Research on Modeling and Simulation of the anti-submarine patrol aircraft taking call search task using the circular sonar buoy array

According to the characteristics of the call search task, this paper analyzes the submarine position dispersion model, and puts forward the evaluation model of the submarine buoy array search probability based on the method of the circular sonar buoy array. The correctness of the model is proved by the computer simulation method, and the results show that the results of the submarine search are closely related to the submarine position, and the method can provide a useful reference for the optimal use of sonar array.


Introduction
Compared with the helicopter, the anti-submarine patrol has the advantages of high speed, long range, long life time, and a variety of anti-submarine combat equipment on board. Therefore, it becomes an important aviation anti-submarine force.
Call anti-submarine warfare is one of the most basic and most commonly used methods. It refers to the antisubmarine patrol aircraft at the airport or designated airspace, flies to the initial position of the submarine. After obtaining information about the activities of the enemy submarine, then takes combat operations about searching, locating, tracking and attacking submarine.
In order to accomplish this task, some submarine detection equipment is needed, such as radar, photoelectric equipment, radio sonobuoy, mad, etc. Among them, the radio sonobuoy is a good and effective equipment, and it is subject to geographical constraints, more use of convenient exploration, so it attaches great importance by navies in most countries.

A distributed model of submarine position for call search 2.1 The initial position of the submarine during the call search
Due to the uncertainty of the submarine position data from other sources, according to the central limit theorem, it can be considered that the initial position of the submarine obeys two-dimensional normal distribution 2 0 (0, ) N  , and the distribution center is at the initial position (coordinate origin), that is, the mathematical expectation value is 0, and the joint probability density function of the initial position (X, Y) is

Location dispersion of submarine after call search
In the time, which is from the initial information of the submarine position to the anti-submarine force arrives at the search area, the position of the submarine continues to expand. It is centered on the initial position, and the speed of the original navigation The size of the dispersion area is related to the speed and the dominant time of the submarine (the call time)Therefore, the current position dispersion of the submarine should contain two parts: the initial dispersion and the uncertainty caused by the motion。 In most cases, the speed of the submarine is unknown. When the submarine speed is unknown and the course Where, the 2 1  is still unknown, and the velocity distribution law of submarine is analyzed as following, in order to solve the 2 1  value: The position probability density function of the submarine, caused by the uncertainty of velocity and course , can be expressed in polar coordinates as following: It can be seen from the formula and  obeys the uniform distribution on the interval.
From the formula (4), we can obtain the probability density function of submarine speed V: , then the upper form can be expressed as following: The economic speed of the submarine se V is taken as the mean of the velocity V distribution function, according to the definition of mean: Thus we can get: So far, the probability density function of the submarine position is obtained when the anti-submarine patrol machine starts to search through time 0 t :

Deployment Circular sonar array to call search the submarine
In the anti-submarine patrol aircraft taking call the search task, for the situation about known the initial target location, it placed a sonar buoy in the general dispersion center as soon as possible, in order to shorten the time of the submarine was found for the first time, and then deploy some sonar buoys around the center by a circular array, and try to cover a large area as large as possible. Considering that there is little difference between the circular and annular array in the actual operation (the main difference is that the annular array can be omitted from the center point buoy), the search efficiency of the call anti-submarine warfare is considered according to the circular array.
When determining the number of buoys to be used, it should be considered that there is a certain spread of each sonar buoy. In order to ensure reliable detection, the distance between two sonar buoys should be less than 2 times buoy's range. The size of the sonar buoys array is related to some elements, such as lost time which the anti-submarine patrol aircraft and submarine formation lost contact (call time), the submarine speed, the number of sonar buoy on the plane, and so on. The typical circular (or annular) search array has 5, 7, 9 or 11 sonar buoys, and 5 and 7 circular search arrays are as shown in Figure 1 and Figure 2.
The flight path is depended on both the distance between two sonar buoy and the plane turning radius. And the deploying order is as much as possible to make the flight route optimization, that is, deploying array time should be short, and the aircraft's large slope turn and maneuver should be less.
Taking 5 passive omnidirectional sonar buoy circular array for example, the deployment sonar buoys process is illustrated as following: When the probe range of sonar buoy is 3.8km and the interval between the buoys is 7km, the first buoy should be layed out in the position deviate from the submarine initial point 5km, and then throw second buoys after fly before 5km, then the third also after fly ahead 5km; After the aircraft to turn left at the 5km radius, when flying through the fourth buoy locations above, it throws the buoy; when the direction of flight and the original direction of 90 degrees and from the first buoys to 5 km, throw the fifth buoys.
The plane flies almost in a great circle in the whole laying array process (see Figure 1), it is mainly due to the aircraft turning limit, and this is one of maneuvering performance difference between the large and medium sized fixed wing anti-submarine patrol aircraft and antisubmarine helicopter. According to the performance of each type of aircraft, there are differences in the corresponding turning radius. For the modern antisubmarine patrol aircraft, when anti-submarine patrol aircraft's command and control system will outputs the flight path of the array for Tactical Navigation System, flight control system can automatically control the plane array process.

Model validation
Assuming that: the initial distribution of the target is 1 nautical miles (mean square error) when in call search submarine, and the speed of the anti-submarine patrol aircraft is about 400 km/h. The detected range of a single passive sonar buoy is calculated according to the hydrological conditions of the search area Because the single sonar buoy's detection range is limited, in the actual aviation anti-submarine search action, it usually lays a set of sonar buoys to form a search array, which is in accordance with the spread law of submarine's position, and cover the search area as much as possible. In this paper, the simulation verification is based on 9 round sonar buoy array.
First of all, according to the submarine distribution law as shown in chapter 1.2, and combination with the characteristics of the search task, the plane lays out a circular search array composed of multiple buoys. For simulation of the search process in computer, the specific steps are as follows: (1) Input the initial condition. Including the submarine speed (0-24kn), each buoy's coordinate parameter which composed of search array.
(2) Generate the initial position of the target, when in call search task.   Figure. 3 The search probability curve of the 9 circular sonar buoy array for the unknown submarine Figure 3 shows the results of the evaluation of the search probability for the circular array of the 9 sonar buoys, using the formula of search probability and the method of computer simulation. The result is shown in figure 3, that the curve with cross mark is the result of the model formula, and the curve of diamond mark is the result of computer simulation.
As is shown in Figure 3, the result of the model calculating is consistent with the simulation results when the anti-submarine patrol machine performs the call search task.
In order to make the calculation results of the model approximate to the simulation results, the corresponding correction coefficients can be added to the model formula. The corrected results are shown in

Conculsion
The efficiency calculation of the search for the sonar buoy array is a complicated problem, The paper established the calculation model of circular sonar buoy array searching submarine, using the proposed submarine position distribution model. The result of respectively assessment model calculation and computer simulation method for searching probability calculation is consistent, which proves the correctness of the model. It also shows that the evaluation results of submarine is closely related to the submarine position distribution, and it should be selected reasonably according to the situation. In the practical use, the correction term can be added to the model, which makes the calculation results of the model and the simulation results closer.