Bias differential game collaborative interception method based on region coverage

Abstract

The invention belongs to the technical field of collaborative interception guidance, and particularly relates to a bias differential game collaborative interception method based on region coverage. The method comprises the steps of establishing a probability distribution model of a target in an interception plane, performing Monte Carlo sampling, and converting a coverage strategy design into an optimal coverage problem of a finite center point in the interception plane; segmenting, iteratively solving the standard maneuver of each interceptor through a Voronoi diagram method to obtain the standard maneuver aTsi of each interceptor; and based on the standard maneuvering aTsi of each interceptor, deploying to a corresponding optimal array position and carrying out offset item design of a differential game guidance law to jointly realize the offset of the coverage range of each interceptor, and finally completing the optimal coverage and interception of the target. According to the method provided by the invention, the defects that an existing disc coverage model has limitation on maneuvering distribution simplification and only a fixed solution can be applied mechanically are overcome; the miss distance and the overload demand of the interceptor are obviously reduced, and the interception efficiency is improved.

Description

Technical Field

[0001] This invention belongs to the field of cooperative interception guidance technology, specifically relating to a bias differential game cooperative interception method based on regional coverage. Background Technology

[0002] Cooperative interception guidance has received considerable attention in recent years and is a hot topic in the aerospace field. To address the threat posed by highly maneuverable targets and improve interception accuracy, it is necessary to employ multiple interceptor missiles working together to achieve cooperative interception. During cooperative interception, each interceptor has a limited interception area. To ensure that the interception areas of multiple interceptors almost or completely cover the areas where the target might appear, efficient area coverage optimization methods need to be studied. Existing optimization methods simplify this to a disk coverage model, which determines the ratio of the smaller circle's radius to the larger circle's radius based on the ratio of the interceptor's maneuverability to that of the target. Given this ratio, there exists a definite minimum number of coverage circles and its optimal spatial configuration. The disk coverage model can directly call the corresponding minimum number of interceptors and optimal distribution pattern based on different ranges of the ratio. However, this model oversimplifies the target's maneuverability distribution and is only applicable to cases where the target's maneuverability is uniformly distributed in a circle. Furthermore, the circular coverage model can only apply a few fixed solutions and cannot optimize the array coverage effect for a given number of interceptors.

[0003] Existing terminal guidance laws are basically based on proportional guidance methods, with bias terms designed to achieve directional shift of the interceptor's coverage area center. However, due to the limited interception performance of proportional guidance methods themselves, they require high maneuverability from the interceptor missile, resulting in a smaller coverage area when maneuverability is the same within the interception plane. Summary of the Invention

[0004] This invention addresses the problems of existing regional coverage optimization methods failing to optimize array coverage for a given number of interceptors and the small coverage area of ​​interceptors designed using existing terminal guidance laws. It provides a bias differential game-based collaborative interception method based on regional coverage.

[0005] The technical solution of the present invention is as follows:

[0006] A biased differential game-based cooperative interception method based on region coverage includes the following steps:

[0007] S1: Within the interception plane, establish a probability distribution model of the target and perform Monte Carlo sampling to transform the coverage strategy design into an optimal coverage problem with a finite number of center points within the interception plane; each center point corresponds to a standard maneuver of an interceptor.

[0008] S2: Solve the standard maneuvers of each interceptor in S1 using the Voronoi diagram method through segmentation and iteration, and obtain the standard maneuver a of each interceptor. Tsi That is, the optimal coverage strategy;

[0009] S3: Standard maneuver based on each interceptor a Tsi They are deployed to their respective optimal positions and bias terms of the differential game guidance law are designed to jointly achieve the offset of the coverage range of each interceptor, and finally achieve optimal coverage and interception of the target.

[0010] Preferably, the intercept plane refers to the plane where the interceptor and the target meet under the linearization assumption. This plane is perpendicular to the line-of-sight direction, and the remaining flight time t between the interceptor and the target is considered. f Similarly, the interception plane is simplified to a unit of acceleration.

[0011] Preferably, the specific process of step S1 is as follows: design a probability distribution model of target maneuver based on the target's reference characteristics and typical combat scenarios; take any point ξ within the target's maximum reachable area, and sample the probability distribution model for point ξ to obtain the sampled points. N ξ Let a represent the total sample size; by using Monte Carlo sampling, the integral operation based on probability density is transformed into a summation operation over the sample points, then the center point a Tsj For ξ i The coverage performance function, measured by the square of the distance, becomes:

[0012]

[0013] For ξ i The expression for the coverage performance of a point is:

[0014]

[0015] The overall coverage performance function for all sample points is:

[0016]

[0017] Therefore, the mathematical description of the area coverage problem is transformed into finding the coordinates a of N center points within the interception plane. Ts1 ,a Ts2 ,…,a TsN This minimizes the overall coverage performance function, i.e.:

[0018]

[0019] In the formula: a Tsj Indicates center point a Tsj Coordinates within the interception plane; ξ i Indicates sampling point ξi Coordinates within the intercept plane.

[0020] Preferably, the process of establishing the probability distribution model is as follows: guidance law assumption a T Follows a pattern with a mean of 0 and a variance of 0. The mutually independent two-dimensional Gaussian distributions are given by the following formula:

[0021]

[0022] In the formula: a Ty This represents the target's acceleration in the y-direction; a Tz This represents the target's acceleration in the z-direction; a Tmax Indicates the target's mobility, a Tmax It is not a hard boundary for target maneuvering, but is only used as a variance parameter.

[0023] Preferably, the specific process of step S2 is as follows: Define the distance from the center point a Tsj All nearest points ξ ij i = 1, 2, ..., N ξj constituting area The performance function is then expressed as:

[0024]

[0025] When considering a single optimization This can be decomposed into N independent optimization problems. Assuming the coverage performance in the y and z directions is independent, it can be further decomposed into 2N independent optimization problems:

[0026]

[0027] To solve J jy For example, for a Tsjy Taking the derivative and setting it to 0, we get:

[0028]

[0029] Similarly, the optimized results for the coordinates of all center points are obtained:

[0030]

[0031] In the formula: when j = 0, =0, =0;

[0032] Repeat the above process, based on the new center point. Forming a new area Another round of optimization is performed until the change in the center point coordinates is less than the set value; finally, the interceptors M are obtained. iThe corresponding standard maneuver a Tsi a Ts1 ,a Ts2 ,…,a TsN This actually corresponds to the center of the coverage area of ​​each interceptor, and is defined as a parameter describing the offset of the interceptor's coverage area, called the standard maneuver 'a' of the interceptor. Tsi .

[0033] Preferably, in step S3, the standard maneuver a based on each interceptor Tsi The specific process of deploying to the corresponding optimal position is as follows: Assuming that the a of each interceptor has been obtained... Tsi And the remaining flight time t f Given that the interception plane coordinates are converted back to distance units, the coordinates of the center of the coverage area of ​​each interceptor are: In reality, it is the projection of the optimal shift change position onto the interception plane, since the vertical distance from the optimal shift change position to the interception plane is V. ix t f Therefore, based on the center coordinates a Tsi and t f The optimal position coordinates at the handover moment can be determined. Assuming the interception plane and origin coordinates are known, the optimal handover position coordinates of the other interceptors in the ballistic coordinate system of a certain interceptor at the moment of collision are as follows:

[0034]

[0035] In the formula: a Tsiy and a Tsiz These represent the coordinates in the y and z directions within the interception plane, respectively.

[0036] Preferably, in step 3, the standard maneuver a based on each interceptor... Tsi The specific process of designing the bias term for the differential game guidance law is as follows: Without considering control costs and with an upper limit to maneuverability, the interceptor and target command accelerations of the traditional differential game guidance law are as follows:

[0037]

[0038] In the formula: ZEM(t) represents the zero-control miss distance; a Tmax Indicates the target's maneuverability; a Mmax Indicates the interceptor's maneuverability;

[0039] To ensure the interceptor's coverage area is truly offset, an improvement on the proportional guidance method is applied to the differential game guidance law, based on existing approaches. This similar application involves designing a bias term ZEMB(t) for the game guidance law, based on the coverage strategy providing ZEM(0). The actual zero-control miss distance is biased using the bias term ZEMB(t) to calculate the command acceleration, as shown in the following formula:

[0040]

[0041] The coverage offset of an interceptor can be adjusted using parameter a. Ts Described as standard maneuvers, ZEM(0) and ZEMB(t) are based on standard maneuver a Ts The settings are as follows:

[0042]

[0043] In the formula: t f Indicates the remaining flight time at the shift changeover time; t go This indicates the remaining flight time at the current moment.

[0044] Beneficial effects

[0045] This invention provides a biased differential game-based cooperative interception method based on region coverage. Within the interception plane, a probability distribution model of the target is established, and Monte Carlo sampling is performed. The coverage strategy design is transformed into an optimal coverage problem with a finite number of center points within the interception plane, overcoming the limitations of existing disk coverage models in simplifying maneuver distributions. The standard maneuver 'a' of each interceptor is obtained through a segmented iterative solution using the Voronoi diagram method. Tsi The optimal coverage strategy, based on the Voronoi diagram segmentation and iteration method, can flexibly determine the optimal array positions according to the number of interceptors and the target maneuver distribution, overcoming the limitation of the disk coverage model, which can only apply fixed solutions. Based on the standard maneuver a of each interceptor... Tsi The interceptors are deployed to their optimal positions and bias terms are designed using a differential game-theoretic guidance law to offset the coverage area of ​​each interceptor, ultimately achieving optimal coverage and interception of the target. This is the first time that a bias term design based on a differential game-theoretic guidance law has been used, reducing the overload requirements on the interceptors and effectively increasing the coverage area of ​​each interceptor. Furthermore, the bias term design based on the game-theoretic guidance law enables active offsetting of the interceptor coverage area, significantly reducing the miss rate and overload requirements compared to existing bias-proportional guidance methods, thus improving interception efficiency. These advantages are supported by theoretical derivation and simulation experiments. Attached Figure Description

[0046] Figure 1This is a flowchart of the biased differential game collaborative interception method based on regional coverage in an embodiment of the present invention.

[0047] Figure 2 This is a schematic diagram illustrating the relative relationship between the target and the interceptor in three-dimensional space in an embodiment of the present invention.

[0048] Figure 3 This is a schematic diagram illustrating the phase division of the guidance process for collaborative interception in an embodiment of the present invention.

[0049] Figure 4 This is a planar schematic diagram of the interceptor encountering the target in an embodiment of the present invention.

[0050] Figure 5 This is a schematic diagram of the target and interceptor coverage area in an embodiment of the present invention.

[0051] Figure 6 This is a schematic diagram of the array coordinates corresponding to the shift handover time in an embodiment of the present invention.

[0052] Figure 7 This is a schematic diagram of a two-dimensional Gaussian distribution in an embodiment of the present invention.

[0053] Figure 8 This is a schematic diagram of the interceptor coverage area after offset in an embodiment of the present invention.

[0054] Figure 9 The corresponding standard maneuver a in the embodiments of the present invention Ts A schematic diagram of the interceptor's coverage in the acceleration plane.

[0055] Figure 10 This is a diagram showing the result of the last iteration of the Voronoi diagram method used in this embodiment of the invention.

[0056] Figure 11 These are partial ballistic diagrams, interceptor overload curves, and deviation distance over time (and partial diagrams) in embodiments of the present invention.

[0057] Figure 12 The contour plots for the minimum miss distance among the five interceptors in this embodiment of the invention represent the minimum miss distance contour plots for the interceptors when deployed to the optimal array position, using the bias game guidance law, the traditional differential game guidance law, and the proportional guidance method (from left to right); where the horizontal and vertical axes represent the constant maneuvers performed by the target. Detailed Implementation

[0058] The present invention will be further described below with reference to the accompanying drawings:

[0059] In a specific embodiment of the present invention, the method provided by the present invention studies the interception of hypersonic glide vehicles as targets, with the interceptors employing a multi-to-one coordinated interception approach. A schematic diagram illustrating the relative relationship between the two in three-dimensional space is shown in Figure 2. Figure 2 In the scenario shown, both sides are in a defensive posture, and N interceptors have the same maneuverability and similar initial velocities, all less than the target. The mathematical description is as follows:

[0060]

[0061] In this scenario, both the hypersonic vehicle and the interceptor employ a three-degree-of-freedom dynamic model, with the acceleration vector serving as the control variable and an ideal control relationship. Their state equations are as follows:

[0062]

[0063] Cooperative interception can be divided into two parts based on the time sequence of the guidance process, including, for example: Figure 3 As shown, one part involves solving the optimal coverage strategy for the possible maneuvering modes of the target when the initial state information of the target is known, and calculating the handover position; the other part involves improving the terminal guidance law so that each interceptor can achieve coverage of the possible maneuvers of the target and achieve interception starting from the optimal handover position.

[0064] The following is a mathematical description of the above problem provided by the method of this invention:

[0065] 1) Coverage strategy problem

[0066] In the method of this invention, the defined optimal coverage strategy refers to the position where each interceptor achieves the best coverage effect on the possible maneuvering patterns of the target. The aim is to allow each interceptor to respond to different target maneuvering situations, ensuring that at least one interceptor can successfully intercept the target. To provide a mathematical description of the coverage problem, the definition of the interception plane is given. Under the assumption of linearization, the plane where the interceptor and the target meet is the interception plane, which is perpendicular to the line-of-sight direction. Figure 4 ).

[0067] Within the interception plane, when overload is limited, the maximum reachable area of ​​the interceptor and the target can be simplified to a circle with the following radius:

[0068]

[0069] In the formula: R T R represents the maximum reachable radius of the target; M Indicates the maximum reachable radius of the interceptor; a T Indicates the target's acceleration; a M t represents the acceleration of the interceptor. fIndicates the remaining flight time at the handover time;

[0070] Therefore, the problem of finding the optimal coverage strategy can be transformed into the problem of optimal coverage of the interceptor's maximum reachable area with respect to the target's maximum reachable area. The target's maximum reachable area is called the area to be covered, and the interceptor's maximum reachable area is called the coverage range. Let t be the remaining flight time at the handover moment. f If the design value is used, the intercepting plane is simplified to a value in units of acceleration, such as... Figure 5 As shown, the radius of the maximum reachable area of ​​the interceptor and the target changes as follows:

[0071]

[0072] The abstract and simplified mathematical description of the coverage problem in the intercept plane is as follows: Define the area to be covered as... Let ξ be any point. φ(ξ) is the probability density function of the target distribution at point ξ, satisfying... Define N center points as a Ts1 ,a Ts2 ,…,a TsN Next, define the center point a. Tsj The coverage performance function for ξ, measured by the square of the distance, is:

[0073] h(a Tsj ,ξ)=||a Tsj -ξ|| 2

[0074] The coverage performance of point ξ depends on the nearest center point, hence the expression is:

[0075]

[0076] The overall coverage performance function for the entire target distribution area is:

[0077]

[0078] Therefore, the mathematical description of the region covering problem is to find the coordinates a of N center points in a plane. Ts1 ,a Ts2 ,…,a TsN This minimizes the aforementioned coverage performance function, i.e.:

[0079]

[0080] Assuming we have already obtained the value of a for each interceptor. Tsi The center of the coverage area of ​​each interceptor is actually the projection of the shift handover position onto the interception plane, based on the center coordinates a. Tsi and t fAble to solve for the array position coordinates corresponding to the shift handover time. Figure 6 ), a Ts1 ,a Ts2 ,…,a TsN The actual center of each interceptor's coverage area can be defined as a parameter describing the offset of the interceptor's coverage area, called standard maneuver 'a'. Tsi .

[0081] 2) Terminal guidance law design problem

[0082] The purpose of terminal guidance law design is to enable the actual coverage range of each interceptor to be offset according to the expected optimal array position. However, simply relying on the deployment of the optimal array position and the combination of traditional guidance law cannot achieve this goal. The terminal guidance law must be improved to achieve the expected coverage effect.

[0083] Whether the terminal guidance law has successfully shifted the coverage area can be determined by the change in the center of the coverage area, i.e., the uncontrolled interception scenario: when the target performs maneuver a Tsi If the corresponding interceptor can achieve uncontrolled interception, it indicates that the interceptor satisfies the coverage offset 'a'. Tsi The necessary condition.

[0084] The necessary condition is described as follows, with time t as the independent variable, and the handover time as the independent variable t. h The collision time is t. f +t h Assume that at the handover moment, all interceptors have reached their optimal handover positions under the guidance strategy and have the same velocity. The terminal guidance law needs to provide the command acceleration control sequence from the handover of the interceptors until the collision. When the target performs a maneuver Tsi When, the following formula holds true.

[0085]

[0086] To address the two key issues mentioned above, this invention provides a biased differential game-based cooperative interception method based on region coverage, such as... Figure 1 As shown, it includes the following steps:

[0087] S1: Within the interception plane, establish a probability distribution model of the target and perform Monte Carlo sampling, transforming the coverage strategy design into an optimal coverage problem with a finite number of center points within the interception plane; each center point corresponds to a standard maneuver of an interceptor; specifically, this includes the following process: based on the target's reference characteristics and typical combat scenarios, establish a probability distribution model of the target maneuver; in a specific embodiment of the present invention, the process of establishing the probability distribution model is as follows: guidance law assumption a T Follows a pattern with a mean of 0 and a variance of 0. mutually independent two-dimensional Gaussian distributions ( Figure 7 The specific formula is as follows:

[0088]

[0089] In the formula: a Ty This represents the target's acceleration in the y-direction; a Tz This represents the target's acceleration in the z-direction; a Tmax Indicates the target's mobility, a Tmax It is not a hard boundary for target maneuvering, but is only used as a variance parameter.

[0090] Take any point ξ within the maximum reachable region of the target, and sample points based on the probability distribution model of point ξ to obtain the sampled points. N ξ Let a represent the total sample size; by using Monte Carlo sampling, the integral operation based on probability density is transformed into a summation operation over the sample points, then the center point a Tsj For ξ i The coverage performance function, measured by the square of the distance, becomes:

[0091]

[0092] For ξ i The expression for the coverage performance of a point is:

[0093]

[0094] The overall coverage performance function for all sample points is:

[0095]

[0096] Therefore, the mathematical description of the area coverage problem is transformed into finding the coordinates a of N center points within the interception plane. Ts1 ,a Ts2 ,…,a TsN This minimizes the overall coverage performance function, i.e.:

[0097]

[0098] In the formula: a Tsj Indicates center point a Tsj Coordinates within the interception plane; ξ i Indicates sampling point ξ i Coordinates within the intercept plane.

[0099] S2: Solve the standard maneuvers of each interceptor in S1 using the Voronoi diagram method through segmentation and iteration, and obtain the standard maneuver a of each interceptor. Tsi The optimal coverage strategy, also known as the optimal coverage strategy, specifically includes the following process:

[0100] Define the distance from the center point a Tsj All nearest points ξ ij i = 1, 2, ..., N ξj constituting area This type of segmentation method is also called a Voronoi diagram, and the performance function can be expressed as:

[0101]

[0102] When considering a single optimization This can be decomposed into N independent optimization problems. Assuming the coverage performance in the y and z directions within the interception plane is independent, it can be further decomposed into 2N independent optimization problems, as shown in the following formula:

[0103]

[0104] In a specific embodiment of the present invention, J is solved. jy For example, for a Tsjy Taking the derivative and setting it to 0, we get:

[0105]

[0106] Similarly, the optimized results for the coordinates of all center points are obtained:

[0107]

[0108] Repeat the above process, based on the new center point. Forming a new area Another round of optimization is performed until the change in the center point coordinates is less than the set value; finally, the interceptors M are obtained. i The corresponding standard maneuver a Tsi a Ts1 ,a Ts2 ,…,a TsN This actually corresponds to the center of the coverage area of ​​each interceptor, and is defined as a parameter describing the offset of the interceptor's coverage area, called the standard maneuver 'a' of the interceptor. Tsi .

[0109] Note that during the iteration process, a Ts1 = (0,0) is set to a fixed value (i.e., when j = 0, =0, The coordinates of the remaining center points are iterated freely (where 0 is the center point coordinate). This setting ensures that the first of the N interceptors is aligned with the target, which not only helps in the design of subsequent guidance strategies, but also ensures that the interception effect of the optimal coverage strategy is better than that of a single interceptor, thus facilitating comparative experiments.

[0110] S3: Standard maneuver based on each interceptor a TsiThey are deployed to their respective optimal positions and bias terms of the differential game guidance law are designed to jointly achieve the offset of the coverage range of each interceptor, and finally achieve optimal coverage and interception of the target.

[0111] In a specific embodiment of the present invention, based on the standard maneuver a of each interceptor Tsi The specific process of deploying to the corresponding optimal position is as follows: Assuming that the a of each interceptor has been obtained... Tsi And the remaining flight time t f Given that the interception plane coordinates are converted back to distance units, the coordinates of the center of the coverage area of ​​each interceptor are: In reality, it is the projection of the optimal shift change position onto the interception plane, since the vertical distance from the optimal shift change position to the interception plane is V. ix t f Therefore, based on the center coordinates a Tsi and t f The optimal position coordinates at the handover moment can be determined. Assuming the interception plane and origin coordinates are known, the optimal handover position coordinates of the other interceptors in the ballistic coordinate system of a certain interceptor at the moment of collision are as follows:

[0112]

[0113] In the formula: a Tsiy and a Tsiz These represent the coordinates in the y and z directions within the interception plane, respectively.

[0114] In a specific embodiment of the present invention, based on the standard maneuver a of each interceptor Tsi The specific process of designing the bias term for the differential game guidance law is as follows: Without considering control costs and with an upper limit to maneuverability, the interceptor and target command accelerations of the traditional differential game guidance law are as follows:

[0115]

[0116] In the formula: ZEM(t) represents the zero-control miss distance; a Tmax Indicates the target's maneuverability; a Mmax Indicates the interceptor's maneuverability;

[0117] To ensure the interceptor's coverage area is truly offset, this method borrows from existing methods that improve upon proportional guidance, applying a similar approach to the differential game guidance law. Based on the coverage strategy providing ZEM(0), a bias term ZEMB(t) is designed for the game guidance law. The bias term ZEMB(t) in the game guidance law is used to bias the actual zero-control miss distance, which is then used to calculate the command acceleration, as shown in the following equation:

[0118]

[0119] The coverage area of ​​an interceptor is offset ( Figure 8 ) can be achieved through parameter a Ts Described as standard maneuvers, ZEM(0) and ZEMB(t) are based on standard maneuver a Ts Set it as follows:

[0120]

[0121] In the formula: t f Indicates the remaining flight time at the shift changeover time; t go This indicates the remaining flight time at the current moment.

[0122] It is proved by mathematical induction that when the target performs a standard maneuver a Ts At this time, the above interceptor has a zero acceleration throughout the entire command process and can achieve unbiased interception, as shown in the following formula:

[0123]

[0124] For ease of description, in the specific embodiments of the present invention, the above-mentioned uncontrolled interception maneuver a is referred to as Ts The target's interceptor is a standard maneuver. Ts The interceptor; for standard maneuvers, a Ts The interceptor, when the target is a constant value a T When maneuvering, ZEM p (t) and (a) T -a Ts Collinearity, guidance law makes ZEM p (t) gradually decreases and stabilizes at 0, considering ZEM p (t)=ZEM(t)+ZEMB(t)andZEMB(t) f Since ) = 0, the final actual miss distance can be stabilized at 0.

[0125] Continuing the discussion of the critical case, let's assume ZEM p (t) The direction does not change throughout the entire journey, i.e., a M (t) Without changing direction, and with the final miss distance being exactly 0, substituting into the miss distance formula yields:

[0126]

[0127] We can conclude that for the standard maneuver, a Ts The interceptor, the target performs a T For constant-valued maneuvers, as long as ||(a T -a Ts )||<a Mmax This allows for bias-free interception.

[0128] Due to the remaining flight time t f Assuming constant design values, the interceptor's coverage area over the target can be described in the acceleration plane as follows:

[0129] (a T -a Ts ) 2 ≤a Mmax (16)

[0130] In the formula: a T Indicates the target acceleration; a Ts Indicates the standard maneuver of the interceptor; a Mmax This indicates the interceptor's maneuverability; corresponding to standard maneuver a. Ts The interceptor's coverage in the acceleration plane is as follows Figure 9 As shown.

[0131] Simulation verification

[0132] A full-process simulation experiment was conducted, with two control groups set up: the proportional guidance method and the traditional game guidance law. The parameter settings are shown in Table 1.

[0133] Table 1. Parameter settings for two control groups: proportional guidance method and traditional game-theoretic guidance law.

[0134] <![CDATA[t f (s)]]> <![CDATA[a Tmax (g)]]> <![CDATA[a Mmax (g)]]> N 10 12 9 5

[0135] First, based on the target maneuver distribution, 10,000 Monte Carlo samplings are performed. Then, the Voronoi diagram method is used to divide and iteratively solve for the standard maneuver 'a' of each interceptor. Tsi The result of the last iteration within the intercept plane is shown in the following figure. Figure 10 As shown, the five black dots represent the five interceptors. Tsi coordinate.

[0136] Table 2 shows the specific values ​​of the standard maneuver coordinates of the interceptors represented by M1-M5 and the optimal position coordinates of interceptor i in the ballistic coordinate system based on the standard maneuver solution. Where: R dikx R diky R dikz This represents the optimal position coordinates of interceptor i in the interceptor's ballistic coordinate system; a Tsix a Tsiy a Tsiz This represents the standard maneuver coordinates of interceptor i in the interceptor's ballistic coordinate system.

[0137] Table 2. Specific values ​​of standard maneuver coordinates and optimal position coordinates in the ballistic coordinate system based on standard maneuver solutions.

[0138] <![CDATA[R dikx (m)]]> <![CDATA[R diky (m)]]> <![CDATA[R dikz (m)]]> <![CDATA[a Tsix (m / s 2 )]]> <![CDATA[a Tsiy (m / s 2 )]]> <![CDATA[a Tsiz (m / s 2 )]]> M1 0 0 0 0 0 0 M2 0 2591.2 0 0 51.82 0 M3 0 -106.5 2594.5 0 -2.13 51.89 M4 0 -2696.3 -179.1 0 -53.93 -3.58 M5 0 119.9 -2651.1 0 2.40 -53.02

[0139] make The state sequence of each interceptor, and the initial states of the target and interceptors are shown in Table 3, where R di The inertial coordinates represent the optimal position.

[0140] Table 3. Initial State of Target and Interceptor

[0141]

[0142]

[0143] All interceptors employed biased game theory guidance, conventional game theory guidance, and proportional guidance methods, respectively, depending on the experimental group, with the target performing constant maneuvers. Taking the experimental group employing the bias game guidance law as an example, the local ballistic plot, interceptor overload curve, and the deviation from the target over time (and local plot) are shown below. Figure 11 As shown in the figure; the off-target amounts of the three methods are shown in Table 4.

[0144] Table 4. Off-target values ​​of biased game guidance law, traditional game guidance law, and proportional guidance method.

[0145]

[0146] The table of miss distances from the comparative experiments of the three methods shows that, under the same optimal array distribution, the biased game theory guidance law performs best, with M2's miss distance within 10m, while the miss distances of other guidance laws without bias design are all on the order of kilometers. Based on the aforementioned theoretical derivation, the reason why M2 can successfully intercept is the target maneuver a T Located within the coverage area of ​​M2, as shown in the following formula:

[0147]

[0148] To further evaluate the performance of this method in dealing with a wider range of target maneuvers, multiple experiments were conducted using uniform grid sampling of the target maneuvers. Control groups were set up for the traditional differential game guidance law and proportional guidance method. The initial states of the interceptors and targets were the same as in the previous single experiment, except for the constant maneuver at the target's final stage. The contour plots of the minimum miss distance among the five interceptors are shown below. Figure 12 As shown.

[0149] From the contour map of the miss distance ( Figure 12In this study, the bias differential game guidance law achieved a miss distance of less than 1m in most constant maneuvering scenarios, essentially achieving the expected coverage effect of the optimal array position. Although both started from the optimal array position, the experimental performance of the traditional differential game guidance law and proportional guidance method without bias term design was significantly inferior, with overlapping coverage areas of multiple interceptors. This set of comparative experiments further verifies that the guidance method designed in this application is not only feasible for intercepting high-speed, constant-maneuvering targets, but also exhibits superior performance compared to traditional guidance methods.

[0150] To further verify the interception performance of this method against unconventional maneuvers such as sinusoidal maneuvers, another set of single-shot simulation experiments was conducted. The optimal array position, initial value settings, and guidance phase were the same as in the first set of single-shot simulation experiments, the difference being that the target underwent the following sinusoidal maneuver during the terminal guidance phase:

[0151]

[0152] The miss distances of the experimental group and the two control groups are shown in Table 5. It can be concluded that, in the case of dealing with the sinusoidal maneuver of the target, the miss distance of the bias game guidance law is still much smaller than that of the game guidance law and the proportional guidance method without bias term design.

[0153] Table 5. Off-target values ​​of the experimental group and two control groups

[0154]

[0155] in conclusion

[0156] The present invention yielded the following conclusions through theoretical derivation and simulation experiments:

[0157] 1) The method provided in this invention overcomes the limitation of existing disk-covering models in simplifying maneuver distribution by establishing a probability distribution model of the target maneuver and performing Monte Carlo sampling. Furthermore, the solution method based on Voronoi diagram segmentation iteration can flexibly determine the optimal array position according to the number of interceptors and the target maneuver distribution, overcoming the shortcoming of the disk-covering model that can only apply fixed solutions.

[0158] 2) The terminal guidance law is based on the game-theoretic guidance law and the bias term is designed to achieve active offset of the interceptor's coverage area. Compared with the existing bias ratio guidance method, it significantly reduces the interceptor's miss distance and overload requirements, and improves the interception efficiency.

[0159] Finally, it should be noted that the above are only specific embodiments of the present invention. Of course, those skilled in the art can make modifications to the present invention. If these modifications and variations fall within the scope of the claims of the present invention and their equivalents, they should be considered as being within the protection scope of the present invention.

Claims

1. A biased differential game-based collaborative interception method based on regional coverage, characterized in that, Includes the following steps: S1: Within the interception plane, establish a probability distribution model of the target and perform Monte Carlo sampling to transform the coverage strategy design into an optimal coverage problem with a finite number of center points within the interception plane; each center point corresponds to a standard maneuver of an interceptor. S2: Solve the standard maneuvers of each interceptor in S1 using the Voronoi diagram method through segmentation and iteration, and obtain the standard maneuver a of each interceptor. Tsi That is, the optimal coverage strategy; S3: Standard maneuver based on each interceptor a Tsi They are deployed to their respective optimal positions and bias terms of the differential game guidance law are designed to jointly achieve the offset of the coverage range of each interceptor, and finally achieve optimal coverage and interception of the target.

2. The biased differential game-based collaborative interception method based on regional coverage according to claim 1, characterized in that, The intercept plane, under the linearization assumption, refers to the plane where the interceptor and the target meet. This plane is perpendicular to the line-of-sight direction. The remaining flight time t between the interceptor and the target... f Similarly, the interception plane is simplified to a unit of acceleration.

3. The biased differential game-based collaborative interception method based on regional coverage according to claim 2, characterized in that, The specific process of step S1 is as follows: Design a probability distribution model of target maneuver based on the target's reference characteristics and typical combat scenarios; take any point ξ within the target's maximum reachable area, and sample the probability distribution model for point ξ to obtain the sampled points. N ξ Let a represent the total sample size; by using Monte Carlo sampling, the integral operation based on probability density is transformed into a summation operation over the sample points, then the center point a Tsj For ξ i The coverage performance function, measured by the square of the distance, becomes: h(a Tsj ,x i )=||a Tsj -x i || 2 (1) For ξ i The expression for the coverage performance of a point is: The overall coverage performance function for all sample points is: Therefore, the mathematical description of the area coverage problem is transformed into finding the coordinates a of N center points within the interception plane. Ts1 ,a Ts2 ,…,a TsN This minimizes the overall coverage performance function, i.e.: In the formula: a Tsj Indicates center point a Tsj Coordinates within the interception plane; ξ i Indicates sampling point ξ i Coordinates within the intercept plane.

4. The biased differential game-based collaborative interception method based on regional coverage according to claim 3, characterized in that, The process of establishing the probability distribution model is as follows: Guidance law assumption a T Follows a pattern with a mean of 0 and a variance of 0. The mutually independent two-dimensional Gaussian distributions are given by the following formula: In the formula: a Ty This represents the target's acceleration in the y-direction; a Tz This represents the target's acceleration in the z-direction; a Tmax Indicates the target's mobility, a Tmax It is not a hard boundary for target maneuvering, but is only used as a variance parameter.

5. The biased differential game-based collaborative interception method based on regional coverage according to claim 4, characterized in that, The specific process of step S2 is as follows: Define the distance from the center point a Tsj All nearest points ξ ij i = 1, 2, ..., N ξj constituting area The performance function is then expressed as: When considering a single optimization This can be decomposed into N independent optimization problems. Assuming the coverage performance in the y and z directions is independent, it can be further decomposed into 2N independent optimization problems: To solve J jy For example, for a Tsjy Taking the derivative and setting it to 0, we get: Similarly, the optimized results for the coordinates of all center points are obtained: In the formula: when j = 0, =0, =0; Repeat the above process, based on the new center point. Forming a new area Another round of optimization is performed until the change in the center point coordinates is less than the set value; finally, the interceptors M are obtained. i The corresponding standard maneuver a Tsi a Ts1 ,a Ts2 ,…,a TsN This actually corresponds to the center of the coverage area of ​​each interceptor, and is defined as a parameter describing the offset of the interceptor's coverage area, called the standard maneuver 'a' of the interceptor. Tsi .

6. The biased differential game-based collaborative interception method based on regional coverage according to claim 5, characterized in that, In step S3, the standard maneuver a based on each interceptor Tsi The specific process of deploying to the corresponding optimal position is as follows: Assuming that the a of each interceptor has been obtained... Tsi And the remaining flight time t f Given that the interception plane coordinates are converted back to distance units, the coordinates of the center of the coverage area of ​​each interceptor are: In reality, it is the projection of the optimal shift change position onto the interception plane, since the vertical distance from the optimal shift change position to the interception plane is V. ix t f Therefore, based on the center coordinates a Tsi and t f The optimal position coordinates at the handover moment can be determined. Assuming the interception plane and origin coordinates are known, the optimal handover position coordinates of the other interceptors in the ballistic coordinate system of a certain interceptor at the moment of collision are as follows: In the formula: a Tsiy and a Tsiz These represent the coordinates in the y and z directions within the interception plane, respectively.

7. The biased differential game-based cooperative interception method based on regional coverage according to claim 6, characterized in that, In step 3, the standard maneuver a based on each interceptor Tsi The specific process of designing the bias term for the differential game guidance law is as follows: Without considering control costs and with an upper limit to maneuverability, the interceptor and target command accelerations of the traditional differential game guidance law are as follows: In the formula: ZEM(t) represents the zero-control miss distance; a Tmax Indicates the target's maneuverability; a Mmax Indicates the interceptor's maneuverability; In order to truly shift the coverage of the interceptor, based on the improvement ideas of existing methods for proportional guidance, a similar treatment is applied to the differential game guidance law; The similar processing is as follows: based on ZEM(0) provided by the coverage strategy, a bias term ZEMB(t) is designed for the game guidance law; based on the bias term ZEMB(t) of the game guidance law, the actual zero-control miss distance is biased and used to calculate the command acceleration, as shown in the following formula: The coverage offset of an interceptor can be adjusted using parameter a. Ts Described as standard maneuvers, ZEM(0) and ZEMB(t) are based on standard maneuver a Ts The settings are as follows: In the formula: t f Indicates the remaining flight time at the shift changeover time; t go This indicates the remaining flight time at the current moment.

Images