the time. The current robot position is described by the token quantity in place robot position. Fig. 12: Instance of a comparator with its interfaces
If the object comparator_I discovers the identity of the positions of the robot and of the workpiece, the transition position achieved? will be enabled. After its firing the place position achieved will be marked. The transition start movement even fires and stores a token quantity, corresponding to the transport rate of the conveyor, into the place robot speed.
Complete hybrid model of the robot: The essential components of the model are the before presented models of the discrete sequencer, the continuous places robot speed and robot position (Fig. 13: places in the middle and right, drawn largely) as well as two instances of the class "comparator" (Fig. 13).
The timed transition TK models the time consumption of the processing task and fires after 3 units of time. Afterwards the place workpiece processed will be marked, the processing is ended. Using the object comparatorj , it will be evaluated, if the robot position is behind or before its initial position. It starts moving with maximal rate until the comparator_2 detect, that the initial position is achieved. This enables the transition initial position achieved - the robot is in its initial position. In order to prevent a multiple processing of a workpiece, the place workpiece processed will be marked.
This model evaluates the signals of the conveyor, compares the current or the initial position of the robot with that of the workpiece and controls the robot for properly processing the workpiece. Interfaces which allow communication with the conveyor are defined (see Fig. 14). If the robot is in its initial position (m2= 1) and a workpiece achieved the conveyor (prs= 1), the transition workpiece achieved will fire and mark the places search processing position as well as P7. By the marking of P7, the object comparator_I determines, if the robot is localised before or behind the current workpiece position. If the robot is in a position xr>xw, the place robot speed will be filled with a token quantity corresponding to -vmax.
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4.3.
Model of the whole system
The total model is made of the composition of each an instance of the class robot as well as for the class
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initial position workpiece ·d ___ - _prs present occuple
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Fig. 13: Object model for the hybrid manufacturing system
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position
conveyor. The objects can communicate with each other by adding three fusion arcs. So both objects describe the hybrid dynamic behaviour of the whole system. The place P demonstrates a possibility to bring workpieces into the system (see Fig. 14).
From the special view on manufacturing systems, the proposed modelling method, based on Petri Nets, will reflect the complex system behaviour in a more realistic way. The main goal hereby is to find an intuitive graphical method to describe complex hybrid systems by allowing the verification of the system model using simulation. All of these aspects are considered by Hybrid Object Nets. All nets are designed and simulated by the tool Visual Object Net ++, which is freely available in an evaluation version in (Drath, 1998b).
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6. ACKNOWLEDGMENT This research is supported by the DFG (Deutsche Forschungsgemeinschaft, (German research association» as a part of the investigation project "Analysis and synthesis of mixed continuous and discrete technical systems" (KONDISK) with the subject "Analysis and synthesis of hybrid subprocesses in flexible manufacturing systems - examinations to an object oriented systems engineering".
Fig. 14: Connections between object for interaction 4.4.
Simulation results
Fig. 15 represents the trajectories for the continuous position of the robot and the values for its rate. It shows the phases for positioning above the workpiece, processing it and return to the initial position of the robot. The system behaviour is context-related and hard to be described using classical continuous or discrete event description methods.
REFERENCES Alia H., David R., (1987). Continuous Petri Nets, Proceedings of the European Workshop on Applications and Theory of Petri Nets, pp. 275-294, Spain,1987. Alia, H., David, R., Bail, J. (1991). Hybrid Petri Nets, ECC European Control Conference, Grenoble, France. Alia, H., David, R., Bail, J. (1992). Asymptotic Continuous Petri-nets: An Efficient Approximation of Discrete Event Systems, International Conference on Robotics and Automation, Nice, France. DiCesare, F., Harhalakis, G., Proth, J.M., Silva, M., Vemadat, F. B. (1993). Practice of Petri Nets in Manufacturing; Chapman & Hall 1993 . Drath, R., Schwuchow, S. (1997): ModeUierung diskret-kontinuierlicher Systeme mit Petri-Netzen. In: Schnieder E. (Hrsg.); EntwurJ komplexer Automatisierungssysteme, 5. Fachtagung, Braunschweig. Drath, R. (1997): Objektorientierte Modellierung hybrider Prozesse - Vorstellung eines neuen Werkzeuges. 42. IWK, TU Ilmenau, 1997; Bd. 3, p. 533-540. Drath, R. (1998a). Hybrid Object Nets: An Object Oriented Concept for Modelling Complex Hybrid Systems. In: Dynamical Hybrid Systems, ADPM98, Reims. Drath, R. (1998b). Tool Visual Object Net ++ , http://www.daimi.au.dklPetriNets/too Is/comp lete_ db.html, section "Visual Object Net ++" Schwuchow, Susan (1997): Petri-Netz-basierte Struktur- und Verhaltensplanung von Automatisierungssystemen der flexiblen Fertigung; 42. IWK, TU Ilmenau, 1997; Bd. 3, p. 520-527.
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Fig. 15 : Simulation results
5. CONCLUSION The integration of different process types in manufacturing systems is rapid increased by the combination of dynamic influences of each process. Therefore the Hybrid Object Nets are proposed. In this way it's possible to get more flexibility compared with the common approximation principles.
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