# Collision Detection for Multi-Robot Motion Planning
with Efficient Quad-Tree Update and Skipping

Abdel Zaro1, Ardalan Tajbakhsh2 and Aaron M. Johnson2 *This work was supported
in part by the National Science Foundation under Grant IIS-1659774 and the
RISS program.1 Department of Mechanical Engineering, UC Berkeley 2 Department
of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA, USA,
(atajbakh<EMAIL_ADDRESS>
###### Abstract

This paper presents a novel and efficient collision checking approach called
Updating and Collision Check Skipping Quad-tree (USQ) for multi-robot motion
planning. USQ extends the standard quad-tree data structure through a time-
efficient update mechanism, which significantly reduces the total number of
collision checks and the collision checking time. In addition, it handles
transitions at the quad-tree quadrant boundaries based on worst-case
trajectories of agents. These extensions make quad-trees suitable for
efficient collision checking in multi-robot motion planning of large robot
teams. We evaluate the efficiency of USQ in comparison with Regenerating Quad-
tree (RQ) from scratch at each timestep and naive pairwise collision checking
across a variety of randomized environments. The results indicate that USQ
significantly reduces the number of collision checks and the collision
checking time compared to other baselines for different numbers of robots and
map sizes. In a 50-robot experiment, USQ accurately detected all collisions,
outperforming RQ which has longer run-times and/or misses up to 25% of
collisions.

###### Index Terms:

Multi-Robot Systems, Motion Planning for Multiple Mobile Robots, Collision
Detection

##  I INTRODUCTION

As robotic systems scale up to many agents operating in a shared environment,
path and motion planning become major challenges in practical deployments due
to the increasing number of states and constraints from other robots and the
environment. In particular, collision checking becomes a major bottleneck in
planning as the number of agents increases, since in the worst-case every
agent usually has to check collisions with every other agent at every
timestep. In receding horizon planning, this becomes even more challenging as
collision checking needs to occur over the entire planning horizon at each
update.

The majority of current multi-robot planning methods use pairwise collision
checking, which involves checking collisions between every pair of robots at
every timestep, as in the conflict-based search algorithm [1]. This can become
quite expensive as the number of robots increases since it scales
quadratically with the number of robots [2]. In addition, many of these
collision checks are practically unnecessary, since some agents will never
collide in the near future, even in the worst case scenario.

Figure 1: Illustration of the conditions required for update and collision
check skipping to take place. In the bottom left quadrant, collision check and
update skipping occur because the robots are far from each other and the
quadrant’s boundary.

Other approaches rely on partitioning the free-space efficiently to quickly
evaluate whether generated trajectories by the motion planner are collision-
free. These methods either approximate the free-space by generating convex
polygons over the entire map, or construct robot-centric pyramids over a
limited horizon [3, 4, 5]. While they are effective for fast robot-environment
collision checking, they do not handle many robot-robot collisions well.
Another class of solutions leverage quad-trees [6, 7, 8]. Quad-trees
iteratively partition the map into quadrants until each quadrant holds at most
a set number of robots. This allows checking inter-robot collisions for only a
subset of robots at any given timestep.

One major challenge with using trees for applications with multiple moving
bodies is efficiently updating the tree in real-time. A few approaches have
been proposed to efficiently address this problem (e.g. [9, 10]), but they
have not been applied to multi-robot collision checking. Furthermore, they
require checking tree updates at every timestep. This can become expensive, as
each update checking requires traversing the quad-tree to a specific element
and looking up its position. Finally, existing methods do not explicitly
reason about agents near boundaries and can be prone to missing critical
collision checks as agents traverse to neighboring boundaries.

We propose a quad-tree update mechanism, Fig. 1, that is time-efficient and is
geared towards multi-robot collision checking, called Updating and Collision
Check Skipping Quad-tree (USQ). Unlike other methods, our approach allows
quad-tree updates to be skipped by leveraging the current state and the
predicted trajectories of robots. In addition, USQ skips collisions with
robots that are not nearby. We propose additional exception handling rules for
robots near boundaries to ensure no critical collision check is skipped. We
demonstrate that these extensions significantly improve collision checking
time compared to other algorithms without missing any critical collisions and
provide a promising approach for collision checking in large robot teams.

This paper is organized as follows. Section II summarizes related works, while
Section III describes the preliminaries. Section IV presents the detailed
design of our USQ algorithm. To showcase the effectiveness of our approach,
Section V discusses experimental results in various environments. Finally,
Section VI concludes the paper and outlines avenues for future work.

##  II RELATED WORKS

A Quad-tree is an indexing structure that can efficiently store and retrieve
data. Quad-trees have many applications such as collision detection [11, 12],
environmental mapping[13, 14], space partitioning [15, 16, 17] and object
tracking [10, 18, 19]. In the context of collision detection, one of their key
advantages lies in their ability to effectively partition the space, thereby
establishing relationships between nearby robots. This partitioning method
facilitates a clear understanding of the proximity between robots, which
allows for collision checking to be limited to robots that are near one
another. The reduced number of collision checks allows them to operate in
O$(n\,log(n))$ time [20]. In [6], a single-robot planning approach is proposed
utilizing a framed quad-tree, which places high resolution cells around the
boundaries of quadrants containing obstacles to improve the quality of the
generated paths. However, this approach increases the memory consumption. [7,
17] use hierarchical map decomposition using quad-trees, which reduces the
number of node expansions in an $A^{*}$ planner without compromising solution
quality. But, this approach does not update the quad-tree online in case of
dynamic obstacles and other environment changes. Similar space-partitioning
approaches have been applied to 3D applications. Octrees, which operate
similar to quad-trees in 3D, have been used for single-robot navigation in
constrained environments [8, 21].

Quad-tree-based approaches have also been used in multi-robot settings. [22]
presents a multi-robot planning algorithm operating in discrete space that
utilizes quad-tree map division. Similar to an occupancy grid, nodes are
labeled as free or occupied. For collision avoidance, at every timestep, the
robots check whether the node they are heading into is occupied. In [11], a
discrete-time multi-robot planning algorithm that uses octrees to partition
the 3D space for planning and collision detection is presented. It precomputes
conflicts and creates safety corridors, which are regions in which collision
avoidance is guaranteed. [12] introduces a flexible octree structure that
permits a less rigid relationship between child and parent nodes. This non-
rigid structure improves collision detection and path planning by employing a
multi-resolution grid. However, the paper focuses on pre-flight planning
rather than real-time execution. In [23], an octree-based 3D occupancy grid is
used for multi-robot planning. The robots do not communicate and collision
avoidance is achieved using buffered Voronoi cells, which do not require
collision checks. In these papers, the standard setup involves one robot per
octree node; however, in this study, two robots are permitted to occupy the
same node. Collision checking is then exclusively performed between robots
that are coexisting within a node or transitioning to a different node.
Moreover, this arrangement offers an additional advantage by providing
proximity information about robots that are close enough to be within the same
quadrant but far enough not to collide within the next timestep. Leveraging
this information, the quad-tree can be optimized by skipping collision checks
and updates when applicable.

Regenerating quad-trees (RQ), which require the tree to be reconstructed at
every timestep, can be computationally expensive. As a result, various methods
have been explored to enable more efficient update mechanisms. [24] uses a
generalization of quad-trees to generate a multi-resolution adaptive grid
online based on the measurement uncertainty. This has shown to be effective in
mapping, but the quad-tree update process is not conditioned on the states of
the involved agents. [10] used a tree structure for object tracking and
assigned each object with an index to simplify the tree search process. The
location of objects is only updated when the object moves into a new node.
However, at each timestep, the new robot position still needs to be compared
with the previous one to determine if the robot has entered a new node. In
addition, several other papers have explored similar approaches for different
applications, updating the tree only when necessary to track changes in the
object’s position [25] or at each timestep [26, 9]. [27] proposed a k-d tree
algorithm to find the nearest neighbors in a set of points. Regions of the
tree are updated when new points are added to the tree. Instead of updating
the entire tree, the algorithm performs a local search around the newly added
point to identify if the nearest neighbors to the other points have
potentially changed.

##  III Preliminaries

###  III-A System Definition

The control objective is to regulate all agents from their initial states to
their desired state in finite time. For our experiments, we use a dynamic
bicycle model (refer to [28]), which is a good approximation for industrial
wheeled robots. The model is discretized using Euler’s method with a timestep
of $\Delta t=0.1$ s. We use a model predictive controller (MPC) similar to
[28] as the motion planner for each agent. MPC only reasons about a single
robot trajectory and does not perform collision avoidance, since this paper is
focused on the collision detection problem. Note, our algorithm does not
account for tracking or sensor errors.

A custom simulation written in Python was used for all the experiments and the
collision checking algorithms. Experiments were done on a laptop with an AMD
Ryzen 7 5700U with 22 GB of RAM. At every simulation step, each agent computes
a sequence of control actions and predicted states using the MPC. Each agent
then executes the next control action and updates its state according to the
described dynamics model. Agents are assumed to have a circular footprint with
a radius of $r=0.5$.

###  III-B Quad-trees

Figure 2: Example illustrating USQ in comparison to RQ. (a) Robots are placed
in the first node of the regular quad-tree. (b) When the number of robots in
the node exceeds the threshold $m$, where here $m=2$, the node is split into
four quadrants. (c) In this step, USQ and RQ behave differently. In USQ,
during the updating process, since robots B and D are far from the boundary
and from each other updating the new robot positions into the quad-tree can be
skipped. Updating the new positions of robots A, C, and E is not skipped
because they are close to the boundary. In RQ, all the robot positions are
updated in the quad-tree at each timestep (not shown in the figure for this
step). (d) Since the robots are no longer far enough from each other to
satisfy the update skipping condition (which is explained in detail in Section
IV), USQ behaves the same as RQ. All the robot positions are updated in the
quad-tree.

The high-level idea of the quad-tree is to perform spatial partitioning of a
given space to enable more efficient collision checking. This is done by
successively dividing each partition into four equal-sized smaller partitions.
These partitions can be described by a tree data structure, where each level
of the tree going down from the root represents one level of division and each
node a partition. This information can be used to restrict collision checking
to only robots that are within close proximity.

The algorithm starts by drawing a bounding box around all the robots. Let $m$
be the maximum number of robots allowed to be present within each partition.
In this paper, this value is set to two robots. When there are more than $m$
robots occupying a partition, the quad-tree algorithm divides it into four new
quadrants. This process is repeated until at most $m$ agents are in any given
partition. Thus, each leaf node of the tree has at most $m$ agents at the end.
An illustration of quad-trees can be seen in Fig. 2. In this paper, a modified
quad-tree data structure based on [29] is used.

##  IV Updating and collision check Skipping Quad-tree Algorithm

###  IV-A Overview

The USQ method is summarized in Algorithm 1 and Fig. 2. At the beginning of
execution, an empty quad-tree is generated (Line 1). It is important to note
that the full quad-tree is generated only once throughout the entire run and
updated incrementally. This is in contrast to the RQ approach of regenerating
the quad-tree at every timestep.

The following operations are then repeated until all the robots reach their
final goal positions (Line 2). First, the MPC controller runs for each robot
to find the next updated position of that robot (Line 4). Second, the
algorithm must update the quad-tree structure (Lines 5-7) and then check for
collisions (Lines 9-24). At the end, the algorithm checks if all the robots
have reached their goal positions (Line 25).

To optimize the efficiency of the algorithm, unnecessary updates and collision
checks within a given quadrant are skipped. This is achieved by selectively
excluding robots from updates and collision checks when they are unnecessary.
Each robot keeps track of a counter called $num\\_skip$ which indicates the
number of consecutive timesteps the robot can proceed without changing its
position within the quad-tree or colliding with another robot. In essence,
this counter determines how many quad-tree and collision check updates can be
skipped in sequence. Skipping updates provides a significant computational
saving over regenerating the quad-tree at every step, as each update of the
tree requires traversing to a specific element. When the number of updates the
robot can skip is 0 (Line 5), the robot position must be updated in the quad-
tree (Line 6).

The $update()$ step could be implemented a few different ways, but our
implementation employs a remove-and-add strategy. This involves removing the
robot from the quad-tree and subsequently placing it back in the updated
location. Re-inserting the robot may require additional partitions to be added
to the tree, which is done as needed. Note that some of the tree nodes may
become empty after removing robots and the original tree could be simplified
by pruning those leaf nodes. However, we have found that the additional
overhead required to perform this simplification outweighs the computational
benefits. As a result, once a node is added to the tree it is kept in the
structure throughout the execution. This effect can be seen in Fig. 6.

Once the quad-tree is updated with the latest locations of all robots,
collision checking is performed. The algorithm iterates through all the leaf
nodes of the quad-tree (Line 9) and checks for collisions between pairs of
robots if any robot in that node has a $num\\_skip=0$ (Line 10). In other
words, collision checks are skipped when $num\\_skip>0$ for all robots in that
node. A collision between robots $a$ and $b$ with radii $r$ occurs when the
following condition is true:

$collision(a,b):\|a.pos-b.pos\|<2r$ (1)

where $a.pos$, $b.pos$ are the robots’ Cartesian positions. A more complicated
collision checking algorithm, especially for non-circular robots, could be
used at this line.

Next, if a robot could be close to a boundary of its quadrant, we perform
additional collision checks with neighboring nodes, since it may cross to a
different quadrant at the subsequent timestep. To do so, we check if the
distance to the quadrant’s border $d\\_border$ is below a specified minimum
threshold $min\\_thres$, (Line 15),

$min\\_thres=2r+max\\_dist\\_traveled+\epsilon$ (2)

$min\\_thres$ takes into account the sum of the radii of two robots ($2r$) and
the distance a robot can travel at maximum velocity in one timestep
($max\\_dist\\_traveled$). $\epsilon$ is a user-defined safety threshold. The
equation represents the minimum distance a robot should maintain from the
boundary to prevent potential collisions with a robot in the next quadrant
during the next timestep. If the robot is within $min\\_thres$ of the border,
collision checks are done with the robots in the neighboring quadrant (Line
16). This process is explained in detail in Section IV-B.

Next, the skipping counter $num\\_skip$ must be updated. If $num\\_skip$ is
zero, then it is recalculated based on the minimum of two distances,
$min\\_d$: the distance of a robot to the border of its quadrant,
$d\\_border$, and the distance to other robots in its quadrant, $d\\_robots$
(Line 18). Based on that minimum distance, $det\\_skip()$ determines the
number of quad-tree updates that can be skipped, if any, by computing the
number of timesteps it would take the robot to collide with another robot or
cross a boundary at maximum velocity (Line 19). If on the other hand
$num\\_skip$ is not yet zero, it is simply decremented by one (Line 21).

It is worth noting that the skipping of quad-tree updates and collision checks
happen together in the USQ method. This is because one of the conditions for
update skipping, robots being far from each other, aligns with the condition
for collision check skipping. Allowing the update and collision check skipping
to occur separately incurred additional tracking costs that outweighed the
benefits gained.

Algorithm 1 USQ

1:$T\leftarrow quad\mathchar 45\relax tree(robot$_$list$)

2:while not $all\\_reached\\_goal$ do

3: for $robot$ $\in$ $robot$_$list$ do

4: $robot.run$_$mpc()$

5: if $robot.num$_$skip$ = 0 then

6: $T.update(robot)$

7: end if

8: end for

9: for $leaf\in T.nodes$ do

10: if $min(leaf.robots.num$_$skip)=0$ then

11: $check$_$collisions(leaf$_$node)$

12: end if

13: for $robot\in leaf$ do

14: if $robot.num$_$skip=0$ then

15: if $d$_$border<min$_$thres$ then

16: $check$_$neighbor$_$collision(robot)$

17: end if

18: $min$_$d\leftarrow min(d$_$border,d$_$robots)$

19: $robot.num$_$skip\leftarrow det$_$skip(min$_$d)$

20: else

21: $robot.num$_$skip\leftarrow robot.num$_$skip-1$

22: end if

23: end for

24: end for

25: $all$_$reached$_$goal\leftarrow check$_$reached(robot$_$list)$

26:end while

###  IV-B Neighbor Checking

Given that robots are positioned in the quad-tree based solely on their center
position it is possible for them to collide at the boundary when their body
extends beyond the edge or when transitioning to a neighboring quadrant. To
address this issue, collision checks are performed on robots within a region
of the upcoming quadrant when a robot is about to cross a boundary, as
illustrated in Fig. 3(a). This region is centered on the point that the
robot’s future trajectory intersects the quadrant border (or, if it will not
cross within two timesteps, the region is centered on the closest point). It
measures $2x$ by $x$, where $x$ is the distance in a given direction that a
collision may occur. The region is $2x$ wide since the robot has the potential
to travel a distance equal to $x$ to the left or right of the quadrant
intersection point. Here we choose,

$\displaystyle x=2r+2max\\_dist\\_traveled+\epsilon$ (3)

since each robot can travel a distance equal to $max\\_dist\\_traveled$.

Furthermore, if the robot’s radius has the potential to extend into adjacent
quadrants while crossing the boundary at a corner, collision checks are
performed with robots present in two additional quadrants as shown in Fig.
3(b). The bottom right search region has a height of $2r$ because, even though
the robot’s trajectory itself does not enter the region, its radius could
potentially extend into that region while transitioning to another quadrant.

Figure 3: (a) When a robot approaches the node boundary, illustrated by its
position beyond the gray dashed line, it does collision checks with robots
situated within the adjacent node’s blue dotted area. (b) When, in addition to
the conditions described in (a), the robot’s future trajectory intersects the
quad-tree boundary at a linear distance of 2 radius units from the corner, it
is identified as a corner case.

##  V Results

Figure 4: (a) Sparse and (b) dense environments with 20 robots.

In this section, we present results comparing the proposed USQ algorithm with
RQ and pairwise collision checking across different numbers of robots and
environments. RQ creates the quad-tree from scratch at each timestep and does
not include any of the skipping logic, while the pairwise method checks for
collisions between every pair of robots at each timestep regardless of their
locations.

We experiment with three separate environments: a sparse 512x512 unit
environment (Fig. 4(a)), a dense 85x85 unit environment (Fig. 4(b)) and a
circular 150 unit radius environment (Fig. 6). For each environment, we
experiment with 5, 20, and 50 robots. The circle environment evaluates the
performance over a particularly difficult single trial, whereas the sparse and
dense environments are used to generate statistical results over 10 randomized
trials for each test case. This results in a total of 60 random trials. For
the randomized trials, the start and goal locations are randomly selected such
that they are inside the map and non-overlapping.

Multiple performance metrics were evaluated at the end of each run. $T_{q}$ is
defined as the time spent creating and updating the quad-tree and $T_{n}$ is
the time spent on neighbor checking, which includes traversing the tree to
find neighboring robots and doing collision checks with them. $T_{n}$ provides
insight into the additional computational cost of enabling quad-tree methods
such as RQ and USQ achieve the same collision detection outcome as the more
computationally expensive pairwise approach. The total collision checking
time, $T_{c}$ encompasses all aspects associated with collision checking
including the time spent on checking collisions between robots, quad-tree time
($T_{q}$), and neighbor checking time ($T_{n})$, when appropriate. $N_{c}$ is
the total number of collision checks, $N_{d}$ is the total number of
collisions detected, and $N_{m}$ represents the number of collisions that were
missed.

###  V-A Randomized Environments

TABLE I: Collision checking results in the sparse environment. Time is measured in seconds. Note: the dash (-) indicates values that are not applicable. For the $N_{m}$ column, it indicates the absence of collisions in the experiment and therefore, there are no collisions to be missed. | $T_{q}$ | $T_{n}$ | $T_{c}$ | $N_{c}$ | $N_{d}$ | $N_{m}$
---|---|---|---|---|---|---
5 Robots |  |  |  |  |  |
Pairwise | - | - | 0.006 | 540 | 0 | -
RQ | 0.015 | - | 0.016 | 77 | 0 | -
USQ | 0.006 | 0.0001 | 0.006 | 5 | 0 | -
20 Robots |  |  |  |  |  |
Pairwise | - | - | 0.083 | 10,260 | 0 | -
RQ | 0.063 | - | 0.065 | 264 | 0 | -
USQ | 0.040 | 0.0008 | 0.043 | 102 | 0 | -
50 Robots |  |  |  |  |  |
Pairwise | - | - | 0.499 | 66,150 | 1 | 0
RQ | 0.146 | - | 0.154 | 768 | 1 | 0
USQ | 0.109 | 0.0072 | 0.121 | 445 | 1 | 0
TABLE II: Collision checking results in the dense environment. Time is measured in seconds. Note: the dash (-) indicates values that are not applicable. For the $N_{m}$ column, it indicates the absence of collisions in the experiment and therefore, there are no collisions to be missed. | $T_{q}$ | $T_{n}$ | $T_{c}$ | $N_{c}$ | $N_{d}$ | $N_{m}$
---|---|---|---|---|---|---
5 Robots |  |  |  |  |  |
Pairwise | - | - | 0.007 | 540 | 0 | -
RQ | 0.019 | - | 0.020 | 56 | 0 | -
USQ | 0.009 | 0.003 | 0.013 | 52 | 0 | -
20 Robots |  |  |  |  |  |
Pairwise | - | - | 0.084 | 10,260 | 8 | 0
RQ | 0.067 | - | 0.071 | 323 | 6 | 2
USQ | 0.047 | 0.014 | 0.063 | 228 | 8 | 0
50 Robots |  |  |  |  |  |
Pairwise | - | - | 0.505 | 66,150 | 56 | 0
RQ | 0.148 | - | 0.160 | 1,090 | 42 | 14
USQ | 0.128 | 0.055 | 0.194 | 928 | 56 | 0

The results for the sparse environment, as shown in Table I, illustrate that
USQ performs better than RQ for all cases in terms of the total number of
collision checks ($N_{c}$) and total collision checking time ($T_{c}$). As
expected, pairwise collision checking does the worst for cases with many
robots. However, for the 5-robot case, pairwise and USQ perform the same. This
is because the additional computational overhead of the quad-tree does not
outweigh the savings in collision checking time. As a result, we can conclude
that pairwise collision checking is still an effective strategy for smaller
robot teams.

In the dense environment, as shown in Table II, the trends are similar to the
sparse environment. In the dense environment with 5 robots, pairwise actually
outperforms both RQ and USQ. For larger numbers of robots, as demonstrated in
Fig. 5, the speed-up in collision checking time between RQ and USQ is reduced
compared to the sparse environment. Both of these trends are because, in the
dense environment, agents are significantly closer to each other. Thus, fewer
updates and collisions can be skipped. Additionally, the quad-tree quadrants
are considerably smaller. This results in a higher number of agents crossing
the boundaries and an increase in the number of collision checks with
neighboring quadrants. For the 50-robot experiment, RQ has lower $T_{c}$ than
USQ, although RQ fails to detect 25% of the total collisions. While USQ still
exhibits lower quad-tree update time ($T_{q}$) since some skipping occurs, the
neighbor checking time ($T_{n}$) is relatively large. Despite this, neighbor
checking is quite important to not miss any collisions.

Fig. 5 shows the speed-up of USQ when compared to RQ, showing the ratio of
average $T_{c}$ for each. The speed-up is the smallest (0.82) in the denser
environment with 50 robots, which has a higher robot density. Additionally, it
is the greatest (2.6) in the sparse environment with 5 robots because the
robots are more spread out, allowing more skips to occur.

Figure 5: Speed-up time in the sparse, dense, and circular environments,
defined as the ratio of $T_{c}$ values for RQ and USQ methods. Bars for the
randomized test cases indicate $\pm$ one standard deviation.

###  V-B Circular Environment

As shown in Fig. 6, for this environment robots are placed in a circle with a
radius of 150 units and are tasked to drive to a position mirrored across the
$x$-axis from their starting position. Table III summarizes the results of
this trial. Similar to the randomized environment cases, USQ and RQ perform
worse than pairwise collision checking for the 5-robot case and USQ is better
than both for the 20- and 50-robot cases. However, the advantage of USQ in the
50-robot case is smaller than in the 20-robot case since there are more robots
close to each other which decreases the number of updates skipped. Fig. 5
shows the speed-up between RQ and USQ in the circle environment.

TABLE III: Collision checking results in the circle environment. Time is measured in seconds. Note: the dash (-) indicates values that are not applicable. | $T_{q}$ | $T_{n}$ | $T_{c}$ | $N_{c}$ | $N_{d}$ | $N_{m}$
---|---|---|---|---|---|---
5 Robots |  |  |  |  |  |
Pairwise | - | - | 0.078 | 8,980 | 8 | 0
RQ | 0.239 | - | 0.251 | 932 | 8 | 0
USQ | 0.130 | 0.007 | 0.143 | 173 | 8 | 0
20 Robots |  |  |  |  |  |
Pairwise | - | - | 1.340 | 167,960 | 33 | 0
RQ | 1.234 | - | 1.294 | 5,973 | 32 | 1
USQ | 0.682 | 0.037 | 0.766 | 3,284 | 33 | 0
50 Robots |  |  |  |  |  |
Pairwise | - | - | 8.511 | 1,097,600 | 87 | 0
RQ | 2.857 | - | 3.007 | 15,085 | 81 | 6
USQ | 2.071 | 0.325 | 2.494 | 9,742 | 87 | 0
Figure 6: The circle environment, showing 20 robots swapping positions with a
quad-tree implemented.

##  VI Conclusion

This paper presents USQ, a novel and efficient collision checking algorithm
for large multi-robot teams. Compared to RQ and pairwise collision checking,
we demonstrate that USQ reduces collision checking time significantly in a
variety of experiments with different numbers of robots and map sizes. USQ is
found to be most effective in sparser scenarios, where robots form distinct
clusters in large maps. However, there is still an improvement in denser
scenarios compared to the other baselines.

This research can be extended in multiple interesting directions. USQ can be
used in multi-agent path finding and motion planning algorithms to speed up
collision checking with other agents and obstacles. In addition, USQ can be
extended to $3$-dimensional spaces to handle systems with higher degrees of
freedom, such as drones.

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# Quasinormal Modes and Phase Structure of Regular $AdS$ Einstein-Gauss-Bonnet
Black Holes

Yerlan Myrzakulov<EMAIL_ADDRESS>Department of General & Theoretical
Physics, L. N. Gumilyov Eurasian National University, Astana, 010008,
Kazakhstan  Ratbay Myrzakulov Eurasian International Centre for Theoretical
Physics Astana 010009, Kazakhstan  Kairat Myrzakulov<EMAIL_ADDRESS>Department of General & Theoretical Physics, L. N. Gumilyov Eurasian National
University, Astana, 010008, Kazakhstan  Sudhaker Upadhyay111Corresponding
author222Visiting Associate, IUCAA Pune, Maharashtra 411007, India
<EMAIL_ADDRESS>Department of Physics, K. L. S. College, Magadh
University,
Nawada 805110, India  School of Physics, Damghan University, P.O. Box
3671641167,
Damghan, Iran  Dharm Veer Singh333Visiting Associate, IUCAA Pune, Maharashtra
411007, India<EMAIL_ADDRESS>Department of Physics, Institute of Applied
Sciences and Humanities, GLA University, Mathura 281406, Uttar Pradesh, India.

###### Abstract

In this paper, we present an exact regular black hole solution in Einstein-
Gauss-Bonnet coupled with nonlinear matter fields. It is a generalization of a
regular Einstein-Gauss-Bonnet black hole in $5D$ $AdS$ spacetime. The causal
structure of the obtained solution identifies with Boulware-Deser black hole
solution, except for the curvature singularity at the center. It incorporates
the Boulware-Deser black holes in the absence of deviation parameters. We also
study the thermodynamic properties of the solution that satisfies a modified
first law of thermodynamics. Furthermore, we discuss the stability of the
obtained black hole solution and, in this regard, a double phase transition
occurs. Within context, we find that phase transition exists at the point
where the heat capacity diverges and, incidentally, the temperature attains
the maximum value. We discuss the fluid nature of the black hole also
exhibiting critical points. The quasinormal modes of the black hole solution
and their dependencies on Gauss-Bonnet coupling and deviation parameters are
also analysed in terms of null geodesics.

Quasinormal modes; $5D$ EGB Black hole; Phase transition.

##  I Introduction

The higher-order curvature theories are useful to explore various (conceptual)
concerns of gravity. Lovelock’s theory of gravity [1, 2, 3] is one of such
higher-order curvature gravity that generalizes the general relativity (GR) to
higher spacetime dimensions. In Ref. [1], Lovelock proposed that GR in $4D$
with a cosmological constant possessing diffeomorphism invariance, metricity
and second-order equations of motion is a unique theory of pure gravity. The
Einstein-Gauss-Bonnet (EGB) gravity is a particular class of Lovelock gravity
that characterizes non-trivial dynamics for the higher-dimensional gravity
having second-order field equations. However, EGB gravity in $4D$ is a
topological theory that, in general, does not contribute to gravitational
dynamics. In the recent past, a lot of efforts have been made to study the EGB
gravity in $4D$. For instance, Glavan and Lin proposed the $4D$ GB theory by
re-scaling the GB coupling constant. But the theory in $D=4$ limit is either
plagued by the partial breaking of diffeomorphism or by additional
gravitational degrees of freedom. The generalization of EGB gravity as F(GB)
gravity in $4D$ was presented in Refs [4, 5]. In another effort, EGB gravity
in $4D$ is constructed that has only two dynamical degrees of freedom but it
breaks the temporal diffeomorphism [6, 7].

Black holes are one of the most fascinating objects and the subjects of active
research. Black holes are either singular or regular solutions of the Einstein
equation of GR. A singular solution for the rotating counterpart of a higher-
derivative theory in Lee-Wick prescription is explored recently [8]. The first
spherically symmetric regular black hole solution was given by Bardeen [9]
that does not hold a strong energy condition. In the recent past, people are
paying much attention to the regular black hole solutions [10, 11, 12, 13, 14,
15, 16, 17]. An exact regular black hole solution for the EGB coupled with
non-Abelian gauge field in 4D $AdS$ spacetime and their thermal properties are
explored recently [18]. Recently, a $4D$ $AdS$ EGB black hole solution with
nonlinear electrodynamics (NLED) is studied [19]. In another recent work, the
EGB massive black hole solution in $4D$ $AdS$ is studied also [20, 21, 22,
23].

The NLED in the context of black hole physics is a more relevant and quite
suitable alternative for Maxwell (linear) electrodynamics as we know that the
real electromagnetic field remains no longer linear at high energy due to the
influence of other physical fields. Since the original consideration of
nonlinear electrodynamics by Born and Infeld [24], extensive progresses on the
subject has been made [25, 26, 27, 28, 29]. Some NLED coupled to GR may
provide a better explanation for the inflation of the universe [30, 31, 32,
33]. Black holes with NLED are quite relevant in astrophysical observations
[34, 35].

The advantage of the study of NLED field coupled with the gravity, we obtained
the regular black hole solution in the particular limits it correctly
retrieves the Reissner-Nordstrom black hole. Another major difference is in
the strong-field limits of Einstein’s gravity, where the exponential mass
function leads to a Minkowski-flat core around, which is in striking contrary
with other regular black holes [36, 37] that generally have de-Sitter core
[38, 39, 40]. Therefore, the curvature of the geometry has some maximal peak
in between spatial infinity and core. Although, in the $4D$ EGB gravity, all
regular black holes have flat core around them [4, 5]. Therefore, this novel
regular black hole share many features with other regular black holes, but
there are also significant differences.

The $AdS/CFT$ correspondence [41, 42, 43] provides a duality between strong
interacting quantum field theory and weakly interacting gravity. This is also
known as the holographic duality or the gauge/gravity duality. GB term in such
correspondence may play an important role, see Refs. [44, 45]. Originally,
this was explored in the context of string theory but further extended to wide
domains, such as the coupling dynamics of QCD and the electroweak theories,
black holes physics, quantum gravity, condensed matter physics, etc. In the
context of black holes, Witten found that black hole thermodynamics in AdS
spaces can get a resemblance with the thermodynamics of dual CFT at the high
temperature [46].

The concept of black hole thermodynamics originated by Bekenstein [47, 48] and
Hawking [49] who realized that entropy is somehow connected to the area of the
Black hole horizon. To a certain extent, it is clear that entropy of the black
holes is proportional to the area of horizon [50, 51, 52, 53]. This subject
was studied further extensively [54, 55, 56, 57, 58, 59] In the context of
black hole thermodynamics (mechanics), it is found that the black hole system
satisfies the first law of thermodynamics. The stability of dS and Nariai
black hole in higher derivative gravity is discussed in Ref. [60]. Here, it is
found that for certain regime Nariai black hole is stable and does not decay
into pure de Sitter space. The this connection of higher derivative gravity,
negative (or zero) Schwarzschild-(Anti)-de Sitter black entropy is found which
depends on the parameters of higher derivative terms [21].

Quasinormal modes (QNMs) have been found an active and wide area of research
[61, 62, 63, 64]. QNMs are found very useful to predict the stability of the
perturbed black holes. Abbott et al. (LIGO scientific collaboration and Virgo
collaboration) detected transient gravitational waves [65]. The images of
Event Horizon Telescope [66] display a prominent ring consistent with the size
and shape of the lensed photon shadow of a supermassive black hole. These
studies hint about the correspondence between QNMs and black hole shadow
radius. The correspondence between QNMs and shadow radius may provide a new
viewpoint for the gravitational waves which are massless particles moving
along an outmost unstable orbit of null geodesics. Recently, the shadow cast
of the charged Reissner-Nordström AdS black hole in both plasma and non-plasma
medium is studied [67].

The rest of the sections are organized as follows. In Sec. II, we consider a
EGB gravity coupled to the NLED in $5D$ $AdS$ spacetime and obtain a new black
hole solution. We discuss the horizon structure of this new black hole
solution in $AdS$ spacetime. The thermodynamics of this black hole along with
stability and phase transition are discussed in section III. The behavior of
black holes as the Van der Waals fluid is reported in section IV. We have
calculated the critical values of pressure, temperature, and horizon radius
and their dependencies on various parameters. The QNMs for the black hole
solution are calculated in section V. Finally, we summarize the results and
make final remarks in the last section.

##  II Action, Black Hole Solutions and Horizon Structure

For the present study, we are interested in the solution of $5D$ EGB gravity
coupled to the NLED in $AdS$ space. The action describing $5D$ EGB gravity
coupled to the NLED in $AdS$ spacetime is written as [68]

$\displaystyle S=\frac{1}{2}\int
d^{5}x\sqrt{-g}\left[{R}-2\Lambda+\alpha(R_{\mu\nu\gamma\delta}R^{\mu\nu\gamma\delta}-4R_{\mu\nu}R^{\mu\nu}+R^{2})-4P\frac{\partial{\cal
H}}{\partial P}+2{\cal{H}}\right],$ (1)

where ${R}$, $R_{\mu\nu}$ and $R_{\mu\nu\lambda\sigma}$ are the Ricci scalar,
Ricci tensor and Riemann tensor, respectively. However, $\Lambda$ and $\alpha$
are the cosmological constant related to $AdS$ length $l$ via relation
$-3/l^{2}$ and the Gauss-Bonnet coupling constant, respectively.
${\cal{H}}(P)$ is the structure- function that depends on the invariant
$P=\frac{1}{4}P_{\mu\nu}P^{\mu\nu}$ of the tensor $P_{\mu\nu}$ which
corresponds to electric induction. The expression for the NLED structure
function ${\cal H}(P)$ is given by

${\cal{H}}(P)=3Pe^{-\frac{q}{M}(-2qP)^{1/3}},$ (2)

where $q$ and $M$ are the free parameters associated with the charge and mass,
respectively. In the weak field limit ($P<<1$), the NLED structure function
(2) corresponds to the linear electrodynamics, i.e. ${\cal{H}}(P)\approx P$.
The requirements to satisfy the weak energy condition are ${\cal H}<0$ and
$\frac{\partial{\cal H}}{\partial P}>0$ [69, 70, 71].

The field equations corresponding to the action (1) for the metric tensor
($g_{\mu\nu}$) and electromagnetic potential ($A_{\mu}$) are

$\displaystyle G_{\mu\nu}+H_{\mu\nu}+\Lambda g_{\mu\nu}$ $\displaystyle=$
$\displaystyle 2\left(\frac{\partial{\cal H}}{\partial
P}P_{\mu\lambda}P^{\lambda}_{\nu}-2P\frac{\partial{\cal H}}{\partial
P}+{\cal{H}}\right),$ (3) $\displaystyle\nabla_{\mu}P^{\mu\nu}$
$\displaystyle=$ $\displaystyle 0,$ (4)

where $G_{ab}$ and $H_{ab}$ are, respectively

$\displaystyle G_{\mu\nu}$ $\displaystyle=$ $\displaystyle
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R,$ (5) $\displaystyle H_{\mu\nu}$
$\displaystyle=$
$\displaystyle-\frac{\alpha}{2}\left[8R^{\rho\sigma}R_{\mu\rho\nu\sigma}-4R_{\mu}^{\rho\sigma\lambda}R_{\nu\rho\sigma\lambda}-4RR_{\mu\nu}+8R_{\mu\lambda}R_{\nu}^{\lambda}\right.$
(6)
$\displaystyle+\left.g_{\mu\nu}\left(R_{\mu\nu\gamma\delta}R^{\mu\nu\gamma\delta}-4R_{\mu\nu}R^{\mu\nu}+R^{2}\right)\right],$

Now, we are interested to obtain a $5D$ EGB black hole solution in the
presence of NED. For this, we first write the static spherically symmetric
metric as follows:

$ds^{2}=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}+\sin^{2}\theta\sin^{2}\phi\,d\psi^{2}),$ (7)

where $f(r)$ is the metric function which will be determined later.

We use the following ansatz for the antisymmetric field:

$P_{\mu\nu}=2\delta^{\theta}_{[\mu}\delta^{\phi}_{\nu]}D(r)\sin^{2}\theta\sin\phi,$
(8)

which, upon integration (4), eventually leads to

$\displaystyle P=\frac{q^{2}}{2r^{6}}.$ (9)

Here, we chose the integration constant as $q$.

With this antisymmetric field $P_{\mu\nu}$ and invariant $P$, the non-
vanishing component of Einstein field equation (3) results

$\displaystyle(4\alpha f^{\prime}-2r)(f-1)-r^{3}f^{\prime}{-\Lambda
r^{2}}=\frac{2Mk}{r^{3}}e^{-k/r^{2}},$ (10)

where the prime (′) is the derivative of the metric function $f(r)$ concerning
$r$ and deviation parameter $k=q^{2}/M$. The solution of Eq. (10) determines
the form of metric function as

$f(r)=1+\frac{r^{2}}{4\alpha}\left(1\pm\sqrt{1+\frac{8M\alpha}{r^{4}}e^{-k/r^{2}}{+}\frac{{8}\Lambda\alpha}{3}}\,\right).$
(11)

We note that the solution (12) has two branches, $+ve$ and $-ve$,
respectively.

For vanishing Mass, the obtained black hole solution (12) becomes

$f(r)=1+\frac{r^{2}}{4\alpha}\left(1\pm\sqrt{1-\frac{8\alpha}{l^{2}}}\right).$
(12)

For $\alpha>0$, $8\alpha/l^{2}\leq 1$ and beyond this, there is no black hole
solution. Thus, the action 1 has two $AdS$ solutions with effective
cosmological constants
$l^{2}_{eff}=\frac{l^{2}}{4}\left(1\pm\sqrt{1-\frac{8\alpha}{l^{2}}}\right)$.
For $8\alpha/l^{2}=1$, both the solutions coincide and, therefore, the theory
has a unique $AdS$ vacuum.

When $\alpha<0$ , the solution (12) still remains $AdS$ for $-ve$ signature
and becomes $dS$ if one takes the $+ve$ signature. From the vacuum case, the
solution (12) with both signs seems reasonable, from which we cannot determine
which sign should be adopted. Then Boulware and Deser showed that the solution
with $+ve$ branch is unstable and the solution is asymptotically an $AdS$
Schwarzschild solution with negative gravitational mass, indicating the
instability. The solution (12) with $-ve$ branch is stable and the solution is
asymptotically a Schwarzschild solution. Therefore the $+ve$ branch is of less
physical interest [45, 44]

This describes a $5D$ $AdS$ regular black hole for EGB gravity coupled with
NLED. The resulting black hole is characterized by parameters like $M$, $k$
and $\alpha$. In the limit, $\alpha\to 0$ and $k=0$, the negative branch of
solution (11) corresponds to the $5D$ Schwarzschild-Tangherlini black hole.
However, in the limit $\alpha\rightarrow 0$, the solution (11) corresponds to
the regular Schwarzschild black hole in $5D$ $AdS$ [72, 71]

$f(r)=\left(1-\frac{2Me^{-k/r^{2}}}{r^{2}}-\frac{\Lambda r^{2}}{3}\right).$
(13)

Here, we remark that the exponential factor present in the solution removes
the curvature singularity. The given metric (11) can also be considered as the
EGB black hole coupled to NLED. It can be checked that solution (11) matches
with the Boulware-Deser black hole provided the mass $(M)$ must be replaced
with $M(r)$:

$M(r)=\frac{\sigma(r)}{\sigma_{\infty}}M,$ (14)

where $\sigma(r)=e^{-k/r^{2}}$ is the probability distribution function
satisfying $\sigma(r)\geq 0$ and $\sigma^{\prime}(r)<0$ for $r\geq 0$. Also,
$\sigma(r)/r\to 0$ for $r\to 0$ and $\sigma_{\infty}$ refers to is the
probability distribution function when $r\rightarrow\infty$. Asymptotically
($(r>>k)$), the metric (11) corresponds to the charged $AdS$ EGB black hole
[73]

$f(r)=1+\frac{r^{2}}{4\alpha}\left(1\pm\sqrt{1+\frac{8\alpha
M}{r^{4}}-\frac{8\alpha q^{2}}{r^{6}}+\frac{8\Lambda\alpha}{3}}\,\right).$
(15)

Henceforth, we end up with a new solution describing an exact regular EGB
black hole coupled with nonlinear matter fields in $AdS$ space. This $AdS$
solution, characterized by the parameter $M$ and $k$, extends the Wiltshire
charged EGB black hole [73] to $AdS$ space.

The horizon of the black hole is described by the following condition:

$1+\frac{r^{2}}{4\alpha}\left(1\pm\sqrt{1+\frac{8M\alpha}{r^{4}}e^{-\frac{k}{r^{2}}}+\frac{8\Lambda\alpha}{3}}\,\right)=0.$
(16)

---
Figure 1:  The metric function $f(r)$ versus $r$ with different value of
deviation parameter $k$ for $\alpha=0.1$ (left panel) and $\alpha=0.2$ (right
panel) with fixed $M$ and $l$.

The plot for $f(r)$ versus $r$ is depicted in FIG. 1. Eq. (16) is a complex
expression that complicates the analysis of the horizon structure
analytically. Henceforth, we prefer numerical analysis of the horizon
condition by varying the deviation parameter $k$. The horizon condition
$f(r)=0$ will find two real roots, namely, $r_{+}$ and $r_{-}$ that correspond
to the event and Cauchy horizon, respectively. The numerical values of $r_{-}$
and $r_{+}$ for different $\alpha$ and $k$ are tabulated in Table 1.

| $\alpha=0.1$ |  |  |  | $\alpha=0.2$ |  |
---|---|---|---|---|---|---|---
$k$ | $r_{-}$ | $r_{+}$ | $\delta$ | $k$ | $r_{-}$ | $r_{+}$ | $\delta$
0.1 |  0.283 | 0.643 | 0.360 | 0.1 |  0.258 | 0.596 | 0.338
0.125 |  0.329 | 0.612 | 0.283 | 0.07 |  0.329 | 0.558 | 0.229
0.17 |  0.483 | 0.483 | 0 | 0.092 |  0.447 | 0.447 | 0
Table 1: Cauchy horizon ($r_{-}$). event horizon ($r_{+}$), and
$\delta=r_{+}-r_{-}$ for the $5D$ $AdS$ EGB Bardeen black hole with
${\alpha}=0.1$ and ${\alpha}=0.2$ with fixed $M$ and $\Lambda$.

The horizons can also be discussed in terms of the deviation parameter $k$.
Now, it is possible to compute the value of $k$ which satisfies the horizon
condition that admits two real roots for $r$. From the table, it is evident
that there exists a critical horizon $r_{c}=r_{\pm}=0.483$ and critical
deviation parameter $k_{c}=0.17$ for $\alpha=0.1$. However, a critical horizon
$r_{c}=r_{\pm}=0.447$ and critical deviation parameter $k_{c}=0.092$ exist for
$\alpha=0.2$. These signify extremal regular $AdS$ black holes. Moreover,
$k<k_{c}$ for $\alpha=0.1$ and $k>k_{c}$ for $\alpha=0.2$, the two different
horizons ($r_{\pm}$) signify the non-extremal black hole. We find that the
size of the black hole decreases with an increase in the value of $\alpha$.

##  III Thermodynamics

Now, we can study the thermodynamic properties of the obtained black hole
solution in terms of horizon radius, which are described by the horizon mass
$(M_{+})$, deviation parameter $(k)$, and the cosmological constant $\Lambda$.
The horizon mass and Hawking temperature ($T_{+}$) is calculated by

$\displaystyle
M_{+}=e^{k/r_{+}^{2}}\left(\frac{r_{+}^{4}}{l^{2}}+(r_{+}^{2}+2\alpha)\right),$
(17) $\displaystyle T_{+}=\left.\frac{1}{4\pi}\frac{\partial}{\partial
r}\sqrt{-g^{rr}g_{tt}}\right|_{r=r_{+}}=\frac{r_{+}^{4}-k(r_{+}^{2}+2\alpha)}{2\pi
r_{+}^{3}(r^{2}+4\alpha)}+\frac{2r_{+}^{6}-kr_{+}^{4}}{2l^{2}\pi
r_{+}^{3}(r_{+}^{2}+4\alpha)}.$ (18)

The Hawking temperature is characterized by $k$, $\alpha$, and $\Lambda$. The
temperature of this regular $AdS$ black hole is plotted in FIG. 2.

---
Figure 2: Temperature $T_{+}$ versus $r_{+}$ for distinct value of deviation
parameter $k$ with $\alpha=0.1$ (left panel) and $\alpha=0.2$ (right panel)
with fixed $M=1$ and $l=10$.

From the figure, we see that the effect of the deviation parameter is more
significant for small black holes. As the value of $k$ increases, the peak of
the temperature decreases and occurs at larger $r_{+}$ as well. The
temperature of the $AdS$ regular black hole also decreases with an increase in
$\alpha$ and shifts toward the large value of the horizon radius.

Being a thermal system, the black hole follows the first law of thermodynamics
given by

$dM_{+}=T_{+}dS_{+}+\phi de,$ (19)

where $S_{+}$ refers to the entropy of the black hole. For the given values of
$M_{+}$ and $T_{+}$, the first law of thermodynamics leads to the following
expression for the entropy:

$S_{+}=\frac{4\pi
r_{+}^{3}}{3}\left[\frac{(2k+r_{+}^{2}+12\alpha)e^{k/r_{+}^{2}}}{r_{+}^{2}}-\frac{2\sqrt{\pi
k}\,(4k+6\alpha)}{r_{+}^{3}}\text{erf}\left(\frac{\sqrt{k}}{r_{+}}\,\right)\right],$
(20)

where erf is the error function. Here, we note that the entropy matches with
the one calculated in Ref. [36] without the cosmological constant. The
deformed entropy from the area-law occurs due to the presence of deviation
parameters and GB parameter.

We know that entropy of the regular black hole does not follow the area law
[74, 75] because the black hole mass is included in the source term. Ma et al
[76] proposed the corrected form of first law black hole thermodynamics for
regular black holes which modifies with the extra factor. The modified first
law is [76, 37, 77]

$C_{M}dM=T_{+}\,dS+\phi\,de,$ (21)

where $C(M,r_{+})$ is

$C(M,r_{+})=1+4\pi\int_{r_{+}}^{\infty}r^{2}\frac{\partial T^{0}_{0}}{\partial
M}dr=2e^{-k/r_{+}^{2}}.$ (22)

For this value of $C(M,r_{+})$ and the obtained black hole solution follows
the area law.

The thermodynamic stability of the given black hole can be explained by the
nature of the heat capacity as the positive and negative signatures of heat
capacity justify the stable and unstable state of the black hole,
respectively. The heat capacity for the black hole solution can be defined as

$\displaystyle C_{+}$ $\displaystyle=$
$\displaystyle\frac{\partial{M_{+}}}{\partial{T_{+}}}.$ (23)

For the given mass (17) and temperature (18), the expression of the heat
capacity reads

$\displaystyle C_{+}=\frac{4e^{\frac{k}{r_{+}^{2}}}\pi
r(r_{+}^{2}+4\alpha)^{2}(r_{+}^{4}(l^{2}+2r_{+}^{2})-k(r_{+}^{4}+l^{2}(r_{+}^{2}+2\alpha)))}{2r_{+}^{6}(r_{+}^{2}+12\alpha)-l^{2}(r_{+}^{6}-4r_{+}^{4}\alpha)+k(r_{+}^{6}-4r_{+}^{4}\alpha+l^{2}(3r_{+}^{4}+14r_{+}^{2}\alpha+24\alpha^{2}))}.$
(24)

From this expression, it is cumbersome to identify the signature and behavior
of heat capacity. Hence, we plot the heat capacity as depicted in diagram 3
for different values of deviation parameter $k$.

---
Figure 3: The heat capacity $C_{+}$ versus $r_{+}$ with different value of
deviation parameter $k$ for $\alpha=0.1$ (left panel) and $\alpha=0.2$ (right
panel) with fixed value of $M=1$ and $l=10$.

To study the nature of the heat capacity, we plotted them for the various
values of $k$ and $\alpha$. From the FIG. 3, we find that there exist double
phase transitions. Firstly, a phase transition occurs from a small stable
black hole to a large unstable black hole and, secondly, from a smaller
unstable black hole to a larger stable black hole. For the fixed value of
$\alpha$ the radii $r_{1+}$ increases and $r_{2+}$ decreases with $k$.

Gibbs free energy also plays an important role in order to discuss the
(global) stability of the black hole. The Gibbs free energy can be calculated
from the standard definition: $G_{+}=M_{+}-T_{+}S_{+}$. This yields

$\displaystyle G_{+}$ $\displaystyle=$ $\displaystyle
e^{k/r_{+}^{2}}\left(\frac{r_{+}^{4}}{l^{2}}+(r_{+}^{2}+2\alpha)\right)-\frac{2}{3}\left(\frac{r_{+}^{4}(l^{2}+2r_{+}^{2})-k(r_{+}^{4}+l^{2}(r_{+}^{2}+2\alpha))}{l^{2}(r_{+}^{2}+4\alpha)}\right)$
(25) $\displaystyle\times$
$\displaystyle\left[\frac{(2k+r_{+}^{2}+12\alpha)e^{k/r_{+}^{2}}}{r_{+}^{2}}-\frac{2\sqrt{\pi
k}\,(4k+6\alpha)}{r_{+}^{3}}\text{erf}\left(\frac{\sqrt{k}}{r_{+}}\,\right)\right].$

The stability can also be explained from the Gibbs free energy plot as
depicted in FIG. 4.

---
Figure 4: The Gibbs free energy $G_{+}$ versus $r_{+}$ with different value of
deviation parameter $k$ for $\alpha=0.1$ and $\alpha=0.2$ with fixed value of
$M=1$ and $l=10$.

From the plot, we observe that the free energy exhibits local minima
($r_{1+}$) and local maxima ($r_{2+}$) for distinct values of $k$ with fixed
value of $\alpha$. For $r>r_{1+}$, the free energy is an increasing function
of $r_{+}$ and remains positive and attains the local maximum value at
$r_{2+}$. After $r=r_{2+}$, the slope of Gibbs free energy turns negative and,
therefore, the theory provides the natural Hawking-Page phase transition. The
various numerical values are tabulated in the TABLE 2. Here, we can also see
that the Gibbs free energy ($G_{+}$) has a minimum and a maximum locally
regarding the extremal points of the temperature where the heat capacity
diverges.

|  | $\alpha=0.1$ |  |  |  |  | $\alpha=0.2$ |  |
---|---|---|---|---|---|---|---|---|---
$r_{1+}$ | $r_{2+}$ | $T_{+}$ | $C_{+}$ | $G_{+}$ | $r_{1+}$ | $r_{2+}$ | $T_{+}$ | $C_{+}$ | $G_{+}$
1.012 | 3.218 | Maximum | Diverge | Loc. Max. | 1.405 | 2.881 | Minimum | Diverge | Loc. Min.
1.251 | 3.148 | Maximum | Diverge | Loc. Max. | 1.714 | 2.754 | Minimum | Diverge | Loc. Min.
1.447 | 3.063 | Maximum | Diverge | Loc. Max. | 2.09 | 2.515 | Minimum | Diverge | Loc. Min.
Table 2: The numerical values of the local maxima and minima to characterize
the nature of $T_{+}$, $C_{+}$, and $G_{+}$.

##  IV Van der Waals Fluid

The aim of this section is to consider the resulting black hole as a Van der
Waals fluid and calculate the $P-v$ criticality. As we know, the negative
cosmological constant induces a thermodynamic pressure (i.e. $\Lambda=-8\pi
P_{+}$ with $G=\hbar=c=1$) in the extended thermodynamics. The thermodynamic
volume $V$ plays the role of conjugate to pressure and can be interpreted as
the change in the mass under the variations in the $\Lambda$ having fixed
horizon area. The mass $M$ is then understood as an enthalpy.

The temperature $T_{+}$ in tandem to the above identifications of
thermodynamic pressure and conjugate volume lead the following equation of
state: Using the and volume $V$, we obtain the following equations of state:

$\displaystyle P_{+}=\frac{1}{4\pi r^{4}(2r_{+}^{2}-k)}\left[6\pi
r_{+}^{3}T(r_{+}^{2}+4\alpha)+3(kr_{+}^{2}-r_{+}^{4}+2k\alpha)\right],\,\,\,\,v=2r_{+},$
(26)

where the $v$ is a specific volume.

The critical points appear at isotherms $T_{c}$ where pressure has an
inflection point at $P_{c}$ and $v_{c}$ satisfying conditions [10,14]

$\displaystyle\left(\frac{\partial P_{+}}{\partial
r_{+}}\right)_{T_{+}}=0,\qquad\left(\frac{\partial^{2}P_{+}}{\partial
r^{2}_{+}}\right)_{T_{+}}=0.$ (27)

The critical radius can be calculated numerically from the relation

$\displaystyle\frac{2r^{8}-24r^{6}\alpha-k^{2}(3r^{4}+16r^{2}\alpha-48\alpha^{2})-12k(r^{6}+9r^{4}\alpha+36r^{2}\alpha^{2})}{4\pi
r^{6}(kr^{2}+2r^{4}-4k\alpha+24r^{2}\alpha)(k-2r^{2})}=0.$ (28)

The fluid behavior of the regular $AdS$ EGB black holes can be seen from the
$P_{+}-r_{+}$ diagram in FIG. 5.

---
Figure 5:  $P_{+}$ versus $r_{+}$ with different value of deviation parameter
$k$ for $\alpha=0.1$ and $\alpha=0.2$ at critical temperature $T_{c}$.

The Eq. (28) can not be solved analytically and, therefore, the critical
radius $r_{c}$, critical pressure $P_{c}$ and the critical temperature $T_{c}$
are obtained numerically. The numerical values are presented in TABLES 3 and 4
for different values of $\alpha$ and $k$.

$k$ | $r_{c}$ | $T_{c}$ | $P_{c}$ | ${P_{c}\,r_{c}}/{T_{c}}$
---|---|---|---|---
0.1 | 1.453 | 0.123 | 0.0225 | 0.2173
0.2 | 1.689 | 0.110 | 0.0176 | 0.2425
0.3 | 1.884 | 0.100 | 0.0146 | 0.2518
0.4 | 2.055 | 0.093 | 0.0125 | 0.2558
0.5 | 2.209 | 0.087 | 0.0107 | 0.2579
Table 3: The critical temperature $T_{c}$, critical pressure $P_{c}$ and $P_{c}\,r_{c}/T_{c}$ corresponding different value of $k=0.1$ with fixed value of $\alpha=0.1$ . $\alpha$ | $r_{c}$ | $T_{c}$ | $P_{c}$ | $P_{c}\,r_{c}/T_{c}$
---|---|---|---|---
0.1 | 1.453 | 0.123 | 0.0225 | 0.2173
0.2 | 1.8434 | 0.0936 | 0.0138 | 0.2617
0.3 | 2.154 | 0.0785 | 0.0094 | 0.3216
0.4 | 2.423 | 0.0690 | 0.0073 | 0.4069
0.5 | 2.662 | 0.0623 | 0.0060 | 0.5381
Table 4: The critical temperature $T_{c}$, critical pressure $P_{c}$ and
$P_{c}\,r_{c}/T_{c}$ corresponding to the different value of $\alpha=0.1$ with
fixed value of $k=0.1$ .

It can be seen that the critical radius $r_{c}$ increases with the increase in
the parameters $k$ and $\alpha$, however, the critical pressure $P_{c}$ and
temperature $T_{c}$ decrease with increase in $k$ and $\alpha$. Incidentally,
the universal ratio $P_{c}r_{c}/T_{c}$ increases with the parameters $k$ and
$\alpha$. It is worth mentioning that the critical radius increases with
decrease in the critical pressure and critical temperature.

##  V QNMs in Ekilon Limit

QNMs are usually predicts the stability of the given black holes perturbed by
an external field or black hole geometry. QNMs also provide the information
regarding gravitational waves. In QNM can be discussed by studying the motion
of photon in the vicinity of the black hole solution (11). The photon motion
limited to equatorial plane $(\theta=\pi/2)$ is described by the following
Lagrangian:

${\cal{L}}=-g_{tt}{\dot{t}}^{2}+g_{rr}{\dot{r}}^{2}+g_{\theta\theta}{\dot{\theta}}^{2}+g_{\phi\phi}{\dot{\phi}}^{2}+g_{\psi\psi}{\dot{\psi}}^{2},$
(29)

where dot denotes the derivative with respect to affine parameter. The
corresponding Hamiltonian is given by

${\cal{H}}=\frac{1}{2}g^{ij}p_{i}p_{j}=0,$ (30)

and the generalized momenta are given by

$\displaystyle p_{t}=\frac{\partial{\cal
H}}{\partial{\dot{t}}}=\text{constant}\equiv E,\qquad
p_{r}=\frac{\partial{\cal H}}{\partial{\dot{r}}}=g_{rr}{\dot{r}},\qquad
p_{\theta}=\frac{\partial{\cal
H}}{\partial{\dot{\theta}}}=g_{\theta\theta}{\dot{\theta}},$ $\displaystyle
p_{\phi}=\frac{\partial{\cal
H}}{\partial{\dot{\phi}}}=\text{constant}\equiv-L,\qquad
p_{\psi}=\frac{\partial{\cal
H}}{\partial{\dot{\psi}}}=g_{\psi\psi}{\dot{\psi}}.$ (31)

Here, $E$ and $L$ refer to the energy and the angular momentum per unit rest
mass of the test particle, respectively. The equations of motion associated
with the photon in the Hamiltonian formalism are given by

$\displaystyle{\dot{t}}=\frac{\partial{\cal H}}{\partial
p_{t}}=-\frac{p_{t}}{g_{tt}},\quad{\dot{r}}=\frac{\partial{\cal H}}{\partial
p_{r}}=-\frac{p_{r}}{g_{rr}},\quad{\dot{\theta}}=\frac{\partial{\cal
H}}{\partial p_{\theta}}=\frac{p_{\theta}}{g_{\theta\theta}},$
$\displaystyle{\dot{\phi}}=\frac{\partial{\cal H}}{\partial
p_{\phi}}=\frac{p_{\phi}}{g_{\phi\phi}},\quad{\dot{\psi}}=\frac{\partial{\cal
H}}{\partial p_{\psi}}=\frac{p_{\theta}}{g_{\psi\psi}}.$ (32)

since the above Hamiltonian does not depend on the coordinates $t$, $\phi$ and
$\psi$. So, the null geodesics equation is written by

${\dot{r}}^{2}+V_{eff}(r)=0,\qquad\text{with}\qquad
V_{eff}=f(r)\left(\frac{L^{2}}{r^{2}}-\frac{E^{2}}{f(r)}\right).$ (33)

For a circular null geodesics, the effective potential must necessarily hold
the following conditions:

$V_{eff}=0,\qquad\text{and}\qquad\frac{\partial V_{eff}}{\partial r}=0.$ (34)

These conditions describe the radius of the photon sphere. These conditions
lead to the equation of the photon radius as

$kM-2Mr^{2}+e^{k/r^{2}}r^{2}\sqrt{1+\frac{8M\alpha
e^{k/r^{2}}}{r^{4}}+\frac{8\Lambda\alpha}{3r^{2}}}=0.$ (35)

This equation can not be solved analytically, so we can calculate the photon
radius $r_{p}$, numerically. The numerical values are presented in TABLE 5.
From this TABLE, we can see that the photon radius increases along with
increasing deviation parameter and GB coupling.

|  | $r_{p}$ |  |
---|---|---|---|---
$\alpha$ | $k=0.1$ | $k=0.2$ | $k=0.3$ | $k=0.4$
0.1 | 1.4381 | 1.4574 | 1.4726 | 1.4840
0.2 | 1.4503 | 1.4783 | 1.5021 | 1.5169
0.3 | 1.4728 | 1.5154 | 1.5479 | 1.5727
0.4 | 1.5298 | 1.6023 | 1.6817 | 1.6954
Table 5: The numerical values of photon radius corresponding to the GB coulpling parameter and devation parameter with $M=1$, where $l=1$ and $n=0$. |  | $r_{p}$ |  |
---|---|---|---|---
$k$ | $\alpha=0.1$ | $\alpha=0.2$ | $\alpha=0.3$ | $\alpha=0.4$
0.1 | 1.438 | 1.450 | 1.472 | 1.529
0.2 | 1.457 | 1.478 | 1.515 | 1.603
0.3 | 1.472 | 1.500 | 1.547 | 1.654
0.4 | 1.484 | 1.516 | 1.572 | 1.695
0.5 | 1.491 | 1.528 | 1.591 | 1.727
Table 6: Values of photon radius corresponding to the GB coulpling parameter
$(\alpha)$ and devation parameter $(k)$ with $M=1$, where $l=1$ and $n=0$.

The QNMs frequency $\omega$ in the eikonal limit can be estimated by the
virtue of the photon sphere as follows

$\omega=l\Omega-i\left(n+\frac{1}{2}\right)|\Lambda|,$ (36)

where $n$ is the overtone number and $l$ is the angular quantum number of
perturbation. However, $\Omega$ is the angular velocity and $\Lambda$ is the
Lyapunov exponent of the photon sphere with following expressions:

$\displaystyle\Omega=\frac{\sqrt{f(r_{p})}}{r_{p}}\qquad\text{and}\qquad\Lambda=\frac{\sqrt{f(r_{p})(2f{r_{p}}-r^{2}_{p}f^{\prime\prime}(r_{p}))}}{\sqrt{2}r_{p}}.$
(37)

Here $r_{p}$ denotes radius of photon sphere (called as photon radius).

The real and imaginary parts of the QNMs of black hole solution (11) for
different values of deviation parameter and GB parameter are depicted in the
FIG. 6 and FIG. 7.

---
Figure 6: The plot of real part (left panel) and imaginary part (right panel)
of QNMs versus GB parameter with different $k$ with fixed $M$ and $l$.
---
Figure 7: The plot of real part (left panel) and imaginary part (right panel)
of QNMs versus deviation parameter with different $k$ with fixed $M$ and $l$.

These diagrams help us to investigate the effects of the black hole parameters
on the QNMs. Here, we see that the real part of the QNMs is a decreasing
function of the GB parameter. However, real part of the QNMs increases with
deviation parameter. On the other hand, the imaginary part of the QNMs with
respect to the GB parameter first decreases very slowly (almost constant) and
then increases sharply. Also, the imaginary part of the QNMs increases with
deviation parameter (more significantly for large $\alpha$).

The signature of the imaginary part of the QNMs characterizes the stability of
black hole. Im $\omega<0$ corresponds to stable modes of black hole and Im
$\omega>0$ corresponds to unstable modes. The imaginary part of the QNMs for
the obtained black hole solution (11) is negative (See Fig. 6 and 7). This
confirms that the modes of the obtained black hole solution (11) are stable.

We list the numerical values of QNMs frequency corresponding various values of
parameters in TABLES 7 and 8.

| $k=1$ | $k=2$ | $k=3$ | $k=4$
---|---|---|---|---
$\alpha$ |  $\omega$=Re $\omega$ $-$ Im $\omega$ |  $\omega$=Re $\omega$ $-$ Im $\omega$ |  $\omega$ =Re $\omega$ $-$ Im $\omega$ |  $\omega$=Re $\omega$ $-$ Im $\omega$
0.1 | 0.71489 - 0.37765 $i$ | 0.721135 - 0.36758 $i$ | 0.726456 - 0.35734 $i$ | 0.73125 - 0.34703 $i$
0.2 | 0.71440 - 0.37984 $i$ | 0.719702 - 0.37125 $i$ | 0.723687 - 0.36266 $i$ | 0.72764 - 0.35300 $i$
0.3 | 0.71376 - 0.32043 $i$ | 0.717979 - 0.32024 $i$ | 0.721104 - 0.31778 $i$ | 0.72375 - 0.31388 $i$
0.4 | 0.71238 - 0.24545 $i$ | 0.714816 - 0.25595 $i$ | 0.714815 - 0.26844 $i$ | 0.71756 - 0.26416 $i$
Table 7: The numerical values of QNMs corresponding to the GB coulpling parameter $(\alpha)$ and devation parameter $(k)$ with $M=1$, where $l=1$ and $n=0$. | $\alpha=0.1$ | $\alpha=0.2$ | $\alpha=0.3$ | $\alpha=0.4$
---|---|---|---|---
$k$ |  $\omega$=Re $\omega$ $-$ Im $\omega$ |  $\omega$=Re $\omega$ $-$ Im $\omega$ |  $\omega$=Re $\omega$ $-$ Im $\omega$ |  $\omega$=Re $\omega$ $-$ Im $\omega$
0.1 | 0.7147 -0.4307 $i$ | 0.7144 -0.3797 $i$ | 0.7137 -0.3202 $i$ | 0.71241 -0.245 $i$
0.2 | 0.7208 -0.4151 $i$ | 0.7197 -0.3712 $i$ | 0.7180 -0.3201 $i$ | 0.7147 -0.2561 $i$
0.3 | 0.7258 -0.4004 $i$ | 0.7239 -0.3621 $i$ | 0.7211 -0.3175 $i$ | 0.7164 -0.2612 $i$
0.4 | 0.7303 -0.3866 $i$ | 0.7277 -0.3528 $i$ | 0.7238 -0.3137 $i$ | 0.7175 -0.2640 $i$
0.5 | 0.7347 -0.3733 $i$ | 0.7312 -0.3435 $i$ | 0.7261 -0.3091 $i$ | 0.7186 -0.2649 $i$
0.6 |  0.7391 -0.3607 $i$ | 0.7348 -0.3341 $i$ | 0.7286 -0.3035 $i$ | 0.7197 -0.2643 $i$
0.7 | 0.7440 -0.3482 $i$ | 0.7386 -0.3245 $i$ | 0.7313 -0.2973 $i$ | 0.7209 -0.2629 $i$
0.8 | 0.7496 -0.3356 $i$ | 0.7432 -0.3143 $i$ | 0.7345 -0.2903 $i$ | 0.7221 -0.2605 $i$
0.9 | 0.7568 -0.3220 $i$ | 0.7492 -0.3028 $i$ | 0.7383 -0.2821 $i$ | 0.7238 -0.2570 $i$
1.0 | 0.7669 -0.3061 $i$ | 0.7572 -0.2891 $i$ | 0.7434 -0.2772 $i$ | 0.7256 -0.2528 $i$
1.1 | 0.7861 -0.2806 $i$ | 0.7716 -0.2674 $i$ | 0.7512 -0.2695 $i$ | 0.7281 -0.2471 $i$
Table 8: The numerical values of QNMs corresponding to the GB coulpling
parameter $(\alpha)$ and devation parameter $(k)$ with $M=1$, where $l=1$ and
$n=0$.

##  VI Results and Conclusion

In this work, we have considered a EGB gravity coupled to the NLED in $5D$
$AdS$ spacetime and constructed a new regular black hole solution in $AdS$
spacetime. The obtained solution is a generalized version of $5D$
Schwarzschild-Tangherlini black hole, $5D$ $AdS$ regular Schwarzschild black
hole and Boulware-Deser black hole. We have found that the black hole solution
exhibits two horizons, namely, the Cauchy and event horizon. There exist
different critical horizons corresponding to different GB parameter that
characterize the extremal/non-extremal nature of black holes. The size of the
black holes decrease with the increasing GB parameter.

Furthermore, we have studied the thermodynamics of the resulting solution by
deriving horizon mass, Hawking temperature and entropy of the black hole. We
have found that the black hole satisfies the modified first law of
thermodynamics. The stabilities of black hole are discussed by estimating both
the heat capacity and Gibbs free energy. The diagrams confirmed that there
exist double phase transitions, one from small stable black hole to large
unstable black hole and other from smaller unstable black hole to larger
stable black hole. The Gibbs free energy analysis confirms the existence of
(local) minimum and maximum associated to the extremal points of the Hawking
temperature. The fluid nature of black hole is also studied. We have observed
that the critical values depend on GB coupling parameter and deviation
parameter considerably. For instance, the critical radius is an increasing
function of the GB coupling parameter and deviation parameter. In contrast,
the critical pressure and critical temperature are decreasing function of the
GB coupling parameter and deviation parameter.

It is worth discussing QNMs for the $5D$ $AdS$ regular EGB black hole coupled
with NLED as QNMs may provide the information regarding gravitational waves.
For this purpose, we studied the motion of photon in the vicinity of the black
hole solution. The effects of GB coupling and deviation parameters on the real
and imaginary parts of the QNMs are also discussed. It will be interesting to
establish a the correspondence between the QNMs in the eikonal limit and the
shadow radii for such black hole solution of the $5D$ EGB gravity coupled to
the NLED in $AdS$ space.

###### Acknowledgements.

This research was funded by the Science Committee of the Ministry of Science
and Higher Education of the Republic of Kazakhstan (Grant No. AP09058240). One
of us (DVS) thanks UGC for the start up grant (Grant No.:
30-600/2021(BSR)/1630).

## Data Availability Statement and Competing Interests

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study. The authors declare no competing interests.

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