Resource-Oriented Petri Nets in Deadlock Avoidance of AGV Systems

Transcription

1 Proceedings of the 2001 IEEE International Conference on Robotics & Automation Seoul, Korea May 21-26, 2001 Resource-Oriented Petri Nets in Deadlock Avoidance of AGV Systems Naiqi Wu Department of Mechatronics Engineering Guangdong University of Technology Guangzhou , P. R. CHINA Mengchu Zhou Dept. of Electrical and Computer Engineering New Jersey Institute of Technology Newark, NJ , USA ABSTRACT-This paper presents a colored resourceoriented Petri net (CROPN) modeling method to deal with conflict and deadlock arising in Automated Guided Vehicles (AGV) systems. The work can be viewed as the continuation of some of the authors previous work. Some unique features in AGV systems require the further investigation into their deadlock avoidance using CROPN models. The proposed approach can easily handle both bidirectional and unidirectional paths. Bidirectional paths offer additional flexibility, efficiency and less cost than unidirectional paths. Yet they exhibit more challenging AGV management problems. By modeling nodes with places and lanes with transitions, one can easily construct a CROPN model for dynamic AGV systems with changing routes. A control policy suitable for real-time control implementation is then proposed. I. Introduction It is well known that deadlock may occur due to limited resources in automated manufacturing systems (AMSs), leading to a system wide standstill. This paper classifies deadlock into two types. One is caused by the competition by parts for such manufacturing resources as machines and buffers. The other type is due to the competition for nodes and lanes by AGVs when multiple AGVs are used in material handling systems (MHS). Several methods are developed to synthesize a live Petri net for AMS so that the resulting PN controllers are deadlock-free [1-2]. They belong to deadlock prevention techniques. Deadlock detection and recovery techniques allow deadlock to occur and then detect and recover from it [3-4]. Deadlock avoidance techniques dynamically assign the resources in the system such that the system is deadlock-free [5-12]. The above mentioned studies addressed the first type of deadlock in AMS but not the second one. When multiple AGVs are used, some problems may arise, e.g., blocking, conflict, deadlock, and collision [ To improve the productivity and resource utilization, it is desired to have as many parts as possible in a system. However, the more parts in AMS, the more likely deadlocked. The parts need to be delivered from a place to another by MHS. Often, only a few AGVs are available in an AGV system mainly due to their high-cost and easily satisfied transportation demand. The total number of AGVs likely remains constant. While the route of each part type is known in advance, the routes of an AGV frequently change based on real-time transportation requests. The theory for handling the first type of deadlock needs to be further studied for the second type. The most widely used technique for vehicle management of AGV systems is zone control. Guided paths are divided into several disjoint zones. A zone can accommodate only one AGV at a time. The guided paths may be unidirectional or bidirectional. The deadlock problem for an AGV system with unidirectional guided paths is studied using the zone control [15-16]. Their algorithms predict deadlock based on the current routings of the AGVs, and then make decisions to avoid deadlocks. However, there is no effective method to deal with the conflict and deadlock in an AGV system with bidirectional paths. AGVs need to compete for not only zones as in unidirectional system but also lanes, increasing the control complexity. One way to solve this problem is to carefully design the configuration of an AGV system so that its management is simplified. For example, tandem configuration [17] partitions all the stations into non-overlapping, single vehicle, closed loops with additional pickup and deposit locations provided as an interface between adjacent loops. Each station is assigned to only one loop, and each loop is served by exactly one AGV. A segmented flow approach [18] partitions the paths into nonoverlapping segments. Each segment is comprised of one or more zones and is served by a single AGV. Transfer buffers are located at both ends of each segment and serve as interface devices between the segments. The buffers are designed to be able to serve both sides of the segments simultaneously. Therefore, the only possible conflicts are the use of the interface buffers. Their drawback is that a load might have to be handled by two or more AGVs before reaching its destination [14]. In addition, extra pickup and deposit stations are required to interface with each other, increasing the system scale and cost. The conflict detection problem is studied for bidirectional systems using colored PN [14]. An approach is presented to detect the competition for a lane by AGVs in the adjacent zones. However, this is not enough to control the system. A conflict may occur when two AGVs in non-adjacent zones compete for some lanes. Furthermore, when such a conflict occurs, it in fact leads to deadlock. Thus, a more comprehensive approach is needed, motivating this present work. A colored resource-oriented Petri net (CROPN) was originally developed to deal with the first type deadlock in AMS [10]. This paper further develops it to model AGV systems to derive a deadlock avoidance policy. The next section presents CROPN modeling of AGV systems. Sections 3-5 present conflict and deadlock-free conditions and control policy in different cases. Section 6 shows the application of the proposed approach. Section 7 concludes the paper. II. SYSTEM MODELING WITH CROPN A. Finite Capacity PN A PN is a particular kind of directed graph containing places and transitions. It can be denoted by a quadruple PN = (P, T, I, O), where P = {,,..., p m } is a finite set of places, T = {, t 2,, t n } is a finite set of transitions, P T, P T =, I: P T {0, 1} is an input function, and O: P T {0, 1} is an output function. We use t ( p) to denote the set of input places of transition t (the set of input transitions of place p) and t (p ) the set of output places of t (the set of output transitions of p). A marking or state M: P N={0, 1, 2, } describes the distribution of tokens in PN. M(P) = (M( ), M( ),, M(p m )) T /01/$ IEEE

2 is an m 1 vector. Marking M 0 is referred to as the initial marking of a PN, representing the initial state of its modeled system. In finite capacity PN, a place is limited to hold a finite number of tokens. They are adopted since a zone (also called node in this paper) and lane allow only a limited number of AGV at a time. Let K(p), p P, denote the capacity of p, which is the maximum number of tokens that p can hold at any time. Definition 2.1: A transition t T in a finite capacity PN = (P, T, I, O) with marking M is said to be enabled if M(p) I(p, t), p P ( 1 ) and K(p) M(p) I(p, t) + O(p, t) ( 2 ) Definition 2.1 means that t is enabled and can fire if all the places in t have enough tokens and all the places in t have enough free spaces. When condition (1) is satisfied, t is processenabled. When (2) is satisfied, t is resource-enabled. Thus, t is enabled only if it is both process and resource-enabled. Firing an enabled transition t T at M changes M into M according to M (p) = M(p) I(p, t) + O(p, t) ( 3 ) A sequence of firings results in a sequence of markings. A marking M is said to be reachable from M 0 if there exists a sequence of firings that transforms M 0 to M. The set of all possible markings reachable from M 0 is denoted by R(M 0 ). B. Modeling Resources in a System One important problem is how to model the resources in a system. Often, places in PN are used to model resources. In order to reduce the complexity of the resultant PN, there is a one-to-one relation between the resources in the system and places in CROPN [10]. There are two types of resources in an AGV system with bidirectional paths: nodes and lanes. An example semiconductor plant s AGV system is shown in Fig. 1 consisting of 11 nodes and 14 bi-directional lanes. This paper models nodes by places and lanes by transitions, respectively although both may be viewed as resources. A node in an AGV system is an intersection of several paths or a station where an AGV can load a part to a machine or pick up a part. Thus, we can regard a node as the resource just as a machine or buffer in CROPN presented in [10]. In Fig. 2, p represents a node, a token in it represents an AGV in the node. Suppose that a node can hold only one AGV at a time. Then K(p) = 1. Place p s multiple inputs and outputs represent that AGV can enter and leave the node along different paths. Since only one AGV can travel on a lane at a time, all the arcs are single if they exist. (I) (O) p t n (I) t n (O) Fig. 2. PN model for a node. t 2 Fig. 3. PN model for a bidirectional lane between two nodes A lane in an AGV system is a path between two adjacent nodes. An AGV can travel on a bidirectional lane in both directions. It is modeled by two transitions as shown in Fig. 3. A token representing an AGV can flow from by firing or by firing t 2, implying that the lane is assigned to an AGV from or. Note that when and are both marked, neither transition is enabled according to enabling condition (2), thereby avoiding any collision on the lane. If a lane is one-way, e.g., from to only, then only is needed to model the lane. C. Modeling the Traveling Processes of AGVs An AGV can pick up a part in a node and travel on a path through one or more nodes. After reaching the destination node, it unloads the part. In general, it begins from a node and stops at another. According to different tasks assigned to AGVs, their routings change dynamically. The process of delivering a part from a node to another can be modeled by a sequence of places and transitions. Figure 4 shows the PN models for two processes of AGVs in Fig. 1 (V 1 : and V 2 : ). We call such a model a subnet that models an AGV travelling from one node to another. Each subnet begins and ends with places (maybe the same place). Let PN i = (P i, T i, I i, O i ) denote the subnet for V i. Only its beginning place is initially marked with a token, i.e., its initial marking M i0 =(1, 0,, 0) T. When its ending place is marked, V i reaches its destination. The final marking, also called destination marking, M id =(0,, 0, 1) T. t 87 t 76 t 6,11 ( a ) 1 1,2 Wafer-Saw Inventory Compound Storage Pack/Ship Die-Attach Bond Mould Routing Station Trim/Form Solder Fig. 1. An AGV system in a semiconductor plant t 67 t 56 1,4 1 t 9,11 0,9 0 ( b ) Fig. 4. Two example subnets for single AGV delivery processes. Definition 2.2: PN = (P, T, I, O) for an AGV system is said to be the union of two subnets PN i = (P i, T i, I i, O i ) and PN j = (P j, T j, I j, O j ) for AGV V i and V j, if P = P i P j, and T = T i T j ( 4 ) M i0 (p), if p P i M 0 (p) = M j0 (p), if p P j ( 5 ) M i0 (p) + M j0 (p), if p P i P j. I i (p, t), if p P i, t T i I(p, t) = I j (p, t), if p P j, t T j 0, otherwise O i (p, t), if p P i, t T i O(p, t) = O j (p, t), if p P j, t T j 0, otherwise

3 If, P i P j, the union of PN i and PN j merges two transitions ( ) in PN i and ( ) in PN j into one because they stand for the same lane. Thus, there is at most one transition between two places along the same direction in the union PN. By defining the union of multiple subnets in the same way, we obtain the PN model for the whole AGV system, which describes the traveling processes of all AGVs. The union PN of the two subnets in Fig. 4 is shown in Fig. 5. Note that the CROPN of an AGV system is constructed dynamically according to the dispatching and routing of the AGVs. Each time when tasks are assigned to AGVs and their routings are determined, a CROPN is constructed. If every time we can control the CROPN such that it is deadlock-free, the system is conflict and deadlock-free. Note that it is easy to construct a CROPN dynamically since the configuration of an AGV system is known in advance. III. DEADLOCK AVOIDANCE IN CIRUITS t 67 t 76 t 56 1,4 p 1 6,11 t 87 p 6 t 9,11 1,2 0,9 0 Fig. 5. The union of two subnets Consider a token, i.e., AGV V 1 or V 2 in 1 (Node 11) in Fig. 5. It enables either 1 2 (V 1 ) or 1 4 (V 2 ) depending on which AGV the token represents. Only one of two transitions should be enabled. Unfortunately, ordinary PN cannot describe this. To model this feature, colors are introduced. Different from most existing work where colors are first defined for tokens in places [19], this work defines the colors for transitions first. Definition 2.3: Define the color of transition t i T as b i. It states that each transition in the model is associated with a unique color, i.e., b i b j if i j. Definition 2.4: If t i p, define the color of a token in p that enables t i as b i. For example, in Fig. 5, if the token in 1 stands for AGV V 1, it enables 1 2. Thus its color is b Note that we do not use colors to identify token types (or AGV), instead to describe the node for each AGV to go to. In this way, we model the dynamical processes of an AGV system. The resulting PN is named colored resource-oriented PN (CROPN). Observing the CROPN shown in Fig. 5, we call U = {1, 1 4,,,, t 56,, t 6 11, 1 } a circuit and Y = {, t 67,, t 76, } a cycle denoted by {, } for short. A cycle is a special circuit which models a bidirectional lane between two nodes as shown in Fig. 3. Let U denote a circuit and Y a cycle. Consider the PN model in Fig. 3. When two tokens with opposite directions are in and, respectively, there is a conflict for the lane and at the same time a circular wait is formed. In other words, in a cycle, a conflict and deadlock occur simultaneously since two AGVs compete for both nodes and lanes in it. It is clear that if there is no deadlock in a cycle, there is no conflict in it either. Dislike a cycle, AGVs in a circuit compete for only nodes. In this sense, if we can avoid deadlocks by using the CROPN, we avoid conflicts as well. Let M id denote the destination marking of a subnet i and M d the destination marking for the union CROPN, respectively. The below equation is similar to (5) for two subnets union and can be easily extended to the union of multiple subnets. M id (p), if p P i M d (p) = M jd (p), if p P j ( 6 ) M id (p) + M jd (p), if p P i P j. Definition 2.5: If a CROPN of an AGV system with initial marking M 0 can reach the destination marking M d, the CROPN is deadlock-free. A. Deadlock-Free Condition in Circuits Let P(U) P denote the set of places on circuit U and T(U) T the set of transitions on U. Further, let K(U) = K(p) p P(U) denote U s capacity. Definition 3.1: If a token in place p P(U) has color b i whose corresponding transition t i T(U) is on circuit U, it is a cycling token, otherwise a leaving token of U. After firing t i, a cycling token remains in U while a leaving token leaves U, releasing a space in U. Let M(U) denote the total number of cycling tokens and M (U) the total number of tokens in U at the current marking M. Clearly, M(U) M (U). Definition 3.2: S(U) = K(U) M (U) is called the numbers of free spaces available in U at the current marking M. Circuits are interactive if they share common places. For example, is shared by U 1 = {,,, t 6,, t 3, } and U 2 = {, t 4,, t 7,, t 5,, t 2, } in Fig. 6(a), and and shared by U 3 = {,,, t 4,, t 6,, t 3, } and U 4 = {, t 4,, t 7,, t 5,, t 2, } in Fig. 6(b). If U i and U j are interactive and there is a free space in the common places, then both S(U i ) > 0 and S(U j ) > 0. Such common space is shared by U i and U j. In Fig. 6(a), if is empty, U 1 and U 2 share a space. Let P ij = P(U i ) P(U j ) be the set of common places of U i and U j, P i = P(U i ) P ij, P j = P(U j ) P ij. t 6 t 3 t 2 t 4 (a) t 5 t 7 Fig. 6. PN models with two circuits. Definition 3.3: Each of U i and U j is said to have a nonshared free space if 1) two spaces are in P ij ; 2) one space in P i and another in P ij ; 3) one space in P i and another in P j ; or 4) one space in P ij and another in P j. By Definition 3.3 we mean that when the condition is met every circuit can be allocated a free space. For example, in Fig. 6(a), if places and are empty and all other places are occupied, then the spaces in and can be allocated to U 1 and U 2, respectively such that each of U 1 and U 2 has a non-shared space. The situation is similar if there are more than two circuits. If the free spaces in the PN in marking M are distributed such that every circuit can be allocated one or more free spaces, then we say that every circuit has at least one non-shared space. Assume that the number of AGVs is less than the number of places in the CROPN of an AGV system since otherwise no vehicle can move. Also assume that no token (an AGV) stays in a place (node) forever since otherwise it should be moved out of t 3 t 6 t 4 (b) t 2 t 7 t 5

4 the system. A sufficient condition for deadlock-free operation of an AGV system in the existence of circuits can be stated below. Theorem 3.1: The PN model for the AGV system is live if every U i in it has at least one non-shared space in any marking M. Proof: See [20]. Note that in the PN in Fig. 6(a) if there is only one free space it may be deadlocked. For example, if only is empty (the two circuits share the only free space) and the token in needs to go to. Firing t 2 leads to deadlock. It is the condition in Thoerem 3.1 that allows the token in in Fig. 6(a) to go to either circuit in the net. Thus the net is made to be live. B. Control Law for Circuits Before we present the control law we need some definitions. Definition 3.4: A transition t in PN is said to be controlled if firing t or not is determined by a control law when t is both process and resource-enabled. When a controlled transition t can fire according to the control law, t is said to be control-enabled. Definition 3.5: A PN is said to be a controlled PN if at least one transition in it is controlled. Therefore, a controlled transition in a PN can fire if it is process, resource and control-enabled. To effectively avoid deadlocks in an AGV system, the resources in the system should be assigned appropriately when more than one AGV complete for the same node. Definition 3.6: t is said to be the input transition of circuit U if t U but t U. A circuit U in the CROPN may have two or more input transitions, we use T I (U) T to denote the set of input transitions of U. Then from Theorem 3.1 we have: Theorem 3.2: The PN model for the AGV system is deadlock-free if the condition given in Theorem 3.1. is satisfied by a) Initial marking, and b) a marking reached due to the firing of t T I (U i ) for any circuit U i. By this control law, we need to observe the state of the system on-line, calculate S(U) for every U, and control the firing of transitions in T I (U). To implement the control law, we need to identify all the circuits in the CROPN of an AGV system and this must be done in real-time. If a PN has many circuits, it may be time consuming to do this. The below result from Theorem 3.1 can be used such that we need identify only a fewer number of circuits. Corollary 3.1: In a CROPN of an AGV system, a necessary condition for a deadlock to occur in circuit U at marking M is S(U) = 0 ( 7 ) Corollary 3.1 means that deadlock can occur in U only when the number of tokens in U equals the number of places on U. This implies that the number of AGVs must be greater than or equal to the number of nodes on a circuit. Considering only fewer AGVs in the system than the number of nodes, deadlock can occur in only those circuits whose place count is less than or equal to the number of AGVs. Because the configuration and the number of AGVs in the system are known in advance, we can identify all these circuits in advance. For example, in the system shown in Fig. 1, a circuit can be deadlocked only if there are at least four AGVs. When there are four AGVs, the possible circuits in which deadlock can occur are { }, { }, { } and { }. Thus, when a CROPN is constructed for a system, we need to ensure only those circuits in the CROPN to be deadlockfree. In fact for any two-agv system, e.g., the one in Fig. 1, a circuit is guaranteed to have three or more places (otherwise becoming a cycle); thereby it will never be deadlocked. IV. DEADLOCK AVOIDANCE IN THE CYCLES A. Single Cycle A single cycle is formed by two AGVs competing for a lane between two adjacent nodes, one AGV travels in one direction, the other in the opposite direction. The CROPN shown in Fig. 5 contains a cycle Y = {, }. If both AGVs take the lane, a collision (conflict) occurs. If both enter the nodes (both and have a token in the cycle Y in Fig. 5), a deadlock occurs. Thus, we have the below result. Lemma 4.1: It is deadlock-free in a single cycle formed by routings of two AGVs in a CROPN iff there is at most one token in the cycle in any marking M. Clearly, if there is no deadlock in a single cycle, two AGVs never occupy the two adjacent nodes in the cycle simultaneously. Thus, the conflict of using the lane never occurs. B. Cycle Chain Definition 4.1: In a CROPN of an AGV system, if and form a cycle, and form a cycle,, and p n-1 and p n form a cycle (n>2), these n places form a cycle chain. A cycle chain example is shown in Fig. 8. It is formed by the AGV routings V 1 : 0 and V 2 : 1. Firing t 82 in Fig. 8 leads the token standing for V 1 in to. Firing 1,5 leads the token standing for V 2 to. Now although either of cycles Y 1 = {, } and Y 2 = {, } has only one token, deadlock is inevitable. To progress, t 23 and, t 54 and t 43 or t 32 and t 54 may fire, leading to two tokens in cycle Y 2, Y 1 or Y 3 = {, }. All these situations are deadlocked. 7 2 t t t 23 t 43 t t Fig. 8. The CROPNs containing a cycle chain. t 49 p 1,5 5 It should be pointed out that a number of cycles form a cycle chain if there exist two AGVs that go through all the cycles in different direction. For example in Fig. 8, if three AGVs have routes, 0, and 1, then Y 1 and Y 3 form a cycle chain; and Y 2 and Y 3 form another. Lemma 4.2: In a CROPN of an AGV system, if there is a cycle chain formed by the routes of two AGVs, the chain is deadlock-free iff there is at most one token in it in any marking. A cycle chain may be formed by routes of more than two AGVs. Assume that there is another AGV V 3 with route. Then two cycle chains H 1 = {Y 1, Y 2, Y 3 } and H 2 = {Y 4, Y 1 } as shown in Fig. 9 are formed by the routes of these three AGVs, where Y 4 = {, }. 7 t82 t 61 2 t 32 t t t 43 0 t 5,10 p 1,5 5 Fig. 9. A cycle chain formed by routes of three AGVs t 54 t 5,

5 It should be pointed out that the routes of more than two AGVs can generate more complicated structures than cycle chains. The cycle chains formed by more than two AGVs may have overlaps. For example, in Fig. 9, H 1 and H 2 overlap at Y 1. Definition 4.2: A subnet of CROPN for AGV systems is called an interactive cycle chain if it is formed by two or more cycle chains and each of the chains overlaps with at least one of the other chains. We call an interactive cycle chain subnet a cycle chain subnet for short and denote it w. Assume that a w is formed by n cycle chains H 1, H 2,, and H n and there are m overlap segments D 1, D 2,, and D m in w. Let F Di (w) = {V fdi, V bdi, V cdi ; D i } and F Hi = {V fhi, V bhi, V chi ; H i }, where V f is the set of AGVs that move forward on D i or H i, V b is the set of AGVs that move backward on D i or H i, and V c is the set of AGVs that cross D i or H i. Further we let 1, if Card(V f ) > 0 and Card(V b ) > 0 δ(f (w)) = ( 8 ) 0, otherwise n and g(w) = δ(f Hi (w)) + i= 1 δ(f Di (w). We assume, i= 1 without loss of generality, that there are no same cycle chains in a w, or H i H j, if i j, then we have the following result. Theorem 4.1: A subnet of cycle chains formed by n cycle chains in a CROPN of an AGV system is deadlock-free iff the following condition holds in any marking M. g(w) = 0 ( 9 ) Notice that for cycle chains Lemmas 4.1 and 4.2 are the special cases of Theorem 4.1. Thus we can use condition (9) in Theorem 4.1 to avoid deadlock and conflict in all cycle chains. A cycle chain can also be formed by the route of a single AGV. In fact, if an AGV goes to some nodes and comes back by the same path (with opposite direction), a cycle chain is formed. Clearly, no conflict and deadlock will occur in such a cycle chain since no other AGV enters the cycle chain. Definition 4.2: t is said to be the input transition of cycle chain H if t H but t H. Let T I (H) T denote the set of input transitions of H. To avoid deadlock and conflict in cycle chains is to control the firings of transitions in T I (H i ) such that condition (9) is always satisfied. It is easy to calculate g(h i ) and thus simple to implement. An algorithm of complexity o( T 2 ) can be derived to identify the set of all single cycles and all cycle chains in the CROPN to facilitate the real-time implementation [20]. V. DEADLOCK AVOIDANCE IN THE OVERALL SYSTEM In a CROPN of an AGV system as the overall system, if there is no interaction between circuits and cycle chains, then we can control the overall system by using the control laws presented in the last two sections to control circuits and cycle chains, respectively. However, the circuits and cycle chains may interact with each other. This makes the problem more complicated. A subnet with interaction of a circuit and cycle chain is shown in Fig. 10. This subnet contains the cycle chain made of places -5 and a circuit U = {, t 2,12, 2, 2,4,, t 43,, t 32, }. Places -4 are on both the cycle chain and U. m 7 t 61 2 t 2,12 t82 t 32 t 23 t t 43 t 4,13 t 54 t5,10 1,5 Fig. 10. A subnet with interaction of a circuit and cycle chain Let us consider condition (9). By this condition, a shared path can be used only in one direction, as analogous to a timeshared system. It requires switching from one direction to another appropriately. In Fig. 10, if the direction from to is active (i.e., some AGV will occupy them at some time), then t 43 and t 32 are active, and so is U. If the direction from to is active, t 23 and are active but this time U is not. It is easy to show that when the circuit is active, circular wait may occur. Otherwise no circular wait will occur. Thus, we can prove the below result using Theorems 3.1 and 4.1. Theorem 5.1: A subnet in a CROPN of an AGV system with interaction of circuits and cycle chains is deadlock-free if the following conditions hold in any marking. 1) The conditions given in Theorem 3.1 and (9) hold, if the circuits are active and 2) The condition (9) holds, if the circuits are not active. Because in a two AGV system a circuit will never be deadlocked we don t need to consider the deadlocks in circuits. However, if there are multiple AGVs and we make the AGV assignments so that there are only cycle chains and circuits, then result given in Theorem 5.1 can be applied to solve the problem. VI. ILLUSTRATIVE EXAMPLE Consider the AGV system in Fig. 1. Two requests and assignments of AGVs are given one after another: 1) V 1 : V 2 : ) V 1 : V 2 : The CROPN for Case 1 is shown in Fig. 11. There are two cycles Y 1 = {, 1 } and Y 2 = {1, }, forming a cycle chain. It is easy to see that the cycle chain is due to the route of AGV V 1. Thus, there will be no conflict and deadlock in it. In fact, even transition 0,9 is fired first and the token representing V 2 enters into in the cycle chain, t 87, t 76 and t 6,11 can still fire and the token in (representing V 1 ) can enter 1. Then we can fire 1,2 and the token leaves 1 for. If the token representing V 2 is in 1, then t 6,11 cannot fire according to the transition enabling and firing rule. Therefore, both AGVs can reach the destination with no deadlock. The CROPN for Case 2 is shown in Fig. 12. There are also two cycles Y 1 = {, 1 } and Y 2 = {1, }, forming a cycle chain H. H is due to the routes of V 1 and V 2. Thus transition firings have to follow the control law specified in Theorem 4.1 to avoid deadlock. Theorem 4.1 requires that g(h)=0. Since V 2 is already in H, we have to limit another token to enter H. This leads to the below transition firing order. Transitions 0,1, 2 and t 23 can fire any time. can fire only after t 9,11, 1,4 and fire. This way no conflict and deadlock will occur. 2,4 1

6 t 87 t 4,11 1,4 t t 6,11 1 1, ,1 t 78 1,9 t 9,11 0,9 Fig. 11. The CROPN for Reques 1,4 t 9,11 t 23 2 t 56 t 67 t 4,11 Fig. 12. The CROPN for Request 2. VII. CONCLUSIONS 1,9,10 1 This paper has developed a PN model, called CROPN, to describe systematically the AGV travelling processes in an AGV system. The conflict and deadlock situations are well modeled by the circuits, cycles, cycle chains, and their interactions in CROPN. An effective control policy for conflict and deadlock avoidance is proposed. It can be used in the real-time control. It should be noted that the literature results, e.g., in [7-8], may apply to the discussed AGV systems but lead to much weaker results, e.g., only one AGV can be allowed in the system [20]. Besides conflict and deadlock, blocking is another important issue to be addressed in the operation in the AGV system. Blocking may be caused by inappropriate resource assignment in the traveling processes of the AGVs. It can also be caused by inappropriate routing. When a vehicle reaches its destination it may stay there until a new task is assigned to it. During its stay, it may block other vehicles. To solve this problem, it may be necessary to do the routing and control concurrently and develop sophisticated techniques for that. This is the future work to be done. The extension of the results should be performed for the cases where a node allows to park multiple AGVs. In CROPN, this implies that a place s capacity is more than one. The theory needs to be extended to the cases where multiple AGV s routes may form very complicated chains. Acknowledgement This work is partially supported by the Chinese NSF under grant and New Jersey State Commission on Science and Technology. Anonymous reviewers helped us improve this paper s presentation. References 1 M. -D. Jeng, and F. DiCesare (1995). "Synthesis Using Resource Control Nets for Modeling Shared Resource Systems," IEEE Trans. Robotics and Automation, 11(3), pp M. C. Zhou, F. DiCesare, and A. Desrochers, Hybrid methodology for synthesis of Petri nets models for manufacturing systems, IEEE Trans. on Robotics and Automation, vol. 8, no. 3, , R. A. Wysk, N. S. Yang, and S. Joshi, Detection of deadlocks in flexible manufacturing cells, IEEE Trans. on Robotics and Automation, vol. 7, , Y. T. Leung and G. Shen, Resolving deadlocks in flexible manufacturing cells, Journal of Manufacturing Systems, vol. 12, no. 4, , Z. A. Banaszak and B. H. Krogh, Deadlock avoidance in flexible manufacturing systems with concurrently competing process flows, IEEE Trans. on Robotics and Automation, vol. 6, , E. Roszkowska and J. Jentink, Minimal restrictive deadlock avoidance in FMS, In: Proc. of The Second European Control Conf. ECC '93, Groningen, Holland, volume 2, pages , Fanti, M. P., B. Maione, S. Mascolo, and B. Turchiano, Event-based feedback control for deadlock avoidance in flexible production systems, IEEE Trans. Robotics Automat., Vol. 13, pp , A. Lawley, Deadlock avoidance for production systems with flexible routing, IEEE Trans. on Robotics and Automation, vol. 15, no. 3, , Viswanadham, Y. Narahari, and T. L. Johnson, Deadlock prevention and deadlock avoidance in flexible manufacturing systems using Petri net models, IEEE Trans. on Robotics and Automation, vol. 6, , N. Q. Wu, Necessary and sufficient conditions for deadlockfree operation in flexible manufacturing systems using a colored Petri net model, IEEE Trans. on Systems, Man, and Cybernetics, Part C, vol. 29, no. 2, , K. Y. Xing, B. S. Hu, and H. X. Chen, Deadlock avoidance policy for Petri net modeling of flexible manufacturing systems with shared resources, IEEE Trans. on Automatic Control, vol. 41, no. 2, , A. Reveliotis and P. M. Ferreira, Deadlock avoidance policies in automated manufacturing cells, IEEE Trans. on Robotics and Automation, vol. 12, no. 5, 1-13, J. Malmbog, A model for the design of zone-control automated guided vehicle systems, Int. J. of Production Research, 28, , L. Zeng, H.-P. Wang, and S. Jin, Conflict detection of automated guided vehicles: a Petri net approach, Int. J. of Production Research, 29, , C.-C. Lee and J. T. Lin, Deadlock prediction and avoidance based on Petri nets for zone-control automated guided vehicle systems, Int. J. of Production Research, vol. 33, , M.-S. Yeh and W.-C. Yeh, Deadlock prediction and avoidance for zone-control AGVS, Int. J. of Production Research, 36(10), , Y. A. Bozer and M. M. Srinivasan, Tandem AGV systems: a partitioning algorithm and performance comparison with conventional AGV systems, European Journal of Operational Research, vol. 63, no. 2, , Sinriech and J. M. A. Tanchoco, An introduction to the segmented flow approach for discrete material flow systems, Int. J. of Production Research, 33(12), , N. Viswanadham and Y. Narahari, Colored Petri net models for automated manufacturing systems, Proc. of IEEE Conference on Robotics and Automation, Raleigh, NC, , N. Wu and M. C. Zhou, A Petri Net Method for Deadlock Avoidance of AGV systems, Working paper, ECE Dept., New Jersey Institute of Technology, September 2000.