Effect of Layout Discretization on the Performance of Zone Control-Based Multi-AGV Traffic Management Systems

Abstract

Automatic Guided Vehicles (AGVs) are widely used in flexible manufacturing systems for material handling inside the factory. Traffic management strategies, required to guarantee a conflict-free operation of the overall fleet, discretize the workspace of the AGVs and use the resulting graph model for route planning and execution. In zone control approaches, AGVs move from node to node on a permit basis, with limitations on the allowed number of AGVs at a time in each area of the graph to prevent and/or resolve deadlocks and conflicts. Hence, for an optimal implementation of traffic controllers in real manufacturing systems, it is essential to understand how the layout discretization influences the performance of the AGV network. This paper analyzes its effect in grid-like shaped workspaces by using a representative zone control algorithm and a recently developed improvement of it. Realistic numerical experiments on different layouts reveal that denser discretizations do not yield faster executions or increase in throughput, while lower control periods in the permit system entail significant performance uplifts.

1. Introduction

The trend of using AGVs for material handling in Flexible Manufacturing Systems (FMS) has recently increased, and it is expected to grow in the coming years [1]. Hence, Multi-AGV (MAGV) fleets are playing a vital role in enhancing the efficiency and robustness of the logistic processes in factories and, consequently, their study has become a major research interest [1,2].

MAGV systems require a coordinated motion of their components to guarantee a timely and deadlock-free completion of all transportation tasks. Those in charge of keeping this so-called liveness [3] of the system are traffic control algorithms, which may achieve this major goal following three basic approaches [4,5]: (i) conflict avoidance, where AGVs plan conflict-free routes; (ii) conflict prevention, where specific rules are applied during route executions to ensure a conflict-free operation, and (iii) detection and resolution, where conflicts are resolved when they occur.

In turn, the traffic control implementation is centralized, when it is managed by a single computer, or decentralized, with decisions taken by the different system components [6]. Centralized strategies may lead to high computational loads on the central controller when the system is scaled up, while decentralized ones may result in sub-optimal solutions [1,6].

Among the different types of traffic control algorithms (see, for example, the thorough classifications in [7,8]), zone control has emerged as the prevailing traffic management technique [9]. In zone control approaches, the workspace is divided into a number of zones, also known as resource points [10] or tiles [11]. Then, AGVs traverse through these zones supervised by the traffic control algorithm, which gives the permissions to move from one zone to the next one, adjacent to it, on their planned paths. This discretization allows to model the layout as a graph, which simplifies not only route planning tasks but also route execution. This is because the traffic controller awards permissions to resume according to the occupancy level of the zones, now graph nodes, in order to prevent collisions and deadlocks.

Zone partitioning in irregular layouts has been studied in [12], where a zone control algorithm considering AGV geometry and guidepath topology that efficiently handles collision and deadlock avoidance is proposed. However, most zone control algorithms are presented for grid-like layouts and assume that AGVs fit within the tiles [5,7,10,13,14,15,16], which is taken as the starting point of this work.

Notice, firstly, that the same grid layout allows different discretizations. This is illustrated in Figure 1: the layout on the top left can be discretized in a basic way, just taking crossings as nodes, as shown in the top-right figure, or using a denser discretization, as depicted in the bottom plots. Therefore, since permits can be neither requested nor given instantaneously, and waiting states may take place farther or closer to troubled nodes depending on the discretization level, the performance metrics of the system may be affected by the type of discretization. This is even more relevant in industrial scenarios, where aisles between crossings may be tens of meters long.

Take, for example, the situation in Figure 2. In the sparse discretization depicted on top, an AGV moving from A to B has to wait at A till B is cleared, and moves straightforward to it when allowed. Instead, in the case of a denser discretization, such as that on the bottom, the motion is from A to C, then from C to D, and, finally, from D to B, which requires calling the traffic controller at each step in order to obtain the corresponding permit. This scales up not only the permit requests, but also the amount of communications in the network and the deceleration and acceleration maneuvers when arriving and leaving a zone, thus yielding extra power consumptions for the AGVs. However, in case B is busy but C and D are available, the AGV can advance till D and wait there for the permit to resume towards B; hence, the resuming motion is initiated much closer to B than in the sparse discretization (top) case, which shortens the overall time taken when moving from A to B.

Therefore, understanding how layout discretizations affect the performance is essential for an efficient implementation of traffic control systems in real industrial scenarios; denser discretizations can improve the time response to certain conflicts, but also result in a software overload that may hinder the overall system performance. Nevertheless, despite its criticality, to the best of the authors’ knowledge, there are no studies in the zone control literature dealing with layout discretization and its effect. In fact, besides assuming that a block can entirely contain an AGV, most works just mention in the simulation section the dimensions of the zones, where squares of side 1 m [5,10,13,14] or 2 m [7] are often taken. Others, instead, indicate the number of nodes of the selected graph, but not the size of the corresponding zones [15]. Finally, ref. [16] uses a basic discretization with nodes representing crossings, workstations, and parking/charging slots, while edges stand for the aisles in between them.

Intrinsically linked to the discretization effects is the control period, i.e., the time interval of activation of the traffic control algorithm loop that issues commands to the AGVs. This is because denser discretizations scale up the number of permit requests from the AGV fleet to proceed; so, dealing promptly with them using lower control periods should have a beneficial influence on the time completion of tasks.

This article deals with the effect of layout discretization on the overall system performance of zone control-based traffic controllers. The analysis is conducted using two representative traffic management strategies: the Dynamic Resource Reservation (DRR) approach [10], and the recently published Improved Dynamic Resource Reservation (IDRR) algorithm [16]. The study takes into account both the density of the discretization and the control period of the traffic manager. Interestingly, realistic numerical simulations run on two different layouts reveal that denser discretizations do not result in significant performance improvements while, as intuitively expected, shorter control periods yield substantial reductions in task completion times.

The remainder of the paper is organized as follows: Section 2 includes basic modeling details and concepts that are common to DRR and IDRR traffic control algorithms. The main features of DRR and IDRR are presented in Section 3 and Section 4, respectively. The experimental setup, including a description of the prescribed task, is presented in Section 5. The numerical results are gathered and discussed in Section 6. The contributions of the paper, together with some suggestions for further research, are discussed in Section 7. Finally, conclusions are drawn in Section 8.

2. Preliminaries

The layout consists of a fully known, grid-like workspace where AGVs carry out point-to-point transportation tasks. Hence, we may find there pick-up stations, drop-off stations, and parking stations, all of them linked by aisles. In turn, aisle intersections are known as crossing points. The following assumptions are considered:

Assumption 1.

Every station is linked to a single crossing point of the grid through an aisle, and every crossing node is linked to, at most, one station.

Assumption 2.

There are, at least, as many parking stations as deployed AGVs.

Assumption 3.

Aisles may be traveled bi-directionally, but only one AGV is allowed at a time in an aisle.

Generically, the workspace is discretized as in Figure 1: Each zone represents a physical area, and it may either contain a station or just be an intermediate point of an aisle. This discretized layout is then modeled as a graph $G=(N,E)$, with nodes representing zones and edges representing links between adjacent zones. Namely, the node and edge sets are denoted, respectively, as $N=\{N_i,1\le i\le n_N\}$, $n_N$ being the total number of nodes, and $E=\{E_{ij};\forall N_i,N_j\mathrm{adjacent}\}$. In turn, $R=\{R_i,i=1\dots n_R\}$ denotes the set of $n_R$ AGVs, $R_i$, deployed in the FMS.

In this framework, a task is considered as a node-to-node displacement, understood as follows: Basic transportation jobs consist of moving loads from pick-up nodes to drop-off nodes. In the case an AGV currently at a parking slot undertakes a job, the assignment is divided into a minimum of two tasks: from the parking node to the pick-up node, and from the pick-up node to the drop-off node. Once this is completed, depending on the specific traffic control algorithm and the pending job list, a final move from the drop-off node to the parking slot might be ordered as well. In any case, once a task is assigned, a shortest route between the initial and final nodes is computed and delivered to the AGV.

Definition 1.

A task for AGV $R_i$ consists of an ordered sequence of $n_{R_i}$ adjacent nodes to be followed, i.e., $T_i={\left(T_{ij}\right)}_{1\le j\le n_{R_i}}$, where each $T_{ij}$ stands for a specific graph node.

Remark 1.

$T_{ij}$ is used to sequentially index the nodes of a task. As an example, if task $T_2$ goes though nodes $N_6$, $N_4$, and $N_9$, $T_2=\left(T_{21},T_{22},T_{23}\right)=\left(N_6,N_4,N_9\right)$.

Definition 2.

The states of $R_i$ can be as follows: (a) idle ($S_i=0$): ready to accept a new task; (b) resuming ($S_i=1$): moving between two adjacent nodes; (c) waiting ($S_i=2$): $R_i$ is at a node, waiting to be allocated to the next one; (d) resolving ($S_i=3$): undergoing a conflict resolution.

AGV motion works on a permit basis as follows: $R_i$ running task $T_i$ and currently located at node $T_{ij}$ requests permission to proceed to $T_{ij+1}$. If granted, $R_i$ leaves $T_{ij}$, which becomes free, reserves $T_{ij+1}$, and proceeds towards it. As conflicts may arise when AGVs compete for the same resources, zone control-based approaches grant permits according to the occupancy of shared routes and/or nodes. This yields the following definitions:

Definition 3.

The state of a node $N_i$, denoted as $S{N}_i$, is given by the number of AGVs occupying or reserving it.

Definition 4.

Assume that $R_i$ is performing task $T_i$ and it is currently located at node $T_{ij}$. Its residual route is the sequence of nodes of $T_i$ that remain to be visited, including $T_{ij}$. Namely, $L_i={\left\{T_{ik}\right\}}_{j\le k\le n_r}$.

Definition 5.

The shared route of $R_i$ is the subset of nodes of $L_i$ that also belong to the residual route of any other AGV. Namely, ${\Sigma }_i=L_i\cap \left(L_1\cup \cdots \cup L_{i-1}\cup L_{i+1}\cup \cdots \cup L_{n_R}\right)$.

Therefore, conflicts arise when ${\Sigma }_i\ne \varnothing$. The next sections detail how this issue is faced by two representative zone control algorithms.

3. The DRR Traffic Control Algorithm

3.1. General Description

The DRR [10] algorithm appears as an evolution of the Chain Of Reservations (COR) strategy introduced in [13], and it relies on a conflict prevention approach.

The basic features of the method are as follows: (i) AGVs have their own parking spot from where they start each assignment and to where they are directed after completing the drop-off; (ii) each resource point, or node of the corresponding graph model, can be reserved, and subsequently occupied, by at most one AGV; (iii) an AGV located at a certain node can reserve at most the next node in the current task.

The allocation mechanism works as follows: Let AGV $R_i$, located at node $N_x=T_{ij}$, request permission to resume to $N_y=T_{ij+1}$; permission is granted if any of the next two conditions is met:

 Condition A. 

$N_y\notin {\Sigma }_i$.

 Condition B. 

$N_y\in {\Sigma }_i$ and $S{N}_z=0$, $\forall N_z\in {\Sigma }_i$, $N_z\ne N_x$.

Essentially, Condition A guarantees that $N_y$ is not in the shared route of $R_i$, which means that $N_y$ is free, i.e., $S{N}_y=0$, and the move is completely conflict-safe. In turn, Condition B allows $R_i$ to access the shared route just in case all its nodes are free. This suggests that, as the cardinals of ${\Sigma }_i$, $i=1,\dots ,n_R$ grow, AGVs will have to wait longer for the allocation of shared resource points, which may have a negative effect on the ending time of the task. A simplified flowchart of DRR is displayed in Figure 3, where $N_x=T_{ij}$, $N_y=T_{ij+1}$ are two adjacent nodes to be visited by AGV $R_i$.

Moreover, the role of discretization seems to be linked to the fact that it bounds the distance between the shared route and the adjacent zone where an AGV is waiting to access it. As pointed out in Section 1, lower distances mean that AGVs would wait nearer and may reach the shared route sooner, but it also scales up permission requests and acceleration/deceleration maneuvers, so the combined effect on the ending time of the task is to be assessed. Finally, if the control period which governs the traffic controller allocation mechanism works faster, the waiting times realized by the AGVs should be shorter than those associated with higher control periods. However, as advanced in the Introduction, the numerical results in Section 6 will show that only the control period plays a significant role.

Remark 2.

Conditions A and B have been adapted to the fact that, differently from [10], the definition of shared route used in this paper considers the current AGV position as a part of it (see Definition 4).

3.2. DRR Model Layout

The continuous AGV road network is discretized in square blocks of the same size (see the bottom-left plot in Figure 1). AGVs have the option to either remain stationary or rotate within designated zones, positioned at the center of these blocks. Moreover, they are restricted from advancing to the next blocks during deceleration intervals. Achieving this limitation is feasible by appropriately designing the AGV speed and block size in practical scenarios. In the simulation experiments presented in [10], the layout is discretized using 1 m side blocks, while the AGV size is 0.7 m × 0.7 m.

4. The IDRR Traffic Control Algorithm

4.1. General Description

The IDRR algorithm [16] presents an AGV fleet control strategy based on conflict prevention and detection and resolution. Hence, differently from DRR, which relies on a conflict prevention approach achieved by restricting the access of shared routes to a single AGV, IDRR also deals with conflict resolutions. This allows multiple accesses to shared routes and, consequently, shortens waiting times. For the sake of brevity, specific details on the IDRR algorithm are omitted in this paper, and the interested reader is referred to [16]. However, a brief description of the rationale behind IDRR follows in the next paragraphs.

IDRR works on a double-layer architecture basis: a graph model, which acts as a symbolic layer mainly used in path planning, and a geometric layer, superimposed to the former and key in path execution. Essentially, a set of geometric positions, termed Control Points (CPs), is identified in the vicinity of each node, and node-to-node motion is effectively conducted between CPs. Then, nodes with several associated CPs are allowed to accommodate more than one AGV and, consequently, they can be used as conflict resolution zones.

Two types of nodes are identified in the layout: (i) those corresponding to pick-up, drop-off, and parking stations, termed Working-Parking (WP) nodes; (ii) crossing nodes, corresponding to aisle intersections, and termed TX nodes according to the specific shapes that can be found in grid-like layouts, namely, 3-way (T) and four-way (X) crossings. WP nodes are linked to the main grid through a TX-crossing, and are allowed single occupancy, i.e., they have an associated unique CP. Instead, T and X nodes have three and four associated CPs, respectively, and it is in the TX nodes where conflict resolutions take place. Figure 4 depicts an X-node with four CPs.

Conflict detection and classification is essential in IDRR to allow an AGV to resume to the next node of its trajectory. This is carried out using information on shared routes with other AGVs. Then, the AGV is either forced to wait at the current node or allowed to go ahead according to the traffic rules. Broadly, the conflicts considered by IDRR are:

• head-on, when two AGVs traveling in opposite directions meet at a node;
• intersection, when the trajectories of two AGVs cross at a node;
• pursuit, when two AGVs traveling in the same direction are located in adjacent nodes;
• loop conflicts, when a closed cycle of pursuit conflicts occurs.

In the case of pursuit and loop conflicts, AGVs are forced to wait at the current node for the situation ahead to clear. Instead, in the case of head-on or intersection conflicts, AGVs are allowed to move to the conflicting node, where they maneuver within its CPs so as to resolve the conflict. Figure 5 illustrates the resolution of a head-on conflict:

• Position 1: initially, AGVs $R_1$, $R_2$ are located in nodes $N_1$ and $N_2$, respectively, facing a head-on conflict;
• Position 2: $R_1$ is moved to CP $N_{22}$ in $N_2$;
• Position 3: $R_2$ is moved to CP $N_{23}$ in $N_2$;
• Position 4: $R_1$ is moved to CP $N_{21}$ in $N_2$;
• Position 5: finally, $R_2$ is moved to CP $N_{22}$ in $N_2$.

Now both AGVs are at the respective exit CPs of $N_2$ according to their future trajectories, and hence are ready to resume when possible according to IDRR’s traffic rules.

Moreover, the possibility of solving head-on or intersection conflicts at a TX node and, at the same time, not inducing deadlocks, is guaranteed by always leaving room enough for it inside a node, and also between adjacent nodes. This is controlled by not permitting more than two AGvs at the same TX-node, which allows maneuvering therein, and more than three AGvs in adjacent TX-nodes, which allows to conduct resolutions in at least one of them; the latter is verified through a path saturation condition [16]. In order to better illustrate the working principle of IDRR, a simplified flowchart of it for a specific case is presented in Figure 6: namely, $N_x=T_{ij}$, $N_y=T_{ij+1}$ are two adjacent TX nodes to be visited by AGV $R_i$, which is assumed to be at the exit CP of node $N_x$, i.e., the CP of $N_x$ that it is closest to $N_y$, waiting to be allocated at $N_y$. The flowchart describes the procedure followed by IDRR to this end. The reader is referred to the original work [16] for a detailed description of the general case.

Notice that allowing multiple AGVs to access shared routes results in lower waiting times when compared to DRR. Another major difference is that IDRR does not need AGVs to head to a parking station after a drop-off and before undertaking a new task. In fact, IDRR moves idle vehicles from drop-off points to parking stations just in case the list of pending tasks is empty. Overall, the combined effect of both features results in major improvements in task completion times [16]. However, IDRR’s traffic management policy outperforms DRR’s regardless of the parking effect; this was shown in [16] using the so-called IDRR with parking (IDRRP) algorithm, which works exactly as IDRR but always heads AGVs to park between task assignments. Hence, IDRRP will also be used in this paper to further enlighten the comparison between DRR and IDRR.

As for the role of discretization and the control period in task completion times, the discussion is the same as for DRR, namely, higher node densities should have a positive effect on reducing waiting times, but also entail a higher number of permission requests and acceleration/deceleration instances, while shorter control periods should result in faster ending times.

4.2. IDRR Model Layout

The layout discretization proposed in [16] for IDRR uses WP stations and TX crossings as nodes, as in the top-right plot of Figure 1. The movement of an AGV can be characterized as transitioning from node to node through aisles, entering each node by the entry CP, and leaving it by the exit CP.

This type of discretization becomes essential in the formulation of the traffic rules of IDRR, because TX nodes are used as conflict resolution areas. Hence, denser discretizations require a reformulation of IDRR by considering that conflict resolutions are only possible in TX-nodes, not in the additional intermediate ones eventually introduced in aisles, as in the bottom-right plot of Figure 1. IDRR’s current traffic rules would be valid under denser discretizations just when these new nodes also have room enough for conflict resolution, but this is not generic; in fact, it could be considered as a change of layout.

5. Experimental Setup

Numerical experiments are carried out in two different layouts. The first one is the benchmark layout from [5], also used in [16] (see Figure 7). It features a square topology measuring 50 m × 40 m, and comprising the following: (i) three origins for work pieces, I1–I3; (ii) three destinations for work pieces, O1–O3; (iii) six machines, M1–M6; (iv) eighteen working (W) stations, S1–S18, with half designated as pick-up stations and the remainder as drop-off stations; and (v) ten parking (P) stations, C1–C10. At the beginning of the simulation, every source point, Ii, is equipped with twenty work pieces, labeled as Part-i type. These pieces are to be individually retrieved from the adjacent W station next to Ii, transported consecutively to two machines, where each piece is deposited, undergoes 20 s processing, is then retrieved, and is ultimately delivered to the W station corresponding to the designated sink point, Oj.

The route to be followed by each type of piece is:

• Part 1: (I1)-S7-S14-(M4)-S13-S5-(M3)-S6-S12-(O3);
• Part 2: (I2)-S10-S3-(M2)-S4-S16-(M5)-S15-S9-(O2);
• Part 3: (I3)-S11-S1-(M1)-S2-S18-(M6)-S17-S8-(O1),

where parentheses indicate sources, sinks, and machines. Therefore, three distinct transportation assignments are linked to each route: from the source W station to the W drop-off station of the first machine, from the W pick-up station of the first machine to the W drop-off station of the second machine, and from the W pick-up station of the second machine to the W drop-off station of the sink. This entails a total of 180 tasks to be allocated to the available AGVs within the layout (comprising 3 types of pieces, 20 pieces of each type, and 3 movements per piece). The AGVs commence from the P stations C1–C10, move at a speed of 1 m/s, and the simulation concludes when the last piece is deposited, and the transporting AGV returns to its designated parking position.

The second layout corresponds to an industrial workspace of 25 m × 30 m with seven pick-up stations, W1 to W7, two drop-off stations, W8 and W9, and ten parking stations, P1 to P10, see Figure 8. Transportation assignments are generated at random and in real time at pick-up stations with time intervals following a uniform distribution. The assignments are subsequently allocated to idle AGVs by searching for the first idle one within the list of AGVs, which is ordered according to the AGV labels, $R_i$. The total simulation time is set to five hours.

For IDRR and IDRRP, both layouts are modeled as in Section 4.2, namely, with WP stations and TX crossings as nodes. Instead, three types of discretization are studied for DRR according to Section 3.2: the same as IDRR and IDRRP, labeled as DRR, and two additional ones with nodes placed at 2 m and 1 m intervals, labeled as DRR-2m and DRR-1m, respectively. In turn, the selected control periods of the allocation procedure are 1 s, 0.5 s, 0.2 s, 0.1 s, 0.05 s, and 0.02 s. In all the cases, the AGVs run at a constant speed of 1 m/s and show equal acceleration and deceleration values. Finally, numerical simulations were carried out using the commercial software Flexsim (version 23.1.4).

6. Numerical Results

In a first test, the effect of layout discretization in the benchmark layout is assessed by setting the control period to $T=1$ s, and recording the ending times as described in Section 5, for IDRR, IDRRP, DRR-2m and DRR-1m, with the number of AGVs varying from 1 to 10. The results are displayed in Figure 9, while the data are collected in

Table 1. Benchmark layout—Test 1: Ending times (in [s]) for $T=1$ s, for 1 to 10 AGVs.

AGVs	IDRR	IDRRP	DRR	DRR-2m	DRR-1m
1	20,139	29,627	29,651	31,166	32,293
2	10,668	15,833	15,973	16,524	16,911
3	7818	10,881	12,992	13,644	13,991
4	6022	8427	13,027	13,812	14,192
5	5249	6989	14,952	14,326	14,203
6	4732	6025	15,118	15,213	15,289
7	4404	5403	14,224	15,438	15,974
8	4068	4985	14,990	14,953	14,941
9	3979	4759	16,080	15,601	15,528
10	3957	4677	15,306	15,889	16,143

.

As already reported in [16], IDRR and IDRRP clearly outperform DRR. Only for 1 and 2 AGVs are the IDRRP and DRR times close, in the former because IDRRP matches DRR, as there are no conflicts, and in the latter because the number of conflicts between only 2 AGVs in this layout is not null but still very low. Moreover, the difference increases with the number of AGVs in the layout, as the number of conflicts scale up and the DRR waiting times increase. This corroborates the superior performance of IDRR with respect to DRR not only because of the parking policy, but also from the pure traffic management side, as the IDRRP and DRR numbers show.

However, it is most interesting that although DRR works with the sparse discretization inherent to IDRR and IDRRP, the use of denser discretizations in DRR-2m and DRR-1m make no significant difference either between them or with DRR. Hence, the fact that denser discretizations entail lower waiting times in the face of conflicts is definitely compensated by the corresponding increase associated with the request of a higher number of permissions to resume. The latter is clearly deduced from the results when there is just one AGV in the layout; the higher ending times observed for denser discretizations can be solely attributed to the additional time required to handle the scaling up of permit requests, because the absence of conflicts excludes the compensating possibility. Therefore, it turns out that (i) DRR works finely with sparse discretizations, and (ii) the use of denser discretizations with IDRR in the general case is not encouraged, as it would complicate the algorithm and there is no expectation of improvement.

A second test is conducted to evaluate the effect of the control period, T, on the time completion of the experiment. Simulations were carried out again for the five traffic management algorithms with $T=1$ s, 0.5 s, 0.2 s, 0.1 s, 0.05 s, 0.02 s, and a number of AGVs ranging from 1 to 4. The results, gathered in

Table 2. Benchmark layout—Test 2: Ending times (in [s]) for different T and 1 to 4 AGVs.

AGVs	T	IDRR	IDRRP	DRR	DRR-2m	DRR-1m
1	1	20,139	29,646	29,651	31,166	32,293
	0.5	19,508	28,713	28,717	29,207	29,394
	0.2	19,428	28,594	28,598	28,690	28,779
	0.1	19,133	28,152	28,155	28,180	28,214
	0.05	19,018	27,978	27,982	27,988	27,998
	0.02	18,980	27,924	27,925	27,924	27,932
2	1	10,668	15,833	15,973	16,524	16,911
	0.5	9961	14,787	15,258	15,335	15,362
	0.2	9642	14,342	15,001	15,067	15,090
	0.1	9587	14,230	14,937	14,975	14,996
	0.05	9563	14,189	14,915	14,939	14,943
	0.02	9560	14,180	14,900	14,904	14,907
3	1	7818	10,881	12,992	13,644	13,991
	0.5	7301	10,180	12,885	13,038	13,098
	0.2	7043	9836	12,798	12,916	12,965
	0.1	7015	9794	12,743	12,772	12,801
	0.05	6993	9752	12,725	12,744	12,755
	0.02	6982	9721	12,713	12,724	12,729
4	1	6022	8427	13,027	13,812	14,192
	0.5	5617	7876	12,979	13,266	13,338
	0.2	5434	7582	12,894	13,105	13,163
	0.1	5398	7564	12,859	12,932	12,979
	0.05	5384	7534	12,789	12,853	12,887
	0.02	5365	7504	12,770	12,789	12,803

, confirm those of Test 1. Namely, no significant differences are observed between the sparse discretization of DRR and the denser ones in DRR-2m, and DRR-1m, while for the 1 AGV case, the mismatch decreases with T; in fact, when $T=0.02$ s, times are almost equal, which indicates that for this control period, the scaling up of resuming requests in denser discretizations has no effect on the time completion of the task.

It is noticeable that the ending times become better as the period of the control loop decreases. AGVs have to wait for the control loop to be activated at the specific interval when the AGVs have requested for allocation between the interval; thus, for lower control periods, the negative cumulative waiting effect in time completions is minor, hence speeding up the whole system. The improvement with respect to the highest ending time, always given by $T=1$ s, increases with lower T’s and tends to stabilize around $T=0.02$ s, irrespective of the algorithm and number of AGVS. For IDRR and IDRRP, the maximums are 10.91% and 10.95%, respectively, while DRR, DRR-2m and DRR-1m score better for 1 or 2 AGVs, the maximum being 13.5% for DRR-1m and 1 AGV, in all cases for $T=0.02$ s. The relative errors are displayed in Figure 10.

These results are further confirmed by the data in

Table 3. Benchmark layout—Test 2: Waiting times (in %) for different T and 1 to 4 AGVs.

AGVs	T	IDRR	IDRRP	DRR	DRR-2m	DRR-1m
1	1	12.5%	15.1%	15.7%	16.7%	21.2%
	0.5	7.7%	8.7%	9.1%	11.6%	15.2%
	0.2	4.2%	4.7%	4.9%	6.2%	9.2%
	0.1	2.7%	2.8%	2.9%	4.2%	5.9%
	0.05	2.0%	2.0%	2.2%	3.0%	3.9%
	0.02	1.6%	1.7%	1.8%	2.4%	2.7%
2	1	13.5%	16.9%	17.2%	19.1%	23.2%
	0.5	8.5%	10.7%	11.4%	13.1%	17.8%
	0.2	4.8%	6.7%	8.0%	9.1%	11.6%
	0.1	3.7%	4.6%	7.1%	8.2%	8.7%
	0.05	3.1%	3.6%	6.7%	7.2%	7.3%
	0.02	2.8%	3.1%	6.5%	6.5%	6.6%
3	1	15.8%	18.7%	26.4%	29.8%	31.6%
	0.5	10.7%	12.6%	25.3%	27.1%	27.8%
	0.2	8.6%	10.2%	24.5%	25.6%	26.0%
	0.1	7.1%	8.6%	23.6%	24.0%	24.5%
	0.05	6.6%	7.9%	23.5%	23.6%	23.6%
	0.02	6.2%	6.5%	23.3%	23.5%	23.6%
4	1	17.2%	20.4%	33.2%	39.4%	39.5%
	0.5	13.1%	16.4%	31.4%	34.2%	35.8%
	0.2	9.3%	11.8%	30.1%	32.7%	33.0%
	0.1	9.1%	11.4%	29.9%	30.9%	31.4%
	0.05	8.0%	10.1%	29.0%	29.7%	30.0%
	0.02	6.8%	8.7%	28.8%	29.0%	29.1%

,

Table 4. Benchmark layout—Test 2: Travel distances (in [m]) for 1 to 4 AGVs.

AGVs	Data	IDRR	IDRRP	DRR	DRR-2m	DRR-1m
1	$R_1$	18,923	27,596	27,589	27,589	27,589
2	$R_1$	9293	13,836	13,899	13,899	13,899
	$R_2$	9248	13,661	13,680	13,680	13,680
	Avg	9271	13,749	13,790	13,790	13,790
	Tot	18,541	27,497	27,579	27,579	27,579
3	$R_1$	6737	9364	9437	9437	9437
	$R_2$	6528	9204	9203	9203	9203
	$R_3$	6543	8978	9030	9030	9030
	Avg	6603	9182	9223	9223	9223
	Tot	19,808	27,546	27,670	27,670	27,670
4	$R_1$	4841	6929	7739	7739	7739
	$R_2$	4848	6832	7036	7036	7036
	$R_3$	4840	6724	6380	6380	6380
	$R_4$	4951	6689	6220	6220	6220
	Avg	4870	6793	6844	6844	6844
	Tot	19,480	27,174	27,375	27,375	27,375

and

Table 5. Benchmark layout—Test 2: Average velocities (in [m/s]) for different T and 1 to 4 AGVs.

AGVs	T	IDRR	IDRRP	DRR	DRR-2m	DRR-1m
1	1	0.940	0.931	0.930	0.885	0.854
	0.5	0.970	0.961	0.961	0.945	0.939
	0.2	0.974	0.965	0.965	0.962	0.959
	0.1	0.989	0.980	0.980	0.979	0.978
	0.05	0.995	0.986	0.986	0.986	0.985
	0.02	0.997	0.988	0.988	0.988	0.988
2	1	0.869	0.868	0.863	0.835	0.815
	0.5	0.931	0.930	0.904	0.899	0.898
	0.2	0.961	0.959	0.919	0.915	0.914
	0.1	0.967	0.966	0.923	0.921	0.920
	0.05	0.969	0.969	0.925	0.923	0.923
	0.02	0.970	0.970	0.925	0.925	0.925
3	1	0.845	0.844	0.710	0.676	0.659
	0.5	0.904	0.902	0.716	0.707	0.704
	0.2	0.938	0.934	0.721	0.714	0.711
	0.1	0.941	0.938	0.724	0.722	0.721
	0.05	0.944	0.942	0.725	0.724	0.723
	0.02	0.946	0.945	0.726	0.725	0.725
4	1	0.809	0.806	0.525	0.495	0.482
	0.5	0.867	0.863	0.527	0.516	0.513
	0.2	0.896	0.896	0.531	0.522	0.520
	0.1	0.902	0.898	0.532	0.529	0.527
	0.05	0.905	0.902	0.535	0.532	0.531
	0.02	0.908	0.905	0.536	0.535	0.535

. Namely, Table 3 contains the waiting time, i.e., the fraction of time (in %) in which AGVs have been in a waiting state during the overall task execution. Higher waiting times indicate a worse utilization of AGVs and, consequently, higher ending times. As expected, IDRR has the lower numbers; the % waiting time increases for IDRRP, as AGVs have to add the extra path from drop-off to parking stations after each trip, but it is still under that of DRR because of the traffic management policy. In turn, the more densely discretized DRR-2m and DRR-1m have slightly worse results than DRR because of the extra permission requests, which are not compensated by the time saved in conflict resolutions. In turn, the waiting time percentage decreases with the control period, T, showing a tendency to stabilization around $T=0.02$ s.

Table 4 shows the distances traveled by the AGVs during the experiment. Besides confirming that IDRR-controlled systems carry out the task with minimal traveled distance, IDRRP and DRR show similar results regardless of the scaling in the number of AGVs, as both algorithms require them to start and finish a task at their own parking places. Of course, the DRR-2m and DRR-1m data match those of DRR, as the traveled distance under the same traffic control algorithm is independent of the discretization density. Instead, it can be observed in Table 5 that the average velocities share the behavior of ending and waiting times already displayed in Table 2 and Table 3, respectively, with respect to discretization and the control period variation. In this case, it is also worth noting that the average velocities of IDRR and IDRRP are very similar, which is due to the fact that, despite the parking policy difference, the traffic control algorithm is the same.

The same tests were run on the industrial layout and for the task rationale also described in Section 5. In this case, the ending time was replaced by the average throughput, i.e., fully completed tasks per AGV and per hour, while the other metrics (waiting time, distance, and velocity) were kept. It is worth highlighting that the results are fully aligned with those already discussed for the benchmark layout.

The average throughput per AGV and per hour for a control period of $T=1$ s is depicted in Figure 11 from the data collected in

Table 6. Industrial layout—Test 1: Average throughput per AGV and per hour for $T=1$ s, from 1 to 10 AGVs.

AGVs	IDRR	IDRRP	DRR	DRR-2m	DRR-1m
1	45.0	36.4	36.4	33.4	33.2
2	43.4	32.4	32.2	30.2	29.6
3	39.6	30.4	29.0	27.0	25.6
4	38.4	29.8	27.8	24.2	22.2
5	34.2	26.2	24.4	20.6	18.4
6	31.4	25.8	24.0	21.4	19.2
7	29.2	22.6	19.6	16.4	13.8
8	27.0	20.2	16.2	12.8	10.0
9	25.8	18.6	16.0	13.6	10.8
10	25.2	17.4	13.8	11.0	8.4

. Notice that, as expected, it decreases as the number of AGVs in the layout increases because of the scaling up of the number of conflicts in the system. Again, IDRR outperforms IDRRP, while the latter does so with DRR. Moreover, in this case, denser discretizations for DRR perform slightly worse than in the benchmark problem, as it stems immediately from the comparison of Figure 9 and Figure 11 or Table 1 and Table 6.

The behavior of the throughput against the control period for 1 to 4 AGVs and $T\in \{1,0.5,0.2,0.1,0.05,0.02\}$ [s] is collected in

Table 7. Industrial layout—Test 2: Average throughput per AGV and per hour for different T and 1 to 4 AGVs.

AGVs	T	IDRR	IDRRP	DRR	DRR-2m	DRR-1m
1	1	45.0	36.4	36.4	33.4	33.2
	0.5	53.4	38.0	38.0	37.6	36.0
	0.2	55.6	39.0	39.0	38.8	37.6
	0.1	56.0	39.0	39.0	39.0	39.0
	0.05	56.2	39.2	39.2	39.2	39.2
	0.02	56.4	39.2	39.2	39.2	39.2
2	1	43.4	32.4	32.2	30.2	29.6
	0.5	51.0	37.2	37.0	34.8	32.0
	0.2	53.0	38.2	38.0	36.0	33.2
	0.1	53.4	38.6	38.0	36.4	34.2
	0.05	53.4	38.8	38.2	36.6	34.4
	0.02	53.6	38.8	38.2	36.6	34.4
3	1	39.6	30.4	29.0	27.0	25.6
	0.5	47.6	34.4	32.8	30.2	27.0
	0.2	52.4	35.8	33.8	31.4	28.6
	0.1	53.0	36.4	33.8	32.2	29.4
	0.05	53.2	36.8	34.0	32.6	30.0
	0.02	53.2	36.8	34.0	32.6	30.0
4	1	38.4	29.8	27.8	24.2	22.2
	0.5	45.0	32.6	31.0	26.8	23.8
	0.2	47.8	33.6	32.4	28.0	25.2
	0.1	48.8	34.0	32.4	28.4	25.4
	0.05	49.2	34.2	32.6	28.6	25.6
	0.02	49.4	34.2	32.6	28.6	25.6

. Once again, significant improvements are shown in all cases as T decreases. The improvement percentages with respect to $T=1$ s, displayed in Figure 12, lie within the interval $\left[4.4,34.3\right]\%$, while for the benchmark problem, it was $\left[0.4,13.5\right]\%$ (recall Figure 10). Finally, the waiting times, traveled distances, and velocities are in accordance with the throughput data; the corresponding tables are omitted for the sake of brevity.

7. Discussion and Future Work

The numerical results show that, although it may be tempting to use denser discretizations hoping that this would yield a reduction in waiting times associated with conflict prevention and detection and resolution, it turns out that this is compensated by the cumulative waiting times associated with the higher number of permission requests inherent to node-to-node displacements, up to the point that no significant benefit is observed from the use of denser discretizations. As denser discretizations also require higher energy consumptions because of the scaling up in communications and acceleration/deceleration maneuvers, sparse discretizations should be prioritised in industrial implementations of zone control approaches.

In turn, a noticeable improvement arises when using faster control periods for the allocation mechanism loop. However, this may increase the computational burden and require better and more expensive hardware, so a trade-off should be required. Interestingly, a practical control period can be determined according to the hardware used for localization of the AGVs; since knowing the position of AGVs is an integral part of the system, the control period can be calculated, setting it some 5 to 10 times slower than the sensor data measuring frequency. Usually, localization sensors work in the range of $\left[20,50\right]$ Hz, so a control period of 0.2 s seems to be practically achievable in real manufacturing systems.

Current limitations when implementing zone control-based approaches stem from the common assumption that the workspace is able to accommodate the AGV kinematic constraints. In real scenarios, issues may arise if the motion trajectories followed by the AGVs do not consider either self-kinematics or those of potential nearby AGVs. This is usually solved when modeling the layout while setting up the system, but a “kinematic-aware” approach [12] can further improve the system performance. Another restriction is linked to the fact that these strategies are mainly devised for point-to-point displacements; in cases where there exist tasks with complex interdependency, specific rules have to be taken into account at the task allocation level without hampering the traffic management policy. Finally, the traffic control approach is closely linked to the geometric layout of the factory. In particular, IDRR is unable to handle non-grid-like layouts, as well as situations where WP stations have multiple CPs or are adjacent to each other.

Further research is oriented to investigating if advantage can still be taken of waiting time reduction in the face of conflicts by using non-uniform discretizations tailored to specific layouts. For example, those with long aisles between crossings might benefit from a discretization of the aisle in two blocks, one that might allow the AGV to leave the intersection but still remain close to it, and another one that would allow the AGV to circulate to the end of the aisle but be required to stop right before the next crossing. The idea is illustrated in Figure 13.

8. Conclusions

This paper analyzed the effect of discretization in grid-like shaped layouts for zone control-based approaches using two representative algorithms. Firstly, it was found that denser layout discretizations hinder the potential improvement in waiting times expected from faster time responses to specific conflict prevention and resolution procedures. Secondly, it was shown that using lower periods in the control loop resulted in a significant increase in the system performance in terms of time completion and average throughput.

To the best of the authors’ knowledge, this is the first study of layout discretization effects on the performance of zone control AGV traffic management systems.