Mathematical Model of Receiver Location with Unknown Transmitter Number

The paper uses the theory of elementary mathematics, uses the angle and side length relationship between multiple UAVs, and applies the sine theorem and cosine theorem to establish the bearings-only passive positioning model. The model can be solved by only two transmitters for accurate positioning, Through practice, the model has a high fitting degree, and finally realizes the positioning of the UAV.


INTRODUCTION 1.1 Background of the problem
When the UAV cluster is flying in formation, in order to avoid external interference, electromagnetic silence should be kept as much as possible and electromagnetic wave signals should be transmitted outward less.][4][5] Bearing-only passive positioning refers to the positioning of UAVs by extracting direction information from the signals transmitted by some UAVs in the formation and passively received by other UAVs.

Questions raised
A circular formation composed of 10 UAVs, of which 9 UAVs (No. FY01-FY09) are evenly distributed on a circle, and the other UAV (No. FY00) is located at the center of the circle.The UAV keeps flying at the same altitude based on its own perception.See Figure1 Schematic diagram of circular UAV formation.

Figure 1 Schematic diagram of circular UAV formation
The UAV with a slight deviation in position received signals from the UAVs numbered FY00 and FY01, and received signals from several UAVs with unknown numbers in the formation.Assuming that there is no deviation in the position of the UAV transmitting the signal, how many UAVs need to transmit the signal in addition to FY00 and FY01 to realize the effective positioning of the UAV?

Core concepts
Suixing formation: "Suixing" means to be able to successfully and successfully perform tasks or carry out certain actions.Suixing formation flight means that UAVs maintain a certain formation flying in Suixing formation.
Passive location: obtain the position of the target without transmitting the electromagnetic wave radiated by the target.UAV uses passive location technology to determine and adjust its own orientation during the flight process, so as to complete the task with a specific flight formation, that is, the bearings-only passive location of UAV.
Transmitter: UAV with known azimuth, used to transmit signals to determine the position of other UAVs.
Receiver: UAV with unknown orientation receives signals from transmitter to determine its position and adjust.

Problem decomposition
According to the requirements of the topic, the transmitter should be determined as little as possible to realize the effective positioning of the UAV, and the receiver may be at any position on the circumference.When the receiver is at different positions, it has different position relations with the transmitter, resulting in different edge and angle relations in the model building, so we will discuss in two cases.Finally, the original problem can be decomposed into the following three small problems: [6][7][8][9][10] (a) When only the positions of FY00 at the center of the circle and FY01 at the circumference are determined, the angle between the two transmitters on the circumference and FY00 becomes unknown.On the premise of ensuring that the number of transmitters determined is as small as possible, how many more UAV positions need to be determined to realize the effective positioning of UAV?That is, how many known conditions need to be created at least to solve the determination of receiver azimuth?(b) If the receiver is on both sides of the transmitter, how to use the relationship between the receiver and the transmitter to establish a mathematical model?(As shown in Figure 2

Problem hypothesis
Only the numbers of FY00 and FY01 transmitters are determined, so the angle between the two transmitters on the circumference and FY00 becomes unknown.Due to the lack of one solution condition (the center angle of the circle is unknown), we need to theoretically select two more transmitters on the circumference, and build a geometric positioning model based on the four-point azimuth angle.According to the angle and side length relationship between the four transmitters (four points) and the receiver, we can use the sine theorem and cosine theorem to establish a nonlinear equation system based on the bearings-only angle to solve the position information of the receiver.

Establishment of mathematical model of the problem 1) Solution of problem (a)
Since only the numbers of FY00 and FY01 transmitters are determined, the angle between the two transmitters on the circumference and FY00 becomes unknown.Due to the lack of one solution condition (the center angle is unknown), we need to select two additional transmitters on the circumference theoretically.