Missile-target distance estimation and guidance law design based on cooperative positioning of missile group

Abstract

The application discloses a method for estimating the distance between a missile and a target and designing a guidance law based on cooperative positioning of a missile group, relates to the field of aircrafts, and comprises the following steps: S1, establishing a relative motion equation of the missile and the target; S2, obtaining a line-of-sight lead angle of a seeker; S3, calculating an included angle between a missile-target connecting line according to the positions of any two missiles and the measurement information of the seeker; S4, calculating the distance between the missile and the target according to the included angle; and S5, designing a guidance law based on a proportional guidance method and an attack time constraint. The distance between the missile and the target is calculated through the interaction information of the missile group, the reasons for the amplification of the estimation error of the distance between the missile and the target are analyzed, the problem of the amplification of the estimation error of the distance between the missile and the target caused by the measurement noise is reduced according to the included angle between the missile-target connecting line, the residual flight time of the missile group is calculated on the basis of the estimation of the distance between the missile and the target, and the residual flight time instruction of the missile group is obtained. A time-constrained lateral guidance law is designed based on the proportional guidance method, the lateral trajectory of the missile in the attack process is adjusted, the attack time constraint is met, and the risk of the missile hitting the ground is reduced.

Description

Technical Field

[0001] This invention relates to the field of aircraft, and in particular to a method for missile-target distance estimation and guidance law design based on missile swarm cooperative positioning. Background Technology

[0002] With the continuous development of defense systems, the threat and strike capability of a single missile is gradually weakening, making coordinated operations with multiple missiles a current research hotspot. Coordinated operations with multiple missiles increase communication between missile groups, particularly enhancing performance in areas such as cluster penetration, coordinated positioning, and saturation strikes, greatly enriching combat tactics. Due to the significant advantages and development potential of coordinated operations, they have received widespread attention both domestically and internationally in both theoretical research and engineering applications. In terms of coordinated strikes, by exchanging information such as the position and velocity of neighboring missiles, and obtaining target distance and remaining flight time based on multi-missile coordinated positioning, it is relatively easy to design guidance laws with angle or time constraints in engineering. This not only improves the destructive capability of a single missile but also enables saturation strikes against targets, which is difficult to achieve in single-missile operations.

[0003] Time-constrained cooperative guidance law designs have emerged in recent years. Among them, the two-layer cooperative guidance architecture is a typical cooperative guidance design scheme, which has the advantages of being intuitive and having clear physical meaning, and has great application prospects. Its top layer is a consistency coordination variable layer, which obtains the remaining flight time command of the missile group through a consistency coordination algorithm, and the bottom layer uses a time-constrained guidance law to achieve cooperative guidance. Therefore, the estimation of the remaining flight time of a single missile is crucial, and its calculation basis is the missile-target distance estimation. On the one hand, the missile-target distance can be located during cooperative guidance based on the spatial triangular relationship formed by any two missiles and the target. However, as the missile group approaches the target, the missile-target line of sight may almost overlap. At this time, under the influence of measurement noise, the missile-target distance estimation error based on cooperative positioning is amplified. On the other hand, the defense system applies interference to the characteristics of cooperative combat, which reduces the communication distance or communication topology connection strength between missile groups, affecting the estimation accuracy of missile-target distance and remaining flight time. Summary of the Invention

[0004] The purpose of this invention is to design a missile-target range estimation and guidance law design method based on missile swarm cooperative positioning in order to solve the above problems.

[0005] The present invention achieves the above objectives through the following technical solutions:

[0006] The missile-target range estimation and guidance law design method based on missile swarm cooperative localization includes:

[0007] S1. Establish the three-dimensional nonlinear equations of motion between the missile and the target;

[0008] S2. Obtain the forward angle of the guide head based on the transformation relationship between coordinate systems;

[0009] S3. Calculate the included angle between the missile and the target line based on the positions of any two missiles and the seeker measurement information;

[0010] S4. Calculate the distance between the projectile and the target based on the included angle;

[0011] S5. Design guidance law based on proportional guidance method and attack time constraint.

[0012] The beneficial effects of this invention are as follows: It calculates the missile-target distance using interactive information from the missile swarm, analyzes the reasons for amplified missile-target distance estimation errors, and reduces the problem of amplified missile-target distance estimation errors caused by measurement noise based on the angle between the missile-target connection lines. Furthermore, it calculates the remaining flight time of the missile swarm based on the estimated missile-target distance, obtaining the remaining flight time command for the missile swarm. A time-constrained lateral guidance law is designed based on the proportional guidance method to achieve lateral trajectory adjustment of the missile during the attack process, thereby meeting the attack time constraint and reducing the risk of missile impact. Attached Figure Description

[0013] Figure 1 This is the process for calculating the target distance of the i-th missile;

[0014] Figure 2 This invention relates to the three-dimensional relative motion geometric relationship between the projectile and the target in the projectile-target distance estimation and guidance law design method based on projectile swarm cooperative positioning.

[0015] Figure 3 It refers to the conversion relationship between the missile system and the guidance head system;

[0016] Figure 4 It refers to the conversion relationship between the seeker head system and the line-of-sight system;

[0017] Figure 5 It is the triangular relationship between any two missiles and the target in space;

[0018] Figure 6 This is a diagram showing the changes in the communication topology of the four missiles;

[0019] Figure 7 This is the simulation result of the flight trajectory in Case 1 of the four-missile coordinated strike;

[0020] Figure 8 This is the simulation result of the remaining flight time in Case 1 of the four-missile coordinated strike;

[0021] Figure 9 This is the simulation result comparing the target distances of missile 1 and missile 2 in Case 1 of the four-missile coordinated strike;

[0022] Figure 10This is the simulation result comparing the target distances of missiles 3 and 4 in Case 1 of the four-missile coordinated strike;

[0023] Figure 11 The simulation results are for the longitudinal acceleration of the target in Case 1 of the four-missile coordinated strike.

[0024] Figure 12 The simulation results are for the lateral acceleration of the target missile in Case 1 of the four-missile coordinated strike.

[0025] Figure 13 This is the simulation result of the flight trajectory in Case 2 of the four-missile coordinated strike;

[0026] Figure 14 This is the simulation result of the remaining flight time in Case 2 of the four-missile coordinated strike;

[0027] Figure 15 This is the simulation result comparing the target distances of missile 1 and missile 2 in Case 2 of the four-missile coordinated strike;

[0028] Figure 16 This is the simulation result comparing the target distances of missiles 3 and 4 in Case 2 of the four-missile coordinated strike;

[0029] Figure 17 The simulation results of the longitudinal acceleration in Case 2 of the four-missile coordinated strike are as follows;

[0030] Figure 18 This is the simulation result of the lateral acceleration of the target missile in Case 2 of the four-missile coordinated strike. Detailed Implementation

[0031] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations.

[0032] Therefore, the following detailed description of the embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the invention without inventive effort are within the scope of protection of the invention.

[0033] It should be noted that similar labels and letters in the following figures indicate similar items. Therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures.

[0034] In the description of this invention, it should be understood that the terms "upper," "lower," "inner," "outer," "left," "right," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings, or the orientation or positional relationship commonly used when the product of this invention is in use, or the orientation or positional relationship commonly understood by those skilled in the art. They are only used to facilitate the description of this invention and to simplify the description, and are not intended to indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on this invention.

[0035] Furthermore, the terms "first," "second," etc., are used only to distinguish descriptions and should not be interpreted as indicating or implying relative importance.

[0036] In the description of this invention, it should also be noted that, unless otherwise explicitly specified and limited, terms such as "set" and "connection" should be interpreted broadly. For example, "connection" can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be a direct connection or an indirect connection through an intermediate medium; it can be a connection within two components. Those skilled in the art can understand the specific meaning of the above terms in this invention according to the specific circumstances.

[0037] The specific embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0038] The missile-target range estimation and guidance law design method based on missile swarm cooperative localization includes:

[0039] S1. Establish the three-dimensional nonlinear equations of motion between the missile and the target; specifically:

[0040] Under the assumption that the target is stationary, according to Figure 2 The relative motion relationship between the missile and the target shown is expressed by the three-dimensional nonlinear relative motion equations of the i-th missile as follows:

[0041]

[0042] Where, r i V i q yi q zi η yi η zi a yi a zi Let be the target range, velocity, line-of-sight elevation angle, line-of-sight azimuth angle, line-of-sight elevation leading angle, line-of-sight azimuth leading angle, normal acceleration, and lateral acceleration of the i-th missile, respectively. r i q yi q zi ηyi η zi The derivative with respect to time, the total leading angle η i Satisfy cosη i =cosη yi cosη zi .

[0043] S2. Obtain the forward angle of the guide head based on the transformation relationship between coordinate systems; specifically:

[0044] When a platform-type seeker tracks a target, the actual line-of-sight (LOS) angle is determined by the frame angle and the misalignment angle, respectively describing the relationship between the missile system and the seeker system, and between the seeker system and the LOS system. Although the misalignment angle is generally small, when the missile-target distance is long, the positioning error caused by the misalignment angle cannot be ignored. Let the elevation frame angle, azimuth frame angle, elevation misalignment angle, and azimuth misalignment angle of the i-th missile after locking onto the target be σ. yi σ zi ε yi and ε zi The transformation relationship between coordinate systems is as follows: Figure 3 and Figure 4 As shown, where Figure 3 and Figure 4 The coordinate system description omits subscripts, and the i-th missile system, seeker system, and line-of-sight system are respectively denoted as Ox. b y b z b , Ox s y s z s and Ox L y L z L Let R be the distance between the missile and the target at a certain moment. i Then we have:

[0045]

[0046] From the above formula, the leading angle of the i-th missile can be obtained, expressed as:

[0047]

[0048] Since the misalignment angle generally satisfies |ε yi ,ε zi | < 1 / 57.3 rad, the leading angle can be approximated as:

[0049]

[0050] S3. Calculate the angle between the missile and the target line based on the positions of any two missiles and the seeker's measurement information; specifically including:

[0051] S31. Passive platform seekers typically mounted on missiles do not have target range measurement capabilities. However, they can calculate the target range based on the positional relationship of the missile group after automatic target locking, and the frame angle and misalignment angle output by the seeker. The R-value of the i-th missile is obtained based on the lead angle. i In Ox b y b z b and ground system Ox g y g z g The components in the equation are represented as:

[0052]

[0053] in, For Ox b y b z b To Ox g y g z g The transformation matrix is ​​represented as:

[0054]

[0055] Where, θ i ψ i γ i These are the elevation angle, azimuth angle, and roll angle of the i-th missile, respectively.

[0056] The elevation and azimuth angles of the line of sight are then expressed as:

[0057]

[0058] S32, according to Figure 5 Any two missiles in the missile swarm shown are at Ox g y g z g The relative relationship diagram in the diagram shows that the i-th and j-th missiles are located at Ox. g y g z g The unit vector pointing towards the target T is represented as:

[0059]

[0060] in, and These are the unit vectors of the line connecting the i-th and j-th missiles, respectively, and are directed from the missiles to the target.

[0061] S33. Analyze the included angle δ between the i-th and j-th missile targets based on the unit vector of the line connecting the target and the i-th missile. ij , is represented as:

[0062] S4. Calculate the target distance based on the included angle; specifically including:

[0063] S41. Determine the included angle δ ij If the angle is greater than the preset threshold δ0, proceed to S42; otherwise, proceed to S43.

[0064] S42. Calculate the distances between the i-th and j-th missiles and the target;

[0065] The line connecting the i-th and j-th missiles and The included angles are denoted as δ. i and δ j ,satisfy:

[0066]

[0067]

[0068] in, Let the vector pointing from the j-th missile to the i-th missile be represented as:

[0069]

[0070] but:

[0071] sinδ ij =sin(δ) i +δ j )=sinδ i cosδ j +cosδ i sinδ j

[0072] According to the law of sines, the distances between the i-th and j-th missiles and the target are expressed as:

[0073]

[0074] Where, r i,j This represents the target distance of the i-th missile obtained based on the information of the j-th missile;

[0075] S43. Calculate the distance between the i-th missile and the target;

[0076] In conjunction with the coordinated attack process of missile swarms, the following two situations should be considered in practical engineering applications. One is due to δ... ij It gradually decreases during the guidance process, when δ ij After being reduced to a certain extent, and The calculation of the projectile-target distance is affected by two main factors: firstly, the projectiles are almost collinear; secondly, the relative motion between the projectiles and the target changes drastically as the distance decreases. Both of these factors lead to significant disturbances in the calculated distance due to measurement noise. To address these issues, the calculation process is improved by setting an angle threshold δ0 and an average distance threshold for the projectile group's distance to the target. δ0 is used for collinearity detection of the line connecting any two missile targets in a missile swarm. Used for detecting the proximity of missile groups to the target. For the i-th missile, the missile-target distance calculation process is as follows: Figure 1 As shown.

[0077] At the τ n-1 At any given moment, if the average target distance of the missile swarm... satisfy Established, Let δ be the threshold value for the average distance between the missile and the target in the missile swarm. Then, for the i-th missile, we need to find the value that makes δ... ij Missile number j that is greater than δ0;

[0078] If it does not exist, then the τth n The target distance of the i-th missile at time i is represented as:

[0079]

[0080] in, For the i-th missile at τ n The velocity at time t, where τ is the time difference between two time points;

[0081] If it exists, let the set formed by missile numbers j be {j1,j2,…,j}. k}, then the τth n The target distance of the i-th missile at time i is represented as:

[0082]

[0083] At the τ n-1 At any given moment, if the average target distance of the missile swarm... satisfy Then at the τth n At time i, the target distance of the i-th missile is expressed as:

[0084]

[0085] S5. Design a guidance law based on proportional guidance method and attack time constraints; specifically:

[0086] according to Figure 2 As shown, the line-of-sight angular velocity at 0x s y s z s The components in the expression are represented as:

[0087]

[0088] Combining Ox s y s z s With Ox b y b z b The transformation relationship between them yields Ox b y b z b The components of the line-of-sight angular velocity in the image are expressed as:

[0089]

[0090] in, and These are the measurements taken by the seeker head;

[0091] Ox can be obtained by proportional derivation. b y b z b The overload instruction in the code is represented as:

[0092]

[0093] Where N is the proportional guidance coefficient, a yci and a zci These are the normal acceleration command and the lateral acceleration command for the i-th missile, respectively.

[0094] Due to Ox s y s z s angular velocity components It is immeasurable and does not affect the attack on the target. Therefore, it is ordered... The overload command can then be represented as:

[0095]

[0096] For a zci Additional Remaining Flight Time Error Compensation Term Ω i To achieve lateral trajectory adjustments to meet attack time constraints and reduce the risk of missile impact due to longitudinal trajectory adjustments, a zci Represented as:

[0097]

[0098] When the error of the remaining flight time between missiles approaches 0, Ω i The lateral acceleration command a should be close to 0. zci Degenerates into proportional guidance;

[0099] The adjacency matrix of the communication topology of n missiles is denoted as A = (aij )∈R n×n When missile i and missile j can communicate, a ij =1; otherwise a ij =0, and a ii =0; Laplace matrix L = (l ij )∈R n×n Represented as:

[0100]

[0101] When the missile uses proportional guidance, the remaining flight time t of the i-th missile goi Represented as:

[0102]

[0103] Based on the Laplace matrix of the missile group communication topology, the relative error ε of the remaining flight time of each missile is... i Represented as,

[0104]

[0105] Based on the relative error of the remaining flight time, the error compensation term Ω is... i Designed as a linear function of the projectile-target distance and relative error, its expression is as follows:

[0106]

[0107] Where r i 0 and denoted as the initial target distance and remaining flight time of the i-th missile, respectively.

[0108] Simulation Case

[0109] Based on a simulation model of a missile swarm guidance and control system, two typical missile-target relationships in four-missile cooperative guidance are set up, such as... Figure 6 As shown, the method provided by this invention is verified. The relevant parameters are set to n = 4, δ0 = 2 / 57.3 rad. N=4, τ=0.02s, missile position standard deviation is 5m, frame angle and misalignment angle standard deviation is 0.1°, line-of-sight angular rate noise standard deviation is 0.05° / s.

[0110] The initial conditions of simulation case 1 are characterized by a large dispersion of the initial positions of the missile swarm. The simulation results are as follows: Figures 7-12 As shown.

[0111] from Figure 7It can be seen that topology changes have little impact on coordinated missile attacks. This is because during the initial guidance phase when communication topology connectivity is strong, the remaining flight time among missiles quickly becomes consistent. Therefore, when the communication topology connectivity strength decreases, the missile group has basically completed the adjustment of its lateral trajectory, and the error compensation term Ω of each missile... i The efficiency is practically zero, and inter-missile communication is even unnecessary;

[0112] from Figure 8 It can be seen that: when the average target distance of the missile swarm is less than At that time, the distance between the target and the target was calculated according to Calculations show that the maximum attack time error is approximately 0.3 seconds, which meets the requirements for coordinated attacks.

[0113] from Figure 9 It can be seen that the noise level of the missile-target distance estimation is positively correlated with the missile-target distance. This is because the farther the missile-target distance is, the greater the positioning deviation of the target caused by the same line-of-sight angle error. This indicates that the misalignment angle of the platform-type seeker should be considered when positioning the target.

[0114] from Figure 10 It can be seen from the overload curve that, due to the error compensation term Ω between the lateral overload command and the estimated remaining flight time and target distance, i Relatedly, when the remaining flight time error between missile groups is not zero, the lateral overload curve exhibits a variation due to Ω. i The noise caused, after the remaining flight time converges, Ω i The lateral overload curve becomes smoother as it approaches zero, which is beneficial for terminal guidance.

[0115] The initial conditions of simulation case 2 are characterized by a relatively small initial dispersion of the missile groups, and its simulation results are as follows: Figure 13-18 As shown. From Figure 13-18 Based on the estimated target distances of missiles 1 and 2, it can be concluded that the condition of collinear lines of sight holds, and the following method is adopted. Updating the missile-target distance reduces the error in the estimated missile-target distance. Other analysis results are consistent with... Figure 7-12 The simulation results are similar, so I will not repeat them further.

[0116] The simulation results of the two typical four-missile cooperative guidance systems demonstrate the engineering feasibility of the methods used in this invention, such as missile-target distance estimation, remaining flight time estimation, and cooperative guidance law design.

[0117] The technical solutions of the present invention are not limited to the specific embodiments described above. Any technical modifications made in accordance with the technical solutions of the present invention fall within the protection scope of the present invention.

Claims

1. A method for missile-target range estimation and guidance law design based on missile swarm cooperative positioning, characterized in that, include: S1. Establish the three-dimensional nonlinear equations of motion between the missile and the target; S2. Obtain the forward angle of the guide head based on the transformation relationship between coordinate systems; S3. Calculate the included angle between the missile and the target line based on the positions of any two missiles and the seeker measurement information; S4. Calculate the distance between the projectile and the target based on the included angle; S5. Design of guidance laws based on proportional guidance methods and attack time constraints; specifically including: Line of sight angular rate in the seeker system The components in the expression are represented as follows: ； Combined with the guidance system With missile systems The conversion relationship between them can be used to obtain the missile system. The components of the line-of-sight angular velocity in the image are expressed as: ； in, and These are the measurements taken by the seeker head; By proportional derivation, we can obtain The overload instruction in the code is represented as: ； Where N is the proportional guidance coefficient. and These are the normal acceleration command and the lateral acceleration command for the i-th missile, respectively. make Therefore, the overload instruction can be represented as: ； right Additional Remaining Flight Time Error Compensation Item Adjusting the lateral trajectory is represented as: ； When the remaining flight time error between missiles approaches 0 The lateral acceleration command should be close to 0. Degenerates into proportional guidance; The adjacency matrix of the communication topology of n missiles is denoted as... When missile i and missile j can communicate, a ij =1; otherwise a ij =0, and a ii =0; Laplace matrix Represented as: ； When the missile uses proportional guidance, the remaining flight time of the i-th missile Represented as: ； Based on the Laplace matrix of the missile group communication topology, the relative error of the remaining flight time of each missile is... Represented as, ； Based on the relative error of the remaining flight time, the error compensation term is... Designed as a linear function of the projectile-target distance and relative error, its expression is as follows: ； in and These represent the initial target distance and remaining flight time of the i-th missile, respectively. and These are the elevation and slenderness of the line of sight, respectively. and line of sight azimuth The derivative with respect to time, For angular velocity components, and These refer to the elevation and azimuth angles of the line of sight, respectively. and Let i and j represent the velocity vector and velocity of the i-th missile, respectively. This is the total leading angle.

2. The method for missile-target range estimation and guidance law design based on missile swarm cooperative positioning according to claim 1, characterized in that, In S1, based on the relative motion relationship between the missile and the target, the three-dimensional nonlinear relative motion equations of the i-th missile are expressed as: ； in, , , , , , , , Let be the target range, velocity, line-of-sight elevation angle, line-of-sight azimuth angle, line-of-sight elevation leading angle, line-of-sight azimuth leading angle, normal acceleration, and lateral acceleration of the i-th missile, respectively. , , , , They are respectively , , , , The derivative with respect to time, the total leading angle satisfy .

3. The method for missile-target range estimation and guidance law design based on missile swarm cooperative positioning according to claim 1, characterized in that, In S2, combining the frame angle and misalignment angle measured by the platform-type seeker, the line-of-sight forward angle is obtained through the transformation relationship between coordinate systems. The i-th missile system, seeker system, and line-of-sight system are denoted as follows: , and Let the distance between the bullet and the target be at a certain moment. Then we have: From the above formula, the leading angle of the i-th missile can be obtained, expressed as: ； in, , These are the line-of-sight height and forward angles, respectively. Forward angle of line of sight The derivative with respect to time, , , and These are the elevation frame angle, azimuth frame angle, pitch misalignment angle, and azimuth misalignment angle, respectively.

4. The method for missile-target range estimation and guidance law design based on missile swarm cooperative positioning according to claim 1, characterized in that, S3 includes: S31. Obtain the i-th missile based on the leading angle. exist and ground system The components in the equation are represented as: ； in, for arrive The transformation matrix is ​​represented as: ； in, , , These are the elevation angle, azimuth angle, and roll angle of the i-th missile, respectively. The elevation and azimuth angles of the line of sight are then expressed as: ； S32, based on any two missiles in the missile swarm... The relative relationship diagram in the diagram shows that the i-th and j-th missiles are in... The unit vector pointing towards the target T is represented as: ； in, and These are the unit vectors of the line connecting the i-th and j-th missiles, respectively, and are directed from the missiles to the target. S33. Analyze the included angle between the i-th and j-th missile targets based on the unit vector of the line connecting the i-th and j-th missile targets. , is represented as: .

5. The method for missile-target range estimation and guidance law design based on missile swarm cooperative positioning according to claim 1, characterized in that, S4 includes: S41. Determine the included angle Is it greater than the preset angle threshold? If yes, proceed to S42; otherwise, proceed to S43. S42. Calculate the distances between the i-th and j-th missiles and the target; S43. Calculate the distance between the i-th missile and the target.

6. The method for missile-target range estimation and guidance law design based on missile swarm cooperative positioning according to claim 5, characterized in that, Specifically in S42: The line connecting the i-th and j-th missiles and The included angles are denoted as follows: and ,satisfy: ； ； in, Let the vector pointing from the j-th missile to the i-th missile be represented as: ； but: ； According to the law of sines, the distances between the i-th and j-th missiles and the target are expressed as: ； in, This represents the target distance of the i-th missile obtained based on the information of the j-th missile.

7. The method for missile-target range estimation and guidance law design based on missile swarm cooperative positioning according to claim 5, characterized in that, Specifically in S43: In the At any given moment, if the average target distance of the missile swarm... satisfy Established, Let be the threshold value for the average distance between missiles and targets in a missile swarm. Then, for the i-th missile, we need to find a threshold value that makes the distance between missiles and targets such that the distance between missiles and targets is ... The missile was designated as missile number j; If it does not exist, then the first The target distance of the i-th missile at time i is represented as: ； in, For the i-th missile at the th... The speed of time, The time difference between two moments; If it exists, let the set formed by missile numbers j be . Then the first The target distance of the i-th missile at time i is represented as: ； In the At any given moment, if the average target distance of the missile swarm... satisfy Then in the first At time i, the target distance of the i-th missile is expressed as: 。

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