2020 China Postgraduate Mathematical Modeling Contest Question D:
Collaborative confrontation of drone clusters. A
new generation of artificial intelligence technology and autonomous technology are rapidly advancing to the battlefield, which will give birth to new combat forces and subvert the traditional war mode. The future war will be intelligent warfare. As an important form of intelligent combat, UAV cluster operations are emerging. Through multiple UAVs coordinated reconnaissance, coordinated detection, coordinated tracking, coordinated attack, coordinated interception, etc., to jointly complete more complex combat tasks.
Now consider the problem of coordinated confrontation between red and blue UAV clusters in a plane area. As the attacker, the blue side hopes to break through the interception of the red drone and successfully reach the destination to carry out military operations; the red side hopes to complete the interception of the blue drone in a given area to prevent the blue side from penetrating defenses . The confrontation area discussed in this question is agreed to be the rectangular area shown in Figure 1. The depth of attack is the distance between them. The flight trajectory of the blue drone cannot go over and on both sides, that is, the attack channel (penetration corridor) is considered. The bandwidth has a limited constraint, the channel bandwidth is the distance between them. The speed of the blue drone is, and the minimum turning radius is; the speed of the red drone is, and the minimum turning radius is; the speed of the red and blue drones remains unchanged, and the direction of movement can be at any time according to the needs of the maneuvering strategy. Change, but limited by the turning radius.
Figure 1 Schematic diagram of two red drone clusters intercepting a blue penetration drone.
This question only discusses the confrontation between red and blue on the plane, that is, the red and blue drones are flying at the same altitude. . Both drones are equipped with detection devices and data links, which can accurately detect the real-time position of the other drone and the own drone. In view of the blue drone's advantage in maneuvering speed, the red side considers the coordination of multiple drones, that is, through the number of advantages to compensate for the performance disadvantages, so as to improve its own countermeasures. The red drones performing interception tasks are carried by the carrier to the attack position, and then a certain number of drones are launched according to the needs of the confrontation to form a drone cluster to confront the blue penetration drones. In the confrontation state shown in Figure 1, the red team is to prevent the blue drone from penetrating. Two drone clusters are formed by the carrier and each launching 5 drones to perform the interception mission. When the distance between the blue penetration drone and at least 2 drones of the red side is less than that, it is considered that the red side successfully intercepted the blue penetration drone. At the same time, according to mission requirements, the blue drone must cross the border inside, otherwise it will be deemed as a failure to penetrate. Both the red and the blue want to make full use of their own advantages and achieve the purpose of penetration and interception through the use of optimal maneuvering strategies.
The red drone cluster can fly in formation according to actual needs. As shown in Figure 2, the red drone cluster adopts one of the formations. The positions of the 5 drones are approximately distributed on a circle. The distance between two adjacent drones is the same. For the needs of control, communication and mutual collision avoidance, the distance between any two drones of the Red Party must be greater than that between each drone and at least two drones in the cluster. The distance between the red carrier and at least one UAV in the UAV cluster should not exceed, and the distance from any UAV must be greater than the distance from the blue penetration UAV for safety reasons. Need to be greater than. The speed of the red carrier is such that the turning radius is not less than. The flight trajectory of the red carrier and UAV is not restricted by the boundary.
Figure 2 The schematic diagram
of the drone cluster formation composed of drones launched by the red carrier. For the convenience of description, the number of the drone launched by the carrier is recorded as, so the numbers of the 5 drones in Figure 2 are marked as,,, ,. The number of the drone launched by the carrier is recorded as, and the rest can be deduced by analogy. The position distribution of the UAV cluster at the initial moment is evenly distributed on a circle. In practice, it takes a certain amount of time for the carrier to deploy the UAV cluster. For the sake of simplicity, this time limit is not considered for the time being in this contest. It can be regarded as an instant deployment of the UAV cluster and the relevant constraints are met. During the confrontation, it is not required to fly in a certain fixed formation throughout the course, and the formation can be adjusted at any time as needed.
Ask your team to study the following problems by establishing a mathematical model:
Problem 1 As shown in Figure 3, the center of the circle of the two UAV clusters against the beginning of the red side are located at and respectively, and the circle radius is, where,. Try to model and analyze where the blue drone is in the rectangular area, no matter what pursuit strategy the red drone adopts, the blue drone can always adopt appropriate strategies to avoid interception by the red drone and achieve success Penetration; discuss the corresponding optimal penetration strategy for the blue UAV.
Figure 3 Schematic diagram of the initial center position of the red drone cluster
Problem 2 As shown in Figure 4, the blue penetration drone is located at the center of the boundary at the beginning of the confrontation, and the circle centers of the two drone clusters on the red side are located at and respectively. The radius of the circle is, where the sum is on the boundary, and the specific position is determined according to needs. Try to model and analyze whether there is a lower limit of channel bandwidth. When the actual channel bandwidth ratio is large, the blue drone can definitely break through the interception of the red drone cluster; in this case, the blue drone has the shortest time Penetration strategy.
Figure 4 Schematic diagram of the initial position of the red and blue UAVs in the confrontation
Problem 3 Each carrier of the red side can launch 10 UAVs in two waves, forming two UAV clusters to carry out the interception mission, each UAV The number of drones in the cluster is not less than three. The initial formation of the UAV cluster in each wave of launch is shown in Figure 2. The distance between the carrier and the center of the circle is as follows. The formation of the UAV cluster can be adjusted as needed, but it must meet the corresponding requirements. Spacing constraints. As shown in Figure 5, at the beginning of the confrontation, the blue drone was located in the center of the boundary with a bandwidth of the channel; the two red carriers were located at points and points on the boundary, and began to launch the first wave of drone clusters, carrying The specific location of the aircraft and UAV cluster center is determined according to needs. When the carrier launches the UAV cluster in the second wave, it must ensure that the carrier and the UAV cluster launched in the first wave meet the constraints on the distance. Discuss the number of drones launched by two red carriers in two waves, the time and position of the second wave launch of each carrier, and the center position of the drone cluster launched by the second wave to achieve the optimal The interception effect; further specific modeling analysis whether there is an upper limit of channel bandwidth, when the actual channel bandwidth is less than, no matter what penetration strategy the blue drone adopts, the red drone cluster has a corresponding interception strategy , Successfully prevented the blue drone from penetrating in the area.
Figure 5 A schematic diagram of the red carrier launching two waves of UAV cluster interception.
Problem 4 As shown in Figure 6, the channel bandwidth, the blue party’s 3 penetration UAVs constitute a penetration cluster starting from the side of the rectangular boundary ( The distance between any two penetration drones must be greater than), and the 5 red carriers each carry 10 drones, and they will simultaneously start the coordinated interception mission from the border side. Each carrier of the Red Party launches drones in two waves to form two drone clusters, each cluster has no less than 3 drones; each carrier launches unmanned in the first wave The time of the aircraft is the initial confrontation time, and the distance from the center of the circle of the UAV cluster geometric configuration is. The initial position of the red carrier, the center position of the UAV cluster of the first wave launched by the red carrier, the time and location of the second wave of the UAV cluster launched by the red carrier, and the launch of the second wave The center position of the drone cluster, the number of drones in two waves, and the initial position of the blue penetration drone are determined as needed. The blue side hopes that as many drones as possible can successfully penetrate the defense, and the red side hopes to successfully intercept as many blue drones as possible. Try to discuss the red best interception strategy and the blue best penetration strategy.
Figure 6 A schematic diagram of confrontation between 5 red carriers carrying drones and intercepting 3 blue drones