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UAVs for formation-only passive positioning in formation flight for full details.
Abstract
Keywords
1. Restatement of the problem
1.1 Background of the problem
Since the UAV cluster should avoid external interference as much as possible when flying in formation, it is necessary to maintain electromagnetic silence as much as possible to reduce the emission of electromagnetic wave signals. In order to maintain the formation formation, a pure The method of azimuth passive positioning adjusts the position of the UAV, that is, some UAVs in the formation transmit signals, and the rest of the UAVs passively receive the signals, and extract the direction information from them for positioning to adjust the position of the UAV. Among them, the direction information received by the UAV is agreed to be: the angle between the UAV and any two UAVs transmitting signals. Each UAV in the formation has a fixed number, and in the formation The relative position relationship with other UAVs remains unchanged.
1.2 Problem restatement
This paper will solve the following problems:
Problem 1: The formation is composed of 10 UAVs, forming a circular formation, of which 9 UAVs are evenly distributed. Distributed on a certain circle, let a UAV be located at the center of the circle. UAVs are kept flying at the same height based on their own perceived height information. Establish a mathematical model to solve the following problems: 1. The UAV at the center of the circle
and The other two UAVs in the formation transmit signals, and the other UAVs with a slight deviation in position receive signals passively. When the position of the UAV that transmits the signal is unbiased and the number is known, establish the positioning of the UAV that passively receives the signal Model.
2. A UAV with a slightly deviated position receives the signals transmitted by the UAVs numbered FY00 and FY01, and also receives the signals transmitted by several UAVs in the formation. If the UAV launched There is no deviation in the position. In addition to FY00 and FY01, how many UAVs need to transmit signals to realize the effective positioning of UAVs? 3.
According to the formation requirements, 1 UAV is located in the center of the circle, and the other 9 UAVs are uniform Distributed on a circle with a radius of 100m. When the position of the drone is slightly deviated at the initial moment, the drone numbered FY00 and at most 3 drones on the circle are selected each time to transmit signals, and the rest of the drones receive Through multiple adjustments, the 9 drones are finally evenly distributed on a certain circle. The initial position of the drones is as shown in the title.
Question 2: When there is no one, the group makes other formation formations, such as conical formation flight, in which the distance between two adjacent UAVs on a straight line is equal. In the case of pure azimuth passive positioning, the position adjustment scheme of UAVs is designed.
2. Problem Analysis
2.1 Analysis of Problem 1 Thoughts
For the problem of UAV positioning proposed in this paper, we do the following analysis one by one:
Problem 1 mainly revolves around ten UAVs forming a circular formation and flying at the same altitude, passing through several UAVs If the signal transmitted by the aircraft is used to realize the pure azimuth passive positioning, then only the circumference problem on the plane needs to be considered at this time. According to the assumption of problem 1, the remaining nine aircrafts are evenly distributed on the circumference as far as possible. The position of the UAV is only related to the number of the UAV, and for the UAV whose position is slightly deviated, the distance and angle from the UAV to the center of the circle are temporarily unknown. We need to analyze the following three Analyze two situations:
1. When the UAV located in the center of the circle (FY00) and the two UAVs in the formation with known numbers and no deviation in position transmit signals, establish a positioning model for UAVs that passively receive signals. For this problem, We have two ideas to solve, one is to establish polar coordinates, according to the known direction information, according to whether the passive UAV is in the middle of the known UAV position, using the sine law of the triangle on the plane, the simultaneous cube The other is to use the triangulation positioning method to obtain the position of the UAV by solving a least squares problem. 2.
Compared with the previous question, we cannot know in this question except FY00 and other UAVs that transmit signals other than FY01, cannot be solved directly by triangulation positioning method. Therefore, the cross positioning method is used to solve the problem, and we increase the number of transmitting UAVs with slightly offset positions one by one until we can determine The position of the UAV.
3. The third question needs to adjust the position of the UAV whose initial position is biased. It is known from the first two questions that to realize the effective positioning of the UAV, at least two unmanned drones on the circumference are required. Since there is a deviation in the position of the drone that sends the signal, we can use the greedy strategy to make the error of the drone's position adjustment the smallest each time.
2.2 Analysis of the second problem
Question 2 is a dynamic programming problem. Different from Question 1, the UAVs in Question 2 must fly at the same height. Therefore, we need to use the dynamic programming model to find the UAV that makes the deviation as small as possible during the adjustment process. Scheme.
Three. Model assumptions
In response to the questions raised in this paper, we made the following model assumptions:
Assumption 1: When the UAV in the title deviates slightly, it is still roughly in the ideal position, and there is no UAV that deviates
beyond XX range ;
Assumption 2: Assume that the label of each UAV is fixed, and it is clear which UAVs transmit signals;
4. Description of symbols
See the table below for the commonly used symbols in this paper, and see the description in the text for other symbols
Meaning of symbols
Independent variable in complex number field Independent
variable in complex number field Complex variable
function
5. Modeling and solution
5.1 Modeling and solution of problem 1
5.1.1 Use polar coordinates to solve the specific position
Assume that the radius of the circle is, . UAV FY00 is the pole, and the polar coordinate system is established with FY00 as the endpoint and the ray passing through FY01 as the polar axis. Without loss of generality, assume that one of the transmitting UAVs on the circumference is FY01. After converting into polar coordinates, its The corresponding polar coordinates are). Since the UAVs that transmit signals are all in accurate positions, the position of another UAV that transmits signals is fixed according to the number, the number is FY0K, because in an ideal state, all UAVs Evenly distributed on the circumference, the corresponding coordinates are, where.
Next, assuming that the polar coordinate position of the UAV receiving the signal is, since the position of the remaining UAV is slightly deviated, both parameters need to be determined Next, let’s assume the included angles between it and the three UAVs that transmit signals. Assume that the included angles with FY00 and FY01 are , the included angles with FY00 and FY0K are , and the included angles with FY01 and FY0K are The included angle is. Next, we need to determine the polar coordinates of the corresponding receiving signal UAV through known information.
We connect the UAV with a slight deviation to the determined UAV position, using the law of sine, through the simultaneous equation group to solve the polar coordinates. Due to the angle involved, it is necessary to classify and discuss the different values of K:
case i) when K=2, there are no other drones between the two peripheral determined drones: distribution The visualization of the situation is shown in Figure 5.1 below:
Figure 5.1 There is no other UAV distribution map between the two peripheral UAVs.
It can be seen from the above Fig. 5.1: if (the position shown in green), we examine the triangle formed by FY00, FY01, FY0K and the composition of FY00, FY02, FY0K According to the sine law, we can get:
, (5-1)
Solve formula (5-1) to get:
; (5-2)
According to Figure 5.1, we can see that if (position shown in blue), we investigate FY00 and FY01 The triangle formed by , FY0K and the triangle formed by FY00, FY02, FY0K can be obtained according to the law of sine:
, (5-3)
is solved by formula (5-3):
; (5-4)
case ii) when K=3 , 4, 5, we need to judge whether the biased UAV is sandwiched between two UAVs at a certain position according to the size of : the visual analysis of the distribution is shown in Figure 5.2
Figure 5.2 Visual analysis of the distribution of drones in case ii
In Figure 5.2: If and at this time the drone is not between two aircrafts, which is consistent with the previous situation, the equation listed is:
(5-5)
is solved :
; (5-6)
In Figure 5.2: If and at this time the UAV is not between two aircrafts, which is consistent with the previous situation, the equation is listed as:
(5-7)
In Figure 5.2: If , the UAV is not between two aircrafts at this time, we examine the triangle formed by FY00, FY01, FY0K and the triangle formed by FY00, FY02, FY0K, according to the sine law: ; (5-8) case
iii
) When, the situation is corresponding to 11-K(1<K5), just replace with in the equation when 1<
K5
. Known information, we can use the angle-based measurement method triangulation positioning method to find the positioning of the drone. The principle of triangulation is mainly based on a measurement target point and two reference points with known coordinates to form a triangle. With the help of the length of the reference side in the triangle, measure the angle formed by the two reference points and the target point to find the distance and distance of the target point. coordinate.
According to the problem setting, the selected coordinates of the three known UAVs that transmit signals are: FY00, FY01, and FY0K, among which, in this question. Then its triangulation model is shown in Figure 5.3 below:
Figure 5.3 UAV distribution triangulation model
Since the angles on the circle opposite the upper chord of the circle are equal, according to the position of FY00 and FY01 and the angle between the point and FY00 and FY01, we can draw the trajectory of the point and uniquely determine a With the center of the circle, set the coordinates to , the radius to , and its equation is
(5-9)
Through FY00, FY01 are all on the circle, and the cosine theorem about the angle at O1 in triangles O1, FY00, and FY11, the following equation can be obtained Group:
(5-10)
so as to be able to solve: and.
Similarly, we can use and establish the following relationship:
(5-11)
(5-12)
to solve and and and, and their analytical solutions are:
(5-13)
( 5-14)
(5-15)
because The point is the intersection point of three circles. As the obtained coordinates, we combine the equations:
(5-16)
Expand the above equations and simplify them to get:
(5-17)
Among them, ,,
record the above equations as:
(5-18)
At this time, solving the coordinates of the target point is transformed into solving the least squares problem:
(5-19)
The corresponding analytical solution is: . In this way, a mathematical model for passively receiving signal UAV positioning is established, and the effective positioning of UAV is determined according to the received direction information.
5.1.3 Analysis of UAV's Effective Positioning
First, when there is only one UAV whose position is not determined to transmit signals, the conditions we need cannot be met. Now we consider two aircrafts.
Since in the second question, the numbers of the UAVs transmitting signals on the circumference are unknown except for FY00 and FY01, therefore, the two-station cross positioning calculation method is used. Assume that the two UAVs transmitting signals on the circumference are: FY0K1, FY0K2, and their coordinates are respectively and, among them, . In order to find the coordinates of the UAV receiving the signal, we consider using the triangulation positioning method for the four UAVs that transmit the signal shown in the table below, and we can obtain the coordinate expressions of the UAV receiving the signal about , , and . Since the three are the same, and can be eliminated, the specific orientation, classification and drone number obtained are shown in the following table:
Table 5.1 Classification