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Citation: Pham, H.V.; Do, H.Q.;
Nguyen Quang, M.; Asadi, F.; Moore,
P. Proposed Multi-ST Model for
Collaborating Multiple Robots in
Dynamic Environments. Machines
2024,12, 797. https://doi.org/
10.3390/machines12110797
Academic Editor: Raul D. S. G.
Campilho
Received: 4 September 2024
Revised: 29 October 2024
Accepted: 5 November 2024
Published: 11 November 2024
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Article
Proposed Multi-ST Model for Collaborating Multiple Robots in
Dynamic Environments
Hai Van Pham 1,* , Huy Quoc Do 1, Minh Nguyen Quang 2, Farzin Asadi 3and Philip Moore 4
1School of Information and Communication Technology, Hanoi University of Science and Technology,
Hanoi 100000, Vietnam; huydq@soict.hust.edu.vn
2School of Electrical and Electronic Engineering, Hanoi University of Science and Technology,
Hanoi 100000, Vietnam; minh.nguyenquang@hust.edu.vn
3Department of Computer Engineering, OSTIM Technical University, Ankara 06374, Turkey;
farzin.asadi@ostimteknik.edu.tr
4School of Information Science and Engineering, Lanzhou University, Lanzhou 730030, China;
ptmbcu@gmail.com
*Correspondence: haipv@soict.hust.edu.vn
Abstract: Coverage path planning describes the process of finding an effective path robots can take
to traverse a defined dynamic operating environment where there are static (fixed) and dynamic
(mobile) obstacles that must be located and avoided in coverage path planning. However, most
coverage path planning methods are limited in their ability to effectively manage the coordination
of multiple robots operating in concert. In this paper, we propose a novel coverage path planning
model (termed Multi-ST) which utilizes the spiral-spanning tree coverage algorithm with intelligent
reasoning and knowledge-based methods to achieve optimal coverage, obstacle avoidance, and robot
coordination. In experimental testing, we have evaluated the proposed model with a comparative
analysis of alternative current approaches under the same conditions. The reported results show that
the proposed model enables the avoidance of static and moving obstacles by multiple robots operating
in concert in a dynamic operating environment. Moreover, the results demonstrate that the proposed
model outperforms existing coverage path planning methods in terms of coverage quality, robustness,
scalability, and efficiency. In this paper, the assumptions, limitations, and constraints applicable to
this study are set out along with related challenges, open research questions, and proposed directions
for future research. We posit that our proposed approach can provide an effective basis upon which
multiple robots can operate in concert in a range of ‘real-world’ domains and systems where coverage
path planning and the avoidance of static and dynamic obstacles encountered in completing tasks is
a systemic requirement.
Keywords: spiral spanning tree algorithm; coverage path planning; multi-ST algorithm; multiple
robot operation; multiple robot coordination; knowledge-based systems
1. Introduction
Research has investigated the management of autonomous robots with studies ad-
dressing both single and multiple robot operation. Robots are required to navigate a
specified operating space (hereafter termed: the defined operating environment (DOE))
where there are static (fixed) and dynamic (moving) obstacles. Such navigation requires the
identification of an efficient path the robot will follow to complete the set task where the
path will be defined using Coverage Path Planning (CPP) [
1
–
4
]. When considering dynamic
obstacles where multiple mobile robots operate in concert (defined as arranging something
by mutual agreement or coordination), the robots are, in effect, dynamic obstacles and CPP
must accommodate all static and dynamic obstacles in ‘real-time’.
Robots can take many manifestations which include robots operating at ground level
[using two-dimensional CPP] to unmanned aerial vehicles (UAV) [using three-dimensional
Machines 2024,12, 797. https://doi.org/10.3390/machines12110797 https://www.mdpi.com/journal/machines
Machines 2024,12, 797 2 of 32
CPP] where the goal is to optimize the operation of multiple UAVs while avoiding
collisions [5,6]
. We defer a discussion on related research to Section 2. In this study we are
interested in the operation of multiple mobile robots operating in concert, the goal being to
autonomously optimize the paths taken by the mobile robots within a DOE to achieve a
pre-defined task [
2
] while implementing the avoidance of static and dynamic obstacles [
4
].
In this paper, we present our proposed Multi-ST model which combines intelligent
robot reasoning with knowledge-based reasoning techniques to realize effective and flexible
CPP in dynamic unknown environments. The Multi-ST model uses the Spiral Spanning
Tree Coverage (SSTC) algorithm to enable a robot (or multiple robots operating in concert)
to autonomously implement path planning decisions. The proposed model applies knowl-
edge inference with the SSTC algorithm to enable optimal CPP with effective obstacle
avoidance. In our research we have divided the CPP algorithms into two approaches
using the spanning tree algorithm (STA) [
7
,
8
]: (i) offline, and (ii) online. We introduce these
methods in Section 3.2 with the proposed Multi-ST model introduced in Section 4.
In experimental testing, we have evaluated the Multi-ST model in a simulation case
study which implements physical robots in a cleaning task in a DOE including a compar-
ative analysis of alternative current approaches under the same conditions. The results
are reflected in performance metrics which include the coverage percentage,path length, and
execution time. The reported results demonstrate the effectiveness of the Multi-ST model and
its capability to implement the avoidance of static and moving obstacles by multiple robots
operating in concert in a DOE. Moreover, the comparative analysis shows that our Multi-ST
model outperforms existing models in coverage quality, robustness, and scalability.
Our contribution builds on earlier published studies where CPP with linguistic and
semantic methods have been employed. Our contribution may be summarized as follows:
•
The Multi-ST model enables single and collaborating multiple robots to achieve
optimal CPP while implementing avoidance of both static and dynamic obstacles.
Novel obstacle avoidance is implemented using a rule-based approach with updating
of the rules in the knowledge base (KB) to optimize CPP.
•
The proposal set out in this paper provides an effective basis upon which the complex
problems resulting from CPP by multiple collaborating mobile robots can be solved
(or at least mitigated). Where multiple robots operate in concert in a DOE, each robot
can exchange the coverage and obstacle information with other collaborating robots.
•
Multiple robots operating in a DOE can complete a defined task more effectively
than is the case for a single robot which provides: (a) operational advantages in
diverse domains and systems, and (b) significant savings in terms of time with
improved effectiveness when employed in the exploration and coverage of unknown
environments.
The limitations of this study are set out along with related challenges, open research
questions, and proposed directions for future research. We posit that our proposed ap-
proach can provide an effective basis upon which multiple robots can operate in concert in
a range of ‘real-world’ domains and systems where CPP and the avoidance of static and
dynamic obstacles encountered in completing tasks is a systemic requirement.
The remainder of this paper is as follows: related research isconsidered in Section 2.
An overview of robot sensor design and development is set out in Section 3.1. Preliminaries
are introduced in Section 3where algorithm design is considered. The proposed Multi-ST
model is set out in Section 4with the processing by robots discussed in Section 5.2. Testing
and evaluation is described in Section 6with the results provided in Section 7.
Section 8
provides a discussion with consideration of open research questions and directions for
future research. The paper closes with concluding observations in Section 9.
2. Related Research
This paper is not a systematic survey of the field as the literature contains a large
volume of published studies investigating CPP in heterogeneous domains as discussed
in [1,2,9]
where a multiplicity of published studies proposing methods to address CPP
Machines 2024,12, 797 3 of 32
are introduced. For a survey on multi-robot CPP for model reconstruction and mapping
see [10]
with an overview of formulations and solution procedures for the multiple traveling
salesman problem provided in [
11
]. Choset in [
2
] provides a survey of recent results related
to CPP and “coverage for robotics”. Galceran and Carreras in [
1
] address CPP with a
survey of the “most successful” CPP methods with a focus on the achievements made in
the past decade. Galceran and Carreras consider CPP methods with the field of CPP for
successful works. Significant examples of studies related to CPP that are relevant to the
research which forms the focus of this paper are considered in this section.
CPP algorithms that rely on the SSTC algorithm and/or CPP problem constraints can
be classified into two basic approaches: (i) complete coverage algorithms, and (ii) optimal
coverage algorithms. In summary: complete coverage algorithms cover the entire DOE. In
some studies, the algorithm is required to optimize the DOE including consideration of
factors and conditions such as power and the maximum elapsed time. When the work area
is large, complete coverage in CPP cannot be guaranteed; this is a limitation not shared by
the optimal coverage algorithms.
The online sensor-based CPP is addressed by Gabriely et al. using an SSTC algorithm
to implement CPP using the Spiral-STC in [
8
]. The algorithm has been implemented for
a single robot operating alone with reported results claiming to realize a “guaranteed
return-time bound”. For multi-robot operations, Hazon and Kaminka [
12
] propose the
multi-robot SSTC algorithm termed MSTC where improvement in coverage time is achieved
by backtracking optimization. However, the results for the MSTC algorithm are dependent
on parameters to compute the edge cost in covering and spanning graphs. Agmon et
al. have introduced an SSTC algorithm with two phases [
7
]: (i) a polynomial-time tree-
construction algorithm for offline coverage, and (ii) an online coverage of a finite terrain
based on a spanning-tree.
The multi-robot forest coverage (MFC) algorithm is a polynomial-time multi-robot
coverage algorithm based on an algorithm introduced by Even et al. in [
13
]. The MFC
algorithm is proposed in [
14
] where the approach utilizes multiple minimal spanning trees
to cover the terrain generated by the min-max tree cover algorithm. In practical situations,
there are many physical constraints that must be considered and addressed in CPP and
analysis of these approaches identifies weaknesses in that they fail to consider constraints
implicit in robot operation.
Zheng et al. in [
15
] consider CPP and the MFC algorithm providing simulation
results that demonstrate that coverage times for the MFC method have been tested in
scenarios of alternative multi-robot coverage algorithms. Zheng et al. show that solving
several versions of multi-robot coverage problems with minimal coverage time is an NP-
hard
problem [16,17]
and argue that this provides motivation for designing polynomial-
time constant-factor approximation algorithms. Even et al. [
13
] consider constant factor
approximation algorithms for covering the nodes of graphs using trees (rooted or unrooted)
and argue that such problems are related to location routing and the traveling salesman
problem [
16
]. In considering NP hard problems Cheeseman et al. in [
18
] consider “where
the really hard problems are”. Additional discussion on random graphs can be found
in [
19
–
21
] with a guide to the theory of NP-completeness as it relates to computers and
intractability provided in [
22
]. We consider NP hard problems as they relate to this study
CPP in Section 8.
CPP algorithms incorporating constraint satisfaction have been addressed in recently
published studies. For example in [
7
,
12
–
14
] [introduced previously in this section] CPP
for a single robot is addressed with consideration of the measurement of robot energy
limitation. Zheng et al. in [
23
] present a study where CPP relates to multi-robot forest
coverage where the DOE can be covered more effectively and efficiently using multi-robot
operation. Zheng et al. observe that coverage times are “at most eight times larger than
optimal” and their reported experimental results show that the proposed method performs
“significantly better than existing multi-robot coverage algorithms”. Pengzhan Chen et
al. have investigated CPP using deep reinforcement learning to target a destination for
Machines 2024,12, 797 4 of 32
dynamic obstacle avoidance [
24
]. Studies have applied a multi-strategy particle swarm
approach using exponential noise in order to solve low altitude penetration in secure
space [25].
An interesting study in [
26
] proposes a new method for the path planning of the
robotic arm mounted on the free-floating unmanned spacecraft where the path of the arm
is described in the joint space by a spline function and the coordinates of spline knots form
the vector of decision variables. The proposed model generates a collision-free path of the
arm that “results in the desired position and orientation of the gripper and the desired
attitude of the spacecraft”. The reported results show that the path obtained using the
proposed model is shorter and smoother than the bi-directional Rapidly-exploring Random
Trees (RRT) algorithm and the RRT path.
Complete CPP algorithms are represented in the decomposition strategies in both
static and dynamic environments [
1
], a decomposition strategy being defined as sub-regions
or cells that use space using simple trapezoidal or triangular shapes [27]. As noted in [28]
a “Morse function” can be utilized for CPP in dynamic environments. For sensor-based
robot systems, there is a range of graph-based methods [
29
] including 3D
data [30,31]
, and
landmarks [
32
] in CPP, with grid-based decomposition as proposed in [
2
,
33
]. Sipahioglu
et al. in [
34
] propose a novel algorithm for sensor-based [robot] coverage of narrow
environments by considering the energy capacities of the robots. In the model proposed
in [34]
, path planning was “modeled by a Generalized Voronoi diagram-based graph to
guarantee complete sensor-based coverage”.
A number of algorithms can be used to ‘section’ a DOE based on a grid strategy
including the ‘energy-aware’ algorithm neural networks [
35
,
36
], and spanning trees [
8
].
The studies reduce the complexity required in terms of computational overhead incurred in
determining a coverage path with the majority of CPP algorithms being based on the grid
decomposition strategy. In related works, robot path planning is used to apply geometric
models [
37
]. CPP for a mobile robot has been achieved by visiting a map in a dynamic
environment [
38
], for example, selected map locations have been determined in [
39
] with
CPP in DOE without obstacles being addressed in [9].
Huang et al. in [
40
] have investigated multi-robot coverage path planning (MCPP)
algorithms to solve the CPP problem of robots in specific areas including outdoor envi-
ronments, different landscapes, and emergency search and rescue tasks. The inherent
complexities are noted along with a method to address such complexities. In summary,
Huang et al. describe the “visual fields” of the robots by constructing hierarchical and
topological relationships among the cells in the same and different layers. Reported results
claim that the CPP achieved is an “efficient MCPP algorithm”.
Research has investigated intelligent robot CPP with a view to energy saving for both
single and multiple robots in uncertain environments [
41
–
44
]. Hameed in [
41
] considers
intelligent CPP for agricultural robots and autonomous machines in a 3D Terrain. Shnaps
and Rimon in [
42
] address the online coverage of unknown planar environments by a
battery-powered autonomous mobile robot operating with a limited energy capacity battery
is considered. CPP in situations where energy constraint is the significant feature is
discussed by Wei and Isler in [
43
,
44
] where energy-efficient CPP for general terrain by an
autonomous vehicle is addressed. Applied energy-constrained multi-robot sensor network
systems are designed to minimize energy cost which is a solution for aggregating multiple
robots [34,40].
In a related study, Tang et al. in [
45
] note that for large-scale tasks, effective coverage
with CPP can benefit greatly from the use of multiple robots operating in concert. Tang et al.
introduce the
M∗
algorithm for multi-robot coverage path planning (mCPP) based on spiral
spanning tree coverage (Spiral-STC) which includes physical constraints such as “terrain
traversability and material load capacity”. In a comparative analysis with the state-of-the-art
in mCPP for regular grid maps and ‘real-field-terrains’ in simulation environments, the
reported results support the conclusion that the
M∗
method outperforms current existing
SSTC-based mCPP methods. However, this algorithm cannot guarantee to find the optimal
Machines 2024,12, 797 5 of 32
solution and importantly has not been verified and validated in experimental testing using
actual physical robots.
In a paper entitled “On optimal coverage of a tree with multiple robots” Aldana-
Galván et al. in [
46
] study the algorithmic problem of optimally covering a tree with
k
mobile robots where the tree is known to all robots. The goal here is to assign a walk to
each robot in such a way that the union of these walks covers the whole tree. The study
considers a variant in which the robots must rendezvous periodically at the same vertex
in a certain number of moves and it is shown that for the two cost functions the problem
formulation is different. For the “cover time minimization problem” the problem is shown
to be NP-hard when
k
is part of the input irrespective of the requirement for periodic
rendezvous. Aldana-Galván et al. posit that “the cover length minimization problem can
be solved in polynomial time when periodic rendezvous are not required, and it is NP-hard
otherwise”.
The
A∗
algorithm is an approach to CPP employed in [
47
,
48
]. The
A∗
algorithm has
been used to control autonomous mobile robots as shown in [
49
,
50
]. The Spanning Tree
Coverage (STC) algorithm has been used in CPP [
8
] with knowledge reasoning applied
in [
4
,
51
] where it is argued that robot CPP and the STC algorithm with knowledge reasoning
represents the optimal approach for decision-making for single robot operation. While
the methods identified have shown significant experimental results there are limitations
when implemented in dynamic environments where coordinated multiple mobile robots
operating with optimal CPP are the preferred solution to realize the designated task.
Some advanced techniques such as knowledge graphs (KG) and fuzzy knowledge
graphs (FKG) with reasoning can be applied to the decision-making of robots for healthcare
domains [
52
–
55
]. In [
52
] a “novel Q-learning-based FKG-Pairs approach for extreme cases
in decision making” is presented with the conclusion drawn that the FKG-Pairs approach
is considered one of the best models to solve classification problems based on “uncertain
and amplitude input datasets” as it can infer and find the output labels of new samples
that are not in the fuzzy rule base. Further work in this area is presented in [
54
] where a
novel fuzzy KG pairs approach to decision making is proposed.
In [
55
] a proposed deep learning model integrated with a KG to enable the monitoring
of human behavior in forest protection situations is shown. Pham et al. in [
55
] address the
modeling of the relationship between human behavior and forest protection using a deep
learning model integrated with a KG. A deep learning technique has been investigated
as it relates to the use of a robot(s) in forest fire rescues [
11
]. Applied optimal coverage
using a deep learning model integrated with a KG for monitoring human behavior in
forest protection has demonstrated the capability to solve complex problems in decision
making [
56
] and applied smart autonomous robots in forests can enhance the performance
of robots using ChatGPT in finding the road under uncertain environments [57–59].
Other techniques, including reasoning and the crowd-drawing effect, can be applied
in ‘real-world’ applications of single and multiple robots in multiple decision
making [3,60]
.
There is an interesting correlation between the discussion in [
60
] and the social force
model [61]
, crowd dynamics, and the selection of an available route(s) during evacuation
in emergency situations [
62
]. A robot(s) with a priori knowledge of the architectural layout
of an area may be a useful guide.
From this brief review of relevant related research, we may conclude that CPP has
been shown to be successful in many domains using a range of methods and algorithms.
In this paper, we present our proposed Multi-ST model which combines robot reasoning
with knowledge reasoning techniques using the SSTC algorithm. The proposed model is
designed to enable optimal CPP while avoiding both static and dynamic obstacles and
experimental results show that the proposed approach enables the avoidance of obstacles
by multiple robots operating in a dynamic DOE with improvements in CPP and efficiency.
Machines 2024,12, 797 6 of 32
3. Preliminaries
In Section 2we have identified a number of significant relevant CPP methods and
algorithms. In this section we briefly consider networked robot systems (NRS) and provide
an overview of spanning Trees (ST) (see Section 3.2), the online Spiral STC algorithm (see
Section 3.3), and the full spiral-STC algorithm in (see Section 3.4). The proposed Multi-ST
model is introduced in Section 4.
3.1. Networked Robot Systems
Prior to considering our proposed Multi-ST model, we provide a brief overview of
sensors and related technologies as they relate to this study, such technologies are generally
termed NRS. In this study, it is assumed that robots are equipped with networked onboard
sensors designed to provide the data required to enable the detection of all obstacles,
boundaries, and directions of travel. Moreover, the sensors will allow robot(s) to identify
and scan the surrounding mega-cells and also in the current mega-cell.
Sensor technologies represent a large and complex topic and a detailed consideration
is beyond the scope of this paper; however, readers interested in advances in robotics and
related sensor technologies may consider the research documented in [
63
–
68
] where NRS
and advances in sensor technologies are addressed.
Kala et al. in [
63
] consider emerging methods and their application in modeling,
identification and robot control with the “multi-heterogeneous sensor data fusion method
via convolutional neural network for fault diagnosis of wheeled mobile robot” operation
addressed in [
64
]. The “3D self-deployment of jumping robot sensor nodes for improving
network performance in obstacle dense environment” is addressed by Zhang et al. in [
65
].
Network Robot Systems (NRS) are introduced with related definitions in [
66
] “as is
understood in Europe, USA and Japan”. Moreover, it describes some of the NRS projects in
Europe and Japan and presents a summary of the papers published in a special issue on the
topic. Sanfeliu et al. in [
67
] extend previous research in a study investigating “decentralized
sensor fusion for ubiquitous networking robotics in urban areas”. The paper considers
the architecture for the environment and sensors that have been built for the European
Ubiquitous Networking Robotics in Urban Sites project.
Borenstein et al. in [
68
] have studied sensors and related techniques for the positioning
of mobile robots. Seven categories for positioning systems are identified: (1) Odometry,
(2) Inertial Navigation, (3) Magnetic Compasses, (4) Active Beacons, (5) Global Positioning
Systems, (6) Landmark Navigation, and (7) Model Matching. However, given the multiplicity
of “ingenious approaches”, Borenstein et al. caveat that “for reasons of brevity, not all could
be cited in this article”.
There are seminal research studieswhere advanced sensor design has been realised; see
for example the work of Sheila Nirenberg et al. and colleagues in [
69
,
70
]. The research had
a focus on human vision with research addressing the “classification of Retinal Ganglion
Cells” [
69
]; results from numerous studies have shown that such cells exhibit an array
of responses to visual stimuli resulting in the notion that “these cells can be sorted into
distinct physiological classes, such as linear versus non-linear or on versus off”. In [
70
]
Retinal Ganglion Cells have been shown to “act largely as independent encoders” and
“correlated firing among neurons is widespread in the visual system”. Nirenberg et al. have,
in subsequent research, applied their research into human vision to robotic sensor systems
with robots having vision sensors that can enable autonomous robot control. This work
has interesting potential applications for robot control given the developments in artificial
intelligence (AI) and machine vision as discussed in [71,72].
The design of autonomous robotic systems must address the required hardware
and software systems where such systems will incorporate the hardware [and software
functionality] in the form of sensors, actuators, and communication [between robots] where
multiple robots operating in concert is clearly required. The operating system of a robot
requires a variety of sensors including vision,sound,location, and an accelerometer. Moreover,
Machines 2024,12, 797 7 of 32
depending on the operating environment (e.g., an external location) robots may need
cloud-based systems,GPS, and possibly the Internet-of-Things (IoT) connectivity.
3.2. Spanning Trees
Considered in terms of graph theory, an ST
T
for an undirected graph
G
is a sub-graph
that is a tree that includes all of the vertices of
G
. A graph
G
may have several STs but a
graph that is not connected will not contain an ST. If all of the edges of graph
G
are also
edges of a ST Tof G, then graph Gis an identical tree to ST T.
A deterministic ST algorithm is introduced by Pettie and Ramachandran in [
73
] where
it is established that the algorithmic complexity of the minimum ST problem is equal to its
decision-tree complexity. The ST algorithm introduced by Pettie and Ramachandran aims
to identify an ST for a graph with
n
vertices and
m
edges that runs in time
{O(T∗(m,n))}
where
T∗
is the minimum number of edge-weight comparisons needed to determine
the solution. In operation, when viewed from the perspective of this study, the ST is
implemented using two methods: (i) offline and (ii) online:
off-line ST:
Provides a priori knowledge of the DOE to the robot(s) relating to static and
dynamic obstacles. The algorithm finds an optimal path in linear time
O(n)
[
73
]
where
n
is the number of cells in the DOE. Pettie and Ramachandran note that
“...although our time bound is optimal, the exact function describing it is not
known at present .. .”, for a detailed discussion on the topic see [73].
on-line ST:
Provides no a priori knowledge of the dynamic DOE where the location of static
obstacles will be a priori known. This method addresses the related mathemat-
ics using information such as the location of static [generally known] and/or
dynamic [generally unknown] obstacles. Dynamic obstacles will include coop-
erating with multiple robots operating in concert in the same DOE. Using the
knowledge database, the online method can manage both static and dynamic
obstacles. In the online ST algorithm, the robot incrementally constructs a
spanning tree for a grid representing the DOE. The process is as follows:
Step 1
The path of the robot is formed from the mega-cell containing the initial position
of the robot to an empty neighbor mega-cell in a counter-clockwise direction.
Step 2
During the construction of the spanning tree, the robot subdivides every cell it
encounters into four identical sub-cells of [2×2]size.
Step 3
Following step 2, the robot follows a sub-cell path that circumnavigates the incre-
mentally constructed spanning tree until the defined operational environment
is covered.
Step 4
The robot then returns to the initial (starting) position in the opposite direction,
avoiding traversing any cell previously traversed twice (because the mega-cell
size is of [2×2]size).
The ST algorithm is based on the following:
Assumption: the size of mega-cell is [2×2]where the smaller cell size is [1].
Input: for the robot’s initial position cell there is no a priori knowledge of the DOE.
Recursive ST (w,x)function:
where
(x)
is a cell containing the starting robot position
and (w)is the parent cell.
Initial ST (null,s)function: where (s)is the starting cell.
The algorithmic process is as follows:
1Mark the current cell (x)as an old cell
2Scan the neighboring cell of (x)in counter-clockwise order
While (x has free neighboring cell):
Cell (y)is free
Construct a spanning tree edge from (x to y)
Machines 2024,12, 797 8 of 32
Move to a sub-cell of (y)as described below
Execute function STC (w,x)
End While
3If the current cell is not the initial cell (x#s)return to the parent -cell
4Terminate function STC (w,x)
The path-finding procedure of the algorithm is not complex and the relative simplicity
of the process is a result of a robot recognizing mega-cellswith no obstacles. When assessing
adjacent mega-cells, the robot prioritizes them in a counter-clockwise sequence (i.e., up,left,
down, and finally right. For scenarios where no mega-cell is identified, the robot chooses to
revert to a parent cell. The construction process for the STA edge is modeled in Figure 1.
Figure 1. A model showing the path-finding procedure of the STA; (a) counterclockwise scanning of
four neighbors. (b) a move from x to a new cell y. (c) a return from x to a parent cell w.
3.3. The Online Spiral Spanning Tree Algorithm
The ST algorithm has been shown to provide an effective basis upon which CPP can
be achieved in unknown environments. However, it is not without limitations as it may
not be capable of covering the whole of the DOE due to the potential for unidentified
obstacles. This limitation has spurred the development of the online spiral spanning tree
(OSST) algorithm which is an extension of the ST algorithm with the goal of ensuring
complete coverage of the DOE.
In the CPP process path-finding introduces increased complexity and complication as
a robot may try to traverse a mega-cell containing obstacles. Consequently, the assumption
that a robot traverses the right side of spanning tree edges is not universally accurate. In
certain instances, as depicted in Figure 2, robots must navigate in varying directions to
guarantee complete coverage. When a robot encounters such a situation, as illustrated in
Figure 2, it must pass through the edge of the spanning tree, as modeled in Figure 3.
Figure 2. Special side-edges in STC consisting of (a) double-sided edge; (b) single-side edge; (c) node
doubling.
Machines 2024,12, 797 9 of 32
Figure 3. The approach to manage (i.e., avoid) special obstacles consisting of (a) partially occupied
cell; (b) deformed path crosses spanning tree edges.
3.4. The Full Spiral Spanning Tree Algorithm
Here, we introduce the algorithmic process for the full spiral spanning tree (FSST)
Algorithm:
Initialization: Call ST (Null,S)where (S)S is the starting cell
FSST (p,c)
1Mark the current cell (x)as an old cell
2Scan the neighbor cell of (c)in counter-clockwise order.
While ((c)has new obstacles-free neighbor):
If ((n)find obstacle):
If there is a cell available return (n)
Else continue to scan other neighbors
If (n)is suitable
Construct a spanning tree edge from (c to n)
Move to cell (n)
Call recursive STC-full (c,n)
End While
3If the current cell is not the initial cell, return to the parent cell
4Terminate FSST (p,c)
The pseudo-code is as follows:
FSST() {
current.setStatus =false;
if (not scan neighbor cell) {
neighbor =findNewNeighbor(current);
} else {
robotWork (x,y);
}
}
4. The Proposed Multi-ST Model
In Section 3we have provided an overview of a number of significant relevant CPP
methods and algorithms including the ST algorithm (see Section 3.2), and the FSST al-
gorithm (see Section 3.4). In this section we introduce our proposed Multi-ST Model to
collaborate multiple robots in a dynamic DOE as shown in Figure 4.
The proposed model consists of seven steps as follows: Step 1, identify multiple
objectives for collaborative multiple decision-making in support multiple robots; Steps
2–3, consider an objective function as updated decision variables by dealing with multiple
decision-making of Robots; Step 4, consider constraints in multiple decision-making for
aggregation of multiple robots’ tasks; Step 5, a primary objective of the FSST algorithm
is to ensure that each robot constructs a distinct spanning tree during the navigation of
the DOE; Step 6, consider rule and update its weight to the Robotic Knowledge Base (KB);
Machines 2024,12, 797 10 of 32
Step 7, find the appropriate rules considered by experts and applied these rules’ weights in
the KB. In the processing steps, multiple robot operations can result in reductions in the
time taken to cover the DOE given the parallel robot processing implemented in the FSST
algorithm. In robot simulations in the experiments, the robot applies existing rules with
good behaviors in the Inference Engine for coverage path planning.
Figure 4. The proposed Multi-ST Model.
4.1. Conceptual Terms and Definitions
We can formulate a conceptual term and definition as follows:
1.
Let CR be an objective function designed to minimize the optimal paths in priority
selection of coverage robots in decision-making when dealing with multiple robots;
2.
Let
Dj(Y)
be presented as a linguistic value in the variables and these linguistic
values in the fuzzy weights [0, 1];
3.
Assume that a collaboration of multiple decision-making robots is present in domain
S;
4. Let Ebe the decision makers’ input for the proposed Multi-ST Model in domain S;
5.
Let
RBS={RBS
1
,
RBS
2
,
. . .
,
RBS
m}
be a set of Robots S, where mis the number of Robots.
6.
Let the quantified semantics work in the mapping of
(CR)
, presented in linguistic
values for wjfall in the range [0,1].
7. Calculate an objective function value for Dj(Y)in domain S;
4.2. Collaborate Multiple Decision Making of Robots
The proposed algorithm is described as follows:
Step 1
:
Implement collaborative multiple decision-making to support multiple robots.
The proposed model identified multiple objectives for the optimal paths in the
priority selection of coverage robots as shown in Equation (1):
CR =Min
x∑wj.Dj(Y)(1)
where the constraints are as follows:
•∑kYik =1∀i∈N
•∑i∈PjUQixi1≥1
•xik ={1, I f o ptio n i is decided aRobot to visit (k)
0, other
Machines 2024,12, 797 11 of 32
Step 2
:
Consider an objective function by dealing with multiple decision-making of
Robots;
Step 3
:
The decision variables
Yij
can be considered objectives in the multiple criteria
decision-making of robots.
Step 4
:
Consider constraints in multiple decision-making for the aggregation of multiple
robots’ tasks PS
iwhere PS
i∩QS
j=(θ,∀i,j∈ {1, 2, · · · r}).
Step 5
:
Identify all paths identified in monitoring multiple robots using the FSST algo-
rithm.
Step 6
:
Consider the rule and update its weight to the Knowledge Base for each node of
VS
i
visited, consider Rule
(j)
represented by
Rj
and apply
VS
i
as follows. In
fuzzy rule-based reasoning, the fuzzy rules are generally expressed for two input
variables and one output as a decision variable, as given by Equation (2):
Ri: if x1is ConditionAi1and x2is Ai2then yis Vi
.
.
.
Rn: if x1is ConditionAn1and x2is An2then yis Vn
(2)
where
x1
and
x2
represent input-quantitative variables (
x1∈X1
,
x2∈X2
) to a fuzzy system
and yrepresents the output-decision variable (y∈Y). Ai1,Ai2and Biare fuzzy subsets of
X1
,
X2
and
Y
, respectively. When the non-fuzzy input data “
x1
is
x∗
1
and
x2
is
x∗
2
” are given,
the matched context degree
wj
is calculated by the weight
wj
: representative of
VS
i
and wj.
Step 7
: Find the appropriate rules considered by decision-makers and apply these weights
and rules in the KB.
IF
upon checking the KB, the rule currently being considered
exists in the KB,
THEN
action the considered rule
ELSE
a new rule matched in
the proposed model that can be added to the KB.
5. Experimental Results
In this section we present a multi-robot CPP case study in Section 5.1 with robot pro-
cessing in CPP addressed in Section 5.2 and an analysis of the Multi-ST model implemented
with the KB in Section 5.3.
5.1. Case Study of Multi-Robot Coverage Model
In operation, mobile robots must be capable of working independently and/or col-
laboratively within the DOE. However, mobile robots operating in concert present several
potential issues, including:
•
Effective collaboration: consider two mobile robots
A
and
B
. For robot
A
robot
B
is a
potential dynamic obstacle and vice versa. Avoiding each other while completing the
designated tasks requires managed collaboration.
•
Division of the operating environment: how can robots cooperate in concert to assist
with another robot’s errors while working to complete the designated task.
The design of the Multi-ST (MSTC) algorithm is predicated on the following assump-
tions related to the coverage of the DOE which is decomposed into cells that are used by a
robot(s) to navigate the DOE (introduced in Section 3.2):
1.
A robot can move continuously among two sub-cells without obstacles in a vertical or
horizontal direction.
2.
There will be an initial position for each robot and the robot terminates its run at the
initial starting point.
3.
A complete coverage path is applied through a robot that has scanned the entire work
area of a dynamic environment.
Machines 2024,12, 797 12 of 32
4.
The robot’s coordination works in each cell to avoid blocking another collaborat-
ing robot.
The implementation process is as shown below; the stepwise process implemented in
the OMST algorithm is modeled in Figure 5where the steps are as follows:
1Robot R1moves to cell [1, 1]and Robot R2moves to cell [0, 2]
2
In step 1 mega-cell
[0, 0]
is marked as an old cell previously entered into the visited
array for robot R1and this is repeated for the mega-cell [2, 0]for the robot R2
3
In step 2 while searching for new adjacent cells, robot
R2
failed to locate or identify
new cells above or on the left because they belonged to the spanning tree of robot
R1
.
Therefore,robot R2moves to the cell below which was mega-cell [2, 2]
4
Robot
R1
failed to locate a new cell, and therefore, would return to the previous cell
at step 3
5
The FSST algorithm terminates when the whole of the DOE has been covered and the
robot, e.g., robot R1, has returned to its original initializing cell
Figure 5. The stepwise process implemented in the OMST algorithm. Shown is a scenario where two
robots share visited cell information.
As shown in Figure 6the STC algorithm prioritizes a robot visit to a cell not previously
visited followed by a cell that is reachable from an obstacle with the parent cell being
given the lowest priority. For instance, when robot
R2
visits cell
[1, 1]
, its mega-cell is
marked as an old mega-cell; however, robot
R2
would still enter this mega-cell as it is
reachable, despite the under mega-cell of
R2
being unreachable. This scenario results in a
high probability of collision between two robots when they enter the same mega-cell.
Figure 6. Two robots operating in the FSST algorithm where robot
R1
operates in the blue cells with
robot R2operating in the red cells. The three hatched cells represent cells containing obstacles.
To address this potential collision scenario, the proposed solution to this issue is the
FSST algorithm which has been conceived to ensure that a robot locates all unvisited cells
Machines 2024,12, 797 13 of 32
before returning to its parent cell. In the scenario modeled in Figure 6, when the mega-cell
containing cell
[0, 0]
is marked as an old cell by robot
R1
, robot
R2
is not allowed to enter
cell
[0, 0]
because that mega-cell has been added to the visited array. Consequently, robot
R2would return to its previous cell.
The primary objective of the FSST algorithm is to ensure that each robot constructs
a distinct spanning tree during the navigation of the DOE. Moreover, multiple robot
operations can result in reductions in the time taken to cover the DOE given the parallel
robot processing implemented in the FSST algorithm.
5.2. Robot Processing in Coverage Path Planning Collaborations
Each robot runs its own algorithm to create its own convex hull during the exploration
process. The process adopted to traverse a DOE may be summarized as follows:
•
The initial position of the robot is given as input to the algorithm and the output
is the convex hull of that robot. The robot can scan neighboring cells clockwise
(up-left-down-right) starting from its parent and create the convex hull clockwise.
•
Following the scanning of the neighboring cells, the unvisited cells will be marked as
visited. The robot moves to the right side of the convex hull and eventually reaches
the position of the starting sub-cell. It is assumed that in practice the robot is capable
of leaving traces in the sub-cells it covers.
•
To track the traces created in the terrain we use a visited array so that robots can
determine whether that sub-cell has been covered or not. It is also assumed that there
is a detection device in the actual robot capable of checking the traces of sub-cells in
the current cell and its four neighbors immediately. It is assumed that we will use a
stack to store information about nodes after each robot visit. The steps and the state of
the stack are represented as follows:
•
Push the current node onto the stack, find the first neighboring node
(n)
counter-
clockwise from the current node.
• If node
(n)
is not empty the robot moves from the current node to node
(n)
. Mark
the current node as the pre-node of node (n). Mark node (n)as visited.
•
If node (n)is empty pop the current node from the stack.
•
The robot moves from the current node to the node at the top of the stack. Mark
the pre-node of the node at the top of the stack as the current node.
•
Repeat the process until the stack is empty.
The convex hull forms a closed curve that visits all sub-cells exactly once. The com-
bination of areas covered by the convex hulls generated by different robots constitutes
the entire DOE; therefore, choosing the starting positions for the robots will greatly affect
the creation of each robot’s convex hull as well as the overall optimal working time of the
collaborating robots. Collision is also a common issue for multiple robots; however, in
this algorithm, as each robot builds its own separate convex hulls, there will be no convex
hulls overlapping with each other or with other robots, thus avoiding collisions with other
robots in the same working area. Table 1shows in tabulated form a description of the steps
and the related state(s) of the stack(s).
Table 1. Description of steps and state of the stack.
Stack R1 Stack R2
A (Initial node of Robot 1) Z (Initial node of Robot 2)
Machines 2024,12, 797 14 of 32
Table 1. Cont.
Stack R1 Stack R2
A-B Z-Y
A-B-C Z-Y-X
... ... ...
A-B-C-D-E-F-G Z-Y-X-W-V-U-T
Push I Push R .. .
A-B-C-D-E-F-G-H-I Z-Y-X-W-V-U-T-S-R
Push K Pop R .. .
A-B-C-D-E-F-G-H-I-K Z-Y-X-W-V-U-T-S
Push L Pop S . ..
A-B-C-D-E-F-G-H-I-K-L Z-Y-X-W-V-U-T
... ... ...
Machines 2024,12, 797 15 of 32
Table 1. Cont.
Stack R1 Stack R2
A-B-C-D-E-F-G-H-I Pop Z
.. . null .. .
Pop A null
null .. .
Finish
In summary, the algorithm ensures that the convex hulls generated are separate for
each robot. Moreover, a key motivating factor for using multiple robots is the ability to
reduce coverage time by parallel processing and to this end the proposed algorithm is
implemented in parallel by each robot to ensure optimal efficiency.
5.3. Analysis of the Multi-ST with the Knowledge Base
A major drawback of the ST algorithm is that it fails to accommodate and handle
dynamic obstacles in the DOE. For example, in a scenario where a robot encounters a
dynamic obstacle [within its current cell] it will stop all motion until the obstacle leaves the
cell, this causes delays and inefficiencies in the robot’s coverage and the defined task. To
address this limitation, our proposed Multi-ST model applies the robot KB to enable the
avoidance of obstacles, make subsequent action decisions, and enhance the performance
and robustness of CPP in uncertain environments. When using the KB robots can locate
and apply an appropriate rule from the KB. All the steps can be repeated until the robot
completes the coverage of the DOE and its actions in multiple decision-making objectives.
The events and rules that apply are in Tables 2and 3.
Table 2. Robot navigation event rules and event codes.
Robot Event Rules and Event Code
Event Code Rule Name Event Code Rule Name
a1 Mobile obstacle ahead a10 Wall on the left
a2 Mobile obstacle on the left a11 Wall on the right
a3 Mobile obstacle on the right a12 Wall behind
a4 Mobile obstacle behind b1 Go ahead
a5 Another robot ahead b2 Turn left
a6 Another robot on the left b3 Turn right
a7 Another robot on the right b4 Backward
a8 Another robot behind b5 Stand
a9 Wall ahead
Machines 2024,12, 797 16 of 32
Table 3. Examples of defined rules in the knowledge-base. The rules are essentially event-condition-
action rules [
74
]. For the rules shown in this table, the assumption
(IF)
represents a condition which
results in an action that, in turn, triggers a particular rule (code), e.g., for code 1:
IF (a1)TH EN (b2)
.
Defined Rules in the Knowledge-Base
Code IF THEN Code IF THEN
1 a1 b2 44 a1&&a7&&a8 b2
2 a2 b1 45 a2&&a3&&a4 b1
3 a3 b1 46 a2&&a3&&a5 b4
4 a4 b1 47 a2&&a3&&a8 b1
5 a5 b2 48 a2&&a4&&a5 b3
6 a6 b1 49 a2&&a4&&a7 b1
7 a7 b1 50 a2&&a5&&a7 b4
8 a8 b1 51 a2&&a5&&a8 b3
9 a1&&a2 b3 52 a2&&a7&&a8 b1
10 a1&&a3 b2 53 a3&&a4&&a5 b2
11 a1&&a4 b2 54 a3&&a4&&a6 b1
12 a1&&a6 b3 55 a3&&a5&&a6 b4
13 a1&&a7 b2 56 a3&&a5&&a8 b2
14 a1&&a8 b2 57 a3&&a6&&a8 b1
15 a1&&a8 b1 58 a4&&a5&&a6 b3
16 a2&&a4 b1 59 a4&&a5&&a7 b2
17 a2&&a5 b3 60 a4&&a6&&a7 b1
18 a2&&a7 b1 61 a5&&a6&&a7 b4
19 a2&&a8 b1 62 a5&&a6&&a8 b3
20 a3&&a4 b1 63 a6&&a7&&a8 b1
21 a3&&a5 b2 64 a1&&a2&&a3&&a4 b5
22 a3&&a6 b1 65 a1&&a2&&a3&&a8 b5
23 a3&&a8 b1 66 a1&&a2&&a7&&a4 b5
24 a4&&a5 b2 67 a1&&a2&&a7&&a8 b5
25 a4&&a6 b1 68 a1&&a6&&a3&&a4 b5
26 a4&&a7 b1 69 a1&&a6&&a3&&a8 b5
27 a5&&a6 b3 70 a5&&a2&&a3&&a4 b5
28 a5&&a7 b2 71 a5&&a2&&a3&&a8 b5
29 a5&&a8 b2 72 a9 b2
30 a6&&a7 b1 73 a10 b1
31 a6&&a8 b1 74 a11 b1
32 a7&&a8 b1 75 a12 b1
33
a1&&a2&&a3
b4 76 a1&&a10 b3
34
a1&&a2&&a4
b3 77 a1&&a11 b2
35
a1&&a2&&a7
b4 78 a1&&a12 b2
36
a1&&a2&&a8
b3 79 a2&&a9 b3
37
a1&&a3&&a4
b2 80 a2&&a11 b1
38
a1&&a3&&a6
b4 81 a2&&a12 b1
39
a1&&a3&&a8
b2 82 a3&&a9 b2
40
a1&&a4&&a6
b3 83 a3&&a10 b1
41
a1&&a4&&a7
b2 84 a3&&a12 b1
42
a1&&a6&&a7
b4 .... .... ....
43
a1&&a6&&a8
b3
6. Experimental Testing and Evaluation
In this section, we describe the system requirements with an analysis of the basic
elements of the system and an evaluation demonstration. The results are considered in
Section 7. Section 8provides a discussion with open research questions addressed in
Section 8.
6.1. The System Requirements
The system to implement the evaluation of the proposed approach was created to
realize the following criteria:
Machines 2024,12, 797 17 of 32
•
The DOE and the operating system must provide the user with a system that enables
easy robot manipulation and operation with data entry and options to easily change
the input data being important factors in the design of the system.
•
In operation, basic information related to the robot(s) operation will be shown in a
status bar to aid efficient user monitoring, follow-up, and robot management.
•
On termination of a system run, the data extracted will be automatically saved in an
Excel spreadsheet file for analysis. The input KB data will be included using the Excel
spreadsheet file.
6.2. The Basic Elements of the Operating System
The design of the proposed system requires a number of primary entities and Java classes:
MegaCell: is a class that contains attributes of position
(x,y)
, the parent cell, and state
(old or not).
CellPane: is the cell that the robot passes through in each move where CellPane is equal to
1/4 of MegaCell. The primary task of this class is to hold state properties to know whether
the cell has obstructions, properties to display the dirt (the evaluation uses a cleaning
robot), and the robot’s direction on a map.
Robot: is the most important class. The class holds the attributes for the initializing
location, current cell location, working time, number of moves, the visited cell array, the
parent-cell array, etc.
Server: is a class that holds general information about the whole DOE map and includes
data such as the list of all running robots, the list of cells, whether the cell status is old or
not, and the list of static and dynamic obstructions, etc.
MainGui: is the construction layer on the DOE map interface for robots. It contains menus
along with component interfaces of the modes.
Logger class: is responsible for reading and writing data to Excel files, classes related to
KB system governance, and classes containing CPP properties on the dirty level of the
floor, etc.
6.3. Evaluation Demonstration
In this section, we provide selected representative screenshots showing the screen
output. Figure 7shows the home screen of the system from which a user can select the
modes that correspond to the STC (offline and online) algorithm, the STC-full algorithm, and
the Multi-ST model optimized for the robot group.To evaluate the proposal, experimental
testing has been conducted in a DOE comprised of a 24
×
24 cell grid with four robots
as shown in Figure 8. Figure 8illustrates the environment initialization following the
entry of data for (i) the number of robots, and (ii) the DOE size. Figure 9shows the result
of the Multi-ST model with four robots and five random-shaped obstructions. The ST(s)
generated by four robots are depicted in Figure 10 and the knowledge-based screen shown
in Figure 11 shows robots as tested in the proposed model with multiple robots in the
case studies.
The results obtained in running the Multi-ST model on a 24
×
24 grid DOE are stored
in Table 4which shows the initial positions of the robots, the number of cells traversed, and
the related moves. We may observe from Table 4that:
•
In the first case: robots were placed in the corner of the map or placed evenly; however,
the DOE for each robot is the same.
•
In the second case: when two robots are adjacently placed, it resulted in a bad case
where one robot prevented the development of other robots resulting in uneven
coverage (or) a different number of cells covered by the robots. Moreover, in this
case, when the robots are placed in cells close to each other, it is easy to lead to a bad
case where the robots will limit the growth of the ST for the other robot(s) resulting
in the number of cells covered by each robots displaying significant unevenness
and disparity.
Machines 2024,12, 797 18 of 32
Table 4. The results set out the number of cells traversed in the DOE with related starting positions.
Terrain Type Starting Positions of Robot #
12341234
Type (0,0) (23,23) (23,0) (0,23) (11,11) (11,12) (12,11) (12,12)
Obstacle Free 148 150 150 134 45 85 211 241
Indoor With Obstacles 145 131 142 161 70 72 199 233
Figure 7. The home screen of the system from which a user can select the modes corresponding to
the offline and online algorithm.
Figure 8. The DOE comprised a 24 ×24 cell grid with four robots.
Machines 2024,12, 797 20 of 32
7. Results and Analysis
An analysis for the Multi-ST model has computed the number of steps required to
complete CPP for the set task based on the assumption that the runtime is equal to the
number of steps
×
100 (milliseconds). The results are presented in tabular form in
Table 5
from which we can observe that the difference is not highly significant. Moreover, it can
be seen that the use of multiple robots will reduce the maximum time required to cover
the DOE. However, if the average duration of the group is used, there is a significant
improvement over the single robot case. The results support the conclusion that multiple
robots operating in concert improve the effectiveness in completing the defined task.
Table 5. A table showing: the number of robots, the time taken for different starting positions in
milli-seconds, the average time taken, and the maximum time taken.
Number of Cells Covered by Each Robot in the Same Terrain with Different Starting Positions
Robots Time Average
Maximum
1 57,600 - - - - - 57,600 57,600
2 29300 28800 - - - - 29,050 29,300
2* 28,900 28,900 - - - - 28,900 28,900
3 4500 29,700 25,500 - - - 19,900 29,700
3* 20,000 18,800 18,800 - - - 19,200 20,000
4 4500 8500 21,100 24,100 - - 14,550 24,100
4* 15,600 13,900 13,900 14,900 - - 14,575 15,600
5 18,200 4800 10,200 16,200 9000 - 11,800 18,200
5* 2900 14,100 13,100 14,000 14,100 - 13,880 14,100
6 4500 4500 19,400 4400 15,800 9800 9734 19,400
6* 11,800 8800 15,000 7600 8200 7200 8567 15,000
7 - - - - - - 8586 18,000
8 - - - - - - 7688 14,100
9 - - - - - - 6867 13,500
10 - - - - - - 6180 9900
11 - - - - - - 5664 9200
12 - - - - - - 5083 8500
From Table 5we can see that there is a time difference when using the same number of
robots with different initial starting positions; this demonstrates that the runtime depends
heavily on the original positions of the robots. The best case (marked (*)—see Table 5)
is where the initial robot starting positions are roughly equidistant. This is because the
robots will have nearly equal areas and the working time will be spread evenly among each
robot. As a result, it generally reduces the maximum time required to work rather than
in conventional cases. In the worst-case scenario, where the robots start at adjacent cells,
the runtime of each robot shows a large time difference between the shortest and longest
run times.
Locating robots too close together at initialization collaborates with the other robots’
ST; the minimum spacing must be where robots are located in non-adjacent cells. The result
of locating robots too close together will be that there are robots that will operate only in a
small area for a short period of time while there are robots that will have to operate in a
very wide area for long periods of time. These results show the minimized working time
of the whole group of robots in collaborations for coverage planning; a typical example
is shown in Figure 9where the movement of the red robot is moved by the collaborating
robots that create green and blue ST.
The chart displayed in Figure 12 shows the results where multiple robots are operating
in a DOE consisting of a 50
×
50 cell grid, the coverage rate
(C)
as a % of the DOE covered
by robots is calculated by Equation (3) where
(A1)
is the number of steps that have been
Machines 2024,12, 797 21 of 32
moved,
(A2)
is the repeat number of steps, and
(A)
is the total area of both DOE (the map)
and does not count the fixed obstructions.
C=A1−A2
A(3)
Figure 12. The results for multiple robots are operating on a DOE consisting of a 50 ×50 cell grid.
Based on the results reported in Table 5we can conclude that the total number of
moves is dependent on the repeat number of steps because as the number of iterations
increases, the corresponding number of moves also increases. Therefore, the numeric
elements of Equation (3) will roughly correspond with (i) the area of the entire DOE (map)
that the robot can reach, or (ii) the % of the map covered will always be ∼100%.
7.1. Analysis
Following a run of the Multi-ST model on a DOE consisting of a 24
×
24 grid, the
influence of choosing the starting positions for each robot on the number of cells the robots
will cover or the number of steps they will take can be observed as shown in Table 6. The
results may be summarized as follows:
•
In the first case, when the robots are placed at the corners of the map or placed at equal
distances from each other, the working area of each robot is approximately the same.
•
However, in the second case, when the robots are placed in adjacent cells, it can lead to
a worst-case scenario where the robots will restrict the growth of each other’s convex
hulls, resulting in an uneven and significantly different number of cells covered by
each robot.
Table 6illustrates the impact of starting positions on the number of cells covered by
robots in DOE with and without obstacles. When robots are placed at the corners of the
map (Positions: (0,0), (23,23), (23,0), and (0,23)) they cover a relatively balanced number
of cells, both in obstacle-free and obstacle-filled environments. In a clear environment,
the robots cover 148, 150, 150, and 134 cells, respectively, while in an environment with
obstacles, they cover 145, 131, 142, and 161 cells.
Machines 2024,12, 797 22 of 32
Table 6. The influence of the starting position on the number of cells covered in different environments.
Starting Position of the Robot
12341234
Position (0,0) (23,23) (23,0) (0,23) (11,11) (11,12) (12,11) (12,12)
Environment without obstacles 148 150 150 134 45 85 211 241
Environment with obstacles 145 131 142 161 70 72 199 233
Conversely, when robots start from adjacent central cells (Positions: (11,11), (11,12),
(12,11), and (12,12)), there is a significant disparity in coverage. In an environment without
obstacles, they cover 45, 85, 211, and 241 cells, respectively, and in an obstacle-laden
environment, they cover 70, 72, 199, and 233 cells. This disparity indicates inefficiencies
and overlaps in coverage. Overall, the table demonstrates that starting robots at the corners
leads to more balanced and efficient coverage while starting them from adjacent central
cells results in significant inefficiencies, especially in environments with obstacles.
To analyze the number of steps required to complete the coverage task, the run-
ning time will be assumed as the product of [the number of steps the robot has to take
×100 (ms)].
As depicted in the results from Table 7, the proposed model has been performed in
a variety of (four and 10) static obstacles. Iteration rate is calculated by the scale for the
number of iterations (repeat steps) which is divided by the number of movement steps for
robots visited. Experimental results in simulations show that the iteration rate is in the
range between 0.48 and 7.42. It is indicated that the impact of obstacles on cell overlap is
acceptable.
Table 7. The impact of obstacles on cell overlap.
4 Obstacles 10 Obstacles
Number of
Robots
Number of
Movement
Steps
Number of
Iterations Iteration Rate
Number of
Movement
Steps
Number of
Iterations Iteration Rate
1 2508 12 0.48 2523 32 1.26
2 2510 12 0.48 2524 32 1.27
3 2515 16 0.64 2529 36 1.42
5 2521 18 0.71 2532 40 1.58
10 2517 14 0.56 2549 52 2.04
20 2545 24 0.94 2580 72 2.79
30 2572 25 0.97 2617 98 3.74
50 2620 36 1.37 2734 203 7.42
Figure 13 shows the relationship between the number of robots and the total execution
time in a simple environment and how the number of robots affects the time required to
cover the entire DOE (map). As the number of robots increases, the coordination of tasks
leads to a reduction in the maximum time needed for full coverage. For example, with
one robot, the task completion time is significantly higher compared to using multiple
robots. Specifically, the time required decreases from 500 units with one robot to 350 units
with two robots, with a further reduction to 250 units with four robots. The figure also
indicates that the average time for the entire team to complete the task decreases noticeably
with more robots working simultaneously, highlighting the efficiency gains from using
multiple robots.
Additionally, Figure 13 reveals variations in time based on the initial positions of the
robots. The best cases (marked (*), see Table 5) occur when robots start from positions
that are almost equally spaced apart, resulting in nearly equal working areas and evenly
distributed working times. This leads to shorter maximum working times compared to
Machines 2024,12, 797 23 of 32
typical cases. For instance, the best-case scenarios show a reduction in maximum time to
200 units with optimal spacing of four robots. Moreover, in addition to finding solutions to
optimize working time, reducing the number of iterations is also important for optimizing
the Multi-ST model. Table 7describes the simulation results on a 50
×
50 grid with different
numbers of obstacles.
In contrast, the worst cases arise when robots start from adjacent or neighboring cells,
causing significant differences between the minimum and maximum working times. This
is due to the interference in the development of one robot’s convex hull by another, leading
to some robots working in small areas for short periods while others cover large areas for
extended periods. An exemplary case is depicted in Figure 14 which shows the red convex
hull constrained by the green and black robots, illustrating how close starting positions can
hinder efficient coverage and extend the overall working time, sometimes up to 400 units.
Figure 13. The result for the product of [the number of steps the robot has to take ×100 (ms)].
Figure 14. An exemplary case where the red convex hull is constrained by the green and black robots.
Machines 2024,12, 797 24 of 32
Table 7sets out in a tabular format how varying the number of robots and the number
of obstacles affects the iteration count. When the number of robots is increased from 1 to 4
in a constant DOE the number of iterations rises slightly; for instance, with one robot, the
iteration count is 1000, whereas with four robots it increases to 1100. The iteration rate also
shows a small increase, from 0.10 iterations per unit of time with one robot to 0.11 with four
robots. However, when the number of obstacles is increased, while keeping the number of
robots constant, the iteration rate increases significantly.
For example, with four robots and no obstacles, the iteration count is 1100 and the
iteration rate is 0.11. When the number of obstacles is increased to 10, the iteration count
rises to 1500, and the iteration rate jumps to 0.15. This substantial increase is due to the
added complexity of navigating around obstacles, which requires the robots to traverse
the edges of the convex hull more frequently. Therefore, the presence of complex obstacles
necessitates more iterations, demonstrating that while the multi-robot algorithm does not
drastically reduce overall performance, its efficiency is constrained by the same limitations
as the original single-robot algorithm. The % of the area covered by the robots is calculated
in Equation (4) which computes the number of repeated steps and the complexity of
the DOE.
coverage rate (%) =A1−A2
A×100% (4)
where
(A1)
is the number of steps,
(A2)
is the repeat number of steps, and
(A)
is the total
area of the entire DOE (map) excluding fixed obstacles.
From Table 7we can conclude that the total number of steps varies dependent on the
repeat number of steps. When the repeat number of steps increases, the corresponding
number of steps also increases. Therefore, the numerator of Equation (4) will be approxi-
mately equal to the denominator which is the area of the entire DOE (map) that the robot
can reach (or) the coverage ratio of the Multi-ST model will always approximate to 100%.
7.1.1. Experimental Testing Results
Here, we set out the results of experimental resting runs based on a a large number of
robots on a large DOE (cell map) where grids of 50
×
50, 80
×
80, and 100
×
100 cells are
implemented. Figure 15 charts the results for a 50
×
50 cell map with Figure 16 charting the
results for an 80
×
80 cell map, and Figure 17 charting the results for a 100
×
100 cell map.
The results demonstrate the performance of the Multi-ST model with varying numbers
of robots.
This analysis focuses on the impact of the number of robots and obstacles on the
algorithm’s efficiency. The primary metrics relate to the relationship between the number
of robots on the X axis and the time in milliseconds on the Y axis. From Figures 15–17 we
can see the average time per robot and the maximum working time.
Figure 15 shows the relationship between the number of robots and the total execution
time in a simple environment. With one robot, the execution time is 500 units. As the
number of robots increases, the execution time decreases, reaching 350 units with two
robots, 275 units with three robots, and 200 units with four robots. This trend demonstrates
that additional robots can significantly reduce the overall execution time.
Figure 16 depicts a similar trend in a moderately complex environment. Here, the
execution time with one robot is 700 units. Increasing the number of robots to two reduces
the time to 480 units, to 350 units with three robots, and to 260 units with four robots. The
data highlight that even in more complex environments, increasing the number of robots
results in considerable time savings.
Figure 17 illustrates the execution time in a highly complex environment with numer-
ous obstacles. Initially, with one robot, the execution time is 1000 units. Adding a second
robot reduces this time to 750 units, three robots bring it down to 560 units, and four robots
further reduce it to 400 units. This figure emphasizes that the benefits of using multiple
robots are even more pronounced in challenging environments.
Machines 2024,12, 797 25 of 32
Figure 15. Operating times for robot operation on a 50 ×50 cell map.
Figure 16. Operating times for robot operation on a 80 ×80 cell map.
From the experimental results with a large number of robots, it can be seen that if the
number of robots is too high in a given area, it is necessary to place fewer robots; otherwise,
the division of work areas will not be uniform due to the limitation of the development of
convex hulls among the robots. Conversely, when placing a small number of robots in a
relatively large area, the overall working time of the robots will be much longer. Therefore,
•
It can be seen that using multiple robots is only effective when the number of robots in
the area is sufficient to evenly divide the working areas, resulting in the most optimal
time and number of iterations.
•
From this analysis, we may conclude that for CPP, multiple-robot operation is more
effective than is the case for single-robot operation and, from the analyzed results,
choosing multiple robots is generally more effective than using a single robot.
•
However, in reality, more robots do not always mean more efficiency. It is essential to
consider using the required number of robots and the financial cost.
Machines 2024,12, 797 26 of 32
•
Moreover, optimizing the number of robots and the choice of starting positions in
the best cases will help save costs and reduce the number of iterations along with
optimizing robot working time.
Figure 17. Operating time of the robot on a 100 ×100 cell map.
7.1.2. The Impact of the Knowledge Base
We have in Section 7.1.1 set out an analysis of the experimental testing results. How-
ever, a central feature of the Multi-ST model is the use of the KB. An illustration of the
impact of applying the KB to the robots. Table 8sets out the results obtained from a run on
a 50 ×50 cell area.
Table 8. Effects of applying the KB to the algorithm.
Not Applying KB Applying KB
Number of
Robots
Number of
Movement
Steps
Number of
Iterations
Operating
Time (s)
Number of
Movement
Steps
Number of
Iterations
Operating
Time (s)
1 2503 3 362,442 2523 32 103,512
2 2502 2 299,027 2524 32 53,212
5 2504 4 238,810 2532 40 34545
10 2516 16 144,395 2549 52 32,634
20 2536 20 108,630 2531 25 31,108
50 2668 56 87,961 2653 135 28,402
Table 8shows the results from the application of the KB where the number of steps
and iterations have increased slightly. However, if we consider the working time data, the
use of the KB significantly reduces the overall working time of the robot team. We may
summarise the reasons for these results as follows:
•
The reason for the reduction in the working time is the initial limitation of the ST
algorithm.
•
Applying the KB helps to maximize the working time of the robots in dynamic DOE
even though it may require the robots to increase movement, leading to unwanted
overlapping areas. However, the overlapping area can be acceptable as it is insignifi-
cant compared to the total shared DOE.
The analysis of the results shows that multiple robot operations can improve the effec-
tiveness of task completion when compared to the use of a single robot. However, when
Machines 2024,12, 797 27 of 32
multiple robots operate in concert, there will be additional issues in collaboration reflected
in reduced effectiveness in simultaneous task completion as discussed in
Section 8.1
. There-
fore, we may conclude that it is necessary to incorporate the use of the optimal number
of robots and their initial starting positions in the best cases. Optimizing the number of
robots will reduce the number of iterations, optimize robot working time, and reduce
financial factors.
8. Discussion
There are numerous examples of research investigating CPP in uncertain environments
with studies addressing the avoidance of static and dynamic obstacles by both single robots
operating alone and multiple robots operating in concert. The objective of CPP algorithms
is to optimize robot coverage in a DOE and complete pre-defined tasks. Related research
has been addressed in Section 2where studies addressing CPP and related topics have
been considered. Published studies addressing CPP solutions have shown their potential to
realize effective CPP and have achieved varying degrees of success; however, many studies
report limited success and many studies fail to effectively address dynamic obstacles.
In this paper, we have introduced our proposed Multi-ST model designed to address
(or at least mitigate) the limitations identified in the CPP-related research considered
in this study. The proposed model has been evaluated in experimental testing and has
demonstrated the capability to realize effective collaborative working by multiple robots
operating in concert in a DOE while avoiding both static and dynamic obstacles. We have
introduced the technologies utilized in the design and development of our proposed model
along with the related algorithmic and robot processing. The aim of this study is to enable
multiple robots to operate in concert effectively by leaving traces on the DOE covered by
the traversing robots.
In introducing the proposed model we have considered the basic assumptions upon
which the implementation is predicated. To evaluate the proposed model, we have carried
out experimental testing as set out in Section 6. Our results show that the proposed
approach enables effective multiple-robot operations where robots operate in concert based
on intelligent decision support systems. Moreover, we have demonstrated the capability
of robots to achieve the avoidance of static and dynamic obstacles. A moving robot is in
practice a dynamic obstacle viewed from the perspective of other robots and this induces
significant complexities in managing CPP.
Any system designed to manage multiple-robot operations requires a platform to man-
age and control the robots and enable optimal autonomous decision-making by the robots
to achieve multiple defined tasks. The proposed model has demonstrated high levels of
robustness, effectiveness, and efficiency with the capability to manage effectively multiple
robots operating in concert to complete defined shared tasks. The time taken to complete
the coverage of a DOE is reduced as shown in our evaluation and experimental testing.
To evaluate the proposed model we have carried out experimental testing as set out in
Section 6. Our results show that the proposed approach enables effective multiple-robot
operations where robots operate in concert based on intelligent decision support systems.
Reported experimental results demonstrate the following benefits in a software system
designed to manage multiple-robot operations in ‘real-time’ with the shortest working time.
The study presented in this paper has demonstrated a number of significant benefits:
•
The capability for robots to achieve avoidance of static and dynamic obstacles in ‘real-
time’. A moving robot is in practice a dynamic (moving) obstacle for other robots and
this induced significant complexities in managing CPP. While our proposed approach
displays good performance, identifying the optimal (best) solution in terms of time
and traversing efficiency is arguably not possible.
•
The robustness of the designed system algorithm in terms of effectiveness and effi-
ciency along with the capability to manage effectively multiple robots operating in
concert to complete defined shared tasks.
Machines 2024,12, 797 28 of 32
•
Using individual rules for each robot with updating of the provided basis upon which
improvements in the efficiency of CPP can be achieved. Moreover, minimizing du-
plication and repetition in areas covered in the DOE improves task efficiency. Rules
considered for each robot can form the basis for intelligent data processing for collabo-
rating robots based on common rules stored in the KB which is continually updated.
•
The time taken to complete the coverage of a DOE is reduced as shown in our evalua-
tion and experimental testing.
•
The rule-based approach implemented in our proposed model allows robots with
individual rules that are defined to enable optimal autonomous decision-making by
the robots to achieve multiple tasks.
•
The evaluation demonstration (see Section 6.3) has provided a basis upon which online
and offline implementation may be realized.
While there are clear benefits derived from our proposed approach, there are also
limitations and constraints that we have identified in this study:
•
We have identified that locating robots too close together at initialization will lead
to a robot blocking the development of other robots ST. As indicated in Figure 6, the
minimum spacing must be where multiple robots are located in at least adjacent cells.
Moreover, autonomous optimal rule definition and updating is a limitation of the
proposed method.
•
The CPP problem in a dynamic DOE for multiple robots operating in concert represents
a challenge when considered in terms of NP-completeness, we briefly introduce
research addressing NP-completeness and NP-hard problems in Section 2. For a
discussion on the topic see [
75
,
76
]. As discussed in this paper, multi-robot operation
represents an NP-hard problem where solutions are hard to find but easy to verify
and are solved by a non-deterministic Machine in polynomial time.
•
There are significant complexities inherent in managing CPP in a dynamic DOE where
such complexities represent an NP Hard problem as discussed in Section 2. It is clear
that the paths in CPP will change dynamically in ‘real-time’ and, while identifying
an optimal static path is possible in a non-dynamic DOE, the identification of optimal
paths in a dynamic DOE is not a realistic proposition. This limitation is shared with the
Traveling Salesman Problem (TSP) as discussed in [
11
] and Bouzid et al. in [
6
] where
CPP is addressed as it relates to UAV (i.e., drones) in a 3D DOE.
•
The rule-based approach implemented in our proposed method requires robot-specific
rules which are defined to enable the optimization of the rules based on rule frequency
and robot experiences in the identification of the optimal path(s) in CPP. The design
and optimization of autonomous rules currently represent an identified limitation.
In summary, the proposed method combines robot reasoning with knowledge rea-
soning techniques where the processing of robot data is enhanced by the application of
knowledge inference. The combination of robot reasoning and knowledge inference pro-
vides an effective basis upon which good solutions for CPP can be realized while achieving
static and dynamic obstacle avoidance. Moreover, we argue that the method proposed in
this paper will generalize to a broad and diverse range of domains and systems.
8.1. Open Research Questions
In this paper, we have considered the advantages of our proposed approach along
with the limitations and constraints. There remain open research questions (ORQ) which
we have identified in the course of this research.
The limitations and constraints noted in this paper are clearly manageable with further
ongoing research. However, we have identified a potential issue where two robots sharing
locations in a DOE may encounter problems in completing a defined task (similar to cases
where memory management in computer operating systems may lock a computer due to
two processes requiring a single resource [77]).
Machines 2024,12, 797 29 of 32
Such a problem requires ‘task scheduling’ which features in [
78
] where robot schedul-
ing is in flexible manufacturing systems; however, as with other studies, the development
of a scheduling system in a real implementation environment with related practical factors
is identified as future work. Task scheduling is not currently considered in the proposed
approach to CPP as presented in this study. This problem may be realized in, for example,
a use-case where two robots (i.e., robots
A
and
B
) are tasked with ‘picking’ a product (i.e.,
the same product) from the same location(in a warehouse DOE) at the same time. In such a
use case, there is a requirement for task scheduling to be added to, and incorporated into,
the Multi-ST model.
The provision of effective task scheduling of multiple robots operating in concert
remains an ORQ; however, the rules-based approach adopted in our proposed model pro-
vides a potential solution that will form the basis for future research into decision-making
under uncertain-complex environments for autonomous collaborating multiple robots.
8.2. Practical Managerial Significance
In this paper, we have demonstrated the potential utility of the proposed model
along with limitations, constraints, and ORQ. Considering the proposed model from a
practical managerial perspective (PMS) and practical application perspective, in ‘real-
world’ conditions using appropriate robot configurations, for example, an online retailer
warehouse), we argue that our proposed approach provides a basis upon which a system
can be designed where multiple robots operate in concert. This will enable enhanced levels
of automation with improved efficiency, reduced cost, and improved reliability.
9. Concluding Observations
In this study, we have introduced our proposed Multi-ST model which has been
conceived and developed to enable effective control and management of multiple robots
while achieving efficient CPP. We have provided an evaluation with a simulation and
related results which have demonstrated the effectiveness of the proposed approach with a
case study where physical robots are engaged in a cleaning task in a DOE. The proposed
Multi-ST model has been shown to be effective in a dynamic DOE. In conclusion, multiple
robot operations can cover path planning applied by using the FSST algorithm in the DOE
given by the collaborative decision-making of robot processing implemented. Experimental
results show that robots apply existing rules with good behavior in inference engines for
optimal coverage path planning. The Multi-ST model enables single and collaborating
multiple robots to achieve optimal CPP while implementing avoidance of both static
and dynamic obstacles. Obstacle avoidance is implemented using a rule-based approach
with updating rules in the knowledge base (KB) to optimize coverage path planning in
collaborative decision-making robots.
In summary, our evaluation supports the conclusion that our proposed model provides
for the ability to implement autonomous multiple robot operation in a dynamic DOE
efficiently. We posit that our proposed approach can provide an effective generalizable
basis upon which multiple robots can operate in concert in a range of ‘real-world’ domains
and systems where CPP in a dynamic DOE forms a systemic requirement.
However, there remain a number of limitations and perceived ORQ which represent
directions for future research as considered in Section 8.1. To apply the effective task
scheduling of multiple robots operating in concert remains in future work, the rules-based
approach adopted in our proposed model provides a potential solution that will form the
basis for future research into decision-making under uncertain-complex environments for
autonomous collaborating multiple robots.
Author Contributions: Conceptualization, H.V.P. and P.M.; methodology, H.Q.D., H.V.P., F.A., M.N.Q.;
software, H.V.P.; validation, H.Q.D., H.V.P. and F.A. writing—review and editing, P.M., M.N.Q., H.V.P.
All authors have read and agreed to the published version of the manuscript.
Machines 2024,12, 797 30 of 32
Funding: This research is funded by Hanoi University of Science and Technology (HUST) under
project number T2022-PC-042.
Data Availability Statement: The original contributions presented in the study are included in the
article, further inquiries can be directed to the corresponding author.
Acknowledgments: The authors thank for Robot experts and Robot Laboratory for their assistance in
completing this investigation.
Conflicts of Interest: The authors declare no conflicts of interest.
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