Research on Passive Location Strategy of UAV

Aiming at the problem of how to reasonably correct the position of UAV formation, we take the circular formation composed of 10 UAVs and the conical formation composed of 15 UAVs in the same plane as examples to study how to adjust the number and angle of UAV signal transmission so that the UAVs in the entire formation are in the designated position. This paper uses geometric model, greedy strategy and other methods, uses python, matlab and other software to obtain a reasonable and effective mathematical model, and makes a test and evaluation. The research results show that no matter circular, conical or other graphics, we can construct a triangle by connecting the transmitting signal and the receiving signal UAV, and solve the problem to obtain the best transmitting signal angle and the minimum number of sorties required; If there is no triangle between the UAV transmitting the signal and each UAV receiving the signal, the UAV receiving the signal will not reach the designated position. Nowadays, the use of UAVs is more and more extensive. The cooperation of multiple UAVs can complete many functions such as detection and performance, which is also the significance of our research on how UAVs can better receive signals.


Introduction
In today's world, remote sensing technology has become deeply integrated into various aspects of life.On the military front, maritime reconnaissance ships employ remote sensing for surveillance [1], and unmanned aerial vehicles (UAVs) monitor sea areas and islands [2].In civilian applications, UAV remote sensing systems play a crucial role in environmental protection initiatives [3].The studies suggest that UAVs have pushed the boundaries of remote sensing technology, enabling single UAVs to achieve precise object positioning.
With the advancements in UAV remote sensing and industrial applications [4], a pertinent question arises: Can an array of UAVs achieve even more accurate positioning?How efficient is the transmission and reception of signals between these UAVs?
Considering that UAV formations are typically distributed in the same plane, we assume both circular and conical formations to be in the same plane.Categorizing UAVs into those transmitting signals and those receiving signals, we draw insights from the multi-station bearings-only passive location algorithm [5] to delve into two key issues: Two significant inquiries are posed.The primary query revolves around a formation consisting of 10 UAVs, shaping a circular configuration.Out of these, 9 UAVs (No. FY01~FY09) are uniformly arranged along the circumference, while the remaining one UAV (No. FY00) is positioned at the center of the circle.Leveraging altitude information sensed by the UAV, it maintains a consistent flying altitude.
In the first query, the UAV stationed at the circle's center (FY00) and the other two UAVs broadcast signals, with the remaining UAVs, slightly offset in positions, passively receiving signals.When the signal-transmitting UAV's position is devoid of deviation, and its number is known, the formulation of a positioning model for the passively receiving UAVs is ascertained.
The second inquiry pertains to the UAV slightly deviated in position, receiving signals from UAVs labeled FY00 and FY01, alongside signals from several UAVs with unidentified numbers within the formation.Assuming no deviation in the signal-transmitting UAV's position, in addition to FY00 and FY01, a determination of the minimum number of UAVs necessary for effective positioning is mandated.
The third question adheres to the formation criteria: one UAV stationed at the circle's center, and the remaining nine UAVs uniformly spread along the circumference with a radius of 100 m.In the event of an initial deviation in the UAV's position, a practical adjustment scheme is sought.This involves, through iterative adjustments, the selection of the UAV labeled FY00 and at most 3 UAVs on the circumference to broadcast signals.The other UAVs subsequently adjust to their ideal positions based on received direction information (disregarding the duration of each adjustment), ultimately achieving an even distribution of the 9 UAVs along a specific circumference.The adjustment of UAV positions is conducted solely based on received direction information, utilizing the provided data.A specific adjustment scheme is requested.
The second issue explores variations in UAV cluster formations during actual flight, such as a conical configuration (with equal straight-line distances, e.g., 50 m, between two adjacent UAVs).Considering the bearings-only passive location scenario, a designed UAV position adjustment scheme is presented.

Analysis of question 1
2.1.1Analysis of the first question.The given scenario outlines the arrangement of UAVs, where 9 UAVs form a circle, and 1 UAV is positioned at the circle's center, creating a circular formation flying at a constant altitude.In this setup, the task is to have the stationary UAV FY00 and two UAVs on the circumference transmit signals.UAVs in other positions may experience positional deviations.In an ideal scenario, UAVs should be uniformly distributed along the circumference, and deviations signify a shift from their original positions, which could occur within, outside, or on the circle.To establish a positioning model with the fixed UAV acting as the signal source and other UAVs as signal receivers, a systematic classification discussion and analysis are essential.

Analysis of the second question.
In the title, it's established that the fixed UAVs FY00 and FY01 transmit signals, with FY00 known to be at the center of the circle formation.The goal is to determine the additional UAVs required for signal transmission, specifically finding the minimum number for effective positioning.Analyzing the problem involves methods like drawing and enumerating, examining if the receiving UAV can effectively determine its location when two or three transmitting UAVs are present.Prior to analysis, the assumption is made that UAVs capable of transmitting signals can also receive signals from each other.The specific approach involves establishing and solving the model.

Analysis of the third question.
In the study of the multi-station bearings-only passive location algorithm [5], the problem involves knowing the polar coordinates of each UAV.The task is to identify no more than three UAVs on the circumference and one UAV at the center of the circle for signal transmission.All UAVs undergo position adjustments based on received direction information, achieving even distribution on the circumference, excluding UAV 0. The approach involves establishing a geometric model of a circle with UAV 0 as the center.To simplify subsequent solutions, set the radius of UAV 0 to 100m.The process involves obtaining original correct key angles through the intersection of two sets of 3-point positioning circles each time, allowing UAVs to calibrate these angles to meet requirements.

Analysis of question 2
This question changes the UAV distribution on the circumference in the first question to the UAV distribution on the three sides and inside of an equilateral triangle.Compared with problem 1, it is more convenient to establish a plane right-angle system in this problem.After calculating the coordinates and distances of each point through special angles such as 30 °, 60 °, 90 ° and 120 °, the angle of the deviated UAV can be adjusted continuously through calculation, so that it can gradually be in the same line with the UAV transmitting the signal or maintain a specific included angle, so as to correct the driving path of the deviated UAV.

Model Assumptions
(1) Assume that the altitude of UAV flight is not considered in all cases (2) It is assumed that the position of the UAV transmitting the signal does not deviate (3) It is assumed that the signals between the receiving UAVs will not affect each other (4) It is assumed that both the transmitting UAV and the receiving UAV can work normally Symbols and descriptions: α_ 1, α_ 2, α_ 3 The included angle generated by transmitting signals between two UAVs ρ Distance from UAV to center of circle θ The polar coordinate angle of the position with FY00 as the center and FY01 as the X axis of the polar coordinate system

Question 1. Research on interference classification and recognition mechanism based on wireless
sensor networks [6], and from the known figure (1) in the title, we can think of FY00 as a fixed signal transmitting UAV, FY01, FY03, and FY02 as fixed UAVs in a circular formation and passive signal receiving UAVs with a slight deviation in position.According to geometry research in middle school [7] and the knowledge of Wireless Sensor Network [8]: We analyze by drawing, and make the center point O, point B and point C of the circle as fixed transmitting signal points, and point A as possible deviation from receiving signal points.∠ BAO is the included angle generated by the stable point transmission signal α 1；∠ OAC is the included angle generated by the stable point transmission signal α 2. ∠ BAC is the included angle generated by the stable point transmission signal α 3. The UAV surrounding model in the topic is converted into abstract geometric figures, and the variation of the included angle of the deviated UAV under different deviation conditions can be obtained after discussion.2), when receiving signal point A slightly deviates from the outside of the circular formation, set its position as point F, connect OF to point M, connect BM, MC, BC, BO, BC, OC, and make ∠ BFO= α 1',∠OFC= α 2',∠BFC= α 3'.According to the theorem of circle: in a circle, the circumference angles of the same chord are equal, so the circumference angles of the same chord BC ∠ BAC can be obtained= α 3=∠BMC.Because F point is outside the circle, ∠ MBF+∠ MCF can be obtained from the "dart" model+ α 3'=∠BMC= α 3, therefore α 3'＜ α 3. It can be obtained in the same way α 1'< α 1, α 2'< α 2'.Through the above analysis, when the UAV receiving the signal passively deviates from the circular formation outside the circle, the relative angle between its latest position and the three UAVs transmitting the signal is smaller than that formed when the UAV receiving the signal passively is on the circle.3), when receiving signal point A slightly deviates from the inside of the circular formation, set its position as point P, connect OP, BP, CP, OB, OC, BC, and still make ∠ BAO= α 1, ∠OAC= α 2,∠BFC= α 3 ', then α 1 'is ∠ BPO, α 2 'is ∠ OPC, α 3 'is ∠ BPC.Using the "dart" model, in ABO, α 1'=∠PBA+∠AOP+ α 1. Therefore α 1< α 1 ', similarly α 2< α 2', α 3< α 3'.Therefore, when the UAV receiving the signal passively is in the circular formation, the relative angle between its latest position and the three UAVs transmitting the signal is larger than that formed when the UAV receiving the signal passively is on the circular formation.4), if the receiving signal point A deviates slightly but is still on the circular track, and the point is outside the arc BC, the point on the circle is set as point E. Connect BE, OB, OC, BC, CE, AE because α 3 corresponds to the superior arc, and α The arc corresponding to 3 'is the inferior arc, so when the UAV receiving the signal passively deviates from the circular formation and is still on the arc, the relative angle between its latest position and the three UAVs transmitting the signal is smaller than the angle formed when the UAV receiving the signal passively is on the circle.To sum up, we summarized all the situations to get the following data, as shown in Table 1.
Table 1.Location and distribution

𝛼𝛼 Location 𝛼𝛼 Comparison between angle and angle without deviation On the circle and the offset position is on the left
On the circle and the offset position is on the right In order to optimize the results, we quantify the range of the deviation, and use Python to iterate the distance formula several times to calculate the position of the final UAV deviation from the received signal, as shown in Table 2.  Finally, only one expression containing a and b can be obtained, and its specific value cannot be obtained, so it cannot be located.Assuming that all UAVs in a perfect situation have no deviation and are uniformly distributed, the angle between any two adjacent UAVs is equal to a when all UAVs in a circular formation are connected to the center of the circle.There are 9 UAVs receiving or transmitting signals on the periphery.If the deviation is uniformly distributed, it is a=360 °/9=40 °.It can be seen from the question that the UAV with a slight deviation in a certain position receives the signals transmitted by the UAV numbered FY00 and FY01, that is, YF00 and FY01 are fixed and known.We assume that the polar coordinates are established.At this time, the polar coordinates of FY00 are set as (0, 0), and the polar coordinates of FY01 are (R, 0).In addition, a UAV with a deviation is required to transmit signals.

Establishment and solution of the third question model: Model geometry
According to the title, the distribution of UAVs is as Table 3: Table 3. UAV number and coordinates The number of the UAV Polar coordinates (m, °) 0 (0, 0) And the radius of the final circular array is not required, so in order to facilitate the positioning and debugging of the UAV with subsequent deviation, we now assume that the position of the UAV with No. 0 as the center of the circle, and the distance from the UAV with No. 0 to No. 1 as the radius, establish the circular model, and then mark the correct position of the UAV after correction and the given position on Figure (8), Among them, the red dot represents the unmanned aerial vehicle formation with deviation, and the blue dot represents the unmanned aerial vehicle formation with uniform distribution without deviation respectively, in order to facilitate the subsequent study of this problem, Assume that the UAV number in the correct position is 1, 2, 9.The points with deviation are 2 ', 3', 9'.Next, we will adjust and position the UAV with number K, and the process is as follows: Step 1: Assuming that the current position of the UAV is at the point K 'with deviation, the connection point K', point 1 and point 2 can get ∠ 1K'2.At this time, we know the size of ∠ 1K2.Refer to the multiple greedy routing algorithm in sensor networks [9] and a vehicle location method based on mobile communication propagation model [10] and related invention patents [11], so we use the greedy strategy to make the UAV adjust the position all the time and finally reach ∠ 1K'2=∠ 1K2, as shown in the following figure(17):

Conclusion
When studying the reception and arrangement of UAV signals, we found that no matter what shape of UAV formation, in order to achieve the positioning of one UAV, at least three UAVs need to send signals to it at the same time, and the distance generated when transmitting the signal can form a triangle through the line to make this UAV in the ideal position.When the UAV formation is circular, the auxiliary circle can be continuously made in the circle to correct the position of the UAV.Our team did not consider the time spent in correcting the position of the UAV group.At the same time, in order to simplify the calculation, we did not consider whether it would take less time for several more UAVs to transmit signals.It needs to be discussed in the follow-up study.In the future, the use of UAVs will certainly increase.Compared with a single UAV, multiple sorties of UAVs can do more sophisticated work.Multiple sorties of UAVs can receive signals by referring to the method adopted in this paper.When the formation shape is irregular, the formation can be divided into several areas using the idea of integration, and the familiar model can be used to debug the UAV to make it reach the designated position.To sum up, based on our current model and conclusions, we can continuously construct a triangle between the distance formed by the transmitting signal and the receiving signal UAV according to the different purposes of UAV use, so that all UAVs can be aligned and positioned to achieve our purpose of use.

Figure 1 .
Figure 1.UAV receiving and transmitting signal diagram

Figure 2 .
Figure 2. The received signal UAV deviates and is outside the circle As shown in Figure (3), when receiving signal point A slightly deviates from the inside of the circular formation, set its position as point P, connect OP, BP, CP, OB, OC, BC, and still make ∠ BAO= α 1, ∠OAC= α 2,∠BFC= α 3 ', then α 1 'is ∠ BPO, α 2 'is ∠ OPC, α 3 'is ∠ BPC.Using the "dart" model, in ABO, α 1'=∠PBA+∠AOP+ α 1. Therefore α 1< α 1 ', similarly α 2< α 2', α 3< α 3'.Therefore, when the UAV receiving the signal passively is in the circular formation, the relative angle between its latest position and the three UAVs transmitting the signal is larger than that formed when the UAV receiving the signal passively is on the circular formation.

Figure 3 .
Figure 3.The received signal UAV deviates and is within the circle

Figure 4 .
Figure 4.The receiving signal UAV deviates and is on the left side of point B

Figure 5 .
Figure 5.The receiving signal UAV deviates and is on the right side of point BTo sum up, we summarized all the situations to get the following data, as shown in Table1.Table1.Location and distribution

4. 1 . 2
The second question of question 1.Since the UAVs numbered FY00 and FY01 in the title are fixed sending signal points, we assume that the UAV FY00 is the center point O, and the UAV FY01 is the fixed point A on the circular formation without deviation.If the circular formation radius is R, then OA=R.4.1.2.1 Situation IAs shown in Figure(6),When 0 transmitting signal UAVs are added, if the receiving signal UAV is unbiased, it is just on the circular formation, marked as point F, and BF=R; If the received signal UAV is biased, the position of the UAV is marked as F 'and ∠ OFA is set= α 1,∠OF'A= α 2. At this time, the angle cannot determine the aircraft positioning.Let OF '=a, AF'=b, known OA=R, ∠ OF'A= α 2.

Figure 6 .
Figure 6.Only two UAVs transmit signals

Figure 7 .
Figure 7. Using cosine theorem to solve the distance generated by three UAV transmitting signals b) Model II Sine theorem to solve geometric modelAssuming that all UAVs in a perfect situation have no deviation and are uniformly distributed, the angle between any two adjacent UAVs is equal to a when all UAVs in a circular formation are connected to the center of the circle.There are 9 UAVs receiving or transmitting signals on the periphery.If the deviation is uniformly distributed, it is a=360 °/9=40 °.It can be seen from the question that the UAV with a slight deviation in a certain position receives the signals transmitted by the UAV numbered FY00 and FY01, that is, YF00 and FY01 are fixed and known.We assume that the polar coordinates are established.At this time, the polar coordinates of FY00 are set as (0, 0), and the polar coordinates of FY01 are (R, 0).In addition, a UAV with a deviation is required to transmit signals.4.1.3Establishment and solution of the third question model: Model geometryAccording to the title, the distribution of UAVs is as Table3:Table3.UAV number and coordinates

Figure 8 .Figure 9 .Figure 10 .Figure 11 .Figure 12 .Figure 13 .Figure 14 .Figure 15 .
Figure 8.Comparison chart of UAV formation with and without deviation Step 1: Determine the position of the UAV transmitting the signalAccording to the assumption that the known position of the UAV numbered 0 and 1 is correct, and the UAV numbered 2 can both transmit and receive signals.If we want to correct the position of the UAV numbered 2 'to the position of the UAV numbered 2, we need the UAV to move and position so that the final position is ∠ 02'1=∠ 021=70 ° and ∠ 2'01=∠ 201=40 °.As shown in Figure(9), figure(10)

4. 2
Question 2 It is assumed that a total of 15 UAVs are in conical formation, and all UAVs have initial time errors.Assuming that the numbers 1, 2 and 3 are UAVs transmitting signals, according to the idea of the third question, we can obtain the formation coordinate diagram of UAVs without errors by calculating and drawing the figure first as follows, As shown in Figure (16):

Figure 16 .
Figure 16.Simplified conical formation diagram First of all, since UAVs 1, 2 and 3 can receive or transmit signals, the correct position of UAVs 1, 2 and 3 can be determined through triangle positioning.Next, we will adjust and position the UAV with number K, and the process is as follows:Step 1: Assuming that the current position of the UAV is at the point K 'with deviation, the connection point K', point 1 and point 2 can get ∠ 1K'2.At this time, we know the size of ∠ 1K2.Refer to the multiple greedy routing algorithm in sensor networks[9] and a vehicle location method based on mobile communication propagation model[10] and related invention patents[11], so we use the greedy strategy to make the UAV adjust the position all the time and finally reach ∠ 1K'2=∠ 1K2, as shown in the following figure(17):

Figure 17 .
Figure 17.Position adjustment of UAV numbered k using greedy algorithm Step 2: The connection points K ', 1 and 3 can get ∠ 1K'3.At this time, we also know the size of ∠ 1K3, so we also use the greedy strategy to make the UAV adjust its position all the time and finally reach the condition of ∠ 1K'3=∠ 1K3, that is, move on the circle.As shown in Figure (18):