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Controller Design for Avoiding Collisions in Automated Guided Vehicle Systems via Labeled Petri Nets
14th 14th IFAC IFAC Workshop Workshop on on Discrete Discrete Event Event Systems Systems 14th IFAC Workshop on Discrete Event Systems May 30 -- June 1, 2018. Sorrento Italy 14th IFAC Workshop on DiscreteCoast, Event Systemsonline at www.sciencedirect.com Available May 30 June 1, 2018. Sorrento Coast, Italy May 30 - June 1, 2018.on Sorrento Coast, Italy 14th IFAC Workshop Discrete Event Systems May 30 - June 1, 2018. Sorrento Coast, Italy May 30 - June 1, 2018. Sorrento Coast, Italy
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IFAC PapersOnLine 51-7 (2018) 139–144
Controller Design for Avoiding Collisions in Controller Design for Avoiding Collisions in Controller Design for Avoiding Collisions in Controller Design for Avoiding Collisions in Automated Guided Vehicle Systems via Automated Guided Vehicle Systems via Automated Guided Vehicle Systems via Automated GuidedPetri Vehicle Systems via Labeled Nets Labeled Petri Nets Labeled Petri Nets Labeled Petri Nets
YaXin Wan, JiLiang Luo, Qi Zhang, WeiMin Wu, YaXin Wan, JiLiang Luo, Qi Zhang, WeiMin Wu, YaXin Wan, JiLiang Luo, Qi Zhang, WeiMin Wu, YaXin YiSheng Wan, JiLiang Luo, Qi Zhang, WeiMin Huang, and MengChu Zhou YiSheng Huang, andQi MengChu Zhou Wu, YaXin YiSheng Wan, JiLiang Luo, Zhang, WeiMin Wu, Huang, and MengChu Zhou YiSheng Huang, and MengChu Zhou YiSheng Huang, and MengChu Zhou Abstract: As for a class class of automated systems (AGVS) that are partially Abstract: guided vehicle vehicle systems Abstract: As As for for a class of of automated automated guided guided systems (AGVS) (AGVS) that that are are partially partially Abstract: As for aa class of automated guidedaa vehicle vehicle systems (AGVS) that are partially controllable and with undeterministic behavior, method is presented to design a maximally controllable and with undeterministic behavior, method is presented to design a maximally Abstract: As for a class of automated guideda vehicle systems (AGVS) that are partially controllable and with undeterministic behavior, method is presented to design a maximally controllable and withthat undeterministic behavior, a method isvia presented to design aIt maximally permissive controller avoids any collision among vehicles labeled Petri nets. is not sure permissive avoids among vehicles labeled nets. is controllable and withthat undeterministic behavior, a method presented to design permissive controller controller that avoids any any collision collision among vehiclesisvia via labeled Petri Petri nets. aIt It maximally is not not sure sure permissive controller that avoids any collision among vehicles via labeled Petri nets. It is not sure which area a vehicle enters after it leaves one because of insufficient sensors. Furthermore, which area aa vehicle enters after it leaves one because of insufficient Furthermore, aaa permissive controller that avoids any collision among vehicles via labeledsensors. Petri nets. It is not sure which area vehicle enters after it leaves one because of insufficient sensors. Furthermore, which area auncontrollably vehicle enters enter after itsome leaves onebecause because of of insufficient insufficient activators. sensors. Furthermore, a vehicle may uncontrollably enter some areas because of insufficient activators. By modeling vehicle may areas By modeling which vehicle enters enter after itsome leaves onebecause because ofof insufficient insufficient activators. sensors. Furthermore, a vehiclearea mayauncontrollably uncontrollably areas By modeling modeling vehicle may enter some areas because of insufficient activators. By the former case by multiple transitions labeled by a same symbol, and the latter case by the former case by multiple transitions labeled by a same symbol, and the latter case by vehicle may case uncontrollably enter some areas because of insufficient activators. By modeling the former by multiple transitions labeled by a same symbol, and the latter case by the former case by multiple an transitions labeled by a by same symbol, andnet thethat latter case by an uncontrollable transition, AGVS is represented aaa labeled labeled Petri is partially an transition, AGVS represented by Petri is the former case by multiple an transitions by a by same symbol, andnet thethat latter case by an uncontrollable uncontrollable transition, an AGVS is is labeled represented labeled Petri net that is partially partially an uncontrollable transition, an AGVS is represented by a labeled Petri net that is partially controllable. The whole net is partitioned into a set of dangerous regions that are controllable controllable. The whole net is partitioned aa set of dangerous regions are controllable an uncontrollable transition, an AGVS is into represented by a labeled Petri that net that is partially controllable. The whole net is partitioned into set of dangerous regions that are controllable controllable. The whole net is partitioned into A a set of dangerous regions that are controllable such that the uncontrollable issue is addressed. method is proposed to compute the consistent such uncontrollable issue is method is to the consistent controllable. whole net is partitioned into A a set of dangerous regions that are such that that the theThe uncontrollable issue is addressed. addressed. A method is proposed proposed to compute compute thecontrollable consistent such that the uncontrollable issue is addressed. A method is proposed to compute the consistent marking set for a given observed sequence. By the resultant consistent marking set, the marking marking set for a given observed sequence. By the resultant consistent marking set, the marking such thatset thefor uncontrollable issuesequence. is addressed. A method is consistent proposed to compute the consistent marking a given observed By the resultant marking set, the marking marking set for a given observed sequence. By the resultant consistent marking set,ttthe marking of each dangerous region is computed, and can be used to design a control action of each dangerous region is computed, and can be used to design a control action marking set for a given observed sequence. By the resultant consistent marking set,tthe marking of each dangerous region is computed, and can be used to design a control action of each dangerous region is computed, anddangerous can be used to design control action prevent each vehicle from entering region where is AGV ooo prevent each from entering each each region where aathere there is one one AGVtt already. already. As As of each dangerous region is computed, anddangerous can be used to design control action prevent each vehicle vehicle from entering each region oaa prevent each vehicle from enteringcontroller each dangerous dangerous region where where there there is is one one AGV AGV already. already. As As result, the maximally permissive controller is obtained. result, the maximally permissive is obtained. oa prevent each vehicle from enteringcontroller each dangerous region where there is one AGV already. As result, the maximally permissive is obtained. a result, the maximally permissive controller is obtained. a©result, the maximally permissive is obtained. 2018, IFAC (International Federationcontroller of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Automated navigation vehicle systems; Collision avoidance; avoidance; Petri Petri nets; nets; Discrete Discrete Keywords: Automated navigation vehicle systems; Collision Keywords: Automated navigation vehicle systems; Keywords: Automated navigation vehicle systems; Collision Collision avoidance; avoidance; Petri Petri nets; nets; Discrete Discrete event systems event systems Keywords: Automated navigation vehicle systems; Collision avoidance; Petri nets; Discrete event event systems systems event systems 1. INTRODUCTION the 1. the safety, safety, the the collision collision avoidance avoidance problem problem is is even even more more 1. INTRODUCTION INTRODUCTION the safety, the avoidance 1. INTRODUCTION the safety, than the collision collision avoidance problem problem is is even even more more important the others. important than the others. 1. INTRODUCTION the safety, than the collision avoidance problem is even more important the others. important than the others. With industry 4.0 and Internet of Things, more attention With industry 4.0 and Internet of Things, more attention With industry industry 4.0 4.0 and and Internet Internet of of Things, Things, more more attention attention important Because their high modeling Because of of than their the highothers. modeling efficiency, efficiency, Petri Petri nets nets (PNs) (PNs) With Because of high efficiency, Petri nets (PNs) has been drawn to the research field about intelligent has been drawn research field about Because of their their high modeling modeling efficiency, Petriand netscontrol (PNs) With industry 4.0to andthe Internet of Things, moreintelligent attention have has been drawn to the research field about intelligent become a successful tool for the analysis have become a successful tool for the analysis and control Because of their high modeling efficiency, Petri nets (PNs) has been drawn to the research field about intelligent have become a successful tool for the analysis and control manufacturing. Different from the traditional manufacmanufacturing. from the traditional manufachave becomeevent a successful tool for the analysis andFerrarini control has been drawnDifferent to the research field about intelligent manufacturing. Different from the traditional manufacof discrete systems Zhou et al. (1993); of discrete event systems Zhou et al. (1993); Ferrarini have become a successful tool for the analysis and control manufacturing. Different from the traditional manufacof discrete event systems Zhou et al. (1993); Ferrarini turing, the intelligent one should be able to quickly and turing, one should able quickly and et of discrete event systems Zhou et Lu al. et (1993); Ferrarini manufacturing. Different thebe manufacturing, the the intelligent intelligent one from should betraditional able to to quickly and al. (2008); Giua et al. (2008); al. (2016); Ru et al. (2008); Giua et al. (2008); al. (2016); Ru of event systems Zhou et Lu al. et (1993); Ferrarini turing, the intelligent one should be able to quickly and et discrete al. (2008); Giua et al. (2008); Lu et al. (2016); Ru flexibly respond to complex orders that tend to be of flexibly respond to complex orders tend to be of al. (2008); Giua et al. (2008); Lu et al. (2016); Ru turing, the intelligent one should be that able to quickly and flexibly respond to complex orders that tend to be of et (2009); Luo et al. (2011, 2017); Cabasino et al. et al. (2009); Luo et al. (2011, 2017); Cabasino et al. et al. (2008); Giua et al. (2008); Lu et al. (2016); Ru flexibly respond to complex orders that tend to be of (2009); Luo et al. (2011, 2017); Cabasino et al. small quantity, personized and multi-batch. This requires small quantity, personized and multi-batch. This requires et al. (2009); Luo et al. (2011, 2017); Cabasino et al. flexibly respondpersonized to complex orders that tend torequires be of (2017). small quantity, and multi-batch. This Therefore, PNs have also been widely used in the (2017). Therefore, PNs have also been widely used in the et al. (2009); Luo et al. (2011, 2017); Cabasino et al. small quantity, personized and multi-batch. This requires (2017). Therefore, PNs have also been widely used in the a control method for automated guided vehicle systems aa control method for automated guided vehicle systems (2017). Therefore, PNs have also been widely used in the small quantity, personized and multi-batch. This requires control method for automated guided vehicle systems research of AGVS research since an AGVS is typically aaa research of AGVS research since an AGVS is typically (2017). Therefore, PNs have also been widely used in the a(AGVS) controlto method for automated guided vehicle systems research of AGVS research since an AGVS is typically (AGVS) to be much improved since AGVS is an imporbe much since AGVS is an imporresearch of AGVS research since an AGVS is typically a a(AGVS) controltomethod for improved automated guided vehicle systems be much improved since AGVS is an impordiscrete event system. Fanti (2002) models an AGVS by discrete event event system. Fantisince (2002) models an AGVS by by of AGVS research an models AGVS an is typically a (AGVS) to be muchtool, improved since AGVS applicable is an impordiscrete system. Fanti (2002) AGVS tant transportation which is broadly in tant transportation which is broadly in research discrete event system. Fanti (2002) models an AGVS by (AGVS) to be muchtool, improved since AGVS applicable is an important transportation tool, which is broadly applicable in a colored timed PN, and presents the control strategy for a colored coloredevent timedsystem. PN, and and presents themodels controlanstrategy strategy for discrete Fanti (2002) AGVS by tant transportation tool, which is broadly applicable in a timed PN, presents the control for modern manufacturing systems. However, as the system modern manufacturing systems. However, the colored collision timed PN, and presents Nishi the control strategy for tant transportation tool, which is broadlyas in aavoiding modern manufacturing systems. However, asapplicable the system system avoiding and deadlock. et al. (2012) and and deadlock. et al. (2012) and aavoiding colored collision timed PN, and presents Nishi the control strategy for modern manufacturing systems. However, as the system collision and deadlock. Nishi et al. (2012) and size increases, the control problems of AGVS become size increases, the control problems of AGVS avoiding collision and deadlock. Nishi et al. (2012) and modern manufacturing systems. However, as the become system Nishi size increases, the control problems of AGVS become et al. (2011) address the dispatching and conflict Nishi et al. (2011) address the dispatching conflict avoiding collision and deadlock. Nishi et al. and (2012) and size increases, thewhich control problems ofdispatching, AGVS become Nishi et al. (2011) address the dispatching and conflict very complicated, which includes task dispatching, path very complicated, includes task path Nishi et al. address the dispatching size thewhich control problems AGVS become very increases, complicated, includes task ofdispatching, dispatching, path problems of an by PN. al. (2015) problems of (2011) an AGVS AGVS by using using PN. Luo Luo et etand al. conflict (2015) Nishi et al. (2011) address the dispatching and conflict very complicated, which includes task path problems of an by Luo al. planning and collision avoidance to planning and collision avoidance according to IFA IFA (2006); (2006); problems ofmethod an AGVS AGVS by using using PN. PN.designing Luo et et the al. (2015) (2015) very complicated, which includesaccording task dispatching, path present planning and collision avoidance according to IFA (2006); the for automatically ladder present the method for automatically designing the ladder problems of an AGVS by using PN. Luo et al. (2015) planning and collision avoidance according to IFA (2006); present the method for automatically designing the Singh et al. (2011); Le-Anh et al. (2006). In order to ensure Singh et al. (2011); Le-Anh et al. (2006). In order to ensure present the method for automatically designing the ladder ladder planning and collision avoidance according to IFA (2006); Singh et al. (2011); Le-Anh et al. (2006). In order to ensure diagram for avoiding collision by ordinary PNs. Nishi et al. diagram for avoiding collision by ordinary PNs. Nishi et al. the method for automatically designing the ladder Singh et al. (2011); Le-Anh et al. (2006). In order to ensure present diagram for avoiding collision by ordinary PNs. Nishi et al. ⋆ diagram for avoiding collision by ordinary PNs. Nishi etthe al. Singh et al. (2011); Le-Anh et al. (2006). In order to ensure ⋆ (2010) develop a Petri net decomposition approach for Y. Wan is with College of Information Science and Engineer(2010) develop a Petri net decomposition approach for the Wan is with College of Information Science and Engineer⋆ Y. diagram for avoiding collision by ordinary PNs. Nishi et al. (2010) develop a Petri net decomposition approach for the Y. Wan is with College of Information Science and Engineer⋆ ing, Huaqiao University, Xiamen 361021 China(e-mail: yaxinmas(2010) develop a Petri net decomposition approach for the Y. Wan is with College of Information Science and Engineeroptimization of conflict-free routing for AGVs. Zhou et al. ing, Huaqiao University, Xiamen 361021 China(e-mail: yaxinmasoptimization ofaconflict-free conflict-free routing for for AGVs. AGVs. Zhoufor etthe al. ⋆ ing, Xiamen 361021 China(e-mail: yaxinmas(2010) develop Petri net decomposition approach optimization of routing Zhou et al. Y. Huaqiao Wan is University, with College of College Information Science and [email protected]). J. Luo is with of Information Science and ing, Huaqiao University, Xiamen 361021 China(e-mail: [email protected]). J. J. Luo Luo is is with with College College of of Information Information Science Science and and optimization of conflict-free routingmotion for AGVs. Zhou et al. (2017) propose that each be modeled as [email protected]). (2017) propose that each robot’s robot’s be modeled as ing, Huaqiao University, Xiamen 361021 China(e-mail: yaxinmasoptimization of conflict-free routingmotion for AGVs. Zhou et al. (2017) propose that robot’s motion be modeled as Engineering, University, Xiamen 361021 and [email protected]). J. Luo is with College of Information Science Engineering, Huaqiao Huaqiao University, Xiamen 361021 China, China, and and al(2017) propose that each each system robot’s and motion becontrolled modeled by as Engineering, Huaqiao University, Xiamen 361021 China, and ala labeled transformation then [email protected]). J. Luo is with College of Information Science and a labeled transformation system and then controlled by so with Fujian Engineering Research Center of Motor Control Engineering, Huaqiao University, Xiamen 361021 China, and al(2017) propose that each robot’s motion be modeled as a labeled transformation system and then controlled by so with Fujian Engineering Research Center of Motor Control so with Fujian Engineering Research Center of China, Motor Control labeled transformation system and then controlled by Engineering, Huaqiao University, Xiamen 361021 and ala distributed algorithm to avoid collisions and deadlocks. and System Optimal Schedule, Xiamen 361021 China(e-mail: jlluaa distributed algorithm to avoid collisions deadlocks. so with Fujian Engineering Research Center of Motor Control and System Optimal Schedule, Xiamen 361021 jllulabeled transformation system and thenand controlled by distributed algorithm to avoid collisions and deadlocks. andwith System Optimal Schedule,Research Xiamen Center 361021 China(e-mail: China(e-mail: jlluso Fujian Engineering of Motor Control aWu distributed algorithm avoid collisions andconflict deadlocks. [email protected]). Q. Zhang is with College of Information Science Wu et propose PN to and and System Optimal Schedule, Xiamen 361021 China(e-mail: [email protected]). Q. is College of Science et al. al. (2005) (2005) proposeto PN method method to the the conflict and [email protected]). Q. Zhang Zhang is with with College of Information Information Science aWu distributed algorithm toaaa avoid collisions andconflict deadlocks. et propose method to and and Engineering, System Optimal Schedule, Xiamen 361021 China(e-mail: jlluand Huaqiao University, Xiamen China(e-mail: [email protected]). Zhang is with College of 361021 Information Science Wu et al. al. (2005) (2005) propose a PN PN method to the the conflict and and Engineering, Engineering,Q. Huaqiao University, Xiamen 361021 China(e-mail: deadlock problems for an AGVS. Hsieh (1998) directly deadlock problems for an AGVS. Hsieh (1998) directly and Huaqiao University, Xiamen 361021 China(e-mail: [email protected]). Q. Zhang is with College of Information Science Wu et al. (2005) propose a PN method to the conflict and deadlock problems for an AGVS. Hsieh (1998) directly [email protected]). W. Wu is with State Key Laboratory of Inand Engineering, Huaqiao University, Xiamen 361021 China(e-mail: [email protected]). W. Wu State Laboratory of deadlock problems for an AGVS. Hsieh (1998) directly converts an AGVS layout into a controlled PN model, [email protected]). W. University, Wu is is with with Xiamen State Key Key Laboratory of InInand Engineering, Huaqiao 361021 China(e-mail: converts an AGVS layout into a controlled PN model, dustrial Control, Institute of Cyber-Systems and Control, Zhejiang deadlock problems for an AGVS. Hsieh (1998) directly converts an AGVS layout into a controlled PN model, [email protected]). W. Wu is with State Key Laboratory of Industrial Control, Institute of Cyber-Systems and Control, Zhejiang dustrial Control, Institute of Cyber-Systems and Laboratory Control, Zhejiang converts an AGVS layout programs into a controlled PNit. model, [email protected]). W. Wu isChina with State Key of Inand designs logic control based on It is and designs logic control based on It is University, Hangzhou 310027, (e-mail: [email protected]). dustrial Control, Institute of Cyber-Systems and Control, Zhejiang University, Hangzhou 310027, China [email protected]). converts an AGVS layout programs into a controlled PNit. model, and designs logic control programs based on it. It is University, Hangzhou 310027, China (e-mail: (e-mail: [email protected]). dustrial Control, Institute of Cyber-Systems and Control, Zhejiang and designs logic control programs based on it. It is possible to construct a feasible solution from a tentative Y. Huang is with Department of Electrical Engineering, National University, Hangzhou 310027, China (e-mail: [email protected]). possible to construct a feasible solution from a tentative Y. Huang Huang is is with with Department Department of of Electrical Electrical Engineering, Engineering, National National and designs logic control programs based on it. It is possible to construct a feasible solution from a tentative Y. University, Hangzhou 310027, China (e-mail: [email protected]). Ilan University, Taiwan (e-mail: [email protected]). M. Zhou possible to construct a feasible solution from a tentative Y. Huang is with Department [email protected]). Electrical Engineering, M. National solution by using blocking and conflict avoidance policies Ilan University, Taiwan (e-mail: Zhou solution by using blocking and conflict avoidance policies IlanHuang University, Taiwan (e-mail:[email protected]). M. Zhou possible to construct a feasible solution from a tentative Y. is with Department Electrical Engineering, National solution by using blocking and conflict avoidance policies is with Department of Electrical and Computer Engineering, New Ilan University, Taiwan (e-mail: [email protected]). M. Zhou is Department of and New solution by using lane blocking andet conflict avoidance policies for unidirectional Tanaka al. (2010). However, such is with with Department of Electrical Electrical and Computer Computer Engineering, Engineering, New for unidirectional Tanaka al. (2010). However, such Ilan University, Taiwan (e-mail: Newark, [email protected]). M. Zhou solution by using lane blocking andet conflict avoidance policies for unidirectional lane Tanaka et al. (2010). However, such Jersey Institute of Technology, NJ 07102 USA (e-mail: is with Department of Electrical and Computer Engineering, New Jersey Institute of Technology, Newark, NJ 07102 USA (e-mail: for unidirectional lane Tanaka et al. (2010). However, such Jersey Institute of Technology, Newark, NJ 07102 USA (e-mail: controller may not be maximally permissive. is with Institute Department of Electrical Newark, and Computer Engineering, New controller may not be maximally permissive. [email protected]). Jersey of Technology, NJ 07102 USA (e-mail: for unidirectional lane Tanaka et al. (2010). However, such controller may not be maximally permissive. [email protected]). [email protected]). Jersey Institute of Technology, Newark, NJ 07102 USA (e-mail: controller may not be maximally permissive. This work was supported in part by National Science Foundation [email protected]). This work was supported in part by National Science Foundation controller may not be maximally permissive. The published studies pay enough This work was supported in part by National Science Foundation [email protected]). The published studies do do not not pay enough attention attention to to of China Grant No. in 61573158 61621002, Fujian ProvinThe published not pay to This workunder was supported part by and National Science Foundation of under Grant 61573158 and 61621002, Fujian Provinof China China Grant No. No. in 61573158 and 61621002, Fujian ProvinThe published studies studies do doand not partially-observable pay enough enough attention attention to This workunder was supported part by National Science Foundation the issues the partially-controllable partially-controllable and partially-observable issues cial Science and Technology Plan Project (2015H0026) and Natuof China under Grant No. 61573158 and 61621002, Fujian Provincial Science and Technology Plan Project (2015H0026) and NatuThe published studies do not pay enough attention to the partially-controllable and partially-observable issues cialChina Science andGrant Technology Plan Project (2015H0026) andProvinNatuof under No. 61573158 and 61621002, Fujian the partially-controllable and partially-observable issues as caused by the lack of sensors and activators or their ral Science Foundation of FuJian Province of China under Grant cial Science and Technology Plan Project (2015H0026) and Natuas caused by the lack of sensors and activators or their ral Science Science Foundation Foundation of of FuJian FuJian Province Province of of China China under under Grant Grant the partially-controllable and partially-observable issues as caused by the lack of sensors and activators or their ral cial Science Foundation and Technology Plan Project (2015H0026) and Grant Natu2017J01117. (Corresponding author: JiLiang Luo) as caused by the lack of sensors and activators or their ral Science of FuJian Province of China under 2017J01117. (Corresponding author: JiLiang 2017J01117. (Corresponding author:Province JiLiang Luo) Luo) as caused by the lack of sensors and activators or their ral Science Foundation of FuJian China under Grant 2017J01117. (Corresponding author: JiLiang of Luo)
2017J01117. author: JiLiang Luo)of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © (Corresponding 2018, IFAC (International Federation Copyright © 2018 IFAC 139 Copyright © 2018 IFAC 139 Peer review under responsibility of International Federation of Automatic Copyright © 2018 IFAC 139 Control. Copyright © 2018 IFAC 139 10.1016/j.ifacol.2018.06.292 Copyright © 2018 IFAC 139
IFAC WODES 2018 140 30 - June 1, 2018. Sorrento Coast, Italy YaXin Wan et al. / IFAC PapersOnLine 51-7 (2018) 139–144 May
failures. These issues may greatly increase the complexity especially when one intends to design the maximally permissive controller. In detail, if all state-transitions in PNs are deterministic, the state can be uniquely identified by an observed sensor signal sequence. Unfortunately, the number of sensors installed in an AGVS is often limited and sometimes malfunction, and, consequently, not all transitions of PN are one-to-one corresponding to the sensor signals, i.e., it is not sure about which transition fires when some sensor signals are observed. This can be described as an undeterministic phenomenon in AGVS. On another hand, a vehicle may uncontrollably enter an area due to a special activation mechanism, and, consequently, this causes uncontrollable events. It is well known that it is not easy to design a maximally-permissive controller for partially controllable PNs. As for the control problem of the class of AGVSs with the undeterminstic and uncontrollable events, a method is proposed by using labeled PNs. First, a labeled PN is constructed by representing the undeterminstic and uncontrollable events by a set of transitions labeled by a same symbol and an uncontrollable transition, respectively. Second, a whole PN is partitioned into a set of regions, called the dangerous regions, that can obtain tokens via only controllable transitions, and, consequently, it is not necessary any more to consider the effect caused by uncontrollable transitions. Since a dangerous region corresponds to an area that is controllable in an AGVS layout, the collision-avoidance problem is transformed into how to ensure that each dangerous region has at most one token(corresponding to an AGV). Third, the method is presented to compute the consistent marking set for a given observed sequence of sensor symbols, and, in turn, the most number of tokens in each dangerous region for this set of consistent markings. Lastly, the control action is to maintain the most number of tokens not more than 1 for any sequence of sensor-symbols, and, resultantly, the maximally-permissive controller is obtained for the collision-avoidance in AGVS. Inally, the proposed method is verified by a simulation experiment. Section II gives a brief introduction to some concepts and properties of labeled PNs. Section III describes a collisionavoidance problem in a class of AGVSs. Section IV gives a method of modeling AGVSs by labeled PNs. Section V presents a method for estimating the state for an observed sensor sequence. Section VI designs a collision avoidance controller. Section VII verifies the proposed method by a simulation experiment. Section VIII concludes the paper. 2. PRELIMINARIES 2.1 Petri net An ordinary PN structure is a four-tuple N = (P, T, F, W ), where P (places) and T (transitions) are non-empty, finite set of places and transition sets respectively, F ⊆ (P × T ) ∪ (T × P ) is a collection of directed arcs that connect places and transitions; W : F → Z+ is a mapping that assigns each arc with a positive integer weight. C − : P × T → Z and C + : T × P → Z are the forward and backward incidence matrices that define the weights of the directed arcs from places to transitions and 140
from transitions to places, respectively. The weight of 0 indicates that there is no directed arc. C = C + − C − is an incidence matrix. In a PN, firing a transition can be prohibited by an external agent. Such transition is called a controllable transition; otherwise, it is uncontrollable. The set of controllable transitions is denoted by Tc , and the uncontrollable transition set, Tc . If firing a transition can be detected, the transition is observable; otherwise,it is unobservable. The token in the PN is represented by a solid black dot and the number of tokens in place p is denoted by m(p). m0 is the initial marking of PN. If and only if m ≥ C − (·, t) (C − (·, t) indicates the column where transition t is in the backward matrix), transition t is enabled under m, denoted as m[t⟩. Only enabled transitions can be fired, and the new marking m′ is generated after t fires such that m′ = m + C(·, t) (C(·, t) indicates the column where transition is in the incidence matrix), which is denoted as m[t⟩m′ . The set of all the transition sequences is denoted as T ∗ . Similarly, if the transition sequence σ ∈ T ∗ is enabled at m, denoted as m[σ⟩, the new marking m′ is reached, denoted as m[S⟩m′ . The marking m is a column vector, m : P → Z is a mapping function that assigns a non-negative integer (tokencount) to each place. 2.2 Labeled Petri net A labeled PN is (N, Σ, L), where N = (P, T, F, W ) is a PN structure; Σ is the alphabet set of letters; and L : T → Σ ∪ {λ} is a label function, which labels a letter for each transition, i.e., ∀t ∈ T, L(t) = e. L−1 (e) = {∀t ∈ T, L(t) = e} is the set of transitions with the same label e. An observed sequence is a sentence S such that ∃σ = tσ1 tσ2 · · · tσn , m0 [σ⟩, S = L(σ) = L(tσ1 )L(tσ2 ) · · · L(tσn ), and σ is a transition sequence associated with S. 3. PROBLEM STATEMENT The class of AGVSs considered in this work is based on the electromagnetic guiding mode, and the example is shown in Fig. 1. z1 ∼ z5 are 5 stations where operations are performed or vehicles can stay; l1 ∼ l7 are seven guiding lines whose arrows denote their directions leading vehicles. In detail, if a guiding line is energized, a magnetic field is generated simultaneously around it, and an vehicle can detect and move along this magnetic field; otherwise, the vehicle stops. s1 ∼ s5 are five sensors installed in z1 ∼ z5 , respectively, and a sensor is used to detect if there is a vehicle in its corresponding station. To avoid collisions among vehicles, it is required that at any time, at most one vehicle can stay in every guiding line or station. How to estimate the distribution of vehicles in an AGVS from a sequence of signals generated by the sensors is the central problem. Its main complexity lies in the undeterministic phenomenon, i.e., the guiding line that this vehicle chooses and the falling edge signal received from the sensor of the station can not be sure when a vehicle leaves a station. Further, it is reasonable that a guiding line should keep powered on if there is a vehicle in it, and, consequently, a vehicle can uncontrollably enter the station toward which the guiding line leads. Evidently, such controllable events further give rise to the complexity of the problem. Accordingly, the labeled PN is to be used as a formal
IFAC WODES 2018 May 30 - June 1, 2018. Sorrento Coast, Italy YaXin Wan et al. / IFAC PapersOnLine 51-7 (2018) 139–144
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Fig. 1. A typical example of the class of AGVSs considered in this work tool to design the maximally permissive controller avoiding collisions. Fig. 2. Labeled PN of the AGVS shown as Fig.1 4. LABELED PN MODEL OF AGVS
transitions increases, so does the computational complexity to estimate the state from an observed sequence.
Given an AGVS, we have to represent it by a formal tool such that the control method can be developed. Hence, the following procedure is presented to do so via labeled PNs.
5. ESTIMATING METHOD OF STATES OF AGVS
(1) For each station or guiding line, design a place representing it. (2) For each pair of station and guiding line, if the guiding line connects this station and the former directs towards the latter, design an uncontrollable transition with the input that is the place representing the former, and the output that is the place representing the latter, respectively; and if the guiding line connects this station and the former directs from the latter, design a controllable transition with the output that is the place representing the former, and the input that is the place representing the latter, respectively. (3) For each sensor si in station zi , i ∈ Z, design two letters ai and bi corresponding to the rising-edge and falling-edge signals, respectively. (4) For each transition t, if it represents a pair from station zi to a guiding line, then L(t) = bi , and, if it represents a pair from a guiding line to zi , L(t) = ai . Evidently, a labeled PN can be constructed according to the above procedure. In this PN, there are uncontrollable transitions, and multiple transitions with a same letter. They can model the uncontrollable and undeterministic phenomenon, respectively. Further, only what is necessary for designing the controller avoiding collisions is captured by this PN model. Example. Taking the AGVS in Fig. 1 as an example. According to the procedure, we obtain the labeled PN, as shown in Fig. 2. P = {z1 , z2 , z3 , z4 , z5 , l1 , l2 , l3 , l4 , l5 , l6 , l7 }, Tc = {t1 , t2 , t3 , t4 , t5 , t6 , t7 }, Tc = {t8 , t9 , t10 , t11 , t12 , t13 , t14 }, and Σ = {a1 , a2 , a3 , a4 , a5 , b1 , b2 , b3 , b4 , b5 }. The arcs and label functions are also clearly shown. t2 , t4 , t5 , t8 , t9 and t11 are deterministic. For instance, when b2 is observed, t2 fires, i.e., the guiding line l2 is energized, and a vehicle leaves zone z2 , and enters l2 . However, transitions t1 , t6 , t3 , t7 , t10 , t13 , t12 and t14 are undeterministic. For instance, when b3 is observed, we are not sure which of t3 and t7 fires. This means that we only know that a vehicle leaves z3 , but we are not sure about which one between l3 and l7 that this vehicle enters. As the number of undeterministic 141
Since a sensor is installed in each station, a symbol (risingedge or falling-edge signal) is observed once a vehicle enters or leaves a station. A particular case is the initial sequence that is observed just at the instant that AGVS is turned on, and this sequence is denoted by w0 . In details, when AGVS is started up, the sensor of a station immediately observes a rising-edge signal if there has been a vehicle parked in this station. Therefore, w0 is composed of a series of rising-edge signals. Further, the initial marking of the PN model can be derived from w0 . Algorithm 1 is used to do so. Algorithm 1 Initial marking of PN model of AGVS Input: labeled PN for AGVS and w0 . Output: C(w0 ) = {m0 }. 1: for each place p do 2: m0 (p) = 0; 3: end for 4: for all 1 ≤ i ≤ |w0 | do 5: ∃t ∈ T (w0 [i]), ∃p ∈ t• , m0 (p) = 1. 6: end for Given an observed sequence w, C(w) = {m m0 [σ⟩m, L(σ) = w} is a set of w, which is called consistent markings . Evidently, C(w) is a set of markings that the PN may reach if w is observed. According to Algorithm 1, the initial marking of the labeled PN model of an AGVS can be accurately deduced from the initial observed sequence. Example. Take AGVS in Fig. 1 as an example. If w0 = a1 a4 , then, according to Algorithm 1, we can deduce C(w0 ) = {m0 } = {(m(z1 ), m(z2 ), m(z3 ), m(z4 ), m(z5 ), m(l1 ), m(l2 ), m(l3 ), m(l4 ), m(l5 ), m(l6 ), m(l7 ))} = {(1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0)}. It means that either of stations z1 and z4 have one vehicle at the initial instance. After AGVS is started, vehicles arrive and leave stations, and, consequently, the sensors in the stations generate signals accordingly. This implies that the length of the observed sequence increases, and that we should estimate the distribution of vehicles once the observed sequence changes. For the convenience to do so, it is assumed that
IFAC WODES 2018 142 30 - June 1, 2018. Sorrento Coast, Italy YaXin Wan et al. / IFAC PapersOnLine 51-7 (2018) 139–144 May
wk , k ∈ {0, 1, 2 · · · }, represents the observed sequence after the k-th observation. Hence, wk is the prefix of wk+1 , and wk+1 = wk δk where δk is the sensed letter or symbol at the k-th observation. Hereafter, we show how to estimate the state after one observation. Algorithm 2 Consistent set for an observed sequence Input: Labeled PN of AGVS, wk−1 , C(wk−1 ), and δk . Output: C(wk ). 1: wk = wk−1 δk ; 2: C(wk ) = ∅; 3: Tk = L−1 (δk ); 4: for all t ∈ Tk do 5: for all m ∈ C(wk−1 ) do 6: if m ≥ C − (·, t) then 7: m′ = m + C(·, t); 8: C(wk ) = C(wk ) ∪ {m′ }; 9: end if 10: end for 11: end for
Suppose that the controller is a function u : Σ∗ → 2T such that it maps each observed sequence to a set of transitions that should be disabled, and uk = u(wk ). Evidently, for the control specification (1), the maximally permissive controller is { � � uk = ∀t ∈ T �∀m ∈ C(wk ), m ≥ C − (·, t), ′
m = m + C(·, t),
∨ ∑
D∈D p∈D
where k = 0, 1, 2, · · · .
′
}
(2)
m (p) ≥ 2 ,
According to (2), Algorithm 3 is designed to compute the control actions for each observed sequence, and, consequently, to prevent collisions among vehicles.
By Algorithm 2, we can estimate the possible markings that the PN reaches once a new observed signal is received. According to Algorithm 1, C(w0 ), can be computed. In turn, C(w0 ) and δ1 are inputed into Algorithm 2, and C(w1 ), i.e., the set of possible markings the net reaches after δ1 is observed, is computed. In the similar way, C(w2 ), C(w3 ), · · · can be obtained one by one. Example. Take AGVS in Fig. 1 as an example. Suppose that w0 = a1 a4 , C(w0 ) = {(1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0)}, and = { δ1 = b1 . Hence w1 = a1 a4 b1 . By Algorithm 2, C(w1 ) } (1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0), (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0) can be computed from C(w0 ) and δ1 . This implies the net may be at one marking in C(w1 ) after δ1 is observed. 6. COLLISION-FREE CONTROLLER SYNTHESIS VIA LABELED PETRI NET The key to prevent any collision is to make sure that at most one vehicle is allowed in each station or guide line at any moment. As for the labeled PN model, it is required that the number of tokens in each place be not more than 1. If there are two or more AGVs in AGVS, then after the previous analysis, the collision never occurs through the collision avoidance system as long as the number of AGVs does not exceed the number of dangerous regions and the number of AGVs in each dangerous region does not exceed one. Theorem 1. Given a labeled PN model of an AGVS and any place p in Ps (Ps is the set of places, called the stationplace set), the dangerous region of p is the set of places that lead an uncontrollable path to p, denoted by D(p), and the set of dangerous regions is D = {D(p)|∀p ∈ Ps }. According to Theorem 1, the collision avoidance problem can be formalized as a conjunction of linear constraints: ∧ ∑ m(p) ≤ 1. (1) D∈D p∈D
Since the net may be at any marking in C(wk ), k = 0, 1, 2, · · · , after the observed sequence wk is received, we should disable any controllable transition whose firing may lead any one in C(wk ) to a marking that violates (1). 142
Algorithm 3 Controller for avoiding collision in AGVS Input: Labeled PN of AGVS, N = (P, T, F, Σ, L), w0 , and δk , k = 1, 2, · · · . Output: uk , k = 0, 1, 2, · · · . 1: compute C(w0 ) by inputting N and w0 into Algorithm 1; 2: let K be a large enough integer; 3: for k = 1 → K do 4: δk is the observed symbol at the kth intance; 5: wk = wk−1 δk ; 6: Compute C(wk ) by Algorithm 2 from C(wk−1 ), wk−1 , and δk ; 7: uk = ∅; 8: for all m ∈ C(wk ) do 9: for all t ∈ L−1 (δk ) do 10: for all m ∈ C(wk−1 ) do 11: if m ≥ C − (·, t) then 12: m′ = m + C(·, t); 13: for all ∑ D ∈ D do 14: if p∈D m(p) > 1 then 15: uk = uk ∪ {t}; 16: end if 17: end for 18: end if 19: end for 20: end for 21: end for 22: end for For k = 0, 1, 2, · · · , uk computed by Algorithm 3 is a set of controllable transitions that should be disabled. Disabling a controllable transition can be implemented by turning off corresponding guiding line. In details, the guiding line, which is corresponding to a controllable transition, is represented by the place whose input is this transition. Example. Taking AGVS in Fig. 1 as an example. In the labeled PN shown in Fig. 2, z1 ’s dangerous region is D(z1 ) = {{l5 , l7 , z1 }, and the set of dangerous } regions is D = D(z1 ), D(z2 ), D(z3 ), D(z4 ),}D(z5 ) = { {l5 , l7 , z1 }, {l1 , z2 }, {l2 , z3 }, {l3 , l6 , z4 }, {l4 , z5 } . According (1), the control specification avoiding collisions is m(z1 ) + m(l5 ) + m(l7 ) ≤ 1; m(l1 ) + m(z2 ) ≤ 1; m(l2 ) + m(z3 ) ≤ 1; (3) m(l3 ) + m(l6 ) + m(z4 ) ≤ 1; m(l4 ) + m(z5 ) ≤ 1.
IFAC WODES 2018 May 30 - June 1, 2018. Sorrento Coast, Italy YaXin Wan et al. / IFAC PapersOnLine 51-7 (2018) 139–144
This means that the collisions among vehicles can be avoided if (3) is enforced on the labeled PN. Further, the control actions can be computed by Algorithm 3. For instance, u0 = {t6 } if w0 = a1 a4 , and C(w0 ) = {(1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0)} from Example 2. u {1 = {t5 } if δ1 = b4 , w}1 = a1 a4 b4 and C(w1 ) = (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0) . 7. SIMULATION RESULTS
In order to verify the proposed method, a simulation experiment is performed. We develop a Matlab program to simulate a labeled PN step by step. Further, a control problem is designed according to Algorithm 3, and can disable controllable transitions such that (1) is fulfilled. In this simulation, we still use AGVS in Fig. 1, which has two AGV. Suppose they are in z1 and z4 initially. The simulation is performed in 200 steps, a transition fires in each step, and a corresponding signal is received by the controller. As a result, the markings of places, and dangerous regions are shown in Figs. 3-4. From them, it can be concluded that at most one token (vehicle) can exist in each place or dangerous region, and, consequently, collision is successfully avoided. Further, the control actions are shown in Fig. 5, which actually disable controllable transitions t1 , t2 , · · · and t7 . For instance, if the signal for t1 is of low level or 0, t1 is disabled in the corresponding period; otherwise, it is allowed to fire. 8. CONCLUSIONS Based on the labeled PN, a control method is proposed to address the collision problem in the class of AGVS that are partially controllable, and have undeterministic behaviors. The dangerous region is defined to deal with uncontrollable events, and a consistent marking set is used for undeterministic behaviors. Further, the presented controller is maximally permissive. Next, we plan to study optimal scheduling problems of this class of AGVSs. REFERENCES Vis IFA, (2006). Survey of research in the design and control of automated guided vehicle systems. Operational Research, Volume 170, 677–709. Singh, N., Sarngadharan, P.V., and Pal, P.K. (2011). AGV scheduling for automated material distribution: A case study. Operational Research, volume 22, 219–228. Le-Anh, T., De Koster MBM(2006). A review of design and control of automated guided vehicle systems. Operational Research, Volume 171, 1–23. Zhou, M., Dicesare, F.(1993). Petri net synthesis for discrete event control of manufacturing systems. Kluwer Academic Ferrarini, L., Piroddi, L.(2008). Modeling and control of fluid transportation operations in production plants with Petri nets. IEEE Transactions on Control Systems Technology, Volume 16, 1090–1098. Giua, A., Seatzu, C.(2008). Modeling and supervisory control of railway networks using Petri nets. IEEE Transactions on Automation Science and Engineering, Volume 5, 431–445. 143
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Lu, X., Zhou, M. and Ammari, A.C.(2016). Hybrid Petri Nets for Modeling and Analysis of Microgrid Systems. IEEE/CAA Journal of Automatica Sinica, Volume 3, 347–354. Yu Ru, Hadjicostis, C.N.(2009). Bounds on the number of markings consistent with label observations in Petri nets. IEEE Transactions on Automation Science and Engineering, Volume 6, 334 –344. Luo, J., Nonami, K.(2011). Approach for transforming linear constraints on Petri nets. IEEE Transactions on Automatic Control, Volume 56, 2751–2765. Luo, J., Zhou, M.(2017). Petri-net controller synthesis for partially controllable and observable discrete event systems. IEEE Transactions on Automatic Control, Volume 62, 1301–1313. Cabasino, M.P., Hadjicostis, C.N. and Seatzu, C. (2017). Marking observer in labeled Petri nets with application to supervisory control. IEEE Transactions on Automatic Control, Volume 62, 1813–1824. Hsieh, S., Kang, M.(1998). Developing agvs Petri net control models from flowpath nets. Manufacturing Systems, Volume 17, 237–250. Fanti, M.(2002). A deadlock avoidance strategy for agv systems modelled by coloured Petri nets. International Workshop on Discrete Event Systems, 61–66. Wu, N., Zhou, M.(2005). Modeling and deadlock avoidance of automated manufacturing systems with multiple automated guided vehicles. IEEE Transactions on Systems Man and Cybernetics Part B Cybernetics, Volume 35, 1193–1202. Nishi, T., Tanaka, Y.(2012). Petri net decomposition approach for dispatching and conflict-free routing of bidirectional automated guided vehicle systems. IEEE Transactions on Systems Man and Cybernetics - Part A Systems and Humans, Volume 42, 1230–1243. Nishi, T., Hiranaka, Y. and Grossmann, I.E.Z.(2011). A bilevel decomposition algorithm for simultaneous production scheduling and conflict-free routing for automated guided vehicles. Operations Research, Volume 38, 876– 888. Luo, J., Ni, H.J., Wu, W., Wang, S. and Zhou, M.(2015). Simultaneous reduction of Petri nets and linear constraints for efficient supervisor synthesis. IEEE Transactions on Systems Man Cybernetics Systems, Volume 45, 44–55. Nishi, T., Maeno, R.(2010). Petri Net Decomposition Approach to Optimization of Route Planning Problems for AGV Systems. IEEE Transactions on Automation Science and Engineering, Volume 7, 523–537. Zhou, Y., Hu, H. and Liu, Y.(2017). Collision and Deadlock Avoidance in Multirobot Systems: A Distributed Approach. IEEE Transactions on Systems Man and Cybernetics Systems, Volume 47, 1712–1726. Tanaka, T., Nishi, T. and Inuiguchi, M.(2010). Dynamic optimization of simultaneous dispatching and conflictfree routing for automated guided vehiclesPetri net decomposition Approach. Jamdsm, Volume , 894–901.
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Fig. 3. Markings of places
Fig. 4. Markings of the dangerous regions
Fig. 5. Controls of transitions
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IFAC PapersOnLine 51-7 (2018) 139–144
Controller Design for Avoiding Collisions in Controller Design for Avoiding Collisions in Controller Design for Avoiding Collisions in Controller Design for Avoiding Collisions in Automated Guided Vehicle Systems via Automated Guided Vehicle Systems via Automated Guided Vehicle Systems via Automated GuidedPetri Vehicle Systems via Labeled Nets Labeled Petri Nets Labeled Petri Nets Labeled Petri Nets
YaXin Wan, JiLiang Luo, Qi Zhang, WeiMin Wu, YaXin Wan, JiLiang Luo, Qi Zhang, WeiMin Wu, YaXin Wan, JiLiang Luo, Qi Zhang, WeiMin Wu, YaXin YiSheng Wan, JiLiang Luo, Qi Zhang, WeiMin Huang, and MengChu Zhou YiSheng Huang, andQi MengChu Zhou Wu, YaXin YiSheng Wan, JiLiang Luo, Zhang, WeiMin Wu, Huang, and MengChu Zhou YiSheng Huang, and MengChu Zhou YiSheng Huang, and MengChu Zhou Abstract: As for a class class of automated systems (AGVS) that are partially Abstract: guided vehicle vehicle systems Abstract: As As for for a class of of automated automated guided guided systems (AGVS) (AGVS) that that are are partially partially Abstract: As for aa class of automated guidedaa vehicle vehicle systems (AGVS) that are partially controllable and with undeterministic behavior, method is presented to design a maximally controllable and with undeterministic behavior, method is presented to design a maximally Abstract: As for a class of automated guideda vehicle systems (AGVS) that are partially controllable and with undeterministic behavior, method is presented to design a maximally controllable and withthat undeterministic behavior, a method isvia presented to design aIt maximally permissive controller avoids any collision among vehicles labeled Petri nets. is not sure permissive avoids among vehicles labeled nets. is controllable and withthat undeterministic behavior, a method presented to design permissive controller controller that avoids any any collision collision among vehiclesisvia via labeled Petri Petri nets. aIt It maximally is not not sure sure permissive controller that avoids any collision among vehicles via labeled Petri nets. 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The whole net is partitioned into a set of dangerous regions that are controllable controllable. The whole net is partitioned aa set of dangerous regions are controllable an uncontrollable transition, an AGVS is into represented by a labeled Petri that net that is partially controllable. The whole net is partitioned into set of dangerous regions that are controllable controllable. The whole net is partitioned into A a set of dangerous regions that are controllable such that the uncontrollable issue is addressed. method is proposed to compute the consistent such uncontrollable issue is method is to the consistent controllable. whole net is partitioned into A a set of dangerous regions that are such that that the theThe uncontrollable issue is addressed. addressed. A method is proposed proposed to compute compute thecontrollable consistent such that the uncontrollable issue is addressed. A method is proposed to compute the consistent marking set for a given observed sequence. By the resultant consistent marking set, the marking marking set for a given observed sequence. By the resultant consistent marking set, the marking such thatset thefor uncontrollable issuesequence. is addressed. A method is consistent proposed to compute the consistent marking a given observed By the resultant marking set, the marking marking set for a given observed sequence. By the resultant consistent marking set,ttthe marking of each dangerous region is computed, and can be used to design a control action of each dangerous region is computed, and can be used to design a control action marking set for a given observed sequence. By the resultant consistent marking set,tthe marking of each dangerous region is computed, and can be used to design a control action of each dangerous region is computed, anddangerous can be used to design control action prevent each vehicle from entering region where is AGV ooo prevent each from entering each each region where aathere there is one one AGVtt already. already. As As of each dangerous region is computed, anddangerous can be used to design control action prevent each vehicle vehicle from entering each region oaa prevent each vehicle from enteringcontroller each dangerous dangerous region where where there there is is one one AGV AGV already. already. As As result, the maximally permissive controller is obtained. result, the maximally permissive is obtained. oa prevent each vehicle from enteringcontroller each dangerous region where there is one AGV already. As result, the maximally permissive is obtained. a result, the maximally permissive controller is obtained. a©result, the maximally permissive is obtained. 2018, IFAC (International Federationcontroller of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Automated navigation vehicle systems; Collision avoidance; avoidance; Petri Petri nets; nets; Discrete Discrete Keywords: Automated navigation vehicle systems; Collision Keywords: Automated navigation vehicle systems; Keywords: Automated navigation vehicle systems; Collision Collision avoidance; avoidance; Petri Petri nets; nets; Discrete Discrete event systems event systems Keywords: Automated navigation vehicle systems; Collision avoidance; Petri nets; Discrete event event systems systems event systems 1. INTRODUCTION the 1. the safety, safety, the the collision collision avoidance avoidance problem problem is is even even more more 1. INTRODUCTION INTRODUCTION the safety, the avoidance 1. INTRODUCTION the safety, than the collision collision avoidance problem problem is is even even more more important the others. important than the others. 1. INTRODUCTION the safety, than the collision avoidance problem is even more important the others. important than the others. With industry 4.0 and Internet of Things, more attention With industry 4.0 and Internet of Things, more attention With industry industry 4.0 4.0 and and Internet Internet of of Things, Things, more more attention attention important Because their high modeling Because of of than their the highothers. modeling efficiency, efficiency, Petri Petri nets nets (PNs) (PNs) With Because of high efficiency, Petri nets (PNs) has been drawn to the research field about intelligent has been drawn research field about Because of their their high modeling modeling efficiency, Petriand netscontrol (PNs) With industry 4.0to andthe Internet of Things, moreintelligent attention have has been drawn to the research field about intelligent become a successful tool for the analysis have become a successful tool for the analysis and control Because of their high modeling efficiency, Petri nets (PNs) has been drawn to the research field about intelligent have become a successful tool for the analysis and control manufacturing. Different from the traditional manufacmanufacturing. from the traditional manufachave becomeevent a successful tool for the analysis andFerrarini control has been drawnDifferent to the research field about intelligent manufacturing. Different from the traditional manufacof discrete systems Zhou et al. (1993); of discrete event systems Zhou et al. (1993); Ferrarini have become a successful tool for the analysis and control manufacturing. Different from the traditional manufacof discrete event systems Zhou et al. (1993); Ferrarini turing, the intelligent one should be able to quickly and turing, one should able quickly and et of discrete event systems Zhou et Lu al. et (1993); Ferrarini manufacturing. Different thebe manufacturing, the the intelligent intelligent one from should betraditional able to to quickly and al. (2008); Giua et al. (2008); al. (2016); Ru et al. (2008); Giua et al. (2008); al. (2016); Ru of event systems Zhou et Lu al. et (1993); Ferrarini turing, the intelligent one should be able to quickly and et discrete al. (2008); Giua et al. (2008); Lu et al. (2016); Ru flexibly respond to complex orders that tend to be of flexibly respond to complex orders tend to be of al. (2008); Giua et al. (2008); Lu et al. (2016); Ru turing, the intelligent one should be that able to quickly and flexibly respond to complex orders that tend to be of et (2009); Luo et al. (2011, 2017); Cabasino et al. et al. (2009); Luo et al. (2011, 2017); Cabasino et al. et al. (2008); Giua et al. (2008); Lu et al. (2016); Ru flexibly respond to complex orders that tend to be of (2009); Luo et al. (2011, 2017); Cabasino et al. small quantity, personized and multi-batch. This requires small quantity, personized and multi-batch. This requires et al. (2009); Luo et al. (2011, 2017); Cabasino et al. flexibly respondpersonized to complex orders that tend torequires be of (2017). small quantity, and multi-batch. This Therefore, PNs have also been widely used in the (2017). Therefore, PNs have also been widely used in the et al. (2009); Luo et al. (2011, 2017); Cabasino et al. small quantity, personized and multi-batch. This requires (2017). Therefore, PNs have also been widely used in the a control method for automated guided vehicle systems aa control method for automated guided vehicle systems (2017). Therefore, PNs have also been widely used in the small quantity, personized and multi-batch. This requires control method for automated guided vehicle systems research of AGVS research since an AGVS is typically aaa research of AGVS research since an AGVS is typically (2017). Therefore, PNs have also been widely used in the a(AGVS) controlto method for automated guided vehicle systems research of AGVS research since an AGVS is typically (AGVS) to be much improved since AGVS is an imporbe much since AGVS is an imporresearch of AGVS research since an AGVS is typically a a(AGVS) controltomethod for improved automated guided vehicle systems be much improved since AGVS is an impordiscrete event system. Fanti (2002) models an AGVS by discrete event event system. Fantisince (2002) models an AGVS by by of AGVS research an models AGVS an is typically a (AGVS) to be muchtool, improved since AGVS applicable is an impordiscrete system. Fanti (2002) AGVS tant transportation which is broadly in tant transportation which is broadly in research discrete event system. Fanti (2002) models an AGVS by (AGVS) to be muchtool, improved since AGVS applicable is an important transportation tool, which is broadly applicable in a colored timed PN, and presents the control strategy for a colored coloredevent timedsystem. PN, and and presents themodels controlanstrategy strategy for discrete Fanti (2002) AGVS by tant transportation tool, which is broadly applicable in a timed PN, presents the control for modern manufacturing systems. However, as the system modern manufacturing systems. However, the colored collision timed PN, and presents Nishi the control strategy for tant transportation tool, which is broadlyas in aavoiding modern manufacturing systems. However, asapplicable the system system avoiding and deadlock. et al. (2012) and and deadlock. et al. (2012) and aavoiding colored collision timed PN, and presents Nishi the control strategy for modern manufacturing systems. However, as the system collision and deadlock. Nishi et al. (2012) and size increases, the control problems of AGVS become size increases, the control problems of AGVS avoiding collision and deadlock. Nishi et al. (2012) and modern manufacturing systems. However, as the become system Nishi size increases, the control problems of AGVS become et al. (2011) address the dispatching and conflict Nishi et al. (2011) address the dispatching conflict avoiding collision and deadlock. Nishi et al. and (2012) and size increases, thewhich control problems ofdispatching, AGVS become Nishi et al. (2011) address the dispatching and conflict very complicated, which includes task dispatching, path very complicated, includes task path Nishi et al. address the dispatching size thewhich control problems AGVS become very increases, complicated, includes task ofdispatching, dispatching, path problems of an by PN. al. (2015) problems of (2011) an AGVS AGVS by using using PN. Luo Luo et etand al. conflict (2015) Nishi et al. (2011) address the dispatching and conflict very complicated, which includes task path problems of an by Luo al. planning and collision avoidance to planning and collision avoidance according to IFA IFA (2006); (2006); problems ofmethod an AGVS AGVS by using using PN. PN.designing Luo et et the al. (2015) (2015) very complicated, which includesaccording task dispatching, path present planning and collision avoidance according to IFA (2006); the for automatically ladder present the method for automatically designing the ladder problems of an AGVS by using PN. Luo et al. (2015) planning and collision avoidance according to IFA (2006); present the method for automatically designing the Singh et al. (2011); Le-Anh et al. (2006). In order to ensure Singh et al. (2011); Le-Anh et al. (2006). In order to ensure present the method for automatically designing the ladder ladder planning and collision avoidance according to IFA (2006); Singh et al. (2011); Le-Anh et al. (2006). In order to ensure diagram for avoiding collision by ordinary PNs. Nishi et al. diagram for avoiding collision by ordinary PNs. Nishi et al. the method for automatically designing the ladder Singh et al. (2011); Le-Anh et al. (2006). In order to ensure present diagram for avoiding collision by ordinary PNs. Nishi et al. ⋆ diagram for avoiding collision by ordinary PNs. Nishi etthe al. Singh et al. (2011); Le-Anh et al. (2006). In order to ensure ⋆ (2010) develop a Petri net decomposition approach for Y. Wan is with College of Information Science and Engineer(2010) develop a Petri net decomposition approach for the Wan is with College of Information Science and Engineer⋆ Y. diagram for avoiding collision by ordinary PNs. Nishi et al. (2010) develop a Petri net decomposition approach for the Y. Wan is with College of Information Science and Engineer⋆ ing, Huaqiao University, Xiamen 361021 China(e-mail: yaxinmas(2010) develop a Petri net decomposition approach for the Y. Wan is with College of Information Science and Engineeroptimization of conflict-free routing for AGVs. Zhou et al. ing, Huaqiao University, Xiamen 361021 China(e-mail: yaxinmasoptimization ofaconflict-free conflict-free routing for for AGVs. AGVs. Zhoufor etthe al. ⋆ ing, Xiamen 361021 China(e-mail: yaxinmas(2010) develop Petri net decomposition approach optimization of routing Zhou et al. Y. Huaqiao Wan is University, with College of College Information Science and [email protected]). J. Luo is with of Information Science and ing, Huaqiao University, Xiamen 361021 China(e-mail: [email protected]). J. J. Luo Luo is is with with College College of of Information Information Science Science and and optimization of conflict-free routingmotion for AGVs. Zhou et al. (2017) propose that each be modeled as [email protected]). (2017) propose that each robot’s robot’s be modeled as ing, Huaqiao University, Xiamen 361021 China(e-mail: yaxinmasoptimization of conflict-free routingmotion for AGVs. Zhou et al. (2017) propose that robot’s motion be modeled as Engineering, University, Xiamen 361021 and [email protected]). J. Luo is with College of Information Science Engineering, Huaqiao Huaqiao University, Xiamen 361021 China, China, and and al(2017) propose that each each system robot’s and motion becontrolled modeled by as Engineering, Huaqiao University, Xiamen 361021 China, and ala labeled transformation then [email protected]). J. Luo is with College of Information Science and a labeled transformation system and then controlled by so with Fujian Engineering Research Center of Motor Control Engineering, Huaqiao University, Xiamen 361021 China, and al(2017) propose that each robot’s motion be modeled as a labeled transformation system and then controlled by so with Fujian Engineering Research Center of Motor Control so with Fujian Engineering Research Center of China, Motor Control labeled transformation system and then controlled by Engineering, Huaqiao University, Xiamen 361021 and ala distributed algorithm to avoid collisions and deadlocks. and System Optimal Schedule, Xiamen 361021 China(e-mail: jlluaa distributed algorithm to avoid collisions deadlocks. so with Fujian Engineering Research Center of Motor Control and System Optimal Schedule, Xiamen 361021 jllulabeled transformation system and thenand controlled by distributed algorithm to avoid collisions and deadlocks. andwith System Optimal Schedule,Research Xiamen Center 361021 China(e-mail: China(e-mail: jlluso Fujian Engineering of Motor Control aWu distributed algorithm avoid collisions andconflict deadlocks. [email protected]). Q. Zhang is with College of Information Science Wu et propose PN to and and System Optimal Schedule, Xiamen 361021 China(e-mail: [email protected]). Q. is College of Science et al. al. (2005) (2005) proposeto PN method method to the the conflict and [email protected]). Q. Zhang Zhang is with with College of Information Information Science aWu distributed algorithm toaaa avoid collisions andconflict deadlocks. et propose method to and and Engineering, System Optimal Schedule, Xiamen 361021 China(e-mail: jlluand Huaqiao University, Xiamen China(e-mail: [email protected]). Zhang is with College of 361021 Information Science Wu et al. al. (2005) (2005) propose a PN PN method to the the conflict and and Engineering, Engineering,Q. Huaqiao University, Xiamen 361021 China(e-mail: deadlock problems for an AGVS. Hsieh (1998) directly deadlock problems for an AGVS. Hsieh (1998) directly and Huaqiao University, Xiamen 361021 China(e-mail: [email protected]). Q. Zhang is with College of Information Science Wu et al. (2005) propose a PN method to the conflict and deadlock problems for an AGVS. Hsieh (1998) directly [email protected]). W. Wu is with State Key Laboratory of Inand Engineering, Huaqiao University, Xiamen 361021 China(e-mail: [email protected]). W. Wu State Laboratory of deadlock problems for an AGVS. Hsieh (1998) directly converts an AGVS layout into a controlled PN model, [email protected]). W. University, Wu is is with with Xiamen State Key Key Laboratory of InInand Engineering, Huaqiao 361021 China(e-mail: converts an AGVS layout into a controlled PN model, dustrial Control, Institute of Cyber-Systems and Control, Zhejiang deadlock problems for an AGVS. Hsieh (1998) directly converts an AGVS layout into a controlled PN model, [email protected]). W. Wu is with State Key Laboratory of Industrial Control, Institute of Cyber-Systems and Control, Zhejiang dustrial Control, Institute of Cyber-Systems and Laboratory Control, Zhejiang converts an AGVS layout programs into a controlled PNit. model, [email protected]). W. Wu isChina with State Key of Inand designs logic control based on It is and designs logic control based on It is University, Hangzhou 310027, (e-mail: [email protected]). dustrial Control, Institute of Cyber-Systems and Control, Zhejiang University, Hangzhou 310027, China [email protected]). converts an AGVS layout programs into a controlled PNit. model, and designs logic control programs based on it. It is University, Hangzhou 310027, China (e-mail: (e-mail: [email protected]). dustrial Control, Institute of Cyber-Systems and Control, Zhejiang and designs logic control programs based on it. It is possible to construct a feasible solution from a tentative Y. Huang is with Department of Electrical Engineering, National University, Hangzhou 310027, China (e-mail: [email protected]). possible to construct a feasible solution from a tentative Y. Huang Huang is is with with Department Department of of Electrical Electrical Engineering, Engineering, National National and designs logic control programs based on it. It is possible to construct a feasible solution from a tentative Y. University, Hangzhou 310027, China (e-mail: [email protected]). Ilan University, Taiwan (e-mail: [email protected]). M. Zhou possible to construct a feasible solution from a tentative Y. Huang is with Department [email protected]). Electrical Engineering, M. National solution by using blocking and conflict avoidance policies Ilan University, Taiwan (e-mail: Zhou solution by using blocking and conflict avoidance policies IlanHuang University, Taiwan (e-mail:[email protected]). M. Zhou possible to construct a feasible solution from a tentative Y. is with Department Electrical Engineering, National solution by using blocking and conflict avoidance policies is with Department of Electrical and Computer Engineering, New Ilan University, Taiwan (e-mail: [email protected]). M. Zhou is Department of and New solution by using lane blocking andet conflict avoidance policies for unidirectional Tanaka al. (2010). However, such is with with Department of Electrical Electrical and Computer Computer Engineering, Engineering, New for unidirectional Tanaka al. (2010). However, such Ilan University, Taiwan (e-mail: Newark, [email protected]). M. Zhou solution by using lane blocking andet conflict avoidance policies for unidirectional lane Tanaka et al. (2010). However, such Jersey Institute of Technology, NJ 07102 USA (e-mail: is with Department of Electrical and Computer Engineering, New Jersey Institute of Technology, Newark, NJ 07102 USA (e-mail: for unidirectional lane Tanaka et al. (2010). However, such Jersey Institute of Technology, Newark, NJ 07102 USA (e-mail: controller may not be maximally permissive. is with Institute Department of Electrical Newark, and Computer Engineering, New controller may not be maximally permissive. [email protected]). Jersey of Technology, NJ 07102 USA (e-mail: for unidirectional lane Tanaka et al. (2010). However, such controller may not be maximally permissive. [email protected]). [email protected]). Jersey Institute of Technology, Newark, NJ 07102 USA (e-mail: controller may not be maximally permissive. This work was supported in part by National Science Foundation [email protected]). This work was supported in part by National Science Foundation controller may not be maximally permissive. The published studies pay enough This work was supported in part by National Science Foundation [email protected]). The published studies do do not not pay enough attention attention to to of China Grant No. in 61573158 61621002, Fujian ProvinThe published not pay to This workunder was supported part by and National Science Foundation of under Grant 61573158 and 61621002, Fujian Provinof China China Grant No. No. in 61573158 and 61621002, Fujian ProvinThe published studies studies do doand not partially-observable pay enough enough attention attention to This workunder was supported part by National Science Foundation the issues the partially-controllable partially-controllable and partially-observable issues cial Science and Technology Plan Project (2015H0026) and Natuof China under Grant No. 61573158 and 61621002, Fujian Provincial Science and Technology Plan Project (2015H0026) and NatuThe published studies do not pay enough attention to the partially-controllable and partially-observable issues cialChina Science andGrant Technology Plan Project (2015H0026) andProvinNatuof under No. 61573158 and 61621002, Fujian the partially-controllable and partially-observable issues as caused by the lack of sensors and activators or their ral Science Foundation of FuJian Province of China under Grant cial Science and Technology Plan Project (2015H0026) and Natuas caused by the lack of sensors and activators or their ral Science Science Foundation Foundation of of FuJian FuJian Province Province of of China China under under Grant Grant the partially-controllable and partially-observable issues as caused by the lack of sensors and activators or their ral cial Science Foundation and Technology Plan Project (2015H0026) and Grant Natu2017J01117. (Corresponding author: JiLiang Luo) as caused by the lack of sensors and activators or their ral Science of FuJian Province of China under 2017J01117. (Corresponding author: JiLiang 2017J01117. (Corresponding author:Province JiLiang Luo) Luo) as caused by the lack of sensors and activators or their ral Science Foundation of FuJian China under Grant 2017J01117. (Corresponding author: JiLiang of Luo)
2017J01117. author: JiLiang Luo)of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © (Corresponding 2018, IFAC (International Federation Copyright © 2018 IFAC 139 Copyright © 2018 IFAC 139 Peer review under responsibility of International Federation of Automatic Copyright © 2018 IFAC 139 Control. Copyright © 2018 IFAC 139 10.1016/j.ifacol.2018.06.292 Copyright © 2018 IFAC 139
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failures. These issues may greatly increase the complexity especially when one intends to design the maximally permissive controller. In detail, if all state-transitions in PNs are deterministic, the state can be uniquely identified by an observed sensor signal sequence. Unfortunately, the number of sensors installed in an AGVS is often limited and sometimes malfunction, and, consequently, not all transitions of PN are one-to-one corresponding to the sensor signals, i.e., it is not sure about which transition fires when some sensor signals are observed. This can be described as an undeterministic phenomenon in AGVS. On another hand, a vehicle may uncontrollably enter an area due to a special activation mechanism, and, consequently, this causes uncontrollable events. It is well known that it is not easy to design a maximally-permissive controller for partially controllable PNs. As for the control problem of the class of AGVSs with the undeterminstic and uncontrollable events, a method is proposed by using labeled PNs. First, a labeled PN is constructed by representing the undeterminstic and uncontrollable events by a set of transitions labeled by a same symbol and an uncontrollable transition, respectively. Second, a whole PN is partitioned into a set of regions, called the dangerous regions, that can obtain tokens via only controllable transitions, and, consequently, it is not necessary any more to consider the effect caused by uncontrollable transitions. Since a dangerous region corresponds to an area that is controllable in an AGVS layout, the collision-avoidance problem is transformed into how to ensure that each dangerous region has at most one token(corresponding to an AGV). Third, the method is presented to compute the consistent marking set for a given observed sequence of sensor symbols, and, in turn, the most number of tokens in each dangerous region for this set of consistent markings. Lastly, the control action is to maintain the most number of tokens not more than 1 for any sequence of sensor-symbols, and, resultantly, the maximally-permissive controller is obtained for the collision-avoidance in AGVS. Inally, the proposed method is verified by a simulation experiment. Section II gives a brief introduction to some concepts and properties of labeled PNs. Section III describes a collisionavoidance problem in a class of AGVSs. Section IV gives a method of modeling AGVSs by labeled PNs. Section V presents a method for estimating the state for an observed sensor sequence. Section VI designs a collision avoidance controller. Section VII verifies the proposed method by a simulation experiment. Section VIII concludes the paper. 2. PRELIMINARIES 2.1 Petri net An ordinary PN structure is a four-tuple N = (P, T, F, W ), where P (places) and T (transitions) are non-empty, finite set of places and transition sets respectively, F ⊆ (P × T ) ∪ (T × P ) is a collection of directed arcs that connect places and transitions; W : F → Z+ is a mapping that assigns each arc with a positive integer weight. C − : P × T → Z and C + : T × P → Z are the forward and backward incidence matrices that define the weights of the directed arcs from places to transitions and 140
from transitions to places, respectively. The weight of 0 indicates that there is no directed arc. C = C + − C − is an incidence matrix. In a PN, firing a transition can be prohibited by an external agent. Such transition is called a controllable transition; otherwise, it is uncontrollable. The set of controllable transitions is denoted by Tc , and the uncontrollable transition set, Tc . If firing a transition can be detected, the transition is observable; otherwise,it is unobservable. The token in the PN is represented by a solid black dot and the number of tokens in place p is denoted by m(p). m0 is the initial marking of PN. If and only if m ≥ C − (·, t) (C − (·, t) indicates the column where transition t is in the backward matrix), transition t is enabled under m, denoted as m[t⟩. Only enabled transitions can be fired, and the new marking m′ is generated after t fires such that m′ = m + C(·, t) (C(·, t) indicates the column where transition is in the incidence matrix), which is denoted as m[t⟩m′ . The set of all the transition sequences is denoted as T ∗ . Similarly, if the transition sequence σ ∈ T ∗ is enabled at m, denoted as m[σ⟩, the new marking m′ is reached, denoted as m[S⟩m′ . The marking m is a column vector, m : P → Z is a mapping function that assigns a non-negative integer (tokencount) to each place. 2.2 Labeled Petri net A labeled PN is (N, Σ, L), where N = (P, T, F, W ) is a PN structure; Σ is the alphabet set of letters; and L : T → Σ ∪ {λ} is a label function, which labels a letter for each transition, i.e., ∀t ∈ T, L(t) = e. L−1 (e) = {∀t ∈ T, L(t) = e} is the set of transitions with the same label e. An observed sequence is a sentence S such that ∃σ = tσ1 tσ2 · · · tσn , m0 [σ⟩, S = L(σ) = L(tσ1 )L(tσ2 ) · · · L(tσn ), and σ is a transition sequence associated with S. 3. PROBLEM STATEMENT The class of AGVSs considered in this work is based on the electromagnetic guiding mode, and the example is shown in Fig. 1. z1 ∼ z5 are 5 stations where operations are performed or vehicles can stay; l1 ∼ l7 are seven guiding lines whose arrows denote their directions leading vehicles. In detail, if a guiding line is energized, a magnetic field is generated simultaneously around it, and an vehicle can detect and move along this magnetic field; otherwise, the vehicle stops. s1 ∼ s5 are five sensors installed in z1 ∼ z5 , respectively, and a sensor is used to detect if there is a vehicle in its corresponding station. To avoid collisions among vehicles, it is required that at any time, at most one vehicle can stay in every guiding line or station. How to estimate the distribution of vehicles in an AGVS from a sequence of signals generated by the sensors is the central problem. Its main complexity lies in the undeterministic phenomenon, i.e., the guiding line that this vehicle chooses and the falling edge signal received from the sensor of the station can not be sure when a vehicle leaves a station. Further, it is reasonable that a guiding line should keep powered on if there is a vehicle in it, and, consequently, a vehicle can uncontrollably enter the station toward which the guiding line leads. Evidently, such controllable events further give rise to the complexity of the problem. Accordingly, the labeled PN is to be used as a formal
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141
Fig. 1. A typical example of the class of AGVSs considered in this work tool to design the maximally permissive controller avoiding collisions. Fig. 2. Labeled PN of the AGVS shown as Fig.1 4. LABELED PN MODEL OF AGVS
transitions increases, so does the computational complexity to estimate the state from an observed sequence.
Given an AGVS, we have to represent it by a formal tool such that the control method can be developed. Hence, the following procedure is presented to do so via labeled PNs.
5. ESTIMATING METHOD OF STATES OF AGVS
(1) For each station or guiding line, design a place representing it. (2) For each pair of station and guiding line, if the guiding line connects this station and the former directs towards the latter, design an uncontrollable transition with the input that is the place representing the former, and the output that is the place representing the latter, respectively; and if the guiding line connects this station and the former directs from the latter, design a controllable transition with the output that is the place representing the former, and the input that is the place representing the latter, respectively. (3) For each sensor si in station zi , i ∈ Z, design two letters ai and bi corresponding to the rising-edge and falling-edge signals, respectively. (4) For each transition t, if it represents a pair from station zi to a guiding line, then L(t) = bi , and, if it represents a pair from a guiding line to zi , L(t) = ai . Evidently, a labeled PN can be constructed according to the above procedure. In this PN, there are uncontrollable transitions, and multiple transitions with a same letter. They can model the uncontrollable and undeterministic phenomenon, respectively. Further, only what is necessary for designing the controller avoiding collisions is captured by this PN model. Example. Taking the AGVS in Fig. 1 as an example. According to the procedure, we obtain the labeled PN, as shown in Fig. 2. P = {z1 , z2 , z3 , z4 , z5 , l1 , l2 , l3 , l4 , l5 , l6 , l7 }, Tc = {t1 , t2 , t3 , t4 , t5 , t6 , t7 }, Tc = {t8 , t9 , t10 , t11 , t12 , t13 , t14 }, and Σ = {a1 , a2 , a3 , a4 , a5 , b1 , b2 , b3 , b4 , b5 }. The arcs and label functions are also clearly shown. t2 , t4 , t5 , t8 , t9 and t11 are deterministic. For instance, when b2 is observed, t2 fires, i.e., the guiding line l2 is energized, and a vehicle leaves zone z2 , and enters l2 . However, transitions t1 , t6 , t3 , t7 , t10 , t13 , t12 and t14 are undeterministic. For instance, when b3 is observed, we are not sure which of t3 and t7 fires. This means that we only know that a vehicle leaves z3 , but we are not sure about which one between l3 and l7 that this vehicle enters. As the number of undeterministic 141
Since a sensor is installed in each station, a symbol (risingedge or falling-edge signal) is observed once a vehicle enters or leaves a station. A particular case is the initial sequence that is observed just at the instant that AGVS is turned on, and this sequence is denoted by w0 . In details, when AGVS is started up, the sensor of a station immediately observes a rising-edge signal if there has been a vehicle parked in this station. Therefore, w0 is composed of a series of rising-edge signals. Further, the initial marking of the PN model can be derived from w0 . Algorithm 1 is used to do so. Algorithm 1 Initial marking of PN model of AGVS Input: labeled PN for AGVS and w0 . Output: C(w0 ) = {m0 }. 1: for each place p do 2: m0 (p) = 0; 3: end for 4: for all 1 ≤ i ≤ |w0 | do 5: ∃t ∈ T (w0 [i]), ∃p ∈ t• , m0 (p) = 1. 6: end for Given an observed sequence w, C(w) = {m m0 [σ⟩m, L(σ) = w} is a set of w, which is called consistent markings . Evidently, C(w) is a set of markings that the PN may reach if w is observed. According to Algorithm 1, the initial marking of the labeled PN model of an AGVS can be accurately deduced from the initial observed sequence. Example. Take AGVS in Fig. 1 as an example. If w0 = a1 a4 , then, according to Algorithm 1, we can deduce C(w0 ) = {m0 } = {(m(z1 ), m(z2 ), m(z3 ), m(z4 ), m(z5 ), m(l1 ), m(l2 ), m(l3 ), m(l4 ), m(l5 ), m(l6 ), m(l7 ))} = {(1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0)}. It means that either of stations z1 and z4 have one vehicle at the initial instance. After AGVS is started, vehicles arrive and leave stations, and, consequently, the sensors in the stations generate signals accordingly. This implies that the length of the observed sequence increases, and that we should estimate the distribution of vehicles once the observed sequence changes. For the convenience to do so, it is assumed that
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wk , k ∈ {0, 1, 2 · · · }, represents the observed sequence after the k-th observation. Hence, wk is the prefix of wk+1 , and wk+1 = wk δk where δk is the sensed letter or symbol at the k-th observation. Hereafter, we show how to estimate the state after one observation. Algorithm 2 Consistent set for an observed sequence Input: Labeled PN of AGVS, wk−1 , C(wk−1 ), and δk . Output: C(wk ). 1: wk = wk−1 δk ; 2: C(wk ) = ∅; 3: Tk = L−1 (δk ); 4: for all t ∈ Tk do 5: for all m ∈ C(wk−1 ) do 6: if m ≥ C − (·, t) then 7: m′ = m + C(·, t); 8: C(wk ) = C(wk ) ∪ {m′ }; 9: end if 10: end for 11: end for
Suppose that the controller is a function u : Σ∗ → 2T such that it maps each observed sequence to a set of transitions that should be disabled, and uk = u(wk ). Evidently, for the control specification (1), the maximally permissive controller is { � � uk = ∀t ∈ T �∀m ∈ C(wk ), m ≥ C − (·, t), ′
m = m + C(·, t),
∨ ∑
D∈D p∈D
where k = 0, 1, 2, · · · .
′
}
(2)
m (p) ≥ 2 ,
According to (2), Algorithm 3 is designed to compute the control actions for each observed sequence, and, consequently, to prevent collisions among vehicles.
By Algorithm 2, we can estimate the possible markings that the PN reaches once a new observed signal is received. According to Algorithm 1, C(w0 ), can be computed. In turn, C(w0 ) and δ1 are inputed into Algorithm 2, and C(w1 ), i.e., the set of possible markings the net reaches after δ1 is observed, is computed. In the similar way, C(w2 ), C(w3 ), · · · can be obtained one by one. Example. Take AGVS in Fig. 1 as an example. Suppose that w0 = a1 a4 , C(w0 ) = {(1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0)}, and = { δ1 = b1 . Hence w1 = a1 a4 b1 . By Algorithm 2, C(w1 ) } (1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0), (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0) can be computed from C(w0 ) and δ1 . This implies the net may be at one marking in C(w1 ) after δ1 is observed. 6. COLLISION-FREE CONTROLLER SYNTHESIS VIA LABELED PETRI NET The key to prevent any collision is to make sure that at most one vehicle is allowed in each station or guide line at any moment. As for the labeled PN model, it is required that the number of tokens in each place be not more than 1. If there are two or more AGVs in AGVS, then after the previous analysis, the collision never occurs through the collision avoidance system as long as the number of AGVs does not exceed the number of dangerous regions and the number of AGVs in each dangerous region does not exceed one. Theorem 1. Given a labeled PN model of an AGVS and any place p in Ps (Ps is the set of places, called the stationplace set), the dangerous region of p is the set of places that lead an uncontrollable path to p, denoted by D(p), and the set of dangerous regions is D = {D(p)|∀p ∈ Ps }. According to Theorem 1, the collision avoidance problem can be formalized as a conjunction of linear constraints: ∧ ∑ m(p) ≤ 1. (1) D∈D p∈D
Since the net may be at any marking in C(wk ), k = 0, 1, 2, · · · , after the observed sequence wk is received, we should disable any controllable transition whose firing may lead any one in C(wk ) to a marking that violates (1). 142
Algorithm 3 Controller for avoiding collision in AGVS Input: Labeled PN of AGVS, N = (P, T, F, Σ, L), w0 , and δk , k = 1, 2, · · · . Output: uk , k = 0, 1, 2, · · · . 1: compute C(w0 ) by inputting N and w0 into Algorithm 1; 2: let K be a large enough integer; 3: for k = 1 → K do 4: δk is the observed symbol at the kth intance; 5: wk = wk−1 δk ; 6: Compute C(wk ) by Algorithm 2 from C(wk−1 ), wk−1 , and δk ; 7: uk = ∅; 8: for all m ∈ C(wk ) do 9: for all t ∈ L−1 (δk ) do 10: for all m ∈ C(wk−1 ) do 11: if m ≥ C − (·, t) then 12: m′ = m + C(·, t); 13: for all ∑ D ∈ D do 14: if p∈D m(p) > 1 then 15: uk = uk ∪ {t}; 16: end if 17: end for 18: end if 19: end for 20: end for 21: end for 22: end for For k = 0, 1, 2, · · · , uk computed by Algorithm 3 is a set of controllable transitions that should be disabled. Disabling a controllable transition can be implemented by turning off corresponding guiding line. In details, the guiding line, which is corresponding to a controllable transition, is represented by the place whose input is this transition. Example. Taking AGVS in Fig. 1 as an example. In the labeled PN shown in Fig. 2, z1 ’s dangerous region is D(z1 ) = {{l5 , l7 , z1 }, and the set of dangerous } regions is D = D(z1 ), D(z2 ), D(z3 ), D(z4 ),}D(z5 ) = { {l5 , l7 , z1 }, {l1 , z2 }, {l2 , z3 }, {l3 , l6 , z4 }, {l4 , z5 } . According (1), the control specification avoiding collisions is m(z1 ) + m(l5 ) + m(l7 ) ≤ 1; m(l1 ) + m(z2 ) ≤ 1; m(l2 ) + m(z3 ) ≤ 1; (3) m(l3 ) + m(l6 ) + m(z4 ) ≤ 1; m(l4 ) + m(z5 ) ≤ 1.
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This means that the collisions among vehicles can be avoided if (3) is enforced on the labeled PN. Further, the control actions can be computed by Algorithm 3. For instance, u0 = {t6 } if w0 = a1 a4 , and C(w0 ) = {(1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0)} from Example 2. u {1 = {t5 } if δ1 = b4 , w}1 = a1 a4 b4 and C(w1 ) = (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0) . 7. SIMULATION RESULTS
In order to verify the proposed method, a simulation experiment is performed. We develop a Matlab program to simulate a labeled PN step by step. Further, a control problem is designed according to Algorithm 3, and can disable controllable transitions such that (1) is fulfilled. In this simulation, we still use AGVS in Fig. 1, which has two AGV. Suppose they are in z1 and z4 initially. The simulation is performed in 200 steps, a transition fires in each step, and a corresponding signal is received by the controller. As a result, the markings of places, and dangerous regions are shown in Figs. 3-4. From them, it can be concluded that at most one token (vehicle) can exist in each place or dangerous region, and, consequently, collision is successfully avoided. Further, the control actions are shown in Fig. 5, which actually disable controllable transitions t1 , t2 , · · · and t7 . For instance, if the signal for t1 is of low level or 0, t1 is disabled in the corresponding period; otherwise, it is allowed to fire. 8. CONCLUSIONS Based on the labeled PN, a control method is proposed to address the collision problem in the class of AGVS that are partially controllable, and have undeterministic behaviors. The dangerous region is defined to deal with uncontrollable events, and a consistent marking set is used for undeterministic behaviors. Further, the presented controller is maximally permissive. Next, we plan to study optimal scheduling problems of this class of AGVSs. REFERENCES Vis IFA, (2006). Survey of research in the design and control of automated guided vehicle systems. Operational Research, Volume 170, 677–709. Singh, N., Sarngadharan, P.V., and Pal, P.K. (2011). AGV scheduling for automated material distribution: A case study. Operational Research, volume 22, 219–228. Le-Anh, T., De Koster MBM(2006). A review of design and control of automated guided vehicle systems. Operational Research, Volume 171, 1–23. Zhou, M., Dicesare, F.(1993). Petri net synthesis for discrete event control of manufacturing systems. Kluwer Academic Ferrarini, L., Piroddi, L.(2008). Modeling and control of fluid transportation operations in production plants with Petri nets. IEEE Transactions on Control Systems Technology, Volume 16, 1090–1098. Giua, A., Seatzu, C.(2008). Modeling and supervisory control of railway networks using Petri nets. IEEE Transactions on Automation Science and Engineering, Volume 5, 431–445. 143
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Fig. 3. Markings of places
Fig. 4. Markings of the dangerous regions
Fig. 5. Controls of transitions
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