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Journal of Manufacturing Systems journal homepage: www.elsevier.com/locate/jmansys

Technical Paper

A fault-tolerant control strategy for multiple automated guided vehicles a,

a

b,c

Marcin Witczak *, Paweł Majdzik , Ralf Stetter , Bogdan Lipiec a b c

a

T

Institute of Control and Computation Engineering, University of Zielona Góra, Zielona Góra, Poland Faculty of Mechanical Engineering, Ravensburg-Weingarten University (RWU), Weingarten, Germany Steinbeis Transfer Center Automotive Systems, Ravensburg, Germany

ARTICLE INFO

ABSTRACT

Keywords: AGVs system Max-plus algebra Fault diagnosis Fault-tolerant control Interval arithmetics

An advanced control of manufacturing and transportation systems forms a prominent research field with powerful algorithms developed in the last decades. Challenges still arise, if several automated guide vehicles (AGV) have to be coordinated. This paper focuses on the modelling and fault-tolerant control of multiple AGVs. The considered application concerns a highly flexible AGV transportation system delivering product items to transfer stations at a high-storage warehouse in a manufacturing system. The research contribution concerns the development of a mathematical description of a set of multiple AGVs along with an algorithm that can generate an optimum sequence of item outlet delivery times. The proposed solution addresses both synchronization and concurrency issues, which are inevitable in this kind of multiple-vehicle systems. Apart from these issues, modelling inaccuracy is also addressed using interval analysis coupled with max-plus algebra. Subsequently, fault diagnosis and fault-tolerant control are also investigated and addressed in the proposed approach. This leads to a fault-tolerant control framework, which is based on a fusion of the predictive control and interval maxplus algebra. The distinct quality of the proposed approach is that the optimization can be carried out in a reliable way and that certain faults and modeling uncertainties can be tolerated. The paper concludes with illustrative examples, which show the performance of the proposed approach using both fault-free and faulty scenarios.

1. Introduction

For the purpose of further deliberations, the fault is described according to the following definition:

In recent years, flexible manufacturing and transportation systems have received a rising attention, because of their numerous advantages such as maintainability and relatively small investment costs. A high production flexibility can be achieved by integrating autonomous guided vehicles (AGVs) within production/transportation lines performing cyclically repeating tasks on some product items. It is straightforward to imagine that such production systems will be capable of processing a large variety of different product families. Flexible manufacturing systems are in the focus of numerous publications [1–4]. Different kinds of flexibility (new product flexibility, product mix flexibility, volume flexibility and delivery flexibility) have an enormous influence on the transportation systems within a flexible production. Consequently, collective efforts of AGVs instead of non-reconfigurable transportation setups can lead to considerable advantages such as flexibility, redundancy and fault-tolerance [5]. In the light of the above introduction, the subsequent points present the fundamentals of AGVs control both for fault-free and faulty cases, which serve as a basis for the subsequent elaborations.

Definition 1. A fault is defined as an unpermitted violation of at least one transportation time of the system from its nominal reference value. It should be noted that Definition 1 is consistent with the classical fault definitions recommended in the state-of-the-art books (e.g., [6]). However, instead of “transportation time” they use “characteristic property”, which is more universal and fits to any kind of systems. 1.1. Control approaches for fault-free cases The initial control tasks for AGV concern the longitudinal and lateral control of a single AGV. These topics were in the centre of research for many years and powerful algorithms were implemented in practice. Current expansions concern the usage of virtual sensors [7], an integrated local trajectory planning and tracking control [8], the outputfeedback triple-step coordinated control for path following [9] and the inclusion of Remaining Useful Life (RUL) considerations [10]. Control

Corresponding author. E-mail addresses: [email protected] (M. Witczak), [email protected] (P. Majdzik), [email protected] (R. Stetter), [email protected] (B. Lipiec). ⁎

https://doi.org/10.1016/j.jmsy.2020.02.009 Received 2 July 2019; Received in revised form 30 January 2020; Accepted 24 February 2020 0278-6125/ © 2020 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.

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tasks on a higher level of abstraction concern the routing and scheduling of AGVs. Routing concerns the pathways of the AGVs (optimization objectives can be a balancing of the routes, overall transportation cost, etc.), whereas the scheduling mainly concerns the time when AGVs reach certain positions (optimization objectives can be the total completion time, the mean waiting time, etc.) [11]. Control strategies for multiple AGVs have been addressed in many papers. An early review was published in [12]. A detailed review study [11] shows many different methods of scheduling and routing. The comprehensive focus of the elaborate review by De Ryck et al. [13] is on control algorithms and techniques for AGVs; additional research concentrates on resource management in decentralized AGV systems [14]. In the reviewed literature, most of the tests and developments were made with an off-line scheduling. A multi-agent based control strategy is presented by Sahin et al. [15]. They present a model, which was designed as an on-line method and disposes of a dynamic structure. The general issue of online scheduling can also be tackled with dynamic routing scheduling, which is based on the number of currently active AGV tasks as well as their priorities [16]. In [17], the genetic algorithm and the ant-colony algorithm are used to find an optimal schedule for two AGVs. A readyto-use computer implementation was proposed in [18]. The particle swarm optimization combined with the memetic algorithm [19] is used to schedule AGVs with multi-loading. However, every single route is planned separately. Skinner et al. [20] used the genetic algorithm to improve container handling operations. Genetic algorithms were also applied in [21] to obtain optimal solutions for flexible manufacturing system (FMS) layout and by Chang et al. for real-time dispatching for integrated delivery [22]. In [23], its combination with fuzzy logic for in-plant transportation control with AGVs is employed. In the paper [24], the authors describe a differential evolution algorithm but the problem was solved for two AGVs, only. Estimated arrival teams in contrast to due arrival times are generated by a heuristic method developed by Chung [25]. Another research group currently develops an innovative method for synthesizing a master assembly network with alternative sequences based on legacy assembly data [26]. In [27], the authors deal with a scheduling problem solved with multi-objective evolutionary algorithms. In [28], a Petri net decomposition was presented that is intended to schedule the AGVs’ route. Similarly, Luo et al. [29] also investigated the control of AGVs via Petri nets. The timewindow method was used in [30] to improve underground parking in China, which can also be applied to AGVs. In [31], traffic managers in an automated warehouse are described. Wang and Zhou [32] presented a scheduling approach for AGVs using two heuristic rules. There are also approaches that investigate scheduling algorithms based on a harmony search [33], the so-called “Jamology” [34], supervisor localization under partial observation [35], switchable languages of discrete event systems with weighted automata [36] as well as delegate multi-agent systems [37]. Finally, practically-oriented research covers the implementation of a radio frequency identification (RFID)-enabled positioning system [38], the simulation of scale-up processes [39] and the evaluation of job dispatching rules [40]. In spite of the incontestable appeal of the approaches listed above, they do not guarantee a feasible solution in faulty situations. Thus, the objective of the subsequent section is to provide a concise review of such approaches.

are sensitive to faults but the impact of faults can be prevented or decreased by means of FTC [43]. In general, passive and active FTC can be differentiated. In the case of passive FTC, the controller is designed in a way that allows to achieve certain specifications both in the fault-free and faulty case. The active FTC approach alters the controller parameters or even the structure in case of faults and needs to include a fault detection and isolation (FDI) [44,45]. A concise overview of FDI approaches is given by Basile [46] as well as Zaytoon and Lafortune [47]. Current research includes system design aspects into FTC investigations and solutions [48]. The most common approach to tackle FDI and FTC is a model-based one. Modern production and transportation systems are often modelled as discrete event systems (DESs). A survey concerning DES for simulation and experimental design methods for job shop scheduling is presented by Xie and Allen [49]. Kampa et al. [50] and Yedukondalu et al. [51] successfully employed discrete event simulation for the improvement of manufacturing systems. A recent overview of fault-tolerant control methods for DES is given by Fritz and Zhang [44]. Having a system model, it is possible to predict its behaviour in the future. Thus, predictive fault-tolerant control uses some kind of a priori knowledge about faults to develop an optimal control and fault accommodation actions [52]. Other branches of current research investigate the application of big data analytics [53] and a fault tree analysis (FTA) for complex manufacturing systems [54]. Although many powerful approaches were developed in the last years (especially obstacle avoidance systems), it is possible to conclude that the development of reliable tools capable of handling fault-tolerance with appropriate fault diagnosis tools may contribute to more powerful and robust control systems. 1.3. Contribution This paper focuses on the predictive FTC of multiple AGVs. The main goal is to obtain an optimal sequence of item production outlet delivery times. Each of them denotes the time of providing the k th item to a given AGV, which is responsible for its transportation from the manufacturing outlet to the warehouse. In nearly all cases, it is profitable to have the k th item at the manufacturing outlet just-in-time. In the presented research, this fact is expressed by a suitable optimization criterion while taking into account inevitable scheduling constraints. A second important aspect of the presented research is that the developed approach is extended to cope with various unappealing phenomena related to AGVs, e.g., delays, AGVs battery premature discharge, sliding surface, etc. All of these phenomena can be summarized under the common name faults. These two aspects can be combined in the objective to obtain a balanced cooperative performance of a set of AGVs allowing satisfaction of scheduling constraints irrespective of faults. The main objective is to realize it in an on-line way by supporting the above optimal sequence calculation strategy with an appropriate fault diagnosis tool. In some realistic cases, it may happen that the faulttolerant capabilities of a given set of AGVs can be insufficient to cope with a given fault scenario. This situation is directly translated to the infeasibility of the optimization problem under a given scheduling constraint. As a third result of the presented research, a suitable relaxation is proposed, which optimally relaxes a given schedule to make the entire optimization process feasible. This guarantees that, irrespective of the present faults, the k th item will always be transferred from the manufacturing outlet to the warehouse. It should also be noted that the schedule computation related to maximum system transportation ability is beyond the scope of this paper. For a comprehensive survey concerning this issue the reader is referred to [55]. Thus, it is assumed that the scheduling constraints are given. Consequently, the central research hypothesis can be formulated: It is possible to design on-line algorithms that allow a predictive fault-tolerant control of multiple AGVs by solving synchronization and concurrency issues as well as addressing modelling inaccuracies.

1.2. Control approaches for faulty cases Faults such as delays, obstacles, slippery surfaces, AGV battery issues, etc. can substantially aggravate the performance of AGV systems. These faults can appear in an unpredicted way. This rises the necessity of developing AGVs control strategies capable of handling this kind of faults, i.e., to develop a certain level of fault-tolerance [41,42]. Indeed, modern production and transportation systems are characterized by a high degree of complexity and need to fulfill enormous requirements concerning safety, reliability and availability. Generally, such systems 57

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Fig. 1. High-storage warehouse with transfer stations. (For interpretation of the references to color in this figure citation, the reader is referred to the web version of this article.)

The paper is organized as follows. In Section 2, the motivation is explained and preliminaries are clarified. Section 3 explains the mathematical description of multiple AGVs. The model predictive control of these AGVs is the focus of Section 4. Section 5 addresses model uncertainty issues and incorporates them into the predictive approach. The fault accommodation is undertaken in Section 6. Section 7 contains the results of the performance evaluation of the proposed predictive FTC.

supervisory control level is in the focus of the research described in this paper. The overall system consists of the production system followed by the transportation system consisting of a set of n v AGVs, which are responsible for delivering given items from the production outlet towards appropriate transfer stations. The items, which are stored on palettes, are available at roller conveyors at the production outlet. An AGV will receive an item at this outlet and will transport it to one of the transfer stations. From these transfer stations, the items will be moved by means of forklift units to be stored in the high-rise shelves. The main assumption underlying the overall system is that the structure and purpose of the warehouse can be flexibly changed. This assumption clearly excludes a conventional conveyor-based transportation lines, which have to be replaced by a set of n v AGVs. Because of safety requirements, AGVs are operating along a dedicated lanes, which are designated to forward and backward communication. This process is exemplified in Fig. 3. In Fig. 3, an AGV (denoted with “1”) can be seen, which returns from the warehouse transfer station. Another AGV (denoted with “2”) comes from the production outlet and drives towards another transfer station. In order to avoid conflicts, AGV 2 will wait until AGV 1 has passed – in this case the well-known right-hand rule will be applied. Another AGV (denoted with “3”) is also visible. This AGV goes from another transfer station to the production outlet. As it was mentioned, possible conflicts between AGVs performing turning-off maneuvers are avoided using a right-hand rule as in the case of conventional cars driving on a road. Also due to the safety requirements, AGVs are operating with a constant velocity. The production outlet provides items in a sequential fashion. Each k th item is labelled by the manufacturing execution system (MES) with the following quadruple:

2. Motivation and preliminaries The considered application is an AGV-based transportation system for a prospective high-storage warehouse portrayed in Fig. 1. Fig. 1 portrays high-rise shelves with packaged goods with regular palettes on them. Between the shelves there are aisles for automated forklifts. Fig. 1 depicts an exemplary warehouse with transfer stations (element with a blue colour). High-storage warehouses allow storing of packaged goods, e.g., on palettes. Such warehouses are characterized by an economical use of floor space, a high storage density and a high flexibility, which is very important in the face of growing product variety in all industrial areas. The advantages are good access to every article, good height utilization, pressure-avoiding storing of the goods and a rational design [56]. The storage and retrieval system is not considered in this paper, but could consist of machines which may move in the aisles. They can dispose of a forklift unit which may retrieve or deliver the goods on palettes. In the application example, these machines receive items from transfer stations or deliver them to transfer stations. These transfer stations dispose of a controlled roller system. In the application example, the items are delivered to the transfer stations by means of transportation systems consisting of AGVs. One example of an AGV which is equipped with a roller conveyor for the transportation of palettes is shown in Fig. 2 on the left side. The AGV shown in Fig. 2 is based on a unique design which allows unlimited manoeuvring possibilities. The design was realized in the form of a multi-purpose production platform (Fig. 2 right side). This kind of a transportation system was chosen because of high flexibility and fault-tolerance. These AGVs can theoretically move freely in the zone in front of the warehouse and can deliver and retrieve items on palettes to and from dedicated transfer stations. The AGVs dispose of a controlled roller system which can move packed items on a palette to transfer stations. The transportation system is composed of a hierarchical control scheme consisting of three control levels. The lowest level is the continuous baseline control including physical and virtual sensors. An intermediate control level is present for detailed path planning. The highest level is responsible for spreading transportation tasks among AGVs. The system being considered is modelled as DES structure as suggested by Moor et al. [57] for flexible systems. This

(k ) = [p (k ), c (k ), b (k ), d (k )],

(1)

where

• p (k ) denotes a number uniquely identifying the transfer station, i.e., p (k ) {1, …, n } with n being the total number of transfer stations; • c (k ) stands for the item packing and transportation time from the outlet of the production system to p (k ) transfer station; b • (k ) is the item unpacking and transportation time from p (k ) transfer station to the production outlet; • d (k ) is the minimum allowable time difference between delivering s

s

(k 1) th item to transfer station and k th item to p (k ) transfer station, respectively.

Note that d (k ) can be selected in such a way as to allow reasonable performance of the forklifts working within the warehouse. However, if there is no need for such a restriction then a judicious choice is d (k ) = 0 . Finally, the MES provides a sequence of the items which have 58

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Fig. 2. AGV for transportation system (left) and multi-purpose production platform (right).

to be transported from the production outlet to the transfer stations:

(0),

(1), …,

(Np

y (0), y (1), …, y (Np

where Np stands for production horizon. Note that each item has to be delivered to the transfer station according to a given MES-based time schedule:

x ref (0), x ref (1), …, x ref (Np

1),

(4)

where y (k ) denotes the time instant at which the k th item is delivered to the AVGs transportation system for k th event counter. Throughout the paper, the performance of AGV-based transportation system is measured as follows:

(2)

1),

1),

Np 1

(3)

J (y ) =

where x ref (k ) denotes the required item delivery time at the p (k ) transfer station for the k th event counter. To achieve this, the work schedule of n v AGVs has to be derived along with a sequence of item outlet delivery times:

y (k ). k=0

(5)

The above function has to be minimized under the scheduling constraint (3) while taking into account the overall performance of the set of n v AGVs. As a result, the largest possible sum of (4) is to be obtained,

Fig. 3. Exemplary scenario of the transportation system. 59

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which guarantees satisfaction of (3). It should be noted that (5) can be, of course, defined in a different fashion, e.g., by allowing maximization of the consecutive differences y (k + 1) y (k ) . This set up will provide the maximum spread between consecutive item outlet delivery times. Taking into account the above preliminary framework, the main motivation behind the developments presented in this paper is to provide the answer to the following questions:

• How to describe mathematically a set of n • • •

if bim (k )

f j, b (k ) = 0,

bj (k ) = 0,

i , j = 1, …, n v .

cj (k ) = 0,

bi (k ) = [ b (k ), b (k )],

bj (k ) = 0,

j

i , j = 1, …, n v .

i

nv + 1

(e = 0 and = of the k th item.

c (k )

) and associating i th AGV with the transportation

Note that vi (k ) = e means that the i th AGV performs a transportation task of the k th item while vi (k ) = denotes an opposite situation. This notation is not accidental and will be exploited within further development of a compact AGVs model. Having the above-defined variables, the time-evolution of x i (k ) for each AGV can be described as follows:

xi (k ) = max( x i (k 1) + bi (k 1) + ci (k y (k ) + vi (k )), i = 1, …, n v ,

1), (13)

with the associated constraints

bi (k ) = max(e, b (k ) + vi (k )),

(14)

ci (k ) = max(e, c (k ) + vi (k ))

(15)

and

(7)

vi (k ) = e

vj (k ) = ,

i

j.

(16)

Note that (14) and (15) corresponds to (6) while (16) exemplifies an obvious fact that only one, i.e., the i th AGV may carry the k th item from the production outlet towards the p (k ) th transfer station. Subsequently, the k th item delivery time at p (k ) th transfer station obeys:

(8)

x nv + 1 (k ) = max( x1 (k ) + c1 (k ) + v1 (k ), x2 (k ) + c2 (k ) + v2 (k ), x3 (k ) + c3 (k ) + v3 (k ), …, x nv (k ) + cn (k ) + vn (k ), x nv + 1 (k 1) + d (k )).

(9)

(17) Taking into account (15), it can be shown that

ci (k ) + vi (k ) = max(e , c (k ) + vi (k )) + vi (k ) = c (k ) + vi (k ),

(18)

and hence (17) boils down to

x nv + 1 (k ) = max( x1 (k ) + c (k ) + v1 (k ), x2 (k ) + c (k ) + v2 (k ),

c (k ) then fi, c (k ) = 0

else fi, c (k ) = cim (k )

(12)

v

i

This fusion constitutes the answer to the third question and allows the possible inclusion of on-lane conflicts of AGVs, short transportation delays, etc. Contrarily, large transportation delays for which the actual measured transportation times cim (k ) and bim (k ) do not belong to the intervals ci (k ) and bi (k ) are defined as faults. This process can be formally written as:

if cim (k )

i, j = 1, …, n v .

• x (k ) – the time instant at which ith AGV (i = 1, …, n ) is ready to transport k th item; ) – k th item delivery time at p (k ) transfer station; • vx (k ) (–kdecision variable taking the value from a two-valued set {e , } •

Subsequently, the above assumption is eliminated using an interval analysis approach. In this case, each ci (k ) and bi (k ) is represented by a predefined interval [ a, a ], which allows rewriting (6) into a less restrictive form:

ci (k ) = [ c (k ), c (k )],

j

The objective of this section is to provide an answer to the first question stated in the preliminary part of the paper. It pertains a mathematical description of multiple AGVs, which allows a real-time determination of their time schedule on a given production horizon Np . Let us start with defining the following variables:

(6)

j

f j, c (k ) = 0,

3. Mathematical description of multiple AGVs

The second question needs an additional explanation. It should be noted that the k th item delivery time, denoted by y (k ) , should be realized just-in-time to guarantee the fulfillment of the assumed schedule formed by (3). In other words, they cannot be provided neither too soon or too late, which is expressed by minimizing (5). The overall answers to the above questions constitute the main contribution of this paper. As it was mentioned in the introduction, there is a spectrum of possible candidate solutions which can provide a partial answer to the first question. However, their real-time performance is usually significantly restricted. Moreover, they frequently use some heuristic strategies, which exclude the possibility of using strict mathematical description while operating on the production horizon Np . In this paper, a new max-plus algebra based strategy is proposed, which eliminates the above unappealing effects. The answer to the second question is provided by applying a model predictive control for the proposed description of a set of n v AGVs. The above approach assumes that the actual transportation times of i th AGV, which carries the k th item are equal to its nominal values while the remaining AGV transportation times are set to zero, i.e.:

cj (k ) = 0,

(11)

Taking into account the above motivations and preliminaries, the objective of the subsequent sections is to provide answers to the abovedefined four questions along with suitable computational models and algorithms.

v

bi (k ) = b (k ),

b (k )

where ¯. denotes an upper bound of the interval. As only i th AGV carry k th item, it is an obvious fact that:

AGVs in such a way as to allow an efficient determination of their work schedule in a realtime? How to obtain a sequence (4) minimizing (5) under scheduling constraint formed with (3)? How to take into account minor inconsistencies between (2) and the real performance of the set of AGVs? How to manage large inconsistencies of the above form, which lead to the significant transportation delays and possible violation of (3)?

ci (k ) = c (k ),

b (k ) then fi, b (k ) = 0

else fi, b (k ) = bim (k )

x3 (k ) + c (k ) + v3 (k ), …, x nv (k ) + c (k ) + vn (k ), x nv + 1 (k 1) + d (k )).

(10)

60

(19)

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It should be stressed that each variable, except for the decision ones vi (k ) , present in the developed model of the multiple AGV system represent a time instant of realizing a given operation. Thus, modelling uncertainties are defined as intervals describing a possible time range of realizing a given operation (cf. bi (k ) , ci (k ) and d (k ) ). Thus, the uncertainty relates to the lack of knowledge pertaining their precise values. Subsequently, any value, which is outside such intervals is perceived as a fault. As it is carefully detailed in Section 6, the proposed approach aims at tolerating such faults assuming that at least one AGV is available. Otherwise, such a situation is perceived as a failure of the multiple AGV system and the proposed approach has to be terminated. Note also that such fault-tolerance is occupied by a certain cost, which may directly translate to the delay with respect to the original schedule.

N

is employed, i.e., {(vi (k ), vi (k 1))} k =p 1. To make the paper self-contained, let us define (max,+) algebraic structure (R max , , ) of the following form [64,65]:

R max a, b a, b

a a

(X

x1 (1) = x1 (0) + b (0) + c (0),

(20)

x2 (1) = max(x2 (0), y (1)),

(21)

(X

= a and a e = a.

1) + c1 (k

1) + c1 (k ) + v1 (k ) b2 (k

Z ) ij = oplus x ik

(25)

zkj = max (x ik + zkj).

(26)

k = 1, … , n

1), v (k ), k )

x (k

1)

B (v (k ), k )

y (k )

(27)

v + 1 × nv + 1 R nmax

x nv + 1 (k ) = max(x1 (k 1) + b1 (k 1) + c1 (k 1) + c1 (k ) + v1 (k ), x2 (k 1) + b2 (k 1) + c2 (k 1) + c2 (k ) + v2 (k ), …, x nv (k 1) + bnv (k 1) + cnv (k 1) + cnv (k ) + vnv (k ), y (k ) + c (k ) + v1 (k ), y (k ) + c (k ) + v2 (k ), y (k ) + c (k ) + v3 (k ), …, y (k ) + c (k ) + vnv (k ), x nv + 1 (k 1) + d (k )). (28) Coupling together (13) and (28) allows deriving matrices A (v (k 1), v (k ), k ) and B (v (k ), k ) , which are given by (29).

1) b2 (k

and Z

where , v (k ) =[v1 (k ), …, vnv (k )] while A (·,·,·) designates v + 1 is the control matrix. the state transition matrix and B(·,·) R nmax Thus, substituting (13) into (19) yields:

The objective of this section is to transform (13)–(19) into a com1) + c1 (k

(24) ×p R nmax

T,

3.2. Compact multiple AGVs model with max-plus algebra

b1 (k

×n Rm max

= ,

yij = max(x ij, yij), n

x (k ) = A (v (k

From the above equation, it is evident that appropriate selection of the outlet delivery time y (1) makes it possible to attain scheduling constraint (3), i.e., x3 (1) x ref (1). Thus, the objective of the subsequent sections is to provide an algorithm allowing optimal selection of vi (k ) and y (k ) with respect to (5). However, the first step is to transform the AGVs model (13)–(19) into a more compact form.

1), v (k ), k ) =

(23)

A more elaborated and comprehensive description of the max-plus algebra can be found in [64,65]. Using the above notation, it is proposed to rewrite the model (13)–(19) into the following form:

(22)

x3 (1) = x2 (1) + c (1) = max(x2 (0), y (1)) + c (1).

Y ) ij = x ij

k=1

and hence (19) obeys:

B (v (k ), k )

R max : a R max : a

For two matrices X , Y

Let us consider n v = 2 set of AGVs and let v1 (0) = e , v1 (1) = and d (k ) = 0 . Thus, from (16) it is evident that v2 (0) = and v2 (1) = e . This setting reflects the situation in which 0 th item is transferred by 1st AGV while 1st item is carried by 2nd one. Therefore, the state evolution model (13) yields:

b1 (k

}, R { b = max(a, b), b = a + b,

with R max as the field of real numbers. Note that and stand for (max,+) addition and (max,+) multiplication with the following interpretation:

3.1. Illustrative example

A (v (k

R max , a R max , a

1) + c2 (k

1) + c2 (k

… …

1)

,

1) + c2 (k ) + v 2 (k ) … d (k )

(29)

= [v1 (k ), v2 (k ), …, vnv (k ), c (k )]T

pact matrix-based form. By analyzing (13)–(19), it can be easily observed that the only mathematical operators being used are + and max . Thus, among available DES modelling techniques [58–60], the maxplus algebra [61,62] seems to be the most suitable one. However, in its classical form, the max-plus-linear discrete-event systems can be used to describe a class of DES in which only synchronization and no concurrency or choice occurs. This constraint can be eliminated using the so-called switching max-plus linear system [63]. The approach proposed in this paper is similar to the one reported in [63] but the difference is that instead of using a single switching variable, a set of them

For the convenience and with a slight abuse of notation the above matrices will be denoted by Av (k ) and B v (k ) . 4. Model predictive control of multiple AGVs The objective of this section is to provide an answer to the second question stated in the preliminary part of the paper. It pertains the determination of the item delivery time sequence (4) on a given production horizon Np , which minimizes (5). It should, of course, be determined by taking into account scheduling constraints (3) and the

61

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performance of a set of n v AGVs. The proposed strategy is based on a general MPC paradigm for max-plus linear systems [62] extended by decision variables vi (k ) , i = 1, …n v . The core of the problem is to find the input sequence y (k ), …, y (k + Np 1) on a moving horizon k, …, k + Np 1. This requires a slight modification of the previously introduced cost function (5):

x v v + 1 (k )

• It is associated with the maximum rate of change of the production outlet delivery time:

y (k + 1)

y (k + j ).

(30)

j=0

y (k ) y (k + 2) y (k + Np

v (k + Np

under (36)–(40). To summarize, the control strategy for multiple AGVs has the structure given by Algorithm 1. Algorithm 1. Multiple-AGV MPC.

1)

• Set k = 1, N , v (0). (k ), …, (k + N 1) , y (k )…,y (k + N 1) and • Get x (k ), …, x (k + N 1) from MES. • Get state measurement x (k 1) and calculate y (k )*, v (k )* by solving (41). • Employ the first vector elements of y (k )* and v (k )* (i.e., y (k )* and v (k )*) and inject them into the system (13). • Set k = k + 1 and go to Step 1. p

[ ()

,

p

( )]T ,

v (k ) = v1 k , …, vnv k

ref

1)

(31)

and a recursive application of (27), which yield

x (k ) = M (v (k ))

x (k

1)

H (v (k ))

(32)

y (k ),

with

H (v (k )) =

H11 (v (k ))

H1Np

1 (v

H21 (v (k ))

H2Np

1 (v (k ))

HNp

M (v (k )) =

(41)

y (k ), v (k )

, x (k + Np

1)

v (k ) v (k + 2)

v (k ) =

x (k ) =

0 is the production performance upper bound.

(y (k )*, v (k )*) = arg min J (y ),

x (k ) ,

(40)

yz (k ),

Having the above constraints, a complete optimization problem boils down to:

Thus, the task is to obtain y (k ), …, y (k + Np 1) for each k . A preliminary step towards the computational framework is derive predictions of x (k + 1), …, x (k + Np 1) . This can be attained by defining

y (k ) =

y (k )

where yz (k )

Np 1

J (y ) =

(39)

x ref (k ).

11 (v

(k ))

M1 (v (k )) M2 (v (k ))

HNp

,

(k ))

z

p

m

This section aims at providing an answer to the third question stated in the preliminary part of the paper. It pertains an uncertainty issue, which is tackled with interval analysis. Firstly, let us note that exact transportation times are never known until an associated transportation event occurs. While describing such an uncertainty, an interval arithmetics can be an efficient alternative to the classical stochastic. Indeed, the latter one requires knowledge about the statistical distribution of the uncertainty bi (k ) and ci (k ) while the former one describes them as intervals. This excludes deliberations on the nature and distribution of these variables. Thus, this point aims at providing an interval max-plus algebra paradigm, and hence, incorporate robustness into (13). Let us start with replacing max by , which is a set of

(33)

where

1)

Av (k + n

2)

Av (k )

(34)

and

Av (·,·,k + n

z

5. Towards robustness with uncertainty intervals

,

MNp (v (k ))

Mn (v (k )) = Av (k + n

p

Apart from an elegant recursive description of (32) and linearity of cost function (30), it can be observed that any optimization constraint of the form a = max(b , c ) can be transformed into a set of equivalent linear constraints, i.e., a b, a c . This clearly indicates that the optimization problem boils down to the combined assignment and linear programming one.

(k ))

1Np 1 (v

ref

1)

B v (k + m Hnm (v (k )) = B v (·,k + n 1) ;

Av (·,·,k + m) if n > m , 1) if n = m , if n < m . (35)

a = [ a , a ] = {x

Before proceeding to the development of the entire algorithm, let us introduce a complete set of constraints, which is required during repetitive optimization cycles on k…,k + Np 1:

| a

x

(42)

a}

extended by , i.e., max tions are defined as follows:

• It is defined by (14)–(15) and pertains transportation times of a set

a, b a, b

of AGV:

max , max ,

a a

{

} . While the max and + opera-

b = [max( a , b), max(a , b )] b = [ a + b , a + b ].

(43)

bi (k ) = max(e, b (k ) + vi (k )),

(36)

Similarly as in the standard max-plus algebra, for matrices m×n n× p X, Y max and Z max :

ci (k ) = max(e, c (k ) + vi (k )).

(37)

(X (X

• It is defined by (16) and concerns selecting an AGV transporting the vj (k ) = ,

i

j.

yij = [max( x ij, y ij), max(x ij, x ij)], n

Z ) ij = oplus x ik k=1

n

zkj = oplus[ x ik + z kj, x ik + zkj]. k=1

Having the above notation it is evident that the domains of some variables and matrices forming (13) have to be redefined as well, i.e., nv + 1 nv + 1 × nv + 1 nv + 1 x (k ) and B (·,·) max , A (·,·,·) max max . As a result, the scheduling constraints have to be redefined as well

k th item:

vi (k ) = e

Y )ij = x ij

(38)

• It is defined by (3) and pertains a required items delivery time:

x nv + 1 (k ) 62

x ref (k ),

(44)

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M. Witczak, et al.

while the rest of the constraints remain the same. Note that above robustness extension does not influence the overall structure of the algorithm, which remains the same as Algorithm 1. In spite of the incontestable appeal of the above approach, it is not well-suited to the critical situations defined as faults and formally written as (10). Thus, the objective of the subsequent part is to extend Algorithm 1 with fault-tolerance capabilities.

Av, i, i (·,·,·)i, i = bi (k

1) + ci (k

ˆ

+ fi, c ,

(51)

Av, nv + 1, i (·,·,·) = bi (k

ˆ ˆ ˆ

1) + ci (k

and the transportation constraints

The objective of this section is to provide an answer to the last question stated in the preliminary part of the paper. It deals with accommodating the possible faults defined by (10). The fundamental effect of the presence of faults (10) could be the violation of scheduling constraints (44) (or (39) in the conventional max-plus algebra form). This may lead to the infeasibility of the overall optimization problem (32). To tackle this problem, it is proposed to incorporate a time varying relaxation variable (k ) 0 into (44) resulting in:

bi (k ) = max(e, b (k ) + fi, b + vi (k )),

x ref (k ) +

ˆ

If fi, b

(k + j ),

min

y (k ), v (k ), ˜ (k )

J (y , ),

(48)

under (36) and (37), (38), (40) and (45). Taking into account the above optimization problem, it is possible to propose the entire FTC algorithm, which updates the matrices Av (·,·,·) and B v (·,·) along with associated constraints depending on fault estimates. To summarize, the FTC strategy is detailed by Algorithm 2. Algorithm 2. FTC for multiple AGVs.

• Set k = 1, N , v (0). • Get (k ), …, (k + N 1), and x (k ), …, x (k + N 1) from the MES. • For an index i of the AGV transporting the (k 1)th item correp

p

•

ref

ref

p

else fi, c (k

if bim (k else fi, b (k

1)

c (k

1) then fi, c (k

1) = cim (k

1)

b (k 1) =

1)

c (k

1) then fi, b (k

bim (k

1)

b (k

(49)

1) = 0

1)

ˆ

and set up fault predictors fi, b = fi, b (k

The main objective of this section is to evaluate the performance of the proposed approach. For that purpose the proposed MPC (Algorithm 1) and FTC (Algorithm 2) will be compared both in fault-free and faulty conditions. The objective of the considered scenarios is to show the performance of the proposed FTC and to compare it with MPC. It should be clearly pointed out that MPC was extended by the schedule relaxation mechanisms detailed in the preceding sections. Indeed, without them it fails to provide appropriate control action due to the violation of scheduling constraints. Generally, the set of AGVs aims at performing in a way presented in Fig. 3. Each scenario is associated with a given transfer station

1) = 0

1)

(50)

ˆ

1) fi, c = fi, c (k

1) .

ˆf 0 and/or ˆf 0, respectively, update the entries of A (·,·,·) • For in (29), corresponding to the state x of i th AGV as follows: i, b

(56)

7. Performance evaluation

sponding to vi (k 1) = e , i = 1, …, n v , measure the actual transportation times bi (k 1) m and cim (k 1) as well x (k 1) . Calculate the faults according to (10), i.e.

if cim (k

(55)

Taking into account the fault definition provided in Section 1 and the introductory background about the fault given in Section 2 (10) and (11), the faults are set to zero if the respective measured operation time (either cim (k 1) or bim (k 1) ) belong to the respective intervals (either c (k 1) or b (k 1) ). If this is not the case then the fault predictors are calculated using (49) and (50). Thus, one can imagine that the fault could be large enough to violate the scheduling constraint (44). This implies that the entire algorithm is infeasible and its operation has to be terminated. To overcome this unappealing phenomenon, the constraint (44) is replaced by (45) with a relaxation parameter (k ) and the associated cost function (46), which has to be minimized. It can be perceived as an entire delay of the transportation tasks within the multiple AGV system. Unfortunately, there is no direct way relating the above faults with alpha(k) because its determination depends on the solution of optimization problem (48). However, it should be noted that the proposed algorithm guarantees that the underlying optimization problem is always feasible assuming that at least one AGV is available. Indeed, in the case that the current performance of an AGV set is insufficient to attain x ref (k ) , it is optimally relaxed. This guarantees that the resulting feasible schedule is the closest to the original infeasible one.

(47)

1 is set by the user. It is adjusted to balance the imwhere 0 portance of either J (y ) or J ( ) , respectively. It should be noted that the optimization problem (32) is feasible if at least one AGV is accessible. Therefore, if it is impossible to handle all outlet production items fulfilling the constraint (44), the solution to be guaranteed the minimal variation from (45) is obtained. Finally, by defining ˜ (k ) = [ (k ), …, (k + Np 1)]T , the optimization problem can be formulated as:

(y (k )*, v (k )*) = arg

0 then matrix B (·,·) should be updated as follows:

• Obtain y (k )* and v (k )* by solving the constrained optimization problem (48) under (53)–(54), (56), (38), (40) and (45) • Use the first vector elements of y (k )* and v (k )* (i.e., y (k )* and v (k )*) and feed them into the system (13). • Set k = k + 1 and go to Step 1.

(46)

) J (y ) + J ( ),

(54)

bv (k ) = max(c1 (k ) + v1 (k ), …, cnv (k ) + vnv (k )).

and hence, a new FTC-oriented cost function can be introduced:

J (y, ) = (1

ˆ

with

Np 1 j=0

(53)

B v, nv + 1 (·,·) = b v (k )

It should be pointed out that (k ) should be as small as possible. Indeed, this exhibit a minor divergence from the desired time schedule. As a results, a cost function associated with j (k ) is proposed:

J( ) =

(52)

ˆ

ci (k ) = max(e, c (k ) + fi, c + vi (k )).

(45)

(k ).

1)+

+ ci (k ) + fi, b + fi, c + fi, c + vi (k )

6. Fault-tolerant control of multiple AGVs

x nv + 1 (k )

ˆ

1) + fi, b +

p = [1, 3, 2, 3, 2, 1, 3, 2, 1, …]T .

i, b

i

(57)

Each element of the sequence denotes the number of station to 63

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M. Witczak, et al.

Fig. 6. SC 2: Gantt chart for MPC and FTC (MPC – top, FTC – bottom).

As it can be noted in Fig. 4 i Fig. 5 the systems are carried out by the same way. In other words the above scenarios show that FTC system without any fault has the same AGVs flow as MPC system. Fig. 4. SC 1: Gantt chart for MPC and FTC (MPC – top, FTC – bottom).

7.2. Scenario 2: 3 AGVs

which the item is transported. Before providing the numerical parameters of the considered system, let us remind that all variables represent certain time instances. In this particular example they are measured in minutes. The unpacking-transportation and transportationpacking times are equal for each transfer station and are defined as b (k ) = [0.8, 1], c (k ) = [0.8, 1]. It is also assumed that d (k ) = 0 and = 0.5. The above data allows defining (1) being a signature of each item:

(k ) = [p (k ), c (k ), b (k ), d (k )].

The second scenario describes a situation with n v = 3. Let us start with the scheduling constraint (45) defined with:

x ref = [1, 2, 3, 5, 6, 7, 8, 9, 11, …]T ,

which is appropriately defined on a given prediction horizon Np = 4 . The fault affecting AGV 3 concerns a 3-minute delay in realizing the packing and a transportation delay, which is defined as:

• AGV 3 fault

(58)

Having a set of above quadruples, it is possible to introduce the production performance upper bound (40), which is yz (k ) = 0.1. Finally, the initial state is x (0) = 0 while the nominal system parameters are b (k ) = 1, c (k ) = 1. This means that such a behaviour corresponds to the worst case (upper bound) of the above intervals. Generally, there is no formal restriction behind any choice of n v but for the purpose of presentations four cases are to be considered, i.e., Fault Free, ?Accident?, 3 AGVs and 5 AGVs, respectively.

f3, c (k ) =

k<1 {30 otherwise.

(61)

The sources of such an unappealing effect could be associated, e.g., with low level of AGV batteries causing significant decrease of the AGV nominal velocity. Note that during the backward phase, the fault is equal to zero, i.e., f3, b = 0 , which clearly means that without the load AGV 3 is able to attain its nominal performance. Fig. 7 exhibits the resulting the k th item delivery time to p (k ) transfer station both for MPC ad FTC. Note that these results were obtained with the control strategies illustrated in Fig. 6. As it can be observed in Fig. 7, the initial behaviour of MPC and FTC is exactly the same. The effect of the fault is obviously the same and causes a significant divergence from the desired schedule. Contrarily to MPC, the proposed FTC scheme detects and appropriately accommodates this fault. To achieve such a behaviour,

7.1. Scenario 1: Fault Free The first scenario presents a situation with n v = 3 and no fault situation during it. The scheduling constraint (45) is described as follows:

x ref = [1, 2, 3, 4, 5, 6, 7, 8, 9, …]T .

(60)

(59)

Fig. 5. SC 1: Item delivery times to transfer stations (MPC – top, FTC – bottom).

Fig. 7. SC 2: Item delivery times to transfer stations (MPC – top, FTC – bottom). 64

Journal of Manufacturing Systems 55 (2020) 56–68

M. Witczak, et al.

Fig. 8. SC 3: Gantt chart for MPC and FTC (MPC – top, FTC – bottom).

Fig. 11. SC 4: Gantt chart for MPC and FTC (MPC – top, FTC – bottom). Table 1 Exemplary statistics of computation times. Np

Mean [s]

Standard deviation [s]

3 5 8

0.0404 0.0701 0.4849

0.004 0.0086 0.092

Fig. 9. SC 3: Item delivery times to transfer stations (MPC – top, FTC – bottom).

Fig. 12. A histogram of computation times for Np = 3.

complete failure concerns the second vehicle and is represented by the time that is equal to 15 times of the nominal delivery time of AGV 2.

Fig. 10. SC 4: Item delivery times to transfer stations (MPC – top, FTC – bottom).

• AGV 2 fault

some operations are realized before time (see k = 4 and k = 6 for FTC case in Fig. 7). In other words, in the counters 4 and 6 the AGVs 1 and 2 start their operations earlier as shown in Fig. 6. This makes it possible to exclude the faulty AGV 3, which can be clearly observed while comparing Gantt charts in Fig. 6.

f2, c (k ) =

(63)

As shown in Figs. 8 and 9 , the application of the FTC strategy allows detecting the complete failure of the second AGV and as a consequence all items from production outlet are handled by only two available AGVs. The flow presented in Fig. 9 shows that the other two AGVs are not able to handle all items according to the assumed schedule (62), but the FTC strategy guarantees the best possible execution, i.e., the obtained schedule is the closest to the assumed one.

7.3. Scenario 3: 3 AGVs and complete failure In the third scenario the scheduling constraint is defined as follows:

x ref = [1, 2, 3, 4, 6, 8, 10, 11, 14, …]T .

0 k < 1, 15*2 otherwise.

(62)

In this case, a complete failure of one of AGVs is taken into account. The 65

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M. Witczak, et al.

the proposed approach, the control variables v (k ) and y (k ) computation times were collected for different production horizons, i.e., Np {3, 5, 8} . To perform a fair comparison 80 computation times where collected for each configuration of production horizon. Finally, it should be pointed out that the proposed algorithm was implemented in MATLAB and tested with MSI Notebook GE72 with processor i7 7700HQ 2.8 GHz, 16 GB RAM and 256 GB SSD. The summary of the obtained results is presented in Table 1. Moreover, the histograms portraying the distribution of the above computation times are portrayed in Figs. 12–14. It can be summarized that from the above results, it is evident that the control variables computation times easily permit an appropriate and on-time realization of the desired schedule (66). Finally, It should be pointed out that the authors performed numerous experiments in which the available performance capability of AGVs was not sufficient to attain a given reference x ref . This, of course, caused a divergence from x ref but at a possibly minimal level. This is guaranteed by the schedule relaxation mechanism (45).

Fig. 13. A histogram of computation times for Np = 5.

8. Conclusion The paper aimed at answering four fundamental questions concerning the development of fault-tolerant control of a multiple AGVs warehouse transportation system. The first answer deals with development of the max-plus algebra-based model of multiple AGVs. The developed model tackles both concurrency and synchronization issues, which are inevitable in such systems. Moreover, it has a strictly analytical structure, and hence, it can be used for predicting the future behaviour of multiple AGVs. The second answer deals with developing model predictive control using such a model. Note that it was shown that the developed MPC algorithm can be implemented using mixed integer-linear programming. This clearly exemplifies a relatively low computational cost of the proposed approach. Subsequently, the third answer concerned robustness issue, which was tackled by incorporating interval arithmetics into MPC. Finally, the last answer pertained a faulttolerance issue, which was tackled by incorporating fault detection and diagnosis into the MPC. The resulting FTC was examined using a set of different scenarios, which clearly show its superiority over a MPC strategy. This constitutes its clear recommendation towards industrial applications. Consequently, the results support the main research hypothesis that it is possible to design a reliable control algorithms that allow a predictive fault-tolerant control of multiple AGVs which solves synchronization and concurrency issues, addresses modelling inaccuracies and leads to an optimal sequence. The industrial applicability is supported by the quality of the approach that the optimization can be carried out on-line and that certain faults and modeling uncertainties can be tolerated. The objective of future developments is to extend the considered transportation system with a last transportation layer involving a set of forklifts. They will perform transportation tasks from transfer stations towards designated storage places in a warehouse. This is, however, beyond the scope of this paper.

Fig. 14. A histogram of computation times for Np = 8.

7.4. Scenario 4: 5 AGVs The last scenario is more complicated, because of n v = 5. It pertains a multiple AGVs fault, i.e., AGV 2 and AGV 3, which are defined as follows:

• AGV 2 and AGV 3 fault f2, c (k ) =

{

f3, c (k ) =

{

0 k < 1, 2 otherwise.

(64)

0 k < 2, 1 otherwise.

(65)

Acknowledgement The work was supported by the National Science Centre, Poland under Grant: UMO-2017/27/B/ST7/00620. Conflict of interest: None declared.

In this case, the transportation schedule is given as follows:

x ref = [11, 12, 13, 15, 16, 17, 19, 21, 22, …]T .

(66)

Similarly as in the case of proceeding section, it can be observed (Fig. 11) that the initial behaviour of MPC and FTC is exactly the same. As previously, FTC accommodates the fault with a better performance. Indeed, it can be observed in Fig. 10 that it yields a smaller divergence from the desired schedule. This is achieved by a suitable work scheduling of the entire set of AGVs (Fig. 11). Note that a perfect delay-free fault accommodation cannot be achieved, which is due to the relatively tight work schedule (66). They show the computational burden behind

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