© 2001 by CRC Press LLC Because of the above differences, the planning knowledge used in assembling mechanical products could not be directly used in planning aircraft frame assemblies. However, the general issues are the same. In developing an assembly planner for aircraft frame assemblies, we still need to address how the assembly model is represented, how the sequence of assembling operations is generated, and how the generated assembly plan is represented. With adequate planning knowledge, it is expected that both the knowledge- and case-based approaches could be used to develop planners for aircraft frame assemblies. There should be no exception for the algorithmic approach. No matter what approach is taken, the unique character- istics of aircraft frame assemblies should be considered in developing the assembly planner. Our assembly planner was developed based on the observation of the layer structure of aircraft frame assemblies [26]. Besides our work, we are not aware of other assembly planners developed for aircraft frame assemblies in the available literature. In the following sections, our methods of assembly model representation, sequence generation, and plan representation for aircraft frame assemblies are discussed. Assembly Model Representation A new representation scheme called LADGA (layered acyclic directed graph with attributes) was devel- oped for aircraft frame assemblies. The LADGA consists of nodes organized into layers with or without attributes attached and directed arcs that connect nodes, as illustrated in Figure 8.3. Each node denotes a part. Different kinds of nodes can be used to differentiate different kinds of parts, if necessary. In this study, circular nodes refer to parts requiring riveting operations only. Triangular nodes indicate that bolting operations are needed (riveting operations might also be needed). The directed arc between two FIGURE 8.2 Spar subassembly. © 2001 by CRC Press LLC 9 Petri Net Modeling in Flexible Manufacturing Systems with Shared Resources* 9.1 Introduction 9.2 Petri Net Models for a Class of FMSs Petri Nets • Using Petri Nets to Model FMSs 9.3 Liveness Conditions for Models 9.4 Deadlock Avoidance Controllers for Models Reducing Models • Optimal Deadlock Avoidance Petri Net Controllers for a Class of s • Deadlock Avoidance Controllers for the s 9.5 Examples 9.6 Conclusion This chapter develops a Petri net model of production routings and resource allocation in a flexible manufac- turing system (FMS) with shared resources, which we call the model, and focuses on the deadlock avoidance problems in the system. We introduce the concept of D-structures and explore some of their basic properties, as well as characterize the liveness of the system in terms of D-structures. We address the deadlock avoidance problems by introducing the restrictive controller. In order to synthesize such a controller, we first reduce the model. The reduced is also an model for which we can present an optimal deadlock avoidance controller. Our deadlock avoidance controller for the consists of two parts: a Petri net controller and a restrictive policy. The Petri net controller can be reduced from the optimal deadlock avoidance controller for the reduced model and can prevent some D-structures from leading to a circular wait relationship which directly causes deadlock in . The restrictive policy restricts the utilization of some key resources. In many cases, the Petri net controller can guarantee not only that the controlled is live, but also that it is an optimal deadlock avoidance controller. And hence, the controlled is an optimal live Petri net model for the system. Some examples are presented to illustrate the results. *This work was supported in part by the National Natural Science Foundation of P. R. China under grant 69574023 and in part by a grant from the Center for Computer Integrated Manufacturing System at Xian Jiaotong University. R 2 PN R 2 PN R 2 PN R 2 PN R 2 PN R 2 PN R 2 PN R 2 PN R 2 PN R 2 PN R 2 PN R 2 PN R 2 PN R 2 PN Ke Yi Xing Xidian University Bao Sheng Hu Xi’an Jiaotong University . call the model, and focuses on the deadlock avoidance problems in the system. We introduce the concept of D-structures and explore some of their basic properties, as well as characterize the liveness. liveness of the system in terms of D-structures. We address the deadlock avoidance problems by introducing the restrictive controller. In order to synthesize such a controller, we first reduce the. taken, the unique character- istics of aircraft frame assemblies should be considered in developing the assembly planner. Our assembly planner was developed based on the observation of the layer