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The Development of the AGVS Model by the Union of the Modulised Floor-path Nets

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Int J Adv Manuf Technol (1994) 9:20-34 9 1994 Springer-Verlag London Limited

The International Journal of

fldvanced

manufacturing

Technolog.q

The Development of an AGVS Model by the Union of the

Modulised Floor-Path Nets

Suhua Hsieh and Ying-Jer Shih

Department of Mechanical Engineering, National Taiwan University, Taipei. ROC

The purpose o f this paper is to propose an easy and quick method f o r the development o f an automated guided vehicle system ( A G V S ) model. It suggests that the A G V S model can be developed directly by the union o f modulised floor-path nets without any modification. Several modulised floor-path nets for both uni- and bi-directional systems are established in this paper. To make sure that the complete model obtained by this method is robust, the modulised floor-path nets are required to possess several essential properties - safeness, boundedness, conservation, reachability and liveness. To keep these properties in the complete model after the union, three union rules and an union procedure are proposed.

Keywords:

A G V S ; Petri-net properties; Robust; Union

1. Introduction

As mentioned in [1,2], to build a robust automated guided vehicle system ( A G V S ) needs very carefully study and design from different aspects (such as the existing plant layout, the manufacturing processes, the cooperate management policies, etc.). In [3], a three-level (vehicle-level, floor-level and management-level) robustness concept for an A G V S has been specified. The technology of the vehicle level is relatively mature when compared to that of the other two levels. This is because most new users usually do not realise that the A G V S design has three levels and the design at the second and third levels depends on the type of application. Therefore, users are often not involved in the design of the A G V S as much as they should be. This results in the failure of the AGVS, The design at the management level is totally case- dependent. Fortunately, the design at shop floor level becomes common to every case, if the floor-path layout is determined.

Correspondence and offprint requests to: Sushua Hsieh, Department

of Mechanical Engineering, National Taiwan University, 1 Roosevelt Road Sec 4, Taipei, Taiwan 10764, Republic of China.

The purpose of this paper is to propose an easy and quick modelling tool which can help users to build a robust system at shop floor level. The modelling method is to establish several basic modulised floor-path nets, and then, by the union of these nets, a complete system model can easily be developed.

A similar idea has been proposed in [1-3]. Four uni- directional basic subnets - line, merge, divide and intersection - for the A G V S have been established in [2]. The A G V S Petri- net structure, the corresponding A G V S Petri-net properties (safeness, boundedness, strict conservation, reachability and liveness), the way to embed those properties into the nets, and the way to identify the robustness of the nets have been discussed thoroughly in [3]. In this paper, based on previous research results, more different types of subnets for both uni- or bi-directional systems will be established, and the robustness of every net will be proved. For the purpose of simplifying the union steps, the submodel modulised concept is adopted in this paper. Therefore, the complete model can be developed straightforwardly by connecting the input part of a subnet with the output part of another subnet. The modulised floor- path net is thus given as the name instead of the subnet.

2. Reviews of AGVS and Petri-nets

Structures

Petri nets were chosen as the modelling language for the development of the A G V S modelling tool. The A G V S robustness properties and its corresponding Petri-net structure and properties will be reviewed here.

2.1 Petri Net Review [3,4]

A Petri net may have safeness, boundedness, strict conser- vation, reachability and liveness properties. By observing its reachability tree, one can examine the safeness, boundedness and conservation properties of a net. It has been proved that the reachability problem is equivalent to the liveness problem. Hence, if a net is live, all the paths of the net are reachable. Although the reachability tree does not necessarily always

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contain enough information to solve the liveness p~oblem, it is the case that the net is live if all separate subtrees are associated with markings which are the same as that of the frontier node, and every transition has been fired at least once. By the use of the incident matrix equation, if every element of the weighting vector is 1, the Petri net is strictly conservative. If a net is conservative, the net must be bounded.

2.2 AGVS and its Petri Nets [3]

According to [3], in order to keep an A G V S functioning well, the A G V S should have four properties, system collision free, constant numbers of vehicles and traffic control signals, paths reachable and system deadlock free, which correspond respectively to Petri-net properties, safeness, boundedness, strict conservation, reachability and liveness. Therefore, the safeness, boundedness, strict conservation, reachability and liveness are the necessary Petri-net properties in the A G V S floor-path net. For the implementation of the zone control function by Petri nets, zones are distiguished as physical zones, pseudo-zones and composite pseudo-zones.

An A G V S floor-path net is composed of a flow-path net and a control-loop net. The flow-path net reflects the physical flow-path layout, and the control-loop net represents the traffic control components. In the flow-path net, each place represents either a physical zone, a pseudo-zone, or a composite pseudo-zone. The direction of the traffic flow is represented by the directed arc. Tokens in places represent vehicles. Transitions represent permissions for vehicles to move from one place to another place. Each place in the control-loop net represents a control node. Tokens in each place represent signals or commands. Directed arcs are used to direct the flow of signals or commands. There is no other set of transitions which belongs to the control-loop net itself. Both the control-loop net and the flow-path net share the same transitions. To fulfil the zone control function, the transition can be fired only when the input places of the flow- path net and control-loop net have enough tokens and the output place of the flow-path net is empty. The complete net can be obtained by placing the control-loop net over the flow- path net.

In an A G V S floor-path net:

1. The flow-path net is not a closed-system net, while the control-loop net is.

2. The necessary properties can be embeded into the nets by adding extra places at proper locations and checked by the use of its reachability tree or incident matrix equations. 3. If those properties exist in both the flow-path net and the

control-loop nets, they also exist in the complete net.

3. Modulised Floor-Path Nets

In this section, several basic floor-path nets will be established. For the purpose of simplifying the union procedure, in this study, not only is the modulised concept adopted, but also

the interface between two nets such as the input of a net and/or the output of a net is clearly defined, so that, at the union stage, one can directly connect the output (or input) zone of a net with the input (or output) zone of the other net. Both input and output zones belong to the physical zone. Since every flow-path net represents a feature of the A G V S floor-path structure, it is not necessary to be a closed system. In order to save the information in the net, the control-loop net is always a closed system. The modulised floor-path net is therefore not a closed system. Because of this, all the property analyses discussed in this paper are within the input and output zones of the net. The discussion of the liveness property for those nets has to be modified. For any flow-path net, if a vehicle at the input zone can arrive safely at one of the output zones, and, as long as that vehicle is removed from the output zone, any other vehicle at an input zone can always move to one of the output zones, and if, by continuously removing vehicles from output zones, it can keep any vehicle (if there is at least one) at an input zone moving to the output zone, it is said that the net has the potential to be live. Therefore, in this paper, if the reachability tree of a non-closed net shows that every individual vehicle at the input zone can move to the output zone, it is said that the net is live within input and output zones.

For the purpose of distinguishing the flow-path net and control-loop net, solid lines are used to draw the flow-path net, and dashed lines are used to draw the control-loop net. Big dark circles represent tokens in the flow-path net, and small dark circles represent tokens in the control-loop net. For analysis purposes, tokens representing vehicles will be placed at the input zones of the flow-path net in the paper. In fact, at the floor-level design stage, vehicles are not there. Hence, in any modulised floor-path net, big circle tokens should be eliminated once the anlayses have been done, while small circle tokens should always be there to ensure the traffic control functions.

3.1 Uni-direcUonal System

Most of the existing A G V S s are uni-directional. This is because the control of the system is much simpler. Vehicles in the system move in direction, which results in the workpart transportation sometimes being very inefficient if the delivery station happens to be next to the pick-up station, but in the opposite direction to the traffic flow. Usually, additional spur track added in a proper way solve the problem. In the following, based on the uni-directional system, several modulised floor- path nets will be established and analysed one by one.

Line

The line structure connects one zone with another (see Fig. 1). Z~ is the input zone and Z2 is the output zone. The flow-

Z l Z2

i i i i

(3)

22 S. Hsieh and Y,-.I. Shih Zl

|

_1

t l

-I

Z2

Fig. 2. The line net.

path net of the line structure which consists o f two places, Zt and Zz, and o n e transition, t~, is established in Fig. 2. Since it is so simple, it d o e s not n e e d any control node. T h e r e f o r e , the control-loop net is not necessary.

For the analysis, a token which r e p r e s e n t s a vehicle is placed in the input zone Z~. T h e initial marking of the net is ( Z I , Z 2 ) = ( l , 0 ) . At this m o m e n t , tj is fireable. A f t e r tt is fired, the marking b e c o m e s (0,1). It is m e n t i o n e d above that the property analyses are done within input and output zones. H e n c e , all the possible states of the net have been presented. The total n u m b e r o f vehicles for each possible state is 1, and the maximum tokens in any place is 1. T h e r e f o r e , the net is safe as well as strictly conservative. In fact, because o f the modified transition firing rule, tokens in the flow-path net can never exceed 1. H e n c e , the safeness property is always true in any flow-path net. From now on, the safeness property will not be c h e c k e d again. A s for the reachability property, in this net, there are only two elements, (1,0) and (0,1), in the reachability set. and only two segments of paths. H e n c e , every path in the net is reachable. It has been m e n t i o n e d above that the reachability p r o b l e m is equivalent to the liveness problem. In the rest o f the paper, as long as the net is live. the reachability property is true too. As for the liveness property, because a vehicle at Z~ can always arrive at Z2 if t~ is fired, hence, the net is live within input and output zones.

Although the line net is so simple, it is the most c o m m o n net in the A G V S . The property analysis o f the line net is as important as those o f o t h e r nets.

Divide

As shown in Fig. 3, in a d i v i d e structure, a vehicle can m o v e from zone Z~ to either zone Z2 or Z3. In this structure, there is no need to have control functions o t h e r than the zone control function. T h e r e f o r e , the control-loop net is not necessary for this structure. A c c o r d i n g to the structure, the divide flow-path net is established in Fig. 4. There are three places Z~, Z2 and Z.~ r e p r e s e n t i n g the three physical zones, and two transitions t~ and t2. Z~ is the input zone and Z2 and Z3 are the output zones o f the net. For analysis purposes, a

Z 1

Z2

_ V " ' [

,,-},,

~_ C - " I

Z~

Fig. 3. The divide floor-path structure.

Z2

_I'

zll

-I

- [ - - Fig. 4. The divide net.

Z! Z3 Z4

[ - - I _ ~ ' - ' r ~ l _ L_.J - / ~ _ _ j i i

r

Fig. g. The merge floor-path structure.

token is placed in Z~. A letter D in a box is the symbol of the modulised divide net. It is e x p e c t e d that a desired net will be automatically built at a desired location if the user places the symbol o f the desired net at the desired location.

In Fig. 4, the initial marking is (1,0,0), A f t e r t~ or t2 is fired, the new marking is (0,1,0) or (0,0,1). Again, property analyses are done b e t w e e n input and output zones. T h e r e f o r e , conclusions can be made regarding the properties. Since the total n u m b e r o f tokens is 1 at any state, the net is strictly conservative. T h e vehicle at the input zones can arrive at either o f the output zones. H e n c e , the net is live within the input and output zones.

Merge

A modified merge structure has been discussed in [3]. H e r e , the merge structure wil be studied again within the input and output zones.

Fig. 5 is the physical merge structure; two lanes merge into one lane. Z~ and Z2 are the input zones, and Z4 is the output zone. The non-closed flow-path net is shown in Fig. 6. Since the control-loop net has been analysed in

[31,

it is not necessary to redraw and analyse it here again.

Z~ t~

|

,

Z3 t a Z4

t2

(i'

(4)

The incident matrix o f the flow-path net is: Zi Z., Z3 Z4

,[i 010

Nf = t2 - 1 1 0 t~ 0 - 1 1

T h e r e exists a weighting vector Wf = (1,1,1,1) T such that N f . Wf = 0. Thus, the flow-path net is strictly conservative.

In Fig. 6, the initial marking is (1,1,0,0). O n c e tt (or t2) is fired, the new marking is (0,1,1,0) ( o r (1,0,1,0)). Since Z3 is a p s e u d o - z o n e , /5 has to be fired immediately after the firing of t~ ( o r t2). A f t e r 15 is fired, the marking b e c o m e s (0,1,0,1) (or (1,0,0,1)). since each individual vehicle at the input zone can safely arrive at the output zone, the net is live within the input and output zones.

To c o m b i n e the flow-path and control-loop nets, the complete merge net can be obtained as is shown in Fig. 7. A n "'M" in a box is its symbol, the places Z3 and C2 are piled one upon another. From Fig. 7, one can see that: 1. W h e n tj is fired, the token in C~ is m o v e d to C2- H e n c e ,

t_, cannot be fired. This g u a r a n t e e s that the vehicle from Zt will not collide with the vehicle from Z2. Because Z3 is a p s e u d o - z o n e , the vehicle from Z~ cannot stay (pseudo- zones do not have enough space to hold a vehicle) at Zs and has to leave immediately, t3 has to be fired immediately after t~ is fired. A vehicle from Zt thus e n t e r s Z4, and C= returns the token to C~ in o r d e r to reset the control status of the net.

2. A f t e r the vehicle from Z~ e n t e r s Z4, t2 satisfies the firing rules. H o w e v e r , since the i m m e d i a t e physical zone Z4 is not free, t2 is not fireable. T h e vehicle at Z2 has to wait until the vehicle at Z4 leaves the net. As long as the vehicle at Z~ leaves the net, the net is live again. 3. Similarly, at this m o m e n t , if a vehicle enters Z~, it has to

wait until the net is live again.

Since both the flow-path net and the control-loop net satisfy the robustness properties, the c o m p l e t e net also has properties. For the p u r p o s e of supporting the s t a t e m e n t "the net is live within the input and output z o n e s " , the e x t e n d e d reachability tree of the net is drawn in Fig. 8. In Fig. 8, those net states above the d a s h e d line are the states within the input and output zones. They indicate that each individual vehicle at the input zone can arrive at the output zone. Those net states below the dash line are the future states where the output

Z l tl

|

,

:

(

t Z3 t3 Z4

F'

(Z~ ,Z2,Z~,Z 4,C~ ,C z ) II (1,1,0,0,1,0) (o,1,1,1,o,1)(1,O,l,O,O.1)

r .OAt.O) (1,o,oJ J,o 2 . . .

, 0 , 0 0 , 0 , , , 0 0 0 , 0 ,

(001001) (001001)

,00:r

, /

\,-

,oooo,o,

,oooo,o,

Fig. 8. The extended reachability tree of the merge net. zone of the net is c o n n e c t e d with the input zone o f a n o t h e r net. T h e firing o f the first t* in Fig. 8 r e p r e s e n t s a vehicle leaving the net from the output zone. A t this m o m e n t , t, can be fired, and /3 has to be fired immediately after t_, is fired. T h e n , if the second t* is fired, the last vehicle in the system leaves the net. Finally, there is only a control signal left in the net. The marking of the control-loop net returns to the original state, and the flow-path net is now ready for a n o t h e r vehicle to pass through.

Thus, the definition of the liveness within input and output zones for those open systems is feasible, and the modulised merge net is robust u n d e r the definition.

Intersection

Fig. 9 is the intersection floor-path structure. T h e r e are four physical zones, Zt, Z2, Z3 and Z4, and a composite pseudo- zone Zs. Zt and Zz are the input zones, and Z3 and Z~ are the output zones. T h e traffic control at the intersection has to include the following functions:

Z4

3

zt

' '

I

I

Y f

.Z2

Z3

(5)

24 S. Hsieh and Y.-J. Shih

Zl

|

Z4

tl

s - I -

Y t 2

2

Fig. 10. The intersection flow-path net.

t3 Z3

_l ~-0

-I

1. T o control the m o v e m e n t s of vehicles f r o m Zt a n d Z2 to Z3 or Z4

2. T o p r e v e n t that the vehicle f r o m Z~ to Z3 b u m p i n g into a n o t h e r vehicle f r o m Z , to Z4

3. T o p r e v e n t a vehicle f r o m Z~ to Za p a s s i n g the vehicle from Z2 to Z3 so closely that t h e y hit e a c h other. Fig. 10 is the flow-path net of the intersection. T h e incident matrix of the n e t is:

Zi Z2 Z3 Z4 Z s

[

q - 1 0 0 0 1 Nr = t2 0 - 1 0 0 1 t3 0 0 1 0 - 1 t4 0 0 0 1 - 1

T h e r e exists a weighting vector Wf = (1,1,1,1,1) T such that Nf 9 Wf = 0. H e n c e , the flow-path n e t is strictly c o n s e r v a t i v e a n d b o u n d e d . T h e liveness p r o b l e m of the flow-path net is the s a m e as that of the c o m p l e t e net, and will be discussed after the c o m p l e t e net is e s t a b l i s h e d .

Fig. 11 is the c o n t r o l - l o o p net of the intersection s t r u c t u r e . Since t h e t h r e e control f u n c t i o n s m e n t i o n e d a b o v e h a v e to be e m b e d e d in the net, two control places Ct and 6"2 are used to achieve t h e m . A t o k e n in C~ r e p r e s e n t s the right of

\

t4

I

,

,

I I

- t - t 2

. . . J

Fig. 11. The intersection control-loop net.

( Ct,

C2)

II

(I

, 0 )

Y

(0,~)

(0,~)

(1 ,o)

(1,0)

(1,0)

(1,0)

Fig. 12. The reachability tree of the intersection control-loop net. way at the intersection. Place C2 is u s e d to p r e s e r v e the t o k e n so that the traffic control signals in the net will n o t be lost.

T h e incident matrix o f the c o n t r o l - l o o p net is: C, C2

[

q - 1 1

Nc = t2 - 1 1 t 3 1 - 1 ta 1 - 1

T h e r e exists a weighting vector Wc = (1,1) T such that N ~ . We = 0. H e n c e , the net is strictly c o n s e r v a t i v e and also b o u n d e d .

Fig. 12 is the reachability tree of the c o n t r o l - l o o p net. T h e initial m a r k i n g is (1,0). A f t e r a series o f transition firings, t h e m a r k i n g o f e a c h s u b t r e e r e t u r n s the initial m a r k i n g , a n d e v e r y transition in t h e net h a s b e e n fired at least o n c e . T h e r e f o r e , the c o n t r o l - l o o p net is live.

By placing the c o n t r o l - l o o p n e t o v e r the flow-path net, the c o m p l e t e intersection n e t is t h u s e s t a b l i s h e d (see Fig. 13). A letter ' T ' in a box is the s y m b o l of the intersection net. Since b o t h the flow-path a n d c o n t r o l - l o o p n e t s are strictly c o n s e r v a t i v e a n d b o u n d e d , the c o m p l e t e net is also.

Fig. 14 is the reachability tree of the i n t e r s e c t i o n net. It s h o w s that e a c h individual vehicle can arrive at each of the o u t p u t z o n e s respectively. T h e r e f o r e , the n e t is live within the i n p u t a n d o u t p u t zones.

T h u s , the m o d u l i s e d intersection net is robust. Z4 \

-~ --t 4

Zl

tl

C2 Z5

t3

Z3

,s169

~

r - - ~ j

-

I I I t 2

(6)

(ZI,Z2,Z3, Z4, Zs ~

CI, C~)

II

(1100010)

y

"x ,

(0100101)

(1000101)

(0110010)

(0101010) (1010010)

(1001010)

Fig. 14. The reachability tree of the merge net.

3.2 BI-dlrectional System

In the design of a bi-directional guide-path system, three alternatives can be adopted. These alternatives include the following [5]:

1. Have parallel wire tracks with reverse orientation on each aisle.

2. Have a single switchable wire-track on each aisle. T h e switching o f the guide-path is d e p e n d e n t on the flow d e m a n d .

3. Have a mixed guide-path that is comprised of both uni- directional and bi-directional aisles, with bi-directional flow paths allowed only on selected aisles.

The alternative 2 is the uni-directional system for a short time period. T h e floor-path model can be built by the union of the uni-directional nets. T h e bi-directional path of the alternative 3 are usually built in those areas w h e r e only o n e vehicle is allowed in and out at one time. T h e alternative 1 is the most c o m m o n one. In this paper, the alternative 1 is a d o p t e d to build the bi-directional A G V S .

Except at points o f intersection or merge, the guide-path system o f the bi-directional system is essentially uni-directional. With sufficient clearance space left b e t w e e n parallel tracks, there is virtually no i n t e r f e r e n c e b e t w e e n vehicles on the same aisle when travelling in opposite directions. T h e r e f o r e , merge and intersection are the two main structures discussed in this section.

It is almost impossible to illustrate all the possible bi- directional system structures here. In the following, several merge and intersection nets are established one by one. Hopefully, one can build a bi-directional A G V S model by the union of the nets d e v e l o p e d here. Since these nets are much more complex when c o m p a r e d with the uni-directional system, a capital " C " which is the a c r o n y m o f " c o m p l e x " is used before the n a m e s o f the structures in o r d e r to distinguish them from the uni-directional structures.

R o t a t i o n c y c l e s

Structures in the bi-directional system are relatively complex when c o m p a r e d with those of the uni-directional system. T h e r e f o r e , the reachability tree is s o m e t i m e s too large to be illustrated in the paper. Instead o f drawing the reachability tree, a "rotation cycle" c o n c e p t is created. T h e concept o f the rotation cycle is based on the reachability tree. In a

closed-loop net, for each sub-branch o f the reachability tree, a sequence o f firing transitions could be found. If the net is live, the last marking of each sub-branch must be the initial marking. H e n c e , if one m o r e transition is fired in each sub- branch, the fired transition must be the first transition o f the sequence o f firing transitions. By putting the first transition as the last of each s e q u e n c e of firing transitions, several transition cycles can thus be formed. Each cycle can be m a p p e d to a sub-branch of the reachability tree, each sub- branch of the reachability tree can form a cycle, and every transition of the net should be included in one or m o r e cycles. A m o n g those cycles, some are long e n o u g h to include o t h e r cycles, and s o m e are not. Those cycles which are too short to include a n o t h e r cycle are n a m e d "'rotation cycles". Definition. A rotation cycle in an A G V S net is the shortest sequence of transitions cr = thti, " . . . tjk such that for each ti, and tj,+, in the sequence there is a place Pi, with Pit E O ( t j , ) a n d pi, E l ( t i , . , ) , t / , = t~k, and Ix' = 6(Ix,tj,) = 6(Ix,o-), where tx is the initial marking of the net.

A rotation cycle is thus the shortest closed path from a transition back to that same transition. A rotation cycle can be p r o v e d to have conservation, b o u n d e d n e s s , reachability and liveness properties. If a net can be d e c o m p o s e d into several rotation cycles, and every transition in the net has been included in one or more rotation cycles, the net is live. C-Line

Observing the line structure of the bi-directional system, except that there are two flows with reverse orientation, each o f the flows is exactly identical to that in the uni-directional line structure. T h e r e f o r e , the modulised C-line net can be established by the union of the two uni-directional line nets. C-Merge I

Fig. 15 is one o f merge structures for the bi-directional A G V S . In the figure, places Z6 and Z7 are pseudo-places which are used to prevent vehicle from Z~ to Z5 and vehicle from Z2 to Z4 or Zs from b u m p i n g against each other. T h e r e are two input zones, Z~ and Zz, and t h r e e o u t p u t zones, Z3, Z4 and Zs. T h e flow-path net o f the C - M e r g e I is shown in the Fig. 16. For analysis purposes, vehicles are placed in the two input

z o n e s .

Z 1

Z3

I ' ~ 1 _

"

F--]

Z4

lZ7

Zf.

D

' '

(7)

26 S. Hsieh and Y.-J. Shih

t6

Z 3

_1 -~0

-I

Z l

|

ts

Z6

t?

5

Z4

t4

t2

Z2

9

,

I.

I-

I-

\

t5

I.

I-

Fig. 16. The flow-path net of the C-Merge I structure.

In a similar way to the conservation analysis for the uni- directional net, the incident matrix Nt of the flow-path net can be obtained, and also it can be proved that there exists a weighting vector Wi = ( 1,1,1, i ,1,1,1 )T such that Nf - W~ = 0. Because every element in Wf is 1, the flow-path net is strictly conservative and bounded 9 By observing the flow-path net, it can be seen that the individual vehicle at the input zone can arrive at the respective output zone and the net is live within input and output zones.

Fig. 17 is the control-loop net of this C-Merge I structure. The token in place Ct is used to prevent vehicles entering Z6 or Z7 at the same time. The token in place C2 is used to prevent vehicles from ZI entering Z4 (a vehicle cannot usually make a turn of more than 90~ 6"3 and 6"4 are used to preserve tokens in the net. By the use of incident matrix equations, it can be proved that there exists a weighting vector We = (1,1,1,1) T such that No- Wc = 0. Hence, the net is strictly conservative and bounded.

Since the reachability tree of the control-loop net is relatively large instead of drawing the reachability tree, the "rotation cycle" concept mentioned above is used to analyse the liveness property. From Fig. 17, it can be found that there are exactly three rotation cycles in the net. These rotation cycles are:

1. tstTt 5 2. 12t4l 2

3. tlt3t7tl

Every transition in the control-loop net has ben inlcuded in one or more rotation cycles, hence, the control-loop net is live. Note: Transition t6 is not in Fig. 17.

Fig. I8 is the complete net of the C-Merge I structure. A symbol "C-M I" in a box represents the modulised C-Merge I net. Since both flow-path and control-loop nets are robust, the complete net is also robust.

••

F . . .

t 4

tl C

t2

//

f s 9 I " I ! G ~

t?

- -- I'~'"

...

J

. . . C l c o n t r o l l i n e . . . C2control l i n e Fig. 17. The control-loop net of the C-Merge I structure.

Z

t5

- '

Z4

t4

Z? C3

!2

f - ' ~ _

I-_----# ~ - - --4-_

[ ; : il~ . . . . L _ Z 3

-I

- Q

F . . . .

Z2

_-:___

.m "11

Fig. 18. The C-Merge 1 net. C-Merge II

Fig. 19 is another merge structure for the bi-directional AGVS. In the figure, Z6 is a composite pseudo-zone, and Z? is a pseudo-zone. There are three input zones, Z~, Z2 and Z3, and two output zones, Z4 and Z5. Vehicles from Z~ can move to Z4 and Zs, vehicles from Z2 can move to Z4, and

(8)

Z s Z7 Z3

I-'1 r - ~ [ ~

Z2

Z6

]

Z4

r - I

- '

m

L..

J

Fig. 19. The C-Merge II structure.

Zs

ts

Z7

t3

Z3

t6 t4

Z4

|

lz, -'

-I

- ~ O

1

Fig. 20. The flow-path net of the C-Merge H structure.

vehicles from Z3 can move to Zs. T h e possible traffic p r o b l e m s o f this structure are:

1. Zt vehicle wants to move to Zs while Z2 vehicle wants to move to Z 4.

2. Z, vehicle wants to move to Z4 while Z2 vehicle wants to move to Z~ too.

3. Z~ vehicle wants to move to Z5 while Z3 vehicle wants to move to Z5 too.

T h e r e f o r e , several, control nodes have to be a d d e d into the structure.

Fig. 20 is the flow-path net o f this structure. By the use o f incident matrix equations, it can be p r o v e d that the flow-path net is strictly conservative and b o u n d e d . By observing the flow-path net, the p r o p e r t y - the liveness within input and output zones - can be seen. T h e flow-path net is robust.

Fig. 21 is the control-loop n e t o f this structure. A token in C~ ( o r C2) is used to p r e v e n t m o r e than o n e vehicles entering Z~ ( o r ZT) at the same time. T h e C3 token is used to restrict the m o v e m e n t of the Zz vehicle so that it does not go to Z~. T h e control place Ca is used to return tokens to C2 or C3. A small circle instead o f an a r r o w h e a d at the transition t7 or t8 is called an inhibitor arc [4]. This notation is b o r r o w e d from switching theory w h e r e the small circle m e a n s " N O T " . T h e firing rule is c h a n g e d as follows: a transition is e n a b l e d when tokens are in all of its inputs and zero tokens are in all of its inhibitor inputs. T h e transition fires by removing tokens from all of its inputs.

Fig. 19 shows that the vehicle from Z~ to Z4 should not interfere with the vehicle from Z3 to Zs. However, Fig. 21 shows that the firings of transition t 3 and t~ are in conflict. This means that when a vehicle moves from Z~ to Z4 and another vehicle moves from Z3 to Z5 they may interfere with each other. This can be improved by the decomposition of Z6 into two pseudo-zones. However, the control-loop net becomes more complex. This is a trade-off between cost and efficiency. C2

I .. . . .

E > . . . .

I

ts

Cs

t3

_1

C6

Fig. 21. The control-loop net of the C-Merge 1II structure.

. . . C 1 c o n t r o l l i n e . . . 0 2 c o n t r o l l i n e 9 C 3 c o n t r o l l i n e

(9)

28 S. Hsieh and Y.-J. Shih

C2

r

. . . .

z5

i

,s

z,

,3

:;_

=

ili

JtYIG

Fig. 22. The C-Merge II net.

By the use o f incident matrix equations, it can be proved that the net is strictly conservative and b o u n d e d . In Fig. 21, there are four rotation cycles in the net. They are: t3tst3, qtdsq, htdTt~ and t2qt~t2. Since the control-loop net can be d e c o m p o s e d into four rotation cycles, and every transition in the net has been included in at least one rotation cycle, therefore, the net is live.

Fig. 22 is the complete net of the C-Merge I structure. A symbol "C-M II'" in a box is given to r e p r e s e n t the net. Since both the flow-path and control-loop nets are robust, the complete net is also robust.

C-Merge II

Fig. 23 is a more complete merge structure for the bi- directional system. In the figure, places Z7, Z. and Z9 are pseudo-zones, and Z~. is a composite pseudo-zone. There are three input zones, Zz, Z2 and Z3, and three output zones, Z4, Z5 and Z6. The possible traffic problems of this structure are: 1. Z~ vehicle wants to move to Z6 while Z2 vehicle wants to

move to Z5 or Z6. Z4 Z~ Z1

Z8

Zs

Z2 ( I - Y \ i r--1 L - f , , ~ _ 5 ~ I I

[-1.

Z6

Z3

[]

]

Fig. 23. The C-Merge III structure.

2. ZI vehicle wants to move to Z4 or Z, while Z~ vehicle wants to move to Z4.

3. Z2 vehicle wants to move to Z~, while Z3 vehicle wants to move to Z4 or Zs.

T h e r e f o r e , a control-loop net is necessary.

Fig. 24 is the complete " C - M e r g e III" net which is the combination of the flow-path net (solid line portion) and the control-loop net (dash line portion). A symbol "C-M l l I " in a box is given to represent the net. T h e strict conservation property of both the flow-path net and control-loop net of this structure can be proved by incident matrix equations. As for the liveness property, the reachability tree or the rotation cycle concept can be used to prove it. Since both the flow- path and control-loop nets can be p r o v e d to be robust, the complete net is also robust.

Inter-Merge

Fig. 25 is one of the possible structures for the bi-directional system. It is named as the Inter-Merge structure. There are three input zones, Z~, Z2 and Z3. and three output zones, Z4, Zs, and Z~,. For the purpose of simplifying the control problem, in the figure, only two composite pseudo-zones, Z7 and Z . are used. In a similar way to the C-Merge I[ structure, the proposed structure of Fig. 25 is a trade-off between cost and efficiency. When the vehicle is moving from Zi through Z7 to Z,, physically, the vehicle at Z2 can move to Z4 or Zs. However, technically, the right (the token) of way will not be issued to the vehicle at Z2 until the former vehicle arrives at Z,. The possible traffic problems of this structure are:

1. Zt vehicle wants to move to Z4 time Z2 vehicle wants to move to 2. Zt vehicle wants to move to Z4

wants to move to Z,.

3. ZI and Z3 vehicles want to move

or Z5 and at the same Z4 or Z~ too.

or Z5 while Z3 vehicle to Z~ at the same time. T h e r e f o r e , the control-loop net is necessary.

Property analyses can be carried out on the flow-path net and control-loop net. T h e analysis results indicate that both nets are robust. H e n c e , the c o m p l e t e net is also robust. T h e modulised "'Inter-Merge" net is shown in Fig. 26. A symbol "'I-M" is given to r e p r e s e n t this net.

C-I-M

Fig. 27 is the floor-path layout o f the C-I-M structure. T h e " C - I - M " , w h e r e " C " is the a c r o n y m of " c o m p l e x , "I" is "intersection", and "'M'" is m e r g e , r e p r e s e n t s a two-way intersection with a merge traffic-flow structure. T h e r e are four input zones (from Zj to Z4) and four output zones (from Z5 to Zs). This is a very complex traffic structure. Many traffic p r o b l e m s could occur in this structure. In the figure, Zg, Z~o, Z ~ and Z~2 are the four pseudo-zones. Because there are too many possible traffic p r o b l e m s , a c o m p o s i t e p s e u d o - z o n e Zt3 is used to control the four-way traffic in the middle. Since the net is too complex to be described and analysed here, only the complete C-I-M net is shown in Fig. 28. F r o m this, it can be seen how difficult it is to a d o p t a structure like this.

(10)

---C 1 control line - - . - - C 2 control line . . . . C 3 control line C3

Q--

I

i

- - ' - - C 4 control line - - , - - C 5 control line - - - - - C 6 control line I,...

i

I

I

I

Fig. 24. The C-Merge Ill net.

Z 5 V--1 k_.l Z3 V--I L_I

E

S

r / ' Za,

i

l Z4

]

7

z2

_ F"'I ._J L..J

z ; - - 1 ,

z~

.y_]

~ L..J 71

]

Fig. 25. The Inter-Merge structure.

3.3 Remarks

Several modulised floor-path nets for both uni- and bi- directional system have been established. The necessary properties for a robust AGVS have been embeded in these nets. Because the control-loop nets established in this paper are closed systems, the traffic control signals in the net are always there (strict conservation property), and, once the

control function of the net is executed, the control tokens return to their initial places and wait for the next through vehicles (liveness properties). The flow-path nets reflect the real floor-path structure, and tokens in the nets reflect system vehicles. Hence, only at the moment when vehicles are in the net, can the net behaviour have these necessary properties. The number of vehicles from input zones to the net always results in the same number of vehicles from output zones out of the net. This ensures that the flow-path net keeps the conservation and liveness properties. Since each control-loop net is already a closed system, when several nets are connected together as a complete system, the connection is done among the flow-path nets only.

4. Union Rules and Procedure

The modulised floor-path nets developed above are satisfied with the system robustness properties - safeness, boundedness, strict conservation, teachability, and liveness within input and output zones. It is now necessary to see by the union of these nets, if a robust system model be built. Several union rules are described as follows:

Rule I: Every output zone of a net has to be connected to the input zone of another net.

(11)

30

S. Hsieh and Y.-J. Shih

Z3

|

I I

C ~ ) _ . . .

C2

@

t,

I

Zs I k

ts

Cs '-8

tz

Zz

",,,it, (:9

r--"4

Fig. 26. The Inter-Merge net.

C6

ZfZ"I F-'! i

I I - "t_,.J~-~. 7

J~IZ6

Fig. 27. The C-I-Mstructure.

Zs

f----] L_.J

Z3

I""1 I I

Rule 2: Similarly, every input zone o f a net has to be c o n n e c t e d to the output zone of a n o t h e r net.

Rule 3: T h e output zone o f a net cannot be c o n n e c t e d with its input zone. Similarly, the input zone of a net cannot be c o n n e c t e d with its output zone.

By following the three rules, all the necessary nets can be united as a c o m p l e t e closed-system model. This is because vehicles from o t h e r net output zones get into input zones of a net, and from output zones of the net get out of the net, and, again, get into a n o t h e r net input zones. This occurs repeatedly, and vehicles are always in the model. T h e r e f o r e ,

the strict conservation property remains in the system after the union. According to the previous study, if a system is conserva- tive, the system should be bounded. The zone control method is used to limit the vehicle number to 1 in a zone at any time. Hence, the safeness property is still there after the union. Since control-loop nets in the system are so independent, as long as every control-loop net can take care its traffic problems, the traffic problems of the whole system should be dealt with. Because each transition of the control-loop net is live and tokens (vehicles) in the flow-path nets keep moving, the deadlock problem never happens in the system; the complete model is live. If a system is live, the reachability property is, of course, there too. Thus, if an A G V S model is built by the union of the modulised floor-path nets developed in this paper, and the union rules are followed, the model will be robust at the floor level if the floor-path layout is robust.

The modelling p r o c e d u r e for a robust A G V S at floor level is described as follows:

1. Obtain a reasonable floor-path layout by any means. 2. C h o o s e p r o p e r modulised floor-path nets according to the

floor-path layout, and place the c h o s e n nets at the p r o p e r location in the layout.

3. C o n n e c t the output ( o r input) zones of a net with the input ( o u t p u t ) zones of the consecutive net.

4. Inspect w h e t h e r the c o m p l e t e model after the union is a closed system or not, if not, go to step 2 or 3.

5. D o qualitative and quantitative analyses on the c o m p l e t e model by either analytical m e t h o d s or simulation. 6. If the results indicate that the model is not robust, modify

(12)

(a)

Ze

C6 Z2

,~C4

I . . . Q l

Z4

Z6

including 4 control lines ahd a physical line

C I (b) C 1 control line

~

- - - ~

C?

Zs C 2 control line

C

3 control

line - - w - - m - - C4 control line --.--.---C 5 control line

C8

C 6

control line . . . . C 7 control line . . . C8 control line . . . C 9 control line

- - ' - < D

--

refer to part (d) ~ Q refer

to psrt

(C)

(c)

-~tzo

(d)

/ 2 2 ~ , ~ , ~ 2 2 .. ". , 9 9 9

tzl ~'~

-t24

/

(13)

32 S. Hsieh and Y.-J. Shih

[]

[]

I t I I

J

Fig. 29. The plant layout,

If the floor-path layout is robust, the A G V S model obtained by following steps 1 to 4 will be robust. If not, o n e can still use the above modelling p r o c e d u r e to d e v e l o p an initial model. Based on the initial model, system analyses and simulations can be p e r f o r m e d so that the model can be modified (step 5 to 7). By repeating steps 1 to 6, very close to an ideal model can be obtained.

5. Application

Fig. 29 is a small manufacturing plant layout. T h e plant includes an a u t o m a t e d storage/retrieval system and several NC or C N C workstations. T h e plant intends to adopt A G V S as their material handling system. A f t e r careful study, the A G V S floor-path layout is d e t e r m i n e d as shown in Fig. 30. T h e floor-path layout shows that there are 2 D, 2 M, 1 I and

several line structures in the system. T h e r e f o r e , by following the u n i o n - p r o c e d u r e steps 1 to 4, the c o m p l e t e A G V S model is obtained (in Fig. 31 ). Then, analytical analyses or simulations can be carried out to verify the robustness o f the net. H e r e , because the anlayses of the c o m p l e t e m o d e l may involve too much for this p a p e r , half o f the model is used to d e m o n s t r a t e how to verify the system robustness. A s u b m o d e l including 1 D, 1 M and a line is c h o s e n (see Fig. 31, cut along line A A and throw away the u p p e r part), and transition tH and

tz,, and tt4 and tzs are o v e r l a p p e d , respectively, in o r d e r to keep the submodel closed (see Fig. 32). T h e safeness, b o u n d e d n e s s , strict conservation and reachability properties will be examined in the following.

1. Safeness: Because o f the modified firing rule that the output place cannot have a token in it, it is g u a r a n t e e d that there will never be more than o n e big token ( r e p r e s e n t i n g vehicles) in any place. T h e r e f o r e , the safeness property is there.

2. B o u n d e d n e s s : if the m o d e l is conservative, the model is b o u n d e d . 3. Strict conservation: /ll ll2 /13 N = t~4 ll5 ll6 t i t IIl,i tlCl

T h e incident matrix of Fig. 32 is Z,2 Z,3 Z,, Z,5 Z,~ Z,7 Z~, Z,9 C3 C, 1 0 0 0 0 0 - 1 0 0 0 - 1 0 1 0 0 0 0 0 0 0 - 1 1 0 0 0 0 0 0 0 0 0 0 - 1 0 0 1 0 0 0 0 0 - 1 0 1 0 0 0 0 0 0 0 0 0 - 1 1 0 0 0 0 0 0 0 0 0 0 - I 0 1 - 1 1 0 0 0 0 0 0 1 - 1 1 - 1 0 0 0 0 - 1 0 0 1 - 1 1 T h e r e exists a weighting vector W = (1,1,1,1,1,1,I.1,1,1) T such that N - W = 0. T h e r e f o r e . the model is strictly conservative and also b o u n d e d .

z11

I Z4

I'-'1

,F--I

" ' ' '

I

1

[

Z14 _ r - - I

Zt 3

L~[

-E

Z l s i i

r

[

I,~1-

I

I

L . . . . I i

t

. _ ~ D Zs I Z , Z l

I--111--1

f~l

f

L_.II L_I

L J

~,

___]Zs

_ ]

I . . i "

-]Z6

"I~ "11

-'~ Zlo Z9

Z22

~ , . _ i _ r---h _

r !-

__1

L_J;

i._l

I

]zB

I

Z19 ZIa i

Z2o

tK,E-'~ _ f"-I ~ P-"l J L _ . I - [ ~ J ' - I I i - / I _lvl I

F 'I z~

-

I

(14)

Zll t~ t= Zz tl Z1

\

\

\

ZG

_I

Zls t~

~.~zte

1

'

8

i l

.

.

.

.

.

J

Fig. 31. The plant AGVS Petri-net model.

t~

Z2z t22

Z~o

/ i

t23

Fig. 32. A submodel of the plant AGVS

model.

i

t13

~'~

_~tls

t.(tzo)

Z12 t12 Z

I

~-0!

I

t14(t=o)

_I

-i

zls!tl i c'

(15)

34 S. Hsieh and Y.-J. Shih

4. Liveness: In Fig. 32, there are two rotation cycles, tl7tlSt 11112/14/'17 and tlgt~st~ lt12t]31lSl16t]9, and all the transitions have been included in one or more rotation cycles. Therefore, the model is live.

5. Reachability: Since the model is live, all the paths are reachable.

The submodel obtained by the union of the divide, merge and line nets is proved to be robust. The complete model (Fig. 31) can be proved to be robust by a similar method. Intuitively, any model obtained by the union of the nets established in this paper should be robust.

6. Conclusions

An intelligent AGVS model tool has been developed in this paper. The AGVS robustness properties at floor level have

been included in this tool. Therefore, the model developed by the tool will automatically be a robust model at floor level. More research regarding the robust AGVS model at other levels will be carried out in the future.

References

1. S. Hsieh and K.-H. M. Lin, "Building AGV Traffic-Control Models With Place-Transition Nets", The International Journal of Advanced Manufacturing Technology, 6, pp. 000-000, 1991. 2. S. Hsieh and K.-H. M. Lin, "AGVS Mechanism", to appear in

the International Journal of Advanced Manufacturing Technology. 3. S. Hsieh and Y.-J. Shih, "AGVS and Its Petri-Net Properties", to appear in The Journal of Intelligent Manufacturing Systems. 4. J. L. Peterson, Petri Net Theory and the Modeling of Systems,

Prentice-Hall, Englewood Cliffs, NJ 07632, 1981.

5. P. J. Egbelu and J. M. A. Tanchoco, "Potentials for Bi-directional Guide-Path for Automated Guided Vehicle Based Systems", International Journal of Production Research, 24(5), pp. 1075-1097, 1986.

數據

Fig.  2.  The  line  net.
Fig. 7.  The  merge  net.  Fig. 9.  The  intersection  floor-path  structure.
Fig.  14. The  reachability  tree  of  the  merge  net.
Fig.  I8  is  the  complete  net  of  the  C-Merge  I  structure.  A  symbol  &#34;C-M  I&#34;  in  a  box  represents  the  modulised  C-Merge  I  net
+7

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