=Paper=
{{Paper
|id=Vol-3806/S_20_Ogurtsov
|storemode=property
|title=
Development of the Local Navigation Algorithm of the Agricultural UAVs During Swarm Movement
|pdfUrl=https://ceur-ws.org/Vol-3806/S_20_Ogurtsov.pdf
|volume=Vol-3806
|authors=Maksym Ogurtsov,Oleh Rybalchenko,Vyacheslav Korolyov,Oleksandr Khodzinsky
|dblpUrl=https://dblp.org/rec/conf/ukrprog/OgurtsovRKK24
}}
==
Development of the Local Navigation Algorithm of the Agricultural UAVs During Swarm Movement
==
https://ceur-ws.org/Vol-3806/S_20_Ogurtsov.pdf
Development of the Local Navigation Algorithm of the
Agricultural UAVs During Swarm Movement
Maksym Ogurtsov1, Oleh Rybalchenko1, Vyacheslav Korolyov1 and Oleksandr Khodzinsky1
1
V.M. Glushkov Institute of Cybernetics of the National Academy of Sciences of Ukraine, Ac. Hlushkova str. 40, building 1, apt.
801, Kyiv, 03187, Ukraine
Abstract
Every year, the scope of unmanned aerial vehicle use is growing, and the amount of such applications is
expanding. To enhance efficiency, the focus of development is shifting from individual unmanned aerial
vehicles to utilizing swarms for various applications such as agricultural surveys, irrigation, etc.
The purpose of this work is to develop an algorithm for building a local positioning system for a swarm of
unmanned aerial vehicles to maintain a stable structure during the swarm movement while performing its
tasks in two-dimensional and three-dimensional settings. At the same time, it should be assumed that the
swarm is controlled by one operator, regardless of the number of swarm elements. An additional limitation
of the developed algorithm should be the prevention of potential collisions of unmanned aerial vehicles
during aerial maneuvers.
As part of the work, the task of forming and maintaining the structure and configuration of an agricultural
unmanned aerial vehicle flock in two-dimensional and three-dimensional settings was set. The
formalization of the given task has been completed. Considered possible methods of mapping the graph
formed by an unmanned aerial vehicle swarm into two-dimensional Euclidean space by forming a basic
triangle method to create a relative coordinate system. To determine the coordinates of the remaining
graph vertices on the plane and correspondingly increase the accuracy of local positioning, the
multilateration method is used and considered simplified options – application of the trilateration
algorithm, triangulation and a simplified algorithm for the degenerate case of two drones.
For the problem in a three-dimensional setting, the possibility of applying the multidimensional scaling
algorithm using the methods of multidimensional scaling/reduction of dimensions is considered. The
developed practical implementation of the created algorithm showed its efficiency during practical
experiments, allowing it to determine the local positioning of elements of a swarm containing three to
twenty elements.
Keywords 1
UAV, drone, swarm, swarm control, local positioning, multilateration, triangulation,
Euclidean space
1. Introduction
Every year, the scope of use of unmanned aerial vehicles (UAVs) is increasing and the spread of
such applications is expanding. At the same time, the emphasis is gradually shifting from the
individual UAVs use to the use of UAV swarms, since for many tasks (for example, surveying
agricultural areas, irrigation, etc.) this is more effective than the use of individual UAVs [1]. The
increase in efficiency is achieved by reducing the number of operators required to operate the same
number of UAVs as swarm elements, instead of using them in the "one operator – one UAV" mode.
In addition, when using swarms, it is possible to use group management with elements of artificial
intelligence, which in many tasks exceeds the capabilities and speed of the human operator's
response [2]. But at the same time, there are several problems. In some cases, the unavailability of
14th International Scientific and Practical Conference from Programming UkrPROG’2024, May 14-15, 2024, Kyiv, Ukraine
*
Corresponding author.
†
These authors contributed equally.
maksymogurtsov@gmail.com (M. Ogurtsov); rv.oleg.ua@gmail.com (O. Rybalchenko); korolev.academ@gmail.com (V.
Korolyov); okhodz@gmail.com (O. Khodzinsky)
0000-0002-6167-5111 (M. I. Ogurtsov); 0000-0002-5716-030X (O. Rybalchenko); 0000-0003-1143-5846 (V. Korolyov); 0000-
0003-4574-3628 (O. Khodzinsky)
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
GPS for agricultural drones that perform the task of processing agricultural land from pests or
monitoring crops [3].
2. Goal
The purpose of this work is to develop an algorithm for building a system of local positioning of a
swarm of UAVs to maintain a stable structure during the movement of the swarm while performing
its assigned tasks in two-dimensional and three-dimensional settings. At the same time, we believe
that the swarm is controlled by one operator regardless of the swarm elements number. The
additional limitation of the developed algorithm: the prevention of potential UAV collisions during
aerial maneuvers.
3. Analysis of previous developments
In the existing scientific works devoted to solving the given problem [1-8], it is necessary to single
out approaches based on the use of the Internet of Things paradigm and artificial intelligence tools
[4], although they are resource-intensive and require significant computing power, which limits the
scope of possible application. The use of parallel genetic algorithms and other methods of
combinatorial optimization [6] is also interesting, although it should be noted that, according to the
authors, solving the given problem in several applications is possible without using similar methods,
in a simpler way.
Another approach is the pre-planning of UAV swarm routes [7], but this significantly narrows
the scope of swarms based on this approach.
The most popular option is the use of particle swarm optimization algorithms [7-8], although this
method is also quite computationally demanding. Usually in similar works, the variant with the
development of a system of swarm elements local mutual positioning, presented in this work, is not
considered.
4. Main part
Let's first consider how the problem can be solved in a simplified, two-dimensional formulation,
which is applicable when the swarm moves at approximately the same height and the height
difference can be neglected. This option is applicable for a swarm of ground drones communicating
in the zone of radio waves direct propagation. We will consider the swarm elements as
homogeneous, although with the additional parameters and restrictions it is possible to use
heterogeneous UAVs.
Meaningfully, we have a set of drones that should form a swarm. In this work, a swarm of up to
20 drones is considered. Each drone is equipped with a set of sensors (GPS, compass, accelerometer,
altimeter) that will al-low determination of mutual position and forming and maintenance of the
swarm structure. The task is to achieve the desired configuration of mutual the swarm elements
placement in the minimum time and further maintenance of this configuration with minimum costs
of the total flight resource of all swarm elements. The algorithm must function iteratively, with a
frequency of repetition at least once every few seconds, to consider possible changes in the swarm
elements relative position. The configuration to be formed by the swarm must provide equal
distances between adjacent swarm elements, i.e. three UAVs will form an equilateral triangle, four –
a square, and so on. Thus, we have:
Input data – data from the sensors of each element of the swarm:
• GPS coordinates.
• the direction of movement.
• speed.
• height.
Parameters:
• maximum UAV movement speed.
• the maximum permissible flight height of the UAV.
• minimum permissible UAV flight height.
• maximum UAV flight range.
• the minimum permissible distance between UAVs.
Restrictions:
Minimization of the total additional movement of all swarm UAVs required to restore the given
configuration.
Output data:
GPS coordinates of the points where each UAV should move to restore the given swarm
configuration.
4.1. Two-dimensional formulation of the problem
The two-dimensional formulation of the problem will involve mapping the graph formed by the
UAV swarm first into three-dimensional Euclidean space [9] (Figure 1), which must then be reduced
to two-dimensional space.
Figure 1: Mapping the graph formed by a swarm of UAVs into three-dimensional Euclidean space.
Let's consider the problem formulation:
is a complete UAV graph, given
is the number of graph vertices, each vertex represents a swarm
element, a separate UAV, which has a set of three coordinates
– coordinates of these vertices, where are unknown, but is known.
– the known lengths of the ribs.
Now consider the mapping of the graph from the three-dimensional Euclidean space to the
problem on the plane (Figure 2).
It is easy to see from the figure that the length of the projection of the edge between the
vertices onto the XOY plane:
To form the desired configuration based on the swarm elements known coordinates, one
must first choose the basic (largest) triangle. Its use will allow the formation of the desired
configuration within the base triangle and will not require an increase in the maximum distance
between any two swarm elements.
The basic triangle in the graph can be chosen according to the following characteristics:
• the largest sum of sides , where are the edges of the graph for some
triplet of vertices (Figure 3):
Figure 2: Mapping Reduction of a three-dimensional problem to a problem on a plane.
Figure 3: Basic triangle by the sum of the largest sides.
• the largest area , where is the semi-perimeter
(Figure 4):
Figure 4: Basic triangle by the largest area.
In this work, the basic triangle with the largest area is used.
When choosing a base triangle, some swarm configurations will cause some swarm elements to
be outside the base triangle (for example, a square for a four-element swarm). This is the standard
situation; in the future, the position of the elements will be adjusted and they will move inside the
base triangle.
4.2. Fixation of the basic triangle on the plane
After selecting the basic triangle, several degrees of freedom are preserved when placing the
projection of the graph on the plane. Let's choose the following: the origin of the coordinate system
(the first vertex of the triangle), the direction of the abscissa axis (the second vertex of the triangle),
the direction of the ordinate axis (the third vertex of the triangle) – Figure 5.
Fix the coordinates of the vertex as the origin of the coordinates:
1. Let the side be part of the axis :
2. – determine the coordinate of the vertex from the area of the triangle:
3. – we determine the coordinate of the vertex according to the Pythagorean
theorem:
Figure 5: Choosing the swarm’s coordinate plane.
To determine the coordinates of the remaining vertices of the graph on the plane, the
multilateration method can be applied [10-13].
Since satellite positioning data is available, the task of maintaining the swarm structure can be
reduced to maintaining the position of the UAV in each location of a geometric figure based on the
coordinates’ absolute values (Figure 6).
Figure 6: Determination of the basic triangle vertices coordinates.
But in the case of small distances be-tween UAVs, the accuracy of determining coordinates may
be insufficient and the mini-mum distance between UAVs, set to prevent collisions, may be violated.
4.3. The problem of increasing the accuracy of local positioning
Global positioning often does not provide a sufficient level of accuracy for the task at hand –
potential errors that, in the case of two fast-moving UAVs, can overlap and add up, causing swarm
elements to collide. One of the options for solving this problem is the multilateration methods use –
local positioning, which does not depend on global positioning satellite systems and is more
accurate than GPS (see Figure 7) [14].
UAV Repeater-Supervisor
UAV 1
UAV 2
Lider
Z
GSC
W N
Figure 7: Illustration for the task of determining the co-ordinates of objects by the distances
between them (Z – height, W – latitude, N – longitude).
Three drones in Figure 7 have their local system used for autonomous flight, and the ground
station control (GSC) is tied to the absolute geographic coordinate system.
A simplified trilateration algorithm consists of the following steps:
Stage 0. Calibration of the distance be-tween nodes by power [15-16].
Input: Calibration based distance table.
Output: distance between nodes.
Stage 1. Each network node will receive information about the signal strength from other nodes
and transmit its data to these nodes.
Stage 2. Each node for all visible nodes calculates distances according to the table and founds
local coordinates by solving the system of trilateration equations. The received values are sent to all
nodes [17].
The next stage is the application of triangulation algorithms.
4.4. A degenerate case of two drones
1. When creating a swarm of two, their mutual location and distance are determined. The
parameters of the mutual local positioning means are stored, which corresponds to the
sensors data for such mutual location, determined by the means of global positioning.
2. One of the drones becomes the main one, the master.
3. The master relays control commands from the operator to the slave (or slave receives
them directly and repeats them). At the same time, a survey is conducted once every ten
seconds (data exchange to determine the signal strength and the corresponding distance
between the drones).
4. If the distance has changed, correction is made by changing the flight direction and re-
measurement.
4.5. The problem in a three-dimensional setting
Let us now consider the three-dimensional formulation of the problem. We have up to twenty
objects, the distances between them are known, and the GSC to which the distance is also known. In
addition, all objects (1-20 pcs.) can be separately controlled remotely any time (that is, telemetry is
received on the GSC and the distance is measured by signal strength and time from
editing/receiving data packets) when the operator needs it. Absolute coordinates are known for
GSC. Thus, we have the following formulation of the problem:
A matrix of distances between two to twenty-one points in three-dimensional space is given. For
one of these points, its absolute coordinates are known. It is necessary to develop an algorithm for
building a relative coordinate system for these points.
4.5.1. Algorithm of multidimensional scaling
The task of determining the coordinates of objects by the distances between them can be reduced
to the well-known task of reducing the dimensionality of input data. A general approach to solving
the problem of data dimensionality reduction consists of performing a sequence of steps to reduce
the number of variables describing this data, and the result of the work of the corresponding
algorithms is an ordered set of main variables.
The methods of multidimensional scaling/reduction of dimensions (Multidimensional Scaling –
MDS) are applied to various types of data [18].
4.5.2. Mathematical formulation of MDS
The basic idea of MDS is to find a general process in a low-dimensional space that minimizes the
variance D in the high-dimensional space and the dependent variance d in the low-dimensional
space. Mathematically, this can be expressed as follows: given a dissimilarity or distance matrix D,
representing the difference between points i and j in space, where Dij is the Euclidean distance, MDS
tries to find a set of coordinates xi and xj. This is done in a low-dimensional space so that the
Euclidean distance between points i and j (denoted dij ) is as close as possible to the difference Dij .
The metric of the MDS algorithm is the stress function S (objective function), which defines the
difference between the distances in the source space and the distances in the subspace. MDS for
dimensionality reduction can be defined as follows:
where Wij is the weight that can be used to emphasize or de-emphasize certain distances.
dij is the Euclidean distance between data points i and j in the lower dimensional space.
Dij is the difference (distance) between data points i and j in a high-dimensional space.
The stress function S is defined as:
where lij is the distance between data points i and j in the original space, is the distance between
data points i and j in the lower dimensional space, and i is the number of data points.
The stress function S is a measure of the deviation of the distances in the lower dimensional
space from the distances in the original space and is used to evaluate the quality of the projection.
To create a relative coordinate system for the number of points from two to twenty-one in three-
dimensional space based on the matrix of distances and absolute coordinates of one point, we used a
modification of the MDS algorithm developed by us. Additionally, the output coordinates had to be
adjusted to include the known absolute coordinates of a single point. At the same time, the emphasis
was on performing calculations on single-board computers of the Raspberry type Pi 4 B onboard
drones [19].
For the practical implementation of the task, a Python script was created based on the developed
algorithm, which uses the Scikit-learn library for the case of 10 UAVs. They are in the local
coordinate system and one control object is tied to the global coordinate system.
The MDS algorithm applied to the drone distance matrix allowed us to find the relative
coordinates of ten points in three-dimensional space (the local coordinate system of the swarm) and
then adjust these relative coordinates to include the known absolute coordinates of a single point
(Figure 8, Figure 9).
Figure 8: Representation of the matrix of distances be-tween eleven objects in the form of an
undirected fully connected graph.
Figure 9: Representation of the distance matrix between eleven objects in the form of an undirected
fully connected graph.
Thus, ten objects are in the local coordinate system without being tied to the absolute geographic
coordinate system.
The developed script applies MDS to a distance matrix to find the relative coordinates of points
(ten in the example) in 3D space and then adjusts those relative coordinates to include the known
absolute coordinates of a single point.
Finally, it plots the points on a 3D plot using Matplotlib, with each point labeled for
identification. You can configure the distances_matrix array and the absolute_coordinates variable
to represent different distances between points and different known absolute coordinates,
respectively (Figure 10, Figure 11).
Figure 10: Representation of the distance matrix be-tween ten objects in the form of an undirected
fully connected graph.
Thus, the modified MDS algorithm applied to the matrix of distances between drones allows
finding the relative coordinates of points in three-dimensional space (local coordinate system of the
swarm) for cases from one to twenty UAVs.
Figure 11: The result of solving the MDS problem for ten objects with the construction of a local
coordinate system.
With its application, the construction of a local system for positioning swarm elements is less
computationally complex than in alternative formulations [1-8].
The MDS algorithm applied to the matrix of distances between drones made it possible to find
the relative coordinates of ten points in three-dimensional space (local coordinate system of the
swarm). Experiments with other well-known dimensionality reduction methods, such as PCA, t-
SNA, did not show satisfactory results for Euclidean metrics, since the idea of construction is based
on the statistical characteristics of the data.
It should be noted that MDS should not be used to obtain the position of drones in the absolute
coordinate system, since MDS does not allow obtaining the correct orientation in space and has
reproduction errors even for the local coordinate system (as it is shown on example – Figure 12,
Figure 13).
Figure 12: An example of a test for ten points.
Figure 13: Example of reproduction of coordinates for the test presented in Figure 12.
4.6. Modeling the use of UWB devices for swarms of agricultural UAVs
The use of Ultra Wideband (UWB) radio modules in agricultural UAVs has the following features:
Using UWB on UAVs increases the communication range compared to indoors, but also
increases the delay in receiving data. For plant and soil monitoring, it is advisable to increase the
distance between swarm drones to more than 100 meters.
The use of radio modules on UAVs raises the question of the effect of Doppler shift on the speed,
distance and stability of data transmission.
The duration of the UWB pulse is much shorter (less than 1 millisecond) [21] than the symbol
transmission time in other technologies, so the UWB technology provides high accuracy
(approximately 0.01 m) and low delay in determining the distance between radio modules.
Positioning accuracy and delay for the most used technologies are [21]: Bluetooth (1-5 m, over 3
sec.), WiFi (5-15 m, over 3 sec.), RFID (1 m., 1 sec.) , GPS (5-20 m., 100 millisec.), 5G (10 m., less than
1 sec.).
UWB radio modules are based on the IEEE 802.15.4a protocol [22]. The Matlab 2024a software
package contains tools for modeling the physical level of this protocol. Modern UWB radio modules
use channel #5 (6489.6 MHz) and channel #9 (7987.2 MHz). Since the 6.5 MHz band is close to WiFi
frequencies that can be suppressed by anti-unauthorized drone protection systems, consider the
effect of Doppler shift on channel #9, which is less busy.
Typical speeds of drones are 50-70 km/h. To assess the influence of the Doppler shift, it is
advisable to consider a three-fold reserve relative to the average speed. For the transmitter carrier
frequency of 7987.2 MHz, the values of the Doppler shift will be as follows: 50 km/h: 370 Hz; 60
km/h: 444 Hz; 70 km/h: 518 Hz; 100 km/h: 740 Hz; 150 km/h: 1110 Hz. The maximum Doppler shift
of 1110 Hz was considered in the Matlab 2024a UWB channel simulator [21] and showed the
applicability of radio modules for the task of positioning a swarm of agricultural UAVs.
4.7. Analysis of advantages and disadvantages of statistical evaluation
methods for improving drone navigation
Kalman filtering methods [23], particle filter methods [24], deep learning of neural networks [25]
and combinations of the listed methods [24] are used to improve the navigation of individual drones
or a swarm of drones.
1. Kalman filter:
• Estimates the state of a dynamic system based on a series of measurements.
• Advantages: accurate, optimal, recursive.
• Disadvantages: not suitable for non-linear systems.
• Possible variations for non-linear non-Gaussian systems.
2. Particle filter:
• Works with nonlinear and non-Gaussian systems.
• Advantages: universal.
• Disadvantages: Less accurate than the Kalman filter.
3. Neural networks:
• Can smooth out noise and emissions.
• Advantages: flexibility, adaptability.
• Disadvantages: difficulty of training, the result may not converge to the true values.
Resume:
• The Kalman filter gives the best results for typical tasks.
• Neural networks can be useful for noise smoothing.
• A particle filter can be used for complex nonlinear systems.
Figure 14 shows the result of positioning data filtering simulation for different types of filters
with a noise variance of 10%. The FilterPy library for the Python language was used to model
the filters, and TensorFlow was used to model the neural network.
Combined filtering of drone positioning data
True positions
Noisy data
Kalman filter
Neural network
Particle filter
Position
Time
Figure 14: The result of modeling noisy navigation data and applying different types of filters.
Simulations have shown that the Kalman filter is the most stable and accurate when using the
FilterPy library for Python and TensorFlow for neural networks. Neural networks and particle filters
also showed significant results, but with some differences in accuracy and stability.
The following combined approaches are also used to improve the positioning and navigation of
drones [26].
1. Integration with simultaneous localization and mapping technologies – Simultaneous
Localization and Mapping (SLAM):
Use of Kalman filtering methods and its variants in combination with SLAM algorithms to
improve simultaneous localization and mapping. This allows drones to navigate more effectively in
unfamiliar or dynamic environments.
2. Use of hybrid methods:
Combination of deep learning methods with traditional filtering methods. For example, learning
neural networks to predict nonlinear system dynamics and using Kalman filters for final correction.
This increases positioning accuracy and stability.
3. Integration of multisensory information:
Fusion of data from different types of sensors: laser location (Light Detection and Ranging –
LiDAR), cameras, inertial measurement devices using filtering methods. This approach allows to
improve navigation accuracy in difficult conditions where one type of sensor may not be effective
enough.
4. Advanced optimization methods:
Use of adaptive algorithms, such as Particle Swarm Optimization (PSO) or genetic algorithms, to
adjust filter parameters in real time. This makes it possible to increase the adaptability and
robustness of the filter in conditions of a changing environment.
Conclusion: The Kalman filter showed the best result, stable over several simulation runs. The
neural network provided little smoothing of the outliers of the random process, while the particle
filter demonstrated over-smoothing for this example. However, in other simulations, its
convergence to true values was at the level of a neural network. The integration of new methods
and technologies, such as multi-sensor fusion, can significantly improve the accuracy and reliability
of drone navigation in various environments.
5. Conclusions
As part of the work, the task of forming and maintaining the structure and configuration of a
flock of agricultural UAVs in two-dimensional and three-dimensional settings was set. The
formalization of the given task has been completed. Considered possible methods of mapping the
graph formed by a swarm of UAVs into a two-dimensional Euclidean space by forming a basic
triangle to create a relative coordinate system. To determine the coordinates of the graph’s
remaining vertices on the plane and correspondingly increase the accuracy of local positioning, the
multilateration method is used and considered simplified options – application of the trilateration
algorithm, triangulation and a simplified algorithm for the degenerate case of two drones.
For the problem in a three-dimensional setting, the possibility of applying the multidimensional
scaling algorithm using the methods of multidimensional scaling/reduction of dimensions is
considered. The developed practical implementation of the created algorithm showed its efficiency
during practical experiments, allowing it to determine the swarm elements local positioning,
containing one to twenty elements.
The direction of further research is the development of an algorithm that allows, based on
determined local coordinates, to eliminate disturbances introduced into the swarm structure due to
internal or external factors, for example, the loss of one of the swarm elements [20].
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