Defense guidance method and system based on three-party differential game of motion camouflage

Abstract

The present invention belongs to the technical field of aircraft guidance methods, and specifically relates to a defense guidance method and system based on a three-party differential game of motion camouflage. The method includes: S1, establishing a three-party game coordinate system; the three parties include a pursuit aircraft P, a target aircraft E and a defense aircraft D; S2, according to the three-party game coordinate system, deriving the maneuvering strategy of each participant and the corresponding required heading angle; S3, according to the result derived in step S2 and the initial conditions of the three-party game in step S1, analyzing the captureability of the pursuit aircraft and the defense aircraft. The present invention has the characteristics of being able to reduce the performance requirements for defense aircraft and can be applied to low-performance defense aircraft such as small drones.

Description

Technical Field

[0001] The invention belongs to the technical field of aircraft guidance methods, and in particular relates to a defense guidance method and system based on a three-party differential game of motion camouflage. Background Art

[0002] Multiplayer pursuit-escape differential games are an important tool for addressing motion strategy challenges in the context of cooperative control of multi-agent systems. A more practical application of pursuit-escape games is the target-attack-defender (TAD) three-party differential game, where the attacker's goal is to capture the target, while the defender's goal is to capture the attacker to protect the target, while the target strives to escape the attacker. In real-world scenarios, such scenarios often occur in the deployment of defensive measures, such as launching weapons to protect aircraft from incoming missiles or protecting important assets from potential damage.

[0003] The saddle point strategy given by the three-party game method based on differential games can achieve better performance when fighting against smarter pursuit aircraft. In addition, in the traditional TAD three-body game, the goal of the defense aircraft is only to capture the pursuit aircraft, so the maneuvering strategy designed is similar to the capture strategy of the pursuit aircraft. In other words, if the captureability needs to be guaranteed, the maneuverability of the defense aircraft must be stronger than that of the pursuit aircraft, which places too high demands on the speed and overload of the defense aircraft.

[0004] Therefore, it is very important to design a defense guidance method and system based on a three-party differential game of motion camouflage that can reduce the performance requirements of defensive aircraft and can be applied to lower-performance defensive aircraft such as small drones. Summary of the invention

[0005] The present invention aims to overcome the problem in the prior art that in the traditional TAD three-body game, since the goal of the defense aircraft is only to capture the pursuit aircraft, the defense aircraft needs to have stronger maneuverability than the pursuit aircraft, and thus the speed and overload requirements of the defense aircraft are too high. A defense guidance method and system based on motion camouflage three-party differential game is provided, which can reduce the performance requirements for the defense aircraft and can be applied to lower-performance defense aircraft such as small drones.

[0006] In order to achieve the above-mentioned object of the invention, the present invention adopts the following technical solutions:

[0007] The defense guidance method based on the three-party differential game of motion camouflage includes the following steps:

[0008] S1, establish a three-party game coordinate system; the three parties include the pursuit aircraft P, the target aircraft E and the defense aircraft D;

[0009] S2, based on the three-party game coordinate system, derive the maneuvering strategy of each participant and the corresponding required heading angle;

[0010] S3, analyzing the captureability of the pursuit aircraft and the defense aircraft based on the results derived in step S2 and the initial conditions of the three-party game in step S1.

[0011] Preferably, step S1 comprises the following steps:

[0012] S11, the motion model of each aircraft in the three-party game is set to be expressed by the following equation:

[0013]

[0014] Among them, X i =[x i ,y i ,v xi ,v yi ] T , represents the kinematic equation of the aircraft; x i ,y i Indicates the coordinate position of the aircraft in the engagement plane, v xi ,v yi represents the velocity vector of the aircraft, θ i ∈[-π , π] represents the instantaneous heading angle, which is specifically expressed as:

[0015]

[0016] Where V i is the speed of each aircraft, a i is the steering overload of each aircraft, a im is the upper limit of the aircraft's steering overload, and |a i |≤a im ; a i The symbol indicates the maneuvering direction of each aircraft, a i >0 means clockwise movement, a i <0 indicates counterclockwise maneuver; the turning radius and angular velocity of each aircraft are represented by r i =V i 2 / a im and ω i =a im / V i ;

[0017] S12, set R PD , R PE and R DE are the distances between each aircraft; α and β are the positive direction of the X-axis and and direction; γ represents the angle between the positive direction of the X axis and The angle between P and φ E are the angles between the sight lines of the pursuit aircraft P and the target aircraft E and the velocity vector respectively; the angle bisector of ∠PDE is recorded as DA, and A is the intersection of the angle bisector and the sight lines of the pursuit aircraft P and the target aircraft E; DP=R PD , DE=R DE , respectively represent the distance between the aircraft in space; AP, AE are the distances between the aircraft and point A; according to the properties of the angle bisector, the following conditions hold:

[0018]

[0019] Among them, μ is a dimensionless proportional parameter;

[0020] Based on the equation of motion (1.1), the state transfer matrix of the system is expressed as follows:

[0021]

[0022] Where Φ(t,t+Δt) is the zero-input state transition matrix; the terminal time t of the three-party game is defined as f The moment when the pursuer captures the target aircraft, the terminal time t is estimated based on the relative speed in the LOS direction f :

[0023]

[0024] v P , v E They represent the velocity vectors of the pursuit aircraft and the target aircraft respectively, and PE is the vector pointing from the pursuit aircraft to the target aircraft.

[0025] Preferably, step S2 comprises the following steps:

[0026] S21, define the profit function of the pursuit aircraft and the target aircraft as J PE :

[0027]

[0028] It means that for the target aircraft, a strategy is sought to maximize the profit, and for the pursuit aircraft, a strategy is sought to minimize the profit;

[0029] The state vector and dynamics are written in relative form, that is,

[0030]

[0031]

[0032] Where X P (t) and X E (t) is the state vector of both the pursuit aircraft and the target aircraft; x P and P Indicates the position coordinates of the pursuit aircraft; and Indicates the speed of the pursuit aircraft; x E and E Indicates the position coordinates of the target aircraft; and Indicates the speed of the pursuit aircraft; a P and a E represents the maneuvering overload value of the pursuit aircraft and the target aircraft; θ P and θ E X is the heading angle of the pursuit aircraft and the target aircraft; PE (t) is the relative state vector of the pursuit aircraft and the target aircraft; f PE (t) represents the relative dynamic model of the pursuit aircraft and the target aircraft;

[0033] Finally, the Hamiltonian function H corresponding to the game is obtained PE for

[0034]

[0035] Among them, λ PE =[λ1,λ2,λ3,λ4] T is the co-state variable vector, λ1~λ4 are co-state variables;

[0036] According to the pursuit-escape game formula (1.6) and the relative dynamic game formulas (1.7) and (1.8), the maneuvering strategies of the pursuit aircraft and the target aircraft are as follows:

[0037]

[0038] Where p1 = x Pf -x Ef ,p2=y Pf -y Ef ;

[0039] is the optimal strategy vector for both parties; Respectively represent the optimal strategies for the pursuit aircraft and the target aircraft; a Pm with a Em are the upper limits of steering overload for the pursuit aircraft and the target aircraft respectively; x Pf ,yPf represents the terminal position of the pursuit aircraft; x Ef ,y Ef Indicates the terminal position of the target aircraft.

[0040] Preferably, step S2 further comprises the following steps:

[0041] S22, define the profit function J of the pursuit aircraft and the defense aircraft PD for:

[0042]

[0043] It means that for the pursuit aircraft, a strategy is sought to maximize the benefits, and for the defense aircraft, a strategy is needed to minimize the benefits;

[0044] The relative state, relative motion equation and co-state variables are written as follows:

[0045]

[0046] Where X P (t), X D (t) are the state vectors of the pursuit aircraft and the defense aircraft respectively; X PD (t) is the relative state vector form; f PD is the relative dynamic form; PD is the co-state variable vector, and λ5~λ8 are the corresponding co-state variables;

[0047] The corresponding Hamiltonian equation H is obtained PD for:

[0048]

[0049] Based on the game formula (1.11) and the relative dynamic formula (1.12), the maneuver strategy of the defensive aircraft is determined as:

[0050]

[0051] in

[0052] where x D ,y D The coordinate position of the defense aircraft; is the optimal strategy for defending aircraft; a Dm To defend the overload limit of the aircraft; θ D is the heading angle of the defense aircraft.

[0053] Preferably, step S2 further comprises the following steps:

[0054] S23, in the pursuit-escape game, for the pursuer, the optimal capture strategy is the parallel strategy, that is, the pursuit aircraft and the target aircraft have a relative speed only in the LOS direction, that is, satisfying

[0055]

[0056] Where V P and V E are the speed values ​​of the pursuit aircraft and the target aircraft respectively; when the speed and direction angle of the pursuit aircraft and the target aircraft are known, the closing speed between the pursuit aircraft and the target aircraft is obtained and the velocity V in the vertical direction along the LOS l ,Right now

[0057]

[0058]

[0059] and are the optimal heading angles of the pursuit aircraft and the target aircraft respectively.

[0060] Preferably, step S2 further comprises the following steps:

[0061] S24, according to the maneuvering strategy of the defensive aircraft in formula (1.13), the heading angle of the defensive aircraft is derived as:

[0062]

[0063] in is the target heading angle of the defense aircraft.

[0064] Preferably, step S3 comprises the following steps:

[0065] S31, analysis of conditions for the pursuit aircraft to capture the target aircraft:

[0066] In order to ensure that the pursuit aircraft captures the target aircraft within a limited time, the following conditions are set:

[0067] V P >V E ,r P <r E (1.19);

[0068] Among them, the condition V P >V E Ensure that the entire strategy space is within the capture domain of the pursuit aircraft; condition r P <r EIt means that the turning radius of the pursuit aircraft is smaller than the turning radius of the target aircraft, so that when the relative positions of the two aircraft are close, the target aircraft cannot escape by continuous maneuvering and turning.

[0069] Preferably, step S3 further comprises the following steps:

[0070] S32, analysis of capture conditions for defensive aircraft:

[0071] Based on the derived maneuver strategy formulas (1.10) and (1.13) and the corresponding geometric meaning formulas (1.14) and (1.18), the game end time T f The exact estimate of is given by:

[0072]

[0073] in and are the optimal heading angles of the pursuit aircraft and the target aircraft, respectively, expressed as:

[0074]

[0075] and They represent the angles between the headings and sight directions of the pursuit aircraft and the target aircraft at the initial moment respectively; and represents the optimal maneuvering value of the pursuit aircraft and the target aircraft; t s It represents the maneuvering time of the pursuit aircraft and the target aircraft in the initial stage, which is specifically expressed as:

[0076]

[0077] Since the initial velocity vector of the defense aircraft is set to V D [cosθ D0 ,sinθ D0 ] T , then solving the capture area is equivalent to finding the speed value V of the defending aircraft D and the initial heading angle θ D0 the boundaries of;

[0078] The sufficient condition for the defensive aircraft to win the game is expressed as:

[0079]

[0080] In the formula, ω D =a Dm / V D The maximum angular velocity of the defense aircraft; To defend the turning radius of the aircraft; is the optimal heading angle of the defending aircraft; γ represents the angle between the line of sight of the pursuing aircraft and the target aircraft and the positive direction of the X axis; V A =|V A |, V A Represents the velocity vector at point A, and the expression is:

[0081]

[0082] Where V P and V E represent the velocity vectors of the pursuit aircraft and the target aircraft respectively.

[0083] The present invention also provides a defense guidance system based on a three-party differential game of motion camouflage, including:

[0084] A model building module is used to establish a three-party game coordinate system; the three parties include a pursuit aircraft P, a target aircraft E and a defense aircraft D;

[0085] The maneuver strategy derivation module is used to derive the maneuver strategy of each participant and the corresponding required heading angle according to the three-party game coordinate system;

[0086] The captureability analysis module is used to analyze the captureability of the pursuit aircraft and the defense aircraft based on the results derived by the maneuver strategy derivation module and the initial conditions of the three-party game.

[0087] Compared with the prior art, the present invention has the following beneficial effects: (1) The present invention establishes a three-body game confrontation model, combines the motion camouflage theory, and mainly designs a cover strategy for the defensive aircraft; different from the traditional defense guidance method that aims to directly capture the pursuit aircraft, under this strategy, the defensive aircraft takes the line of sight of the target aircraft and the pursuit aircraft as the target, and covers the target aircraft by moving itself in the line of sight of both parties; (2) Compared with the design of the maneuvering strategy for the defensive aircraft in the traditional three-body game, the method of the present invention reduces the demand for overload of the defensive aircraft; and the designed strategy is in analytical form, so it can be applied to low-performance defensive aircraft such as small drones; (3) Since the maneuvering strategy of the present invention is not aimed at intercepting the pursuit aircraft as the main goal, it is a non-suicidal defense strategy, and therefore it is expected to be used in the pursuit and escape game of recoverable intelligent aircraft or low-cost satellites and space orbit game scenarios in the future. BRIEF DESCRIPTION OF THE DRAWINGS

[0088] Figure 1 A coordinate system diagram of a three-party game established in an embodiment of the present invention;

[0089] Figure 2 A schematic diagram of a strategy adopted for defending against an aircraft in an embodiment of the present invention;

[0090] Figure 3 A schematic diagram of the initial game situation of the defense aircraft in an embodiment of the present invention;

[0091] Figure 4 A schematic diagram of a game situation during the movement of a defensive aircraft in an embodiment of the present invention;

[0092] Figure 5 A schematic diagram of the defense aircraft capture area of ​​Example 1 in an embodiment of the present invention;

[0093] Figure 6 A schematic diagram of the three-party trajectory of Example 1 in an embodiment of the present invention;

[0094] Figure 7 A schematic diagram of the angle change of Example 1 in an embodiment of the present invention;

[0095] Figure 8 A schematic diagram of the defense aircraft capture area of ​​Example 2 in an embodiment of the present invention;

[0096] Fig. 9 A schematic diagram of the three-party trajectory of Example 2 in an embodiment of the present invention;

[0097] Fig.10 A schematic diagram of the angle change of Example 2 in an embodiment of the present invention;

[0098] Fig.11 It is a schematic diagram of the change of motion camouflage parameters in Example 1 and Example 2 in the embodiment of the present invention. DETAILED DESCRIPTION

[0099] In order to more clearly illustrate the embodiments of the present invention, the specific implementation methods of the present invention will be described below with reference to the accompanying drawings. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For those of ordinary skill in the art, other accompanying drawings and other implementation methods can be obtained based on these accompanying drawings without creative work.

[0100] Example:

[0101] The present invention provides a defense guidance method based on a three-party differential game of motion camouflage, comprising the following steps:

[0102] S1, establish a three-party game coordinate system; the three parties include the pursuit aircraft P, the target aircraft E and the defense aircraft D;

[0103] S2, based on the three-party game coordinate system, derive the maneuvering strategy of each participant and the corresponding required heading angle;

[0104] S3, based on the results derived in step S2 and the initial conditions of the three-party game in step S1, analyze the captureability of the pursuit aircraft and the defense aircraft.

[0105] The TAD game and coordinate system of the present invention are as follows Figure 1 As shown in Figure 1. The pursuit aircraft (P), target aircraft (E) and defense aircraft (D) are all simple turning movements with constant speed. Their velocity vectors are represented by V i =[v xi ,v yi ] T Indicated by, where i = {P, E, D}, their speed is represented by V i =|V i | indicates. The control variables are the accelerations perpendicular to their velocity vectors. Therefore, the motion model of each aircraft in this three-body game can be described by the following equation:

[0106]

[0107] Where X i =[x i ,y i ,v xi ,v yi ] T ,θ i ∈[-π , π] represents the instantaneous heading angle, expressed as

[0108]

[0109] a i is the acceleration value of each aircraft, and |a i |≤a im .a i The sign indicates the maneuvering direction, a i >0 means clockwise movement, a i <0 means the opposite. Therefore, the turning radius and angular velocity of each aircraft can be expressed as and ω i =a im / V i .

[0110] like Figure 1 As shown, let R PD , R PE and R DE are the distances between each aircraft. α and β are the positive direction of the X-axis and and γ represents the angle between the positive direction of the X axis and The angle between them. P and φ Eare the angles between their line of sight and velocity vectors. The angle bisector of ∠PDE is denoted as DA. Therefore, according to the properties of the angle bisector, the following conditions hold:

[0111]

[0112] Based on the equation of motion (1.1), the state transfer matrix of the system can be expressed as follows:

[0113]

[0114] Where Φ(t,t+Δt) is the zero-input state transition matrix. Define the terminal time t of the TAD game f The moment when the target aircraft is captured by the pursuer can be estimated based on the relative speed in the LOS direction:

[0115]

[0116] In the present invention, the defense aircraft adopts the MC strategy. Figure 2 As shown in Figure 1, the MC strategy is described as the defense aircraft first moving to the CL (Constant Line), that is, the line of sight between the pursuit aircraft and the target aircraft. And it remains on the CL until the end of the three-body game. Under this strategy, the defense aircraft will not actively approach the pursuit aircraft unless the pursuit aircraft takes the initiative to pursue the target aircraft. Therefore, Z = R PD +R DE -R PE To measure whether the defensive aircraft has reached MC. According to the triangle inequality theorem, it is obvious that only when the defensive aircraft reaches PE, Z will reach the minimum value of 0. This just meets the requirements of the defensive aircraft on CL. Figure 2 As shown, the defense aircraft is at the reference point. In addition, in order to simplify the derivation process, the following assumptions are made:

[0117] A1) The speed and maneuverability of the pursuit aircraft are better than those of the target aircraft, while the defense aircraft, although slower than the pursuit aircraft, has a slight advantage in maneuverability, i.e.

[0118]

[0119] A2) The initial distance between the three aircraft is significantly larger than the turning radius. Therefore, in the initial stage, the effect of the turning radius on the angle in the derivation process can be ignored;

[0120] A3) The speeds of the three aircraft remain constant throughout the TAD game;

[0121] A4) All aircraft know each other's real-time status. Under the dynamics and the above assumptions, the following two issues are mainly discussed in the following:

[0122] Problem 1: Find a strategy for each aircraft with its own goal, considering the PE game and the MC strategy of the defending aircraft.

[0123] Question 2: Given the initial conditions and the given maneuvering strategy, determine the defensive aircraft's advantageous area to ensure that the defensive aircraft protects the target aircraft from being captured by the pursuer.

[0124] Furthermore, the present invention divides the TAD problem into two zero-sum sub-games, and derives the maneuvering strategy of each aircraft therefrom.

[0125] The strategy for pursuing the aircraft and the target aircraft is to set the profit function of the pursuing aircraft and the target aircraft as:

[0126]

[0127] The state vector and dynamics can be written in relative form, that is

[0128]

[0129]

[0130] So the Hamiltonian function of the game can be written as:

[0131]

[0132] Among them, λ PE =[λ1,λ2,λ3,λ4] T is a covariate variable.

[0133] Lemma 1: Considering the pursuit-escape game (1.6) and the relative dynamic game (1.7) and (1.8), we can derive the maneuvering strategies of the pursuit aircraft and the target aircraft as follows

[0134]

[0135] Where p1 = x Pf -x Ef ,p2=y Pf -y Ef .

[0136] Proof: Because the Hamiltonian (1.9) and the dynamic control a p and a e is decoupled. Therefore, the Isaac condition holds, and the optimal strategy of each participant satisfies the Nash solution. Based on Pontryagin's maximum / minimum principle, the optimal maneuver strategy can be expressed as follows

[0137]

[0138] In order to solve (1.11), the co-state equation and terminal co-state variables are expressed as follows:

[0139]

[0140]

[0141] Among them, x Pf ,x Ef ,y Pf ,y Ef Represent the terminal positions of the pursuit aircraft and the target aircraft respectively. Therefore, the co-state variables can be derived from equations (1.12) and (1.13), namely

[0142]

[0143] The optimal control equation can be expressed as:

[0144]

[0145] According to (1.4), the zero-input terminal state can be calculated as follows:

[0146] X PE (t f )=Φ(t,t f -t)X PE (t) (1.16)

[0147] Substituting equations (1.5) and (1.16), we can derive the maneuver strategy in equation (1.10).

[0148] The derived maneuver strategy (1.10) is actually a strategy solution given by (1.16) with the current state and the inferred final state. The maneuver strategy of the pursuit vehicle aims to adjust its terminal time t f The target aircraft’s maneuver strategy is to move in a direction that hinders the pursuer’s approach. As the three-body game begins, the aircraft is required to perform specific maneuvers that minimize the final distance between the two aircraft. f The estimated value of is less accurate than the actual value. However, due to f -t’s non-negativity and the convergence relationship in the LOS direction allow t to be updated in real time during the game. f Replace t f -t and calculate the corresponding terminal state, thereby updating the strategies of both aircraft in real time during the game.

[0149] The strategy for the defending aircraft is still determined by finding the saddle point solution through a two-player game between the defending aircraft and the pursuing aircraft.

[0150] from Figure 2 It can be seen that the MC strategy not only needs to capture the pursuit aircraft in terms of terminal conditions, but also has certain constraints in the pursuit process. In order to avoid the highly nonlinear and high-dimensional two-point boundary value problem brought by the process reward function, the present invention discretizes the game time and adopts a saddle point solution in each time interval. Through this transformation, the problem becomes a terminal reward game problem involving the terminal conditions of each time interval. The reward functions of the pursuit aircraft and the defense aircraft are set as follows:

[0151]

[0152] like Figure 2 As shown in Figure 1, the defense aircraft has two purposes: one is to maintain the geometric constraints of motion camouflage, and the other is to capture the pursuer. Geometrically, PD+ED≥PE holds, and this inequality is minimized when the defense aircraft is located on the line segment PE, which means that when the MC constraint is satisfied, the payoff function should be minimized to 0.

[0153] At the same time, V P >V E = means that the pursuer can win the game only when the speed of the pursuer in the LOS direction is higher than the speed of the target aircraft in that direction. In other words, when the defending aircraft appears at a specific position on the LOS, the pursuer will actively approach the defending aircraft in order to approach the target aircraft. Therefore, in general, the payoff function of the MC game can be expressed as (1.17).

[0154] The relative motion equations and solution process are very similar to the strategy derivation process for the pursuit vehicle and the target vehicle. However, there are differences in solving for the co-state variables. Therefore, the main focus will be on clarifying the method used to solve for the co-state variables. The relative state, relative motion equations, and co-state variables can be written as follows:

[0155]

[0156] Therefore, the Hamiltonian equation is

[0157] Theorem 1: Based on the game (1.17) and relative dynamics (1.18), the maneuver strategy of the defensive aircraft can be determined as:

[0158]

[0159] in

[0160] Proof: The proof is similar to Theorem 1. The main differences are the payoff function and the terminal time. The optimal control equation can be expressed as

[0161]

[0162] The form of the co-state equation is the same as (1.12). When solving the co-state equation, the time series is discretized and each discrete point is regarded as the terminal time of a game. Therefore, at each time interval (t, t+τ), the solution of the co-state equation is:

[0163]

[0164] The terminal state in each time interval can be obtained similarly to formula (1.16):

[0165] X PD (t+τ)=Φ(t,t+τ)X PD (t) (1.22)

[0166] Substituting (1.21) and (1.22) into the co-state equation, we can obtain the co-state variable. Then, according to Pontryagin's maximum / minimum principle and the non-negativity of τ, the maneuver strategy of the defensive aircraft can be derived similarly to the previous section, expressed as (1.19).

[0167] Furthermore, the target heading angle derivation process of each aircraft is as follows:

[0168] For pursuit and target aircraft:

[0169] Lemma 2: In the pursuit-escape game, the optimal capture strategy for the pursuer is the parallel strategy, that is, the two aircraft have a relative speed only in the LOS direction, that is, satisfying

[0170]

[0171] This conclusion is also consistent with other papers using different kinetic models.

[0172] Proof: According to the maneuver strategy in (1.10), it is observed that under certain conditions, It is established, indicating that both aircraft stop maneuvering and continue to move forward at the current speed direction. Therefore, it is necessary to identify the situation that satisfies this condition. According to the maneuvering strategy in (1.10), let

[0173]

[0174] Substituting (1.2) and (1.16) and expanding them, we have

[0175]

[0176] Adding both sides of the equation and combining them, we have

[0177]

[0178] According to (1.24) and the rules of vector algebra, the right side of the equation becomes 0, and the left side becomes the following form:

[0179] R PE (V P sinφ P -V E sinφ E )=0 (1.27)

[0180] Therefore, formula (1.23) is proved.

[0181] Note 2: According to the conclusions drawn from Theorem 1 and Lemma 1, the control input in the present invention is the steering acceleration, but there is a certain similarity with the research results when the control input is the heading angle, where the goal of both strategies is to ensure that V P and V E Satisfying (1.23) is also called a parallel strategy. This also means that, despite the maneuverability of the pursuer, the performance of these strategies is similar to the results when the input control is the instantaneous heading. In addition, given the speed and direction angles of both aircraft, it is possible to obtain the closing speed between the two aircraft and the velocity V of LOS in the vertical direction l ,Right now

[0182]

[0183]

[0184] For defensive aircraft:

[0185] Theorem 2: According to the maneuvering strategy of the defensive aircraft in equation (1.19), its target heading angle is:

[0186]

[0187] This shows that under the MC strategy, the saddle point maneuver strategy of the defending vehicle aims to orient its heading angle toward the bisector of the angle ∠PDE, thereby reaching and remaining on the CL.

[0188] Proof: According to the maneuver strategy in (1.19), the following equation holds:

[0189] p3sinθ D =p4cosθ D (1.31)

[0190] according to Figure 1, let p3=cosα+cosβ, p4=sinα+sinβ. Then, according to the relationship between trigonometric functions, we can deduce

[0191]

[0192] Therefore, formula (1.31) is proved.

[0193] Note 3: According to assumption A3), within a certain time interval t∈[t0,t′], the defensive aircraft can be determined according to the initial position relationship.

[0194]

[0195] Where α * =∠XDP and β * =∠XDE respectively represent the angles between DP and DE and the positive direction of the X axis at the initial moment. The defense aircraft will maneuver and turn along this angle, move forward in this direction, and reach CL.

[0196] Furthermore, a capture area analysis is performed. The capture capabilities of the pursuit aircraft and the defense aircraft when adopting strategies (1.10) and (1.19) are analyzed in detail. Based on assumptions A1)-A3), according to the maneuvering strategies, geometric relationships and initial conditions of the pursuit aircraft and the defense aircraft, the conditions for the pursuit aircraft and the defense aircraft to win the game are given. The winning conditions of the pursuit aircraft only consider the game between the pursuer and the target aircraft, and do not consider the influence of the defense aircraft. In addition, the present invention also makes the following assumptions:

[0197] A4) At the beginning of the engagement, the defending aircraft is located between the pursuing aircraft and the target aircraft, forming an obtuse triangle △PDE, where D is the obtuse angle.

[0198] Analysis of the pursuer's capture conditions:

[0199] Lemma 3: As long as the following conditions are met, the pursuit aircraft can always catch the target aircraft in a limited time. In order to ensure that the pursuit aircraft captures the target aircraft in a limited time, the following conditions should be met:

[0200] V P >V E ,r P <r E (1.34)

[0201] Discussion 1: The first condition ensures that the entire strategy space is within the capture domain of the pursuit aircraft. The second condition means that the turning radius of the pursuit aircraft is smaller than the turning radius of the target aircraft, so that when the relative positions of the two aircraft are close to a certain distance, the target aircraft cannot escape by continuously maneuvering and turning. Without the second condition, the pursuit aircraft will not be able to use the above strategy to further capture the target aircraft.

[0202] Analysis of capture conditions for defensive aircraft:

[0203] Based on the initial conditions, maneuverability and previously derived maneuvering strategies of the three aircraft, the winning conditions for the defensive aircraft are derived.

[0204] Gaming time

[0205] Lemma 4: Based on the derived maneuvers (1.10), (1.19) and their geometric implications (1.23), (1.31), a more accurate estimate of the game time can be expressed as:

[0206]

[0207] in and It is expressed as:

[0208]

[0209] And t s represents the maneuvering time of the pursuit aircraft and the target aircraft in the initial stage, which can be expressed as

[0210]

[0211] Proof: This terminal time can be regarded as the time it takes for the pursuit aircraft to capture the target aircraft in the direction of the optimal speed. According to the strategy in equation (1.10), the control input is 0 when the following conditions are met:

[0212] p1sinθ p -p2cosθ p =0

[0213] p1sinθ e -p2cosθ e =0

[0214] According to Lemma 1, this situation can only occur if condition (1.23) is satisfied. The maneuvering direction will remain unchanged until both parties have adjusted to the desired heading angle. Based on the initial conditions, it is straightforward to deduce a P and a E . Therefore, condition (1.23) can be rewritten as the time-dependent equation in (1.37).

[0215] Once you get t s , the velocity direction angle can naturally be expressed as (1.33). The estimated pursuit time inferred from this heading angle and initial distance is less than the actual terminal time. f It is mainly used for the subsequent derivation of the capture area of ​​the defensive aircraft. Although the smaller terminal time sacrifices a part of the capture area of ​​the defensive aircraft, it will produce stronger capture conditions and ensure the performance of the boundary.

[0216] Advantageous range of the initial velocity vector of the defensive aircraft:

[0217] Since the initial velocity vector of the defense aircraft is expressed as V D [cosθ D0 ,sinθ D0 ] T , so solving for the capture region is equivalent to finding V D and θ D0 the border.

[0218] Theorem 3: The sufficient condition for the defending aircraft to win the game is expressed as:

[0219]

[0220] Where V A represents the velocity vector at point A, V A =|V A |, its expression is:

[0221]

[0222] Proof: First, the velocity vectors of each point on the line PE must change linearly. Therefore, the velocity vector of point A can be calculated using linear interpolation. From equation (1.3), we can get DP / PE=1 / (μ+1), so according to the linear change relationship, we can get equation (1.39). Next, (1) in (1.38) provides a lower limit for the initial velocity vector of the defending aircraft. This requirement ensures that the defending aircraft turns to the direction indicated by (1.33) (the first term) and reaches CL (the second term) before the pursuing aircraft catches up with the dodger.

[0223] like Figure 3 As shown, the actual path of the defense aircraft is represented by the line segment Therefore, the initial velocity vector of the defensive aircraft must meet the following conditions:

[0224]

[0225] where t nis the turning time, which is equal to the first term. Because the calculation of D′A′ is relatively complicated, a slightly stricter lower limit can be derived based on a certain approximate relationship. With the help of geometric configuration, the following inequality holds

[0226] DA+AA′>DA′,DA′>D′A′;

[0227] Since V D >V A , to ensure that the defense aircraft can reach CL in the specified direction and maintain it. Otherwise, the defense aircraft will not be able to reach CL. Therefore, it can be deduced that

[0228]

[0229] The time term on the left side of (1) in formula (1.38) will always exceed the actual time required, ensuring that within this boundary, the defending aircraft can reach the CL before the pursuing aircraft wins the game.

[0230] Finally, (2) in formula (1.38) provides an upper limit for the initial velocity vector of the defensive aircraft. When the defensive aircraft reaches CL, the velocity vector of the defensive aircraft is directed toward In addition, since V D >V A , the defender aircraft will inevitably make another steering adjustment to ensure that it is as aligned with the CL as possible. Therefore, (2) in equation (1.38) outlines the conditions under which the defender aircraft can remain on the CL.

[0231] Figure 4 The relative trajectory diagram of the defense aircraft after reaching CL is shown. Because when the defense aircraft reaches CL, V perpendicular to the CL direction D Must be greater than V l Therefore, a secondary directional adjustment is required to keep the aircraft on CL. If after the defense aircraft reaches CL, V D ·V l If V is less than 0, the defensive aircraft may easily deviate from CL, resulting in failure in the competition. Therefore, the boundary condition is V D ·V l =0, established Figure 4 The geometric relationships shown.

[0232] Therefore, the turning time t of the defending aircraft is d and in V l The displacement in is as follows:

[0233]

[0234] During the turning process, the displacement of CL is td ·V l , when (1.38) holds, it means that when the defense aircraft reaches CL for the second time, it moves in the same direction as CL, that is, V D ·V l ≥ 0. Therefore, Theorem 5 is proved.

[0235] In addition, the present invention also provides a defense guidance system based on a three-party differential game of motion camouflage, including:

[0236] A model building module is used to establish a three-party game coordinate system; the three parties include a pursuit aircraft P, a target aircraft E and a defense aircraft D;

[0237] The maneuver strategy derivation module is used to derive the maneuver strategy of each participant and the corresponding required heading angle according to the three-party game coordinate system;

[0238] The captureability analysis module is used to analyze the captureability of the pursuit aircraft and the defense aircraft based on the results derived by the maneuver strategy derivation module and the initial conditions of the three-party game.

[0239] In order to verify the technical effect of the present invention, simulation is performed according to the following example:

[0240] The turning maneuverability of the three aircraft is set as a Pm =100m / s 2 ,a Em =50m / s 2 and a Dm =120m / s 2 . And τ in Theorem 2 is set to 0.5s.

[0241] Example 1: The remaining initial conditions of the three participants in this example are shown in Table 1:

[0242] Table 1 Initial condition data of three aircraft

[0243] parameter value Pursuit aircraft location [3500,-1000]m Target aircraft position [0,0]m Defense aircraft position [1000,-1000]m Pursuit vehicle initial speed [-180,90]m Target aircraft initial speed [120,120]m

[0244] According to the initial conditions, several basic parameters can be obtained: μ = 1.76, DA = 691.32m, V A =109.77m / s, γ = -0.278rad. Using equations (1.23) and (1.37), we can calculate t s =0.92s. In addition, according to Lemmas 2, 4 and Theorem 4, and T f They are determined to be 2.219 rad, 0.5136 rad, 1.178 rad and 12.99 s respectively. After substituting the above parameters into (1.38), the area of ​​the initial velocity vector of the defense aircraft is as follows Figure 5 Therefore, the following simulation selects V D =[127,140]m / s. The pursuit trajectories of the three participants are as follows Figure 6 As shown in Figure 2, the pursuer will win the three-body game at 14.59s, while the defense aircraft will catch the pursuer at 12.38s. Fig.11 As shown in (a), the defending aircraft reaches CL at 10.22 seconds, and then keeps CL close to the pursuing aircraft, eventually winning the game. Figure 7 As shown, the above theorem is verified.

[0245] Example 2: The remaining initial conditions of the three participants in this example are shown in Table 2:

[0246] Table 2 Initial condition data of three aircraft

[0247]

[0248]

[0249] According to the initial conditions, several basic parameters can be obtained: μ = 3.256, DA = 1092 . 5m, V A =175.77m / s, γ = 0rad. Using equations (1.23) and (1.37), we can calculate t s =0.43s. In addition, according to Lemmas 2, 4 and Theorem 4, and T f They are determined to be 2.142 rad, 1.729 rad, 1.156 rad and 28.49 s respectively. After substituting the above parameters into (1.38), the area of ​​the initial velocity vector of the defense aircraft is as follows Figure 8 Therefore, the following simulation selects V D =[-200,280]m / s. The pursuit trajectories of the three participants are as follows Fig. 9 As shown in Figure 2, the pursuit aircraft will win the three-body game at 36.19s, while the defense aircraft will catch the pursuit aircraft at 13.8s. Fig.11 As shown in (a), the defending aircraft reaches CL at 7.42s, and then keeps CL close to the pursuing aircraft, eventually winning the game. Fig.10 As shown, the above theorem is verified.

[0250] In the simulation, it is worth noting that, in addition to T f Except for a certain difference from the actual terminal time, the errors of other parameters calculated based on the initial conditions are relatively negligible compared with the actual data.f The value is smaller than the actual three-body game termination time, and this conclusion also enhances the robustness of the lower bound condition of the defense aircraft. Figure 6 and Fig. 9 As shown, although the initial stage involves maneuvers of three parties, the theoretical angle change caused by the relative position change is minimal due to the considerable distance involved. This result further illustrates the rationality of the assumptions made in the present invention.

[0251] The present invention establishes a three-body game confrontation model and combines the motion camouflage theory to mainly design a cover strategy for the defensive aircraft. Different from the traditional defense guidance method that aims to directly capture the pursuit aircraft, under this strategy, the defensive aircraft takes the line of sight between the target aircraft and the pursuit aircraft as the target and covers the target aircraft by moving itself in the line of sight of both parties.

[0252] Compared with the maneuver strategy design for defensive aircraft in the traditional three-body game, the method of the present invention reduces the demand for overload of the defensive aircraft. And the designed strategy is in analytical form, so it can be applied to low-performance defensive aircraft such as small drones. In addition, since the maneuver strategy is not aimed at intercepting the chasing aircraft, it is a non-suicidal defense strategy. Therefore, it is expected to be used in the chase and escape game of recoverable intelligent aircraft or low-cost satellites and space orbit game scenarios in the future.

[0253] The above description is only a detailed description of the preferred embodiments and principles of the present invention. For ordinary technicians in this field, according to the ideas provided by the present invention, there will be changes in the specific implementation methods, and these changes should also be regarded as the protection scope of the present invention.

Claims

1. A three-party differential game-based defense guidance method based on motion camouflage, characterized in that: The steps include: S1, establish a three-party game coordinate system; the three parties include the pursuit aircraft P, the target aircraft E and the defense aircraft D; S2, based on the three-party game coordinate system, derive the maneuvering strategy of each participant and the corresponding required heading angle; S3, analyzing the captureability of the pursuit aircraft and the defense aircraft based on the results derived in step S2 and the initial conditions of the three-party game in step S1; Step S1 includes the following steps: S11, the motion model of each aircraft in the three-party game is set to be expressed by the following equation: Among them, X i =[x i ,y i ,v xi ,v yi ] T , represents the kinematic equation of the aircraft; x i ,y i represents the coordinate position of the aircraft in the engagement plane, v xi ,v yi represents the velocity vector of the aircraft, θ i ∈[-π , π] represents the instantaneous heading angle, which is specifically expressed as: Where V i is the speed of each aircraft, a i is the steering overload of each aircraft, a im is the upper limit of the aircraft's steering overload, and |a i |≤a im ; a i The symbol indicates the maneuvering direction of each aircraft, a i >0 means clockwise movement, a i <0 indicates counterclockwise maneuver; the turning radius and angular velocity of each aircraft are expressed as and ω i =a im / V i ; S12, set R PD , R PE and R DE are the distances between each aircraft; α and β are the positive direction of the X-axis and and direction; γ represents the angle between the positive direction of the X axis and The angle between P and φ E are the angles between the sight lines of the pursuit aircraft P and the target aircraft E and the velocity vector respectively; the angle bisector of ∠PDE is recorded as DA, and A is the intersection of the angle bisector and the sight lines of the pursuit aircraft P and the target aircraft E; DP=R PD , DE=R DE , respectively represent the distance between the aircraft in space; AP, AE are the distances between the aircraft and point A; according to the properties of the angle bisector, the following conditions hold: Among them, μ is a dimensionless proportional parameter; Based on the equation of motion (1.1), the state transfer matrix of the system is expressed as follows: Where Φ(t,t+Δt) is the zero-input state transition matrix; the terminal time t of the three-party game is defined as f The moment when the pursuer captures the target aircraft, the terminal time t is estimated based on the relative speed in the LOS direction f : v P , v E denote the velocity vectors of the pursuit vehicle and the target vehicle, respectively. is the vector pointing from the pursuit aircraft to the target aircraft; Step S2 includes the following steps: S21, define the profit function of the pursuit aircraft and the target aircraft as J PE : It means that for the target aircraft, a strategy is sought to maximize the profit, and for the pursuit aircraft, a strategy is sought to minimize the profit; The state vector and dynamics are written in relative form, that is, Where X P (t) and X E (t) is the state vector of both the pursuit aircraft and the target aircraft; x P and P Indicates the position coordinates of the pursuit aircraft; and Indicates the speed of the pursuit aircraft; x E and E Indicates the position coordinates of the target aircraft; and Indicates the speed of the pursuit aircraft; a P and a E represents the maneuvering overload value of the pursuit aircraft and the target aircraft; θ P and θ E X is the heading angle of the pursuit aircraft and the target aircraft; PE (t) is the relative state vector of the pursuit aircraft and the target aircraft; f PE (t) represents the relative dynamic model of the pursuit aircraft and the target aircraft; Finally, the Hamiltonian function H corresponding to the game is obtained PE for Among them, λ PE =[λ1,λ2,λ3,λ4] T is the co-variable vector, λ1~λ4 are co-variables; According to the pursuit-escape game formula (1.6) and the relative dynamic game formulas (1.7) and (1.8), the maneuvering strategies of the pursuit aircraft and the target aircraft are as follows: Where p1 = x Pf -x Ef ,p2=y Pf -y Ef ; is the optimal strategy vector for both parties; Respectively represent the optimal strategies for the pursuit aircraft and the target aircraft; a Pm with a Em are the upper limits of steering overload for the pursuit aircraft and the target aircraft respectively; x Pf ,y Pf represents the terminal position of the pursuit aircraft; x Ef ,y Ef Indicates the terminal position of the target aircraft; Step S2 also includes the following steps: S22, define the profit function J of the pursuit aircraft and the defense aircraft PD for: It means that for the pursuit aircraft, a strategy is sought to maximize the benefits, and for the defense aircraft, a strategy is needed to minimize the benefits; The relative state, relative motion equation and co-state variables are written as follows: Where X P (t), X D (t) are the state vectors of the pursuit aircraft and the defense aircraft respectively; X PD (t) is the relative state vector form; f PD is the relative dynamic form; PD is the co-state variable vector, and λ5~λ8 are the corresponding co-state variables; The corresponding Hamiltonian equation H is obtained PD for: Based on the game formula (1.11) and the relative dynamic formula (1.12), the maneuver strategy of the defensive aircraft is determined as: in where x D ,y D The coordinate position of the defense aircraft; is the optimal strategy for defending against aircraft; a Dm To defend the overload limit of the aircraft; θ D The heading angle of the defense aircraft; Step S2 also includes the following steps: S23, in the pursuit-escape game, for the pursuer, the optimal capture strategy is the parallel strategy, that is, the pursuit aircraft and the target aircraft have a relative speed only in the LOS direction, that is, satisfying Where V P and V E are the speed values ​​of the pursuit aircraft and the target aircraft respectively; when the speed and direction angle of the pursuit aircraft and the target aircraft are known, the closing speed between the pursuit aircraft and the target aircraft is obtained and the velocity V in the vertical direction along the LOS l ,Right now and are the optimal heading angles of the pursuit aircraft and the target aircraft respectively; Step S2 also includes the following steps: S24, according to the maneuvering strategy of the defensive aircraft in formula (1.13), the heading angle of the defensive aircraft is derived as: in The target heading angle for the defending aircraft; Step S3 includes the following steps: S31, analysis of conditions for the pursuit aircraft to capture the target aircraft: In order to ensure that the pursuit aircraft captures the target aircraft within a limited time, the following conditions are set: V P >V E ,r P r E (1.19); Among them, the condition V P >V E Ensure that the entire strategy space is within the capture domain of the pursuit aircraft; condition r P <r E It means that the turning radius of the pursuit aircraft is smaller than that of the target aircraft, so that when the relative positions of the two aircraft are close, the target aircraft cannot escape by continuous maneuvering and turning; Step S3 also includes the following steps: S32, analysis of capture conditions for defensive aircraft: Based on the derived maneuver strategy formulas (1.10) and (1.13) and the corresponding geometric meaning formulas (1.14) and (1.18), the game end time T f The exact estimate of is given by: in and are the optimal heading angles of the pursuit aircraft and the target aircraft, respectively, expressed as: and They represent the angles between the headings and sight directions of the pursuit aircraft and the target aircraft at the initial moment respectively; and represents the optimal maneuvering value of the pursuit aircraft and the target aircraft; t s It represents the maneuvering time of the pursuit aircraft and the target aircraft in the initial stage, which is specifically expressed as: Since the initial velocity vector of the defensive aircraft is set to V D [cosθ D0 ,sinθ D0 ] T , then solving the capture area is equivalent to finding the speed value V of the defending aircraft D and the initial heading angle θ D0 the boundaries of; The sufficient condition for the defensive aircraft to win the game is expressed as: In the formula, ω D =a Dm / V D The maximum angular velocity of the defense aircraft; To defend the turning radius of the aircraft; is the optimal heading angle of the defending aircraft; γ represents the angle between the line of sight of the pursuing aircraft and the target aircraft and the positive direction of the X axis; V A =|V A |, V A Represents the velocity vector at point A, and the expression is: Where V P and V E represent the velocity vectors of the pursuit aircraft and the target aircraft respectively.

2. A defense guidance system based on a three-party differential game of motion camouflage, used to implement the defense guidance method based on a three-party differential game of motion camouflage according to claim 1, characterized in that: The defense guidance system based on the three-party differential game of motion camouflage includes: A model building module is used to establish a three-party game coordinate system; the three parties include a pursuit aircraft P, a target aircraft E and a defense aircraft D; The maneuver strategy derivation module is used to derive the maneuver strategy of each participant and the corresponding required heading angle according to the three-party game coordinate system; The captureability analysis module is used to analyze the captureability of the pursuit aircraft and the defense aircraft based on the results derived by the maneuver strategy derivation module and the initial conditions of the three-party game.

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