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Multi-Robot Directed Coverage Path Planning in
Row-based Environments
Tingjun Lei, Student Member, IEEE, Pradeep Chintam, Student Member, IEEE,
Chaomin Luo, Senior Member, IEEE, and Shahram Rahimi
Abstract—Multiple autonomous robots are deployed to fulfill
tasks collaboratively in real-world applications with row-based
settings as found in precision agriculture, warehouses, factory
inspections, and wind farms. One batch of robots are assigned
to explore, search and localize objects in large-scale row-based
environments, while the other batch of robots move directly to
the detected targets to retrieve the objects. In this paper, a multi-
robot collaborative navigation framework with two different
batches of robots is proposed to explore the environment and
achieve the obtained targets, respectively. The first batch of
robots act as detection robots, which are driven by a pro-
posed informative-based directed coverage path planning (DCPP)
through a multi-robot minimum spanning tree algorithm. It
refines and optimizes the coverage path based on the information
gained from the environment. The second batch of robot reaches
the multiple targets by guidance from a hub-based multi-target
routing (HMTR) scheme, which is applicable to row-based
environments. The feasibility and effectiveness of the proposed
methods are validated by simulation and comparison studies.
Index Terms—Multi-robot directed coverage path planning
(DCPP), informative planning protocol, robot exploration, hub-
based multi-target routing (HMTR) scheme
I. INTRODUCTION
IN recent years, autonomous robots have been deployed to
numerous fields. As one of the effective working modes
of autonomous robots, a mission of searching the row-based
environment and reaching the targets is extensively undertaken
in numerous row-based fields, such as warehouse management,
search and rescue (SAR), precision agriculture, factory inspec-
tion, etc. Among them, robot navigation is one of the most
essential portions to determine the robot route effectively and
efficiently in the row-based workspace [1, 2].
Many approaches have been proposed to achieve reliable
autonomous robot navigation, such as ant colony optimiza-
tion (ACO) [3], fireworks algorithm (FWA) [4], bat-pigeon
algorithm (BPA) [5], graph-based method [6, 7], and neural
network models [8, 9, 10, 11]. Lei et al. [3] proposed a
hybrid model to optimize trajectory of the global path using a
graph-based search algorithm associated with an ant colony
optimization (ACO) method. A hybrid fireworks algorithm
with LiDAR-based local navigation was developed capable
of generating short collision-free trajectories in unstructured
environments [4, 12, 13]. A bat-pigeon algorithm was devel-
oped in crack detection-driven autonomous vehicle navigation
T. Lei, P. Chintam, and C. Luo are with the Department of Electrical
and Computer Engineering, Mississippi State University, Mississippi State,
MS 39762, USA Chaomin.Luo@ece.msstate.edu; S. Rahimi is
with the Department of Computer Science and Engineering, Mississippi State
University, Mississippi State, MS 39762, USA
and mapping, in which a local search-based bat algorithm and
a global search-based pigeon-inspired optimization algorithm
are effectively fused to improve the speed performance of
robot path planning and mapping [5]. A biologically motivated
neural network model using a shunting equation is proposed
by Yang and Luo [14] for real-time path planning with obstacle
avoidance. Luo et al. [15] extended this model to a trajectory
planning with safety consideration in conjunction with the
virtual obstacle algorithm [16].
In search and exploration tasks, especially when the
workspace is sizable, robots navigation requires the coverage
path planning (CPP) algorithms to assist robots. Luo and
Yang [8] developed a bio-inspired neural network (BNN)
method to navigate robots to perform complete coverage path
planning (CCPP) while avoiding obstacles under dynamic
environments [17]. The robot is attracted to unscanned areas
but repelled by the previous scanned areas or obstacles based
on the neural activity through the BNN model. The next
position of the robot depends on the current position of the
robot and neural activity associated with its current position.
Unlike the BNN approach, the boundary representation model
that defines the workspace is adopted by the Boustrophedon
cellular decomposition (BCD) method and deep reinforcement
learning approach (DRL) [18]. The BCD method proposed
by Acar and Choset [19] decomposes the environment into
many line scan partitions explored through a back-and-forth
path (BFP) in the same direction. In trapezoidal decomposition
as a cell, it is covered in back-and-forth patterns. For a
complex configuration space with irregular-shaped obstacles,
BCD needs to construct a graph that represents the adjacency
connections of the cells in the Boustrophedon decomposition.
Similarly, Lei et al. [20] proposed a deep learning method
to detect the workspace and generate turning waypoints for
the robot to complete the coverage of the entire workspace.
The trajectory generated by the above presented algorithms
completely covers the workspace in light of the size of the
robot, which is more suitable for a large-sized workspace with
a broad sensing range. However, with the immense workspace,
complete coverage trajectory requires a large amount of time
and energy, which may not be applicable with the limited
robot’s on-board energy.
Therefore, a directed coverage path planning (DCPP) fused
with an informative planning protocol is proposed in this
paper to efficiently search the entire workspace by the first
batch of robots. After locations of targets are precisely and
efficiently identified, the second batch of robots are deployed
to reach these targets identified by the first batch of robots.
114
2022 IEEE Fifth International Conference on Artificial Intelligence and Knowledge Engineering (AIKE)
978-1-6654-7120-6/22/$31.00 ©2022 IEEE
DOI 10.1109/AIKE55402.2022.00025
2022 IEEE Fifth International Conference on Artificial Intelligence and Knowledge Engineering (AIKE) | 978-1-6654-7120-6/22/$31.00 ©2022 IEEE | DOI: 10.1109/AIKE55402.2022.00025
To save time and energy, the total length of the path to
reach multiple targets in succession is minimized. With that
regard, a new hub-based multi-target routing (HMTR) scheme
is developed in this paper, which is originated from the row-
based environments. By introducing hub grids, computational
efficiency is improved for distance calculations between tar-
gets. On the basis of the locations of the objects, the best
visiting sequence is determined while planning collision-free
trajectories simultaneously. This is of great assistance in robot
safety and obstacle avoidance, and prolongs the life of the
robot. Overall, we propose a framework to search and visit
targets with two batches of robots. An informative directed
coverage path planner is developed for target detection utilized
in the first batch of robots. The second batch of robots
reach targets by hub-based multi-target path routing (HMTR)
scheme, which is utilized for the mission of collection and
rescue in row-based environments.
II. PROPOSED MODEL
The workspace is decomposed into a grid-based working
map and fed into the robots. The first batch of robots take
the map information and historical target distribution into
consideration while it outputs a coverage trajectory. It mainly
consists of two parts: (i) multi-robot directed coverage path
planning (DCPP) and (ii) informative planning protocol.
The second batch of robot determines the final trajectory
in real-time by utilizing the coordinates of targets throughout
th workspace. It consists of an HMTR scheme. Details of
the corresponding algorithms are described in the following
sections. The overall workflow is illustrated in Fig. 1.
The first batch of robots
Multi-robot
directed coverage
path planning
Informative
planning
protocol
Target
coordinates
The second batch of robot
Hub-based multi-target
routing (HMTR) scheme
Row-
based
workspace
Robots execution
Robot execution
Fig. 1. Overall workflow of the proposed path planning framework.
A. Multi-robot directed coverage path planning
For the sake of modesty, the area to be covered by multiple
robots is assumed to be a 2D state space [21]. Trivial changes
in altitude are ignored assuming the planning domain is a
known environment. The planning area can be considered to
be a rectangle with dimensions of L×Wunits. Each unit or
grid is the area within sensing range of the robot, and L,W
are the length and width of the workspace, respectively.
Our optimal coverage path planning is implemented in
two steps. First, Divide Areas Algorithm for Optimal Multi-
Robot Coverage Path Planning (DARP) [22] has been used
to equally divide obstacle free grids among all the available
robots. Second, a modified node level mixed linear integer
programming model (MILP) based minimum spanning tree
(MST) is used by each robot to circumnavigate all the grids
assigned to it by the first step [23]. DCPP decomposes the
overall area into Nnumber of mutually exclusive regions, one
per each robot.
MST has been utilized to ensure that each robot completely
covers it’s distributed area. Since MST based approach uses
circumnavigating the actual tree to achieve complete coverage,
the map has been partitioned such that there is one node in
the MST for every 4 grids in the map. This is explained in
Fig. 2.
(a) (b)
Obstacle-free node
Obstacle node
Obstacles
Fig. 2. Illustration of the cell decomposition method to identify nodes that
are occupied by obstacles and obstacle-free space. Grey area represents the
obstacles. A grid is considered to be inaccessible only if the entire grid
is covered by an obstacle. The red nodes represent inaccessible grids, and
the green nodes represent accessible grids. (a) Node representation after the
regular cell decomposition. One node per grid has been used. (b) There is
only one node for every 4 grids. This helps the MST circumnavigate all the
grids around a node once the MST is identified.
The state space of L×Wgrids is defined as S, where each
grid is defined as Sl,w,∀Sl,w ∈S,∀l∈L, and ∀w∈W. All the
grids occupied by obstacles O, and all free grids F, are defined
as O⊂S,∀Ol,w ∈O∈Sand F=S\O, respectively. It has
been assumed that the each robot Ribegins in an obstacle-
free grid, and each robot can localize itself constantly in the
workspace.
P[Ri(τ)] ∈F,τ ∈{0,1,2, ..., T},i ∈{1,2, ..., N}(1)
where P[Ri(τ)] is the position of the robot Riat time τ,N
is the number of robots and Tis the total time to complete
the navigation by all robots. It has also been assumed that
each robot can be only navigated from its current grid to its
adjacent grid as long as the adjacent grid is obstacle free.
P[Ri(τ+1)]=(lj,w
j),P[Ri(τ)] = (lk,w
k); (2)
(lj,w
j)/∈O,(lk,w
k)/∈O
|wk−wj|+|lk−lj|≤1.(3)
Let Pibe the path taken by robot Ri,Pi∈F. Thus, the
formal definition of the CCPP is defined as
n
i=0
Pi=F|i∈{1,2,3, ..., N}(4)
In our approach, each robot is expected to follow the 4
principles. First, each robot covers all the obstacle-free grids
in its assigned area. Second, each robot adheres to the non-
backtracking property (does not visit the same grid more than
once). Third, it depends on no human effort to position the
robot to a particular grid before starting the coverage plan.
Fourth, each robot is excepted to move back to its starting
position after covering all its assigned grids, i.e., Pi(T−1)
and Pi(0) must be adjacent grids, and this is achieved using
a modified circumnavigating MST [24].
115
The optimization problem in the complete coverage path
planning can now be defined using 2 parameters. First, the total
number of obstacle-free grids covered by each robot should be
approximately same as any other robot in the group. Second,
the overall time taken by each robot should be approximately
same as any other robot in the group.
L(Ph)N
i=1 L(Pi)
N;T(Ph)N
i=1 T(Pi)
N(5)
where his an arbitrarily chosen robot, Lis the length of
path and Tis the time taken by the robot Rito navigate in
it’s path. However, it has to be ensured that no obstacle-free
grid is visited more than once. Though by using MST it is
guaranteed that a single robot does not revisit the grids in its
path. It has to be considered that a robot in the group should
not visit the grids in the path of another robot.
P1∪P2∪P3... ∪PN=F(6)
Pi∩Pj=∅;i=j(7)
DCPP employs the DARP for the optimal allocation of cells
among all robots. It is an iterative process, which appropriately
modifies an evaluation matrix Eiwhere i={1,2,3, ...., N}for
each robot in a coordinated fashion. Eiexpresses the distance
between all the grids in Fand the ith robot’s initial position
Ri(τ0). Since our target area is a 2D space, Euclidean distance
function has been used to construct the evaluation matrix.
The evaluation matrix at iteration zfor each robot shows the
distance between the robot position at time τzand all other
obstacle-free grids excluding the robot’s current position.
Ei|l,w =d(Ri(τ0),F\Pi(τ0)) ∀i∈{1,2,3, ..., N}(8)
The evaluation matrix for each robot is corrected after
each iteration using a scalar correction factor, ηi,∀i∈
{1,2,3, ..., N}, such that each robot achieves a fair share of
grids fifrom the total number of obstacle-free grids, i.e.,
fiF/N∀i∈{1,2,3, ..., N}. A standard gradient descent
method based on a cost function Cis used to update η.
C=1/2
N
i=0
(L(Pi)/fi)2(9)
Once the grids are optimally assigned to each robot, respective
MSTs must be created using the nodes assigned to each
robot. Searching for a minimum spanning tree in a graph
is a complex task, though it can be solved in polynomial
time, the aim is to find the MST in optimal time or find
the optimal overall distance from start to sink nodes in the
tree. Assume a connected graph G=(N,J)where Nis
the set of nodes and Jis the set of edges. Let cjbe the
cost of edge j,∀j∈J. MST is a subset of edges M⊂J
which connects all the nodes, N, in the graph such that
the total distance from start node to sink node is minimum,
i.e., j∈Mcjis minimum. There are various integer and
mixed integer linear programming (MILP) models to solve the
MST problem. Some of them use set packing formulations,
node level formulations, cut formulations and network flow
formulations. In our approach, a node level MILP model is
utilized to solve the MST problem for each robot [24].
Assuming that the graph is a connected graph, MST has
3 primary requirements along with other conditions. (i) All
the nodes in the graph should be connected by the MST. (ii)
There should not be cycles in the tree. (iii) The tree should
be optimal. In node level formulations, cycles are eliminated
by assigning level decision variables for the nodes. Node level
models are similar to water flows; water always flows from a
higher level to lower levels and not backwards. Likewise, the
sink node in the tree is assumed to have the lowest level and
the root (or the starting node) is assumed to have the highest
level. Cycles are automatically eliminated by ensuring that no
directed edge is connected from a lower level node to a higher
level node. Assume that the decision to include or exclude an
edge from the MST can be fulfilled using a binary variable
bj, where bj=0when the edge is excluded and bj=1when
the node is included. If yis the total number of nodes in N,
then the goal is defined as below.
j∈J
bj=y−1(10)
arg min
S
j∈S
cjbj,b
j∈{0,1},∀j∈J (11)
where Srepresents all possible combinations of subset of
edges in the graph which can create a spanning tree. One such
subset of edges with minimum cost will be considered as the
MST. While Equation 10 ensures that cycles are prevented,
Equation 11 ensures that an optimal tree is chosen.
i
j
2
k
l
m
13
4
5
6
4
i
j
2
k
l
m
3
6i
j
2
k
l
m
13
5
4
i
j
k
l
m
3
4
5
6
4
i
j
2
k
l
m
13
5
i
j
2
k
l
m
13
4
(a) (b) (c)
(d) (e) (f)
Fig. 3. The difference among various tree types are shown, and also the
requirement for a tree to be considered as Minimum Spanning Tree is
presented. (a) It shows a weighted and connected graph with 5 nodes and 7
edges. (b) This cannot be a tree as not all the nodes are connected. (c) Though
all the nodes in this are connected, thus there is a cycle between nodes j,
k,l. (d) Though there are no immediate cycles connected to adjacent nodes,
nodes j,k,l,mform a cycle. (e) There are no cycles here, which can be
considered as a tree. However, the overall path length is not the minimum. (f)
In this graph, this is the MST as all nodes are connected, there are no cycles
and the over all length is the minimum possible value in this graph.
Our approach is based on a Digraph, G=(N,D), which
is constructed by replacing every edge j=(l, w)∈J by
directed edges (l, w)and (w, l)∈J, both having a distance
of cj. For every pair of edges (l, w)and (w, l)∈D, variables
d(l, w)and d(w, l)are defined, respectively. An edge j∈
Jis included in the tree whenever either d(l, w)or d(w, l)
equals to 1. Fig. 3 explains the difference between various tree
types and shows the requirement for a tree to be considered as
minimum spanning tree. Constraints to eliminate cycles and
ensure validity of the trees of this model is provided in [24].
116
B. Informative planning protocol
The informative planning protocol further refines and opti-
mizes the coverage path based on obtained information. In the
general case, the informative planning protocol incorporates
the multi-objective informative planning by using DCPP and
obtained information that guide the robots to visit those places
more frequently with higher information gain. This multi-
objective optimization problem is stated in Eq. (12).
m∗=argmax
m∈M
{D(m),I1(m),I2(m),··· ,In(m)},
s.t. Cm≤B
(12)
where mis a sequence of robot motion planning, m∗is
the selected robot motion with maximum information gain,
Mis the list of possible robot motion sequences, Cmis the
cost budget and Bis a budget with regard to time, memory,
energy or maximum iteration times, etc. Ii(m),i=1,2, ..., n,
represents the information gathered by the robot motions and
nis the number of the information source in the grid. D(m)
represents the information from the planned DCPP directions.
Selection path
Backpropogation pathOptimal node
Expansion node
New node for simulation Robot
(a) (b)
Reward VectorRRRD
R
R
R
R
Informative
Path Planner
Primitive
Paths
Actions Awards
Historical
Data
Coverage
Directions
Workspace Learner
Fig. 4. Illustration of the informative planning protocol. (a) General process
of the informative planning protocol, and (b) A specific example of the
informative planning protocol. The trees in figure (b) are mirrored for readable
presentation, and the two robots are the same ones.
The informative DCPP is shown in Fig. 4. In Fig. 4 (a),
the planner weighs different objectives (e.g., information gain,
DCPP, primitive path, and reward) and optimizes the multi-
objective problem to choose a preferred solution from a given
set of optimal solutions [25]. The preferred solution is then
reflected as actions planned into the workspace, and it is fur-
ther improved by the learner in a back-propagation direction.
Finally, the most promising solution is determined based on
rewards, and a mortality-based coverage path is constructed
accordingly. A specific example of the informative planning
protocol is depicted in Fig. 4 (b). The major component is a
primitive path which is built with robot kinematic constraints
and decomposed into five sub-branches, each connected by
two nodes [26]. The path is built accumulatively during each
iteration, and the most informative nodes are explored and
selected at first for the area with the highest probability of
target appearance. The rewards are only utilized in the selected
nodes for optimization. The framework consists of selection,
expansion, simulation, and back-propagation in each iteration.
During selection, from the initial position, target nodes are
recursively selected until an undependable node is met. During
expansion, each expandable node is subdivided into five sub-
branches. During simulation, the multiple-objective function
is operated in each expandable node to gain a reward vector.
Finally during back-propagation, the rewards are backed up in
each selected and expandable node.
Historical data
Coverage direction
Predict actions
Informative directed coverage
trajectory in grid
(b)(a)
Fig. 5. Illustration of the informative directed coverage path planning in the
grid. (a) Heatmap with information and DCPP direction information. (b) The
final planned informative directed coverage trajectory in grid.
Note that the information in this paper is defined by the
historical data of the targets in the workspace. Since historical
data as a priori map also has uncertainty, we utilize Gaussian
Process (GP) to transform it into a continuous information
map. The amount of information in each grid can be varied
with number of targets detected. To ensure the robot’s path
tends to the next planned grid, we set the D(m)as the
average value of the current grid information at the adjacent
edge, which proportionally decreases with the distance in the
opposite direction of DCPP in the grid. The D(m)is shown
in the dashed red rectangle in Fig. 5 (a). The planned path
tends to be a compromise between the length and information
gain along it. The resultant trajectory is shown in Fig. 5 (b).
C. Multi-target routing scheme
The detected objects in the coverage path of the workspace
are located as multiple targets discovered by the first batch
of detection robots. After the targets are detected by the first
batch of robots, a multi-target path planning is required to
navigate a robot to connect those targets with total minimized
distance cost. The multi-robot should start from and move
back to the same point (entrance/exit gate of the workspace)
as addressed in Section 2. Meanwhile, the multi-target path
planning of the robot should have two major functions: ob-
stacle avoidance and minimal travel distance regarding time
and energy consumption. With those regards, the multi-target
path planning consists of two steps. The first step is point-to-
point navigation, which generates the shortest collision-free
path between target points. The second step is to navigate the
robot to the targets in sequence, so as to minimize the total
length of the trajectory.
To achieve the optimal visiting sequence of Ntargets, it
is necessary to define the cost of path lengths between target
points, that is, the shortest distance with obstacle avoidance
between target points. Dijkstra’s algorithm is utilized to min-
imize the length of point-to-point navigations. However, the
Dijkstra’s algorithm needs to build a Ns×N
sadjacent
117
matrix. The Nsis the total number of grids in a decomposed
workspace, resulting in highly expensive computation. There-
fore, in light of the row-based environment layout, we design
hub grids to decrement the computational effort. Note that the
proposed hub-based multi-target routing (HMTR) scheme is
applicable to a variety of row-based environments for multi-
target navigation, such as storage buildings, warehouses, crops,
power station, etc.
Hub grids
Hub bridge
path
Hub connection path
Hub Ports
Corridor
F
Inaccessible
ports
Obstacle
T
T
T
T
T
F
F
FF
FFFF
Fig. 6. Example illustration of the HMTR scheme.
The hub grid refers to the grid at the two extremes of the
rows in workspace. For instance, in Fig. 6, seven hub grids
are shown in the red dashed-line rectangle. It has the charac-
teristics of unobstructed connection of targets in the corridor
formed by the obstacles, such as the hub grids connecting
targets T1,T2,T4and T5with no collision. It also features
collision-free connection to targets outside the corridor, such
as the hub grid that can connect target T3without collision.
When the connecting lines between target points are forbidden
by obstacles, as shown by the black dotted lines in Fig. 6, the
corresponding hub grids are initially obtained for connection.
The selection of the corresponding hub grids to the targets i
and jis made by the total distance. Let FAi be the distance
of one side of target point to the grid hub. Let FBi be the
distance from the target point ito the centerline of the hub
grid on the different side and FAi be the distance from the
target point ito the centerline of the hub grid on the other
side. If (FAi +FAj )>(FBi +FBj), the hub grid near the A
side is selected. While (FAi +FAj )≤(FBi +FBj), the hub
grid near the Bside is selected.
To further optimize the path selection, we decompose the
hub grid into nine ports that can be connected. When the port is
blocked by an obstacle in row-based workspace, it is regarded
as inaccessible. The total path length between target points i
and jis Lij , which is the addition of the length of the hub
bridge path and the length of the hub connection path shown in
Fig. 6. Dijkstra’s algorithm is utilized to minimize the length
of point-to-point navigations. The trajectory is created between
two points, which can be selected from each accessible port
or from the targets. Each point is recursively connected with
the other points, and the distances of connection lines passing
through obstacles are assigned with infinite number. As a
result, point-to-point navigation with obstacles is excluded,
and the feasible solutions are retained. The shortest paths
between each pair of points are selected from those feasible
solutions. Each target skips intermediate connections between
adjacent grids, which is directly connected with the nearest
port in the hub grid or another target, thereby attaining to avoid
obstacles with the shortest traveling distance and improving
computational efficiency, revealed in Algorithm 1.
The Hopfield neural network (HNN) is introduced for
sequencing the target points [27]. We define the connection
variable as Cij and path lengths between target points iand j
as Lij . The objective function is then given by
min
i
j
Lij Cij (13)
To ensure that the result is a valid tour, several constraints
must be added Eq. (14)
i∈V Cij =1,∀j∈V,i=j
j∈V Cij =1,∀i∈V,i=j(14)
where Vis the set of number of target points. The first set
of equalities is a move-forwarding constraint, which requires
that each target point is visited only once and different from
another one. The second set of equalities is a backwarding
constraint, which requires that the departure of each target
point also occurs once and different from another one.
HNN utilizes analog circuits (resistors, capacitors, and
amplifiers) to describe the neurons in the networks. Let the
internal membrane potential state of neuron i(i=1,2, ...n)be
denoted by Ui, the input capacitance of the cell membrane is
Ci, the resistance of the membrane is Ri, the output voltage is
Vi, and the external input current is denoted by Ii. The parallel
connection of Ri, and Ci, simulates the time constant of
biological neurons, ωij simulates the synaptic characteristics
between neurons, and the amplifier simulates the nonlinear
characteristics of neurons, and Iirepresents the threshold.
CjdUj(τ)
dτ=m
i=1 ωij Vi(τ)−Uj(τ)
Rj+Ij
Vj(τ)=sj(Uj(τ)) j=1,2,··· ,m
(15)
where mis the number of neurons in the neural network;
Vj(τ)and Uj(τ)are output and input potential, respectively. sj
is the transfer function of the neuron. ωij is weight coefficient
between neuron iand j.
The energy function of the designed HNN corresponds to
the objective function (i.e., the shortest path). Additionally, the
meaning of the effective solution should be considered, that
is, each row and each column of the transposition matrix can
only have one 1. Therefore, the energy function of the network
includes two parts: the objective term (objective function) and
the constraint term (transposition matrix). Here, the energy
function of the network is defined as
E=A
2
N
x=1 N
i=1
Vxi −12
+A
2
N
i=1 N
x=1
Vxi −12
+
D
2
N
x=1
N
y=1
N
i=1
Vxidxy Vy,i+1 (16)
118
where the first two items are the constraints of the problem
and the third item is the objective item to be optimized. The
dynamic equation of the neural network is then defined in view
of HNN. We finally obtain a visiting sequence of targets. The
robot with that sequence starts and ends at the gate of the
workspace with shortest collision-free paths.
Algorithm 1: Hub-based multi-target routing (HMTR)
scheme
Load Nobtained target locations as T1, ..., TNand
robot initial location as T0.
Initialize the connection variable matrix C;
for i=0:Ndo
for j=0:Ndo
if Tican directly connect to Tjthen
Lij =(xTi−xTj)2+(yTi−yTj)2;
else
if (FAi +FAj )>(FBi +FBj)then
Select corresponding Aside hub grids;
else
Select corresponding Bside hub grids;
end
Select the best ports in the hub grids;
Use Dijkstra algorithm to obtain Lij
end
end
end
Use Hopfield neural networks to find the optimal
visiting sequence.
III. SIMULATED EXPERIMENTS AND COMPARISON
STUDIES
In this section, simulation and comparative studies were
carried out to validate the effectiveness and efficiency of the
proposed methods. The environments referred to are those in
a typical row-based warehouse environment in [28, 29].
Gate
Obstacles
Obstacles
(a) (b)
Robot 1
coverage path
Robot 5
coverage path
Robot 4
coverage path
Robot 3
coverage path
Robot 2
coverage path Robot 1
coverage path
Robot 3
coverage path
Robot 2
coverage path
Fig. 7. Our model in various environments completely covers the warehouse.
(a) Fig. 7 of [29] has been used to show that not all robots need to start at the
same location, but can still effectively cover the entire warehouse. (b) Fig. 5
of [29] has been used to show that all robots can start at the same location
(charging location), and can return to the same location effectively covering
the entire warehouse.
DCPP has been first implemented in various warehouse-
like environments to partition area through all robots, each of
which circumnavigates its respective subregion of grids. Fig. 7
TABLE I
COMPARISON OF PATH LENGTH ACROSS DIFFERENT ENVIRONMENTS AND
DIFFERENT NUMBER OF ROBOTS.
Free Space
(grids)
Number of
Robots
Min Length
(grids)
Time
(minutes)
Warehouse Environment 1
Figure 5 of [29] 8456
2 4228 70.46
3 2816 46.93
4 2112 35.2
5 1688 28.13
Warehouse Environment 2
Figure 7 of [29] 6060
2 3028 50.46
3 2020 33.66
4 1512 25.2
5 1212 20.2
reveals simulation results in various scenarios. Fig. 7 (a) shows
multiple robots starting at different positions in the warehouse
but returning to their respective starting positions after cov-
ering their respective assigned areas. Fig. 7 of [29] has been
used to depict this environment. However, some warehouses
may have a dedicated charging station for all robots, which
are required to start at the same location and return to their
charging station. Fig. 7 (b) depicts such a scenario. Fig. 5
of [29] has been used to depict this environment. Also, there
could be a need to have varying number of robots within a
warehouse for various reasons. Table I compares the minimum
path length per robot and navigation time required to cover
the entire area with different number of robots. It can be
observed that the path length and navigation time are reduced
as the number of robots increases. However, trade-offs must be
considered with respect to the real-estate available for robots
and the charging stations.
With the assistance of the DCPP, the informative planning
protocol is shown in Fig. 8. The robot accounts for the overall
DCPP direction and historical data depicted as heat maps
with red color areas indicating higher possibility of target
appearance. Cyan areas indicate multiple branches subdivided
by the informative planning protocol. A path indicated as a red
line designates a maximal information gain. In the case of Fig.
8 (a), the informative planning protocol generates a trajectory
with the most information gain for exploration in search
of a relatively large region, which remarkably decrements
worthless travel distance and saves time as well as energy. The
case of Fig. 8 (b) is executed in a complicated environments
with cluttered obstacles. Once the robot meets a random
obstacle (indicated as the black objects in Fig. 8 (b), the
information gain of informative planning protocol is optimized
with the historical data and DCPP direction while avoiding the
obstacles. Fig. 8 (c) is a part of scenario in Fig. 9 (c).
By leveraging DCPP and informative planning protocol, the
path lengths and the navigation time are effectively reduced.
Fig. 9 shows the DCPP implementation in a warehouse with
different grid sizes. Fig. 11 from [28] has been used to
compare and validate our approach. Fig. 9 (a) shows that 3
robots start at the entry point into the warehouse and cover
the entire area effectively where the grid size is approximately
1.53 meters. In Fig. 9 (b), the same environment has been
decomposed into grids of approximately 6.25 meters, i.e.,
considering one grid for every four grids from Fig. 9 (a). By
fusing informative planning protocol with DCPP, grid size was
119
Predicted actions
Historical data
Coverage direction Informative
directed coverage
trajectory in grid
Informative
directed coverage
trajectory in grid
Predicted actions
Historical data
Coverage direction
(a)
(b) (c)
Informative planning in grids
Informative planning in grids
Robot 1
Robot 2
Fig. 8. Informative planning protocol in the grid. (a) The informative directed coverage trajectory and predicted actions in light of heatmap with historical
data and coverage direction in obstacle-free workspace. (b) The informative directed coverage trajectory and predicted actions in light of historical data and
coverage direction in a complicated environments with cluttered obstacles. (c) A part of scenario in Fig. 9 (c).
Multiple robots start and
end at the gate
Small size of the grids
Robot 1 coverage path
Robot 2 coverage path
Robot 3 coverage path
Multiple robots start and
end at the gate
Large size of the grids
Robot 3 coverage path
Robot 2 coverage path
Robot 1 coverage path
(a)
Storage Door
Distribution Area
Inspection Area
Packing Area
Elevator
Elevator
Passageway
Main
Weighting Area
Fast Distribuary Area
Passageway
150150150150150150150 150 150 150 150 150
150150150150150 150 150150 150 150 150
4476
4000
200
Exit
Main Passageway
Main
Fire Door
Warehouse
Entry Inspetion
Temparay Area
Inspection
Distrbutary Area
Shooting Area
150150 150 150 150
200 650
Cleaning Tools Resting Area
455 933 904
62
62
324
200
1480
Package Area
(b) (c)
Details in Fig. 8
Fig. 9. Illustration of grids distribution among robots. Path length changes as the grid size changes. (a) Schematic diagram of warehouse layout (redrawn from
Fig. 8 of [28] ). It is approximately 400 meters long and 460 meters wide. (b) The warehouse has been completely covered by 3 robots. Different colored
paths show paths of different robots used. Grid size was approximately 1.53 meters. (c) The same warehouse with scaled grid sizes. The new grid size was
approximately 6.25 meters. The overall path lengths are much smaller when the grid sizes are relatively big.
increased without losing the robot’s capabilities to completely
cover its assigned areas. Table II shows the path length of
each robot and the total navigation time in the environment
depicted by Fig. 9. It can be observed that the path lengths and
thereby the navigation times are greatly reduced by integrating
informative planning protocol with DCPP.
With the AI-based sensing techniques, locations of targets
are obtained in an overall trajectory for the second batch of
robots. The HMTR receives information of targets and gener-
ates a collision-free route to reach the targets in an efficient
sequence so that the total traveling distance is minimized.
To validate the adaptability and efficiency of our algorithms
in various number of targets, three datasets are selected for
simulation and comparative studies [30]. In each dataset, we
iteratively perform 30 executions to compute the mean and
standard deviation of path length. The qualitative comparison
results between the features of our algorithm and state-of-art
models, such as Genetic Algorithm (GA) and Particle Swarm
Optimization (PSO) are outlined in Table III.
IV. CONCLUSION
A multi-robot autonomous robot system for detecting,
searching and achieving targets in a row-based environment
TABLE II
COMPARISON OF THE PROPOSED MODEL WITH DARP. FIGURE 11 FROM
[28] HAS BEEN UTILIZED FOR COMPARISON.GRID SIZES OF DARP AND
THE PROPOSED MODEL ARE SET IN THE WAY ENVIRONMENTALLY
.
DARP Proposed Model
Grid Size (m) 1.53 6.25
Path Length of Robot 1 (m) 14,129.1 7562.5
Path Length of Robot 2 (m) 14,129.1 8073.2
Path Length of Robot 3 (m) 14,116.8 8251.4
Total Path Length (m) 42,375.1 23887.1
Total Time (min) 235.4 137.5
TABLE III
COMPARISON OF MINIMUM PATH LENGTH,AVERAGE PATH LENGTH,
STANDARD DEVIATION (STD) OF PATH LENGTH,MINIMUM TIME,
AVERAGE TIME AND STD OF TIME WITH OTHER MODELS.THE VALUES
ARE REPORTED FOR 30 EXECUTIONS.
Test Data Set Model Min Length(m) Average Length(m) Length STD(m)
bays29
GA 9.47e+03 9.98e+03 4.46e+03
PSO 9.07e+03 9.25e+03 2.38e+03
Our Model 9.07e+03 9.14e+03 0.57e+03
KroA200
GA 2.39e+05 2.57e+05 1.15e+04
PSO 1.09e+05 1.17e+05 5.50e+03
Our Model 1.06e+05 1.09e+05 0.82e+03
PA561
GA 1.37e+05 1.93e+05 6.30e+04
PSO 1.11e+05 1.14e+05 1.60e+03
Our Model 1.03e+05 1.10e+05 1.07e+03
120
is a promising direction for warehouse managements. Aiming
at this vision, we have developed a multi-robot informa-
tive planning protocol-based navigation system through two
batches of robots. The first batches of robots consist of multi-
robot DCPP for constructing overall coverage trajectory and
informative planning protocol for detailing the trajectory based
on historical data, DCPP direction, and obstacles. The second
batches of robot receives the information of target locations
and plans an optimal ordered route by the HMTR scheme.
The comparison and simulation results demonstrated a great
potential of the proposed methods for robot navigation, being
useful tools for supporting row-based warehouse management.
REFERENCES
[1] L. Liu, C. Luo, and F. Shen, “Multi-agent formation control
with target tracking and navigation,” in 2017 IEEE International
Conference on Information and Automation (ICIA). IEEE,
2017, pp. 98–103.
[2] Q. Qu, H. Zhang, C. Luo, and R. Yu, “Robust control design
for multi-player nonlinear systems with input disturbances via
adaptive dynamic programming,” Neurocomputing, vol. 334, pp.
1–10, 2019.
[3] T. Lei, C. Luo, G. E. Jan, and K. Fung, “Variable speed robot
navigation by an ACO approach,” in International Conference
on Swarm Intelligence, 2019, pp. 232–242.
[4] T. Lei, C. Luo, J. E. Ball, and Z. Bi, “A hybrid fireworks
algorithm to navigation and mapping,” in Handbook of Research
on Fireworks Algorithms and Swarm Intelligence, 2020, pp.
213–232.
[5] T. Lei, C. Luo, T. Sellers, and S. Rahimi, “A bat-pigeon algo-
rithm to crack detection-enabled autonomous vehicle navigation
and mapping,” Intelligent Systems with Applications, vol. 12, p.
200053, 2021.
[6] T. Lei, C. Luo, J. E. Ball, and S. Rahimi, “A graph-based ant-
like approach to optimal path planning,” in IEEE Congress on
Evolutionary Computation (CEC), 2020, pp. 1–6.
[7] J. Chen, C. Luo, M. Krishnan, M. Paulik, and Y. Tang, “An
enhanced dynamic Delaunay triangulation-based path planning
algorithm for autonomous mobile robot navigation,” in Intel-
ligent Robots and Computer Vision XXVII: Algorithms and
Techniques, vol. 7539. SPIE, 2010, pp. 253–264.
[8] C. Luo and S. X. Yang, “A bioinspired neural network for
real-time concurrent map building and complete coverage robot
navigation in unknown environments,” IEEE Transactions on
Neural Networks, vol. 19, no. 7, pp. 1279–1298, 2008.
[9] T. Lei, T. Sellers, C. Luo, and L. Zhang, “A bio-inspired neural
network approach to robot navigation and mapping with nature-
inspired algorithms,” in Advances in Swarm Intelligence, 2022,
pp. 3–16.
[10] Z. Chu, F. Wang, T. Lei, and C. Luo, “Path planning based
on deep reinforcement learning for autonomous underwater
vehicles under ocean current disturbance,” IEEE Transactions
on Intelligent Vehicles, 2022.
[11] C. Luo, S. X. Yang, and M. Q.-H. Meng, “Neurodynamics
based complete coverage navigation with real-time map building
in unknown environments,” in 2006 IEEE/RSJ International
Conference on Intelligent Robots and Systems. IEEE, 2006,
pp. 4228–4233.
[12] E. Jayaraman, T. Lei, S. Rahimi, S. Cheng, and C. Luo,
“Immune system algorithms to environmental exploration of
robot navigation and mapping,” in International Conference on
Swarm Intelligence, 2021, pp. 73–84.
[13] T. Lei, T. Sellers, S. Rahimi, S. Cheng, and C. Luo, “A nature-
inspired algorithm to adaptively safe navigation of a Covid-19
disinfection robot,” in International Conference on Intelligent
Robotics and Applications. Springer, 2021, pp. 123–134.
[14] S. X. Yang and C. Luo, “A neural network approach to complete
coverage path planning,” IEEE Transactions on Systems, Man,
and Cybernetics, Part B (Cybernetics), vol. 34, no. 1, pp. 718–
724, 2004.
[15] C. Luo, J. Gao, Y. L. Murphey, and G. E. Jan, “A computation-
ally efficient neural dynamics approach to trajectory planning
of an intelligent vehicle,” in International Joint Conference on
Neural Networks (IJCNN), 2014, pp. 934–939.
[16] C. Luo, S. X. Yang, X. Li, and M. Q.-H. Meng, “Neural-
dynamics-driven complete area coverage navigation through
cooperation of multiple mobile robots,” IEEE Transactions on
Industrial Electronics, vol. 64, no. 1, pp. 750–760, 2016.
[17] C. Luo, S. X. Yang, M. Krishnan, and M. Paulik, “An effective
vector-driven biologically-motivated neural network algorithm
to real-time autonomous robot navigation,” in IEEE Interna-
tional Conference on Robotics and Automation (ICRA), 2014,
pp. 4094–4099.
[18] Z. Chu, B. Sun, D. Zhu, M. Zhang, and C. Luo, “Motion
control of unmanned underwater vehicles via deep imitation
reinforcement learning algorithm,” IET Intelligent Transport
Systems,vol. 14, no. 7, pp. 764–774, 2020.
[19] E. U. Acar and H. Choset, “Sensor-based coverage of unknown
environments: Incremental construction of morse decomposi-
tions,” The International Journal of Robotics Research, vol. 21,
no. 4, pp. 345–366, 2002.
[20] T. Lei, C. Luo, G. E. Jan, and Z. Bi, “Deep learning-based
complete coverage path planning with re-joint and obstacle
fusion paradigm,” Frontiers in Robotics and AI, vol. 9, p.
843816, 2022.
[21] T. Lei, C. Luo, T. Sellers, Y. Wang, and L. Liu, “Multi-
task allocation framework with spatial dislocation collision
avoidance for multiple aerial robots,” IEEE Transactions on
Aerospace and Electronic Systems, 2022.
[22] A. C. Kapoutsis, S. A. Chatzichristofis, and E. B. Kosmatopou-
los, “DARP: Divide areas algorithm for optimal multi-robot
coverage path planning,” Journal of Intelligent & Robotic
Systems, vol. 86, no. 3, pp. 663–680, 2017.
[23] Y. Xing, F. Shen, C. Luo, and J. Zhao, “L3-SVM: a lifelong
learning method for SVM,” in 2015 international joint confer-
ence on neural networks (IJCNN). IEEE, 2015, pp. 1–8.
[24] T. F. Abdelmaguid, “An efficient mixed integer linear pro-
gramming model for the minimum spanning tree problem,”
Mathematics, vol. 6, no. 10, p. 183, 2018.
[25] W. Zhao, R. Lun, C. Gordon, A.-B. M. Fofana, D. D. Espy,
A. Reinthal, B. Ekelman, G. D. Goodman, J. E. Niederriter,
C. Luo et al., “Liftingdoneright: A privacy-aware human mo-
tion tracking system for healthcare professionals,” International
Journal of Handheld Computing Research (IJHCR), vol. 7,
no. 3, pp. 1–15, 2016.
[26] W. Zhao, R. Lun, C. Gordon, A.-B. Fofana, D. D. Espy, M. A.
Reinthal, B. Ekelman, G. Goodman, J. Niederriter, C. Luo et al.,
“A privacy-aware Kinect-based system for healthcare profes-
sionals,” in 2016 IEEE International Conference on Electro
Information Technology (EIT). IEEE, 2016, pp. 0205–0210.
[27] C. Zhong, C. Luo, Z. Chu, and W. Gan, “A continuous hopfield
neural network based on dynamic step for the traveling salesman
problem,” in International Joint Conference on Neural Networks
(IJCNN), 2017, pp. 3318–3323.
[28] Y. Liu, R. Sun, T. Zhang, X. Zhang, L. Li, and G. Shi,
“Warehouse-oriented optimal path planning for autonomous
mobile fire-fighting robots,” Security and Communication Net-
works, vol. 2020, 2020.
[29] N. Baras, A. Chatzisavvas, D. Ziouzios, and M. Dasygenis,
“Improving automatic warehouse throughput by optimizing task
allocation and validating the algorithm in a developed simula-
tion tool,” Automation, vol. 2, no. 3, pp. 116–126, 2021.
[30] B. Bixby and G. Reinelt. Traveling salesman problem
library (on line). [Online]. Available: http://elib.zib.de/pub/mp-
testdata/tsp/tsplib/tsp/index.html
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