Three-dimensional target defense differential game method based on motion camouflage

Abstract

The invention discloses a three-dimensional target defense differential game method based on motion camouflage. The method comprises the following steps: establishing a three-dimensional coordinate system and three-dimensional dynamic models of a target, a pursuer and a defender; decomposing a target-pursuer-defender three-party game into two sub-games; aiming at each sub game, constructing a Hamiltonian containing a relative kinetic equation, solving a co-state equation by applying a Pontryagin maximum / minimum principle, deducing a guidance strategy of each party, and updating the strategy in real time through a discretization time sequence; based on initial conditions and the guidance strategy, the saddle point solution of each time period is solved through a simulation environment, and a dynamic guidance instruction is generated to complete the capture task of an attacker. According to the method, aiming at a target-pursuer-defender (TAD) three-party game, an equilibrium strategy in a three-dimensional space is designed by decomposing a game problem, and in combination with motion camouflage constraints of biological mimicry, efficient interception of a defender to a pursuer and dynamic protection of a target are realized.

Description

Technical Field

[0001] The present invention relates to the field of intelligent body cooperative control and differential game technology, and specifically refers to a three-dimensional target defense differential game method based on motion camouflage. Background Art

[0002] Pursuit and escape games have attracted much attention in real-world scenarios such as drone coordination and target defense by simulating strategic interactions between players. Existing research has mostly focused on the problem of coordinated pursuit by multiple players in two-dimensional space. However, such methods are difficult to apply directly to actual scenarios—players usually need to move in three-dimensional space using three-degree-of-freedom control. This invention attempts to explore the equilibrium strategy of a type of realistic pursuit and escape game: players use three-degree-of-freedom control. A more practical application of the pursuit and escape game is the target-pursuer-defender differential game (TAD), in which the goal of the pursuer is to capture the target, the goal of the defender is to capture the pursuer to protect the target, and the target tries its best to avoid the pursuer.

[0003] Motion camouflage is a concealment strategy used by many visually perceptive insects to capture prey, first proposed by Srini Vasan et al. It usually refers to the fact that from the perspective of the avoider, the pursuer always stays in a straight line between the reference point and the avoider and maintains a constant azimuth.

[0004] In the prior art, linear quadratic differential games (LQDGs) are widely used in multi-agent confrontation problems, but they are mostly limited to two-dimensional space and do not effectively combine dynamic constraints with real-time strategy updates. For example, the three-party game model of Rusnak et al. does not involve three-dimensional maneuvering strategies, and the state feedback optimal strategy proposed by Garcia is complex to calculate and difficult to apply in real time. In terms of biomimicry strategies, the motion camouflage proposed by Srinivasan is only for single tracking scenarios and has not been extended to three-party games.

[0005] The existing technology has the following defects: 1. Dimension limitation: Traditional methods are mostly limited to two-dimensional space and are difficult to directly apply to actual three-dimensional scenes; 2. Poor real-time performance: Offline calculations cannot adapt to dynamic game states, and strategy updates lag behind; 3. Low defense efficiency: Defenders do not utilize motion camouflage constraints, maneuvering energy consumption is high and the interception success rate is limited; 4. High computational complexity: It is difficult to directly solve the three-party game and there is a lack of effective decomposition methods. Summary of the invention

[0006] To solve the above problems, the present invention proposes a three-dimensional target defense differential game method based on motion camouflage. For the target-pursuer-defender (TAD) three-party game, the defender can efficiently intercept the pursuer and dynamically protect the target by combining the motion camouflage constraints of bio-mimicry by decomposing the game problem, designing an equilibrium strategy in three-dimensional space, and combining it with the motion camouflage constraints of bio-mimicry.

[0007] In order to achieve the above object, the specific technical solution adopted by the present invention is as follows:

[0008] A differential game method for three-dimensional target defense based on motion camouflage includes the following steps:

[0009] Step A: Establish a three-dimensional coordinate system and a three-dimensional dynamic model of the intelligent agent

[0010] Step B: Decompose the three-party game into the game between the target and the pursuer and the game between the defender and the pursuer under motion camouflage

[0011] Step C: Based on the objective function, the Hamiltonian is constructed in combination with the relative dynamics equation, the Pontryagin maximum / minimum principle is applied to solve the co-state equation, and then the guidance strategy of the three parties is derived. The strategy is updated in real time through the discretized time series.

[0012] Step D: According to the initial conditions and the derived guidance strategy, a MATLAB simulation environment is built, and the saddle point solution of each time period is solved through the ODE function to provide a guidance strategy for the three parties of TAD to complete the capture task.

[0013] The target-pursuer-defender (TAD) three-party game is decomposed into two sub-games. That is, the game between the target and the pursuer and the game between the defender and the pursuer. The game between the defender and the pursuer uses the motion camouflage index as its objective function. By solving the saddle point solution for each time interval, it is ensured that the game constraints are satisfied. The guidance strategy can be predicted based on the current game state. We give specific initial conditions and verify the effectiveness of the strategy through simulation. The present invention provides some theoretical support and algorithm framework for intelligent agent confrontation tasks (such as air combat interception and anti-UAV defense) in complex three-dimensional environments.

[0014] The present invention has the following characteristics and beneficial effects:

[0015] Using the above method, the present invention proposes a new target defense differential game strategy based on bionic guidance, and uses differential game theory to find the saddle point solution. The simulation results show that the method we proposed can enable the defender to achieve motion camouflage conditions and complete the task of capturing the pursuer. It provides inspiration for non-suicidal escort missions or interception missions with speed restrictions. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the prior art descriptions are briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention, and for ordinary technicians in this field, other drawings can be obtained based on these drawings without creative labor.

[0017] Figure 1 Schematic diagram of a flow chart of an embodiment of the present invention.

[0018] Figure 2 It is the coordinate system of TAD game in the embodiment of the present invention.

[0019] Figure 3 Schematic diagram of motion camouflage in an embodiment of the present invention.

[0020] Figure 4 This is a trajectory diagram in simulation example 1 in an embodiment of the present invention.

[0021] Figure 5 Schematic diagram of the angle change of the agent in simulation example 1 according to an embodiment of the present invention.

[0022] Figure 6 Schematic diagram of distance transformation between agents in simulation example 1 according to an embodiment of the present invention.

[0023] Figure 7 Schematic diagram of the change of the motion camouflage index in simulation example 1 according to an embodiment of the present invention.

[0024] Figure 8 This is a trajectory diagram in simulation example 2 in an embodiment of the present invention.

[0025] Fig. 9 Schematic diagram of the angle change of the agent in simulation example 2 in an embodiment of the present invention.

[0026] Fig.10 Schematic diagram of distance transformation between agents in simulation example 1 according to an embodiment of the present invention.

[0027] Fig.11 Schematic diagram of the change of the motion camouflage index in simulation example 1 according to an embodiment of the present invention. DETAILED DESCRIPTION

[0028] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments may be combined with each other.

[0029] In the description of the present invention, it should be understood that the terms "center", "longitudinal", "lateral", "up", "down", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inside", "outside" and the like indicate positions or positional relationships based on the positions or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore cannot be understood as limiting the present invention. In addition, the terms "first", "second", and the like are only used for descriptive purposes, and cannot be understood as indicating or implying relative importance or implicitly indicating the number of technical features indicated. Thus, features defined as "first", "second", and the like may explicitly or implicitly include one or more of the features. In the description of the present invention, unless otherwise specified, "multiple" means two or more.

[0030] In the description of the present invention, it should be noted that, unless otherwise clearly specified and limited, the terms "installed", "connected", and "connected" should be understood in a broad sense, for example, it 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 it can be indirectly connected through an intermediate medium, or it can be the internal communication of two components. For ordinary technicians in this field, the specific meanings of the above terms in the present invention can be understood by specific circumstances.

[0031] Example 1

[0032] The present invention provides a three-dimensional target defense differential game method based on motion camouflage, such as Figure 1 shown.

[0033] Step A, establishing a three-dimensional coordinate system and a three-dimensional dynamic model of the target, the pursuer, and the defender, wherein the three-dimensional dynamic model includes the position coordinates, heading angle, pitch angle, and corresponding control parameters of each intelligent agent;

[0034] Specifically, in this embodiment, consider the following scenario:

[0035] In the present invention, TAD game and coordinate system are as follows Figure 2 The pursuer (P), target (E), and defender (D) exhibit a simplified motion characterized by constant speed and turning radius. Their velocity vectors are denoted by V i =[V xi ,V yi ,V zi ] T , where i∈{P,E,D}, the corresponding speed is V i =|V i|. The state vector of the three-dimensional dynamics is defined as X i =[x,y,z,ψ,θ] T , where x, y and z represent the coordinates of the drone, which are longitude, latitude and altitude respectively, ψ represents the heading angle, and θ represents the pitch angle. The three-dimensional dynamic model is described by the following equation:

[0036]

[0037] where ψ∈[0,2π], ω Ψ ,ω θ are control parameters, representing the angular velocity in the heading and pitch directions respectively. Therefore, the control vector can be defined as U i =[0,0,0,ω Ψ ,ω θ ].

[0038]

[0039] Assume R PD ,R PE ,R DE They are the distances between the pursuer and the defender, the pursuer and the target, and the defender and the target, respectively.

[0040] Step B: decompose the target-pursuer-defender three-party game into two sub-games, wherein the two sub-games are the pursuit-escape game between the target and the pursuer, and the defense game between the defender and the pursuer under motion camouflage conditions.

[0041] Specifically, based on the three-dimensional dynamic model (1), the state transition equation of the system, which is used to predict the state of the system after Δt, can be expressed as follows:

[0042] X i (t+Δt)=Φ(t,t+Δ·)X i (t) (3)

[0043] where Φ(t,t+Δt) is the zero-input state transition matrix.

[0044] The end time t of TAD game f , defined as the moment when the pursuer captures the target, can be estimated based on the relative velocity in the line of sight direction:

[0045]

[0046] In the present invention, the defender adopts a motion camouflage (MC) strategy, which is described as the pursuer approaching the target along the line connecting the target and a reference point. Figure 3As shown, the MC strategy is described as the defender first moving along the line of sight (also called the concealment line, CL) between the pursuer and the evader, and maintaining this position until the end of the game. PD +R DE -R PE To estimate whether the defender has reached MC. According to the triangle inequality theorem, it is obvious that Z will reach the minimum value 0 only when the defender is on the PE segment. This just meets the requirement that the defender is on the hidden line (CL).

[0047] In addition, the following assumptions help to simplify the derivation process.

[0048] A1) The pursuer has better speed and maneuverability than the evader, while the defender, although slower than the pursuer, has a slight advantage in maneuverability.

[0049]

[0050] A2) The initial distance between the three players is much larger than the turning radius. Therefore, in the initial stage, the turning radius can be ignored in the derivation process.

[0051] A3) The speeds of the three players remain constant throughout the TAD game.

[0052] A4) All players know each other's states accurately.

[0053] Under the above dynamics and assumptions, the following two main issues are discussed in the subsequent sections.

[0054] Problem 1: Considering the P, E game and the defender’s MC strategy, find the strategy for each player to achieve their respective goals.

[0055] Problem 2: Given the initial conditions and the derived maneuver strategy, determine the conditions necessary for the pursuer and defender to achieve their respective capture objectives.

[0056] Step C: For each sub-game, construct a Hamiltonian containing the relative dynamics equation, apply the Pontryagin maximum / minimum principle to solve the co-state equation, derive the guidance strategies of each party, and update the strategies in real time through the discretized time series;

[0057] Specifically, in this embodiment, the TAD problem is decomposed into two sub-games. The first sub-game is a pursuit-escape game involving a target and a pursuer. The second sub-game is a defense game between a defender and a pursuer. From this, the maneuver strategy of each player is derived.

[0058] For the game strategy between the target and the pursuer:

[0059] The objective functions of the pursuer and the target are:

[0060]

[0061] The state vector and the relative dynamics equations can be expressed in relative form as follows:

[0062]

[0063] The Hamiltonian of this subgame can be written as:

[0064]

[0065] Among them, λ PE =[λ1,λ2,λ3,λ4,λ5] T is the costate vector.

[0066] Considering the pursuit-escape game objective function (6) and the state vector and relative dynamics equations (7)(8), the strategies derived for both players are as follows:

[0067]

[0068] Where p1 = x Pf -x Ef , p2=y Pf -y Ef , p3=z Pf -z Ef

[0069] x Pf ,y Pf 、z Pf Indicates the terminal position information of the pursuer, x Ef ,y Ef 、z Ef The terminal location information of the target is obtained by predicting its state and terminal time, a Pψ ,a Pθ ,a Eψ ,a Eθ They represent the angular velocity control quantities of the chaser and target in the heading and pitch directions, respectively. Sgn() is the sign function, and v P ,v E denote the speeds of the pursuer and target, respectively, sinψ P ,cosψ P ,sinθ P ,cosθ P The sine and cosine components of the chaser’s heading and pitch angles, sinψ E ,cosψ E ,sinθ E ,cosθ EThey represent the sine and cosine components of the heading and pitch angles of the target respectively.

[0070] Since the Hamiltonian and the dynamic equations of formula (9) are controlled by the variable a P and a E Therefore, the Isaac condition holds, and the optimal strategy of each player satisfies the Nash solution. According to the Pontryagin maximum / minimum principle, the optimal maneuver strategy can be expressed as follows:

[0071]

[0072] To solve equation (11), the costate equation and terminal costate are as follows:

[0073]

[0074] where x Pf ,x Ef ,y Pf ,y Ef ,z Pf ,z Ef represents the terminal positions of the pursuer and target, predicted by their states and terminal times.

[0075] X PE (t f )=Φ(t,t f -t)X PE (t) (14)

[0076] Therefore, the comorphism can be derived from (12) and (13):

[0077]

[0078] The control equation can be expressed as:

[0079]

[0080] The extreme value of , that is, the Nash equilibrium, is obtained by The final result is shown in formula (10).

[0081] The guidance strategy derived from formula (10) is actually the result of a chase-escape game that focuses on the current state and the final relative position, which are derived from formula (14). The pursuit strategy aims to adjust its f In contrast, the target's guidance strategy is designed to move in a direction that hinders the pursuer's approach.

[0082] When the game starts, participants need to perform specific manipulation actions, which makes t f The estimated value of is not as accurate as the actual value. However, due to t f – The non-negativity of t and the proximity in the line of sight, during the game, t can be updated in real time f , to replace t f –t and calculate the corresponding terminal state. This allows the strategies of both parties to be updated in real time during the game.

[0083] Further, for the defender's strategy:

[0084] Here, in this example, the defender's strategy will be derived. Although the pursuer adopts the strategy derived in the game strategy between the target and the pursuer, the defender's strategy is still determined by finding the saddle point solution in the two-player game between the defender and the pursuer:

[0085] like Figure 3 As shown in Figure 2, the motion camouflage (MC) strategy not only needs to capture the pursuer, but also requires that some constraints be satisfied during the game. In order to avoid the highly nonlinear and high-dimensional two-point boundary value problem in the objective function, this study discretized the game time and adopted a saddle point solution in each time interval.

[0086] With this transformation, the problem becomes a game involving terminal conditions at each time interval.

[0087] The objective functions of the pursuer and defender are set as follows:

[0088]

[0089] like Figure 3 As shown in Figure 1, the defender has two goals: one is to maintain the geometric constraint of motion camouflage, and the other is to capture the pursuer. Geometrically, PD+ED≥PE, and this inequality is minimized when the defender is located on the PE segment. That is, when the MC constraint is satisfied, the objective function should be minimized to 0.

[0090] At the same time, V P >V E It means that the pursuer can win the game only when the speed of the pursuer relative to the escaper in the same direction is higher than that of the escaper. In other words, when the defender is at a specific position on the LOS (line of sight), the pursuer will actively approach the defender to get closer to the escaper. In general, the objective function of the MC game can be expressed as (17).

[0091] The relative dynamics equations and solution process are similar to those described in the game strategy between the target and the pursuer, but differ in solving for the co-state variables. Therefore, the main focus is on explaining how to deal with the co-state variables.

[0092] The relative state, relative dynamic equations and co-states can be written as follows:

[0093]

[0094] Therefore, the Hamiltonian is

[0095] Based on the subgame (17) and the relative dynamics equation (18), the defender's guidance strategy can be determined as follows:

[0096]

[0097] Among them, a Dψ ,a Dθ represents the defender’s angular velocity control in the heading and pitch directions, Sgn() is a sign function, and v P ,v D are the speeds of the pursuer and defender, respectively, sinψ P ,cosψ P ,sinθ P ,cosθ P The sine and cosine components of the chaser’s heading and pitch angles, sinψ D ,cosψ D ,sinθ D ,cosθ D denote the sine and cosine components of the defender’s heading and pitch angles, respectively, and

[0098]

[0099] Among them, x P ,y P 、z P represents the position information of the pursuer after each time interval, x E ,y E 、z E Represents the location information of the target after each time interval, x D ,y D 、z D Represents the location information of the target after each time interval.

[0100] The proof process is similar to the game strategy between the target and the pursuer, the main difference lies in the objective function and the terminal time.

[0101] The control equation can be expressed as:

[0102]

[0103] Based on the subgame (17) and the relative dynamics equation (18), the defender's guidance strategy can be determined as follows:

[0104] The co-state equation has the same form as (12). When solving the co-state equation, the time series is discretized and each discrete point is regarded as the terminal time of the game. Therefore, in each time interval (t, t+τ), the solution of the co-state equation is:

[0105]

[0106] The terminal state of each time interval can be similarly obtained by equation (14):

[0107] X PD (t+τ)=Φ(t,t+τ)X PD (t) (22)

[0108] By substituting (21) and (22) into the co-state equation, the co-state variables can be obtained. Then, according to the non-negativity of Pontryagin's maximum / minimum principle, the defender's guidance strategy can be derived similarly to the pursuit-escape game and expressed as (19).

[0109] By analyzing the objective function (17), it is obvious that the Nash strategy of the pursuer here is not to capture the escaper as in section (10). Instead, its strategy is to destroy the MC condition set by the defender. Therefore, the strategy of the pursuer, as discussed in the subsequent content of the present invention, is achieved through the capture strategy, as derived in the game strategy between the target and the pursuer.

[0110] Step D: Based on the initial conditions and the guidance strategy, the saddle point solution of each time period is solved through the simulation environment, and dynamic guidance instructions are generated to complete the capture task of the attacker.

[0111] Example 2

[0112] The difference between this embodiment and embodiment 1 is that in this embodiment, the turning maneuverability of the three players is set to ω Pm =0.2rad / s,ω Em =0.1rad / s and ω Dm =0.3rad / s, τ is set to 0.5s.

[0113]

[0114] Table 1 is the data of Example 2

[0115] The trajectories of the three players in the chase-escape game are as follows: Figure 4As shown in Figure 1, the chaser will win at 16.9 seconds, while the defender will catch the chaser at 14.7 seconds. The angle parameters are as follows: Figure 5 As shown, the above theorem is verified. The change of distance between players and motion camouflage index is shown in Figure 6 , 7 shown.

[0116] Example 3

[0117] The difference between this embodiment and embodiment 1 is that in this embodiment, the turning maneuverability of the three players is set to ω Pm =0.2rad / s,ω Em =0.1rad / s and ω Dm =0.3rad / s, τ is set to 0.5s.

[0118]

[0119] Table 2 is the data of Example 3

[0120] The trajectories of the three players in the chase-escape game are as follows: Figure 8 As shown in Figure 1, the chaser will win at 15.9 seconds, while the defender will catch the chaser at 8.5 seconds. The angle parameters are as follows: Fig. 9 As shown, the above theorem is verified. The change of distance between players and motion camouflage index is shown in Fig.10 , 11 shown.

[0121] The present invention proposes a TAD game in which the defender adopts a motion camouflage strategy to move to the camouflage line (CL) and maintain this state until the pursuer is captured. The three-party differential game is decomposed into two zero-sum subgames: one between the pursuer and the escaper, and the other between the pursuer and the defender, where the defender operates under the motion camouflage constraint. The final open-loop strategy is derived through Isaacs' differential game theory and based on real-time state prediction over the game duration.

[0122] The heading and pitch angle control proposed in this invention is derived from the maneuver strategy.Based on these results, we analyze the necessary conditions for the pursuer and defender to achieve their capture goals respectively.

[0123] Finally, we verify the theoretical findings through simulation results. The proposed defender strategy has potential application value in non-suicidal cover missions or limited speed interception missions.

[0124] The embodiments of the present invention are described in detail above with reference to the accompanying drawings, but the present invention is not limited to the described embodiments. For those skilled in the art, various changes, modifications, substitutions and variations of these embodiments including components are made without departing from the principles and spirit of the present invention, and still fall within the scope of protection of the present invention.

Claims

1. A differential game method for three-dimensional target defense based on motion camouflage, characterized in that: The steps include: Step A, establishing a three-dimensional coordinate system and a three-dimensional dynamic model of the target, the pursuer, and the defender, wherein the three-dimensional dynamic model includes the position coordinates, heading angle, pitch angle, and corresponding control parameters of each intelligent agent; Step B, decomposing the target-pursuer-defender three-party game into two sub-games, the two sub-games are respectively a pursuit-escape game between the target and the pursuer, and a defense game between the defender and the pursuer under motion camouflage conditions; Step C: For each sub-game, construct a Hamiltonian containing the relative dynamics equation, apply the Pontryagin maximum / minimum principle to solve the co-state equation, derive the guidance strategies of each party, and update the strategies in real time through the discretized time series; Step D, based on the initial conditions and the guidance strategy, solving the saddle point solution of each time period through the simulation environment, and generating dynamic guidance instructions to complete the attacker's capture task; The motion camouflage constraint requires the defender to always be located in the direction of the line connecting the pursuer and the target during the game, and optimizes the interception path by minimizing the motion camouflage index.

2. A three-dimensional target defense differential game method based on motion camouflage according to claim 1, characterized in that: The state vector of the three-dimensional dynamic model is defined as: X i [x,y,z,ψ,θ] T Where x, y and z represent the coordinates of the agent, which are longitude, latitude and altitude respectively, ψ represents the heading angle, and θ represents the pitch angle; The control vector of the three-dimensional dynamic model is defined as: U i =[0,0,0,ω Ψ Oh, oh θ ] Among them, ω Ψ ,ω θ are control parameters, representing the angular velocity in the heading and pitch directions respectively.

3. The differential game method for three-dimensional target defense based on motion camouflage according to claim 1 is characterized in that: In step B, the motion camouflage constraint is achieved through the following conditions: Z=R PD +R DE -R PE Among them, R PD , R PE , R DE They are the distances between the pursuer and the defender, the pursuer and the target, and the defender and the target, respectively.

4. The differential game method for three-dimensional target defense based on motion camouflage according to claim 1 is characterized in that: The objective function of the game strategy between the target and the pursuer in the guidance strategy is: The objective function of the defense game strategy for the defender and the pursuer in the guidance strategy is: Among them, R PE (t f ) is the terminal distance between the pursuer and the target, R PD (t f ) and R DE (t f ) are the terminal distances between the defender and the pursuer, and between the defender and the target, respectively.

5. The differential game method for three-dimensional target defense based on motion camouflage according to claim 4 is characterized in that: The game strategy between the target and the pursuer is expressed as follows: Where p1 = x Pf -x Ef , p2=y Pf -y Ef , p3=z Pf -z Ef ; x Pf ,y Pf 、z Pf represents the terminal position information of the pursuer, x Ef ,y Ef 、z Ef The terminal location information of the target is obtained by predicting its state and terminal time, a Pψ , a Pθ , a Eψ , a Eθ They represent the angular velocity control quantities of the chaser and target in the heading and pitch directions, respectively. Sgn() is a sign function, and v P , v E denote the speeds of the pursuer and target, respectively, sinψ P , cosψ P , sinθ P , cosθ P The sine and cosine components of the chaser’s heading and pitch angles, sinψ E , cosψ E , sinθ E , cosθ E They represent the sine and cosine components of the heading and pitch angles of the target respectively.

6. The differential game method for three-dimensional target defense based on motion camouflage according to claim 5 is characterized in that: The defensive game strategy for the defender and the pursuer in the guidance strategy is expressed as follows: Among them, a Dψ , a Dθ represents the defender’s angular velocity control in the heading and pitch directions, Sgn() is a sign function, and v P , v D are the speeds of the pursuer and defender, respectively, sinψ P , cosψ P , sinθ P , cosθ P The sine and cosine components of the chaser’s heading and pitch angles, sinψ D , cosψ D , sinθ D , cosθ D Represent the sine and cosine components of the defender’s heading and pitch angles, Among them, x P ,y P 、z P represents the position information of the pursuer after each time interval, x E ,y E 、z E Represents the location information of the target after each time interval, x D ,y D 、z D Represents the location information of the target after each time interval.

7. The differential game method for three-dimensional target defense based on motion camouflage according to claim 4 is characterized in that: The saddle point solution includes: predicting the terminal state in each time interval by discretizing the time series, and updating the guidance strategy in real time based on the co-state equation and the terminal co-state condition.

8. The differential game method for three-dimensional target defense based on motion camouflage according to claim 1 is characterized in that: In the step D, the simulation environment is built by MATLAB, and the ODE function is used to solve the differential equation and verify the effectiveness of the guidance strategy, and generate the dynamic trajectory of the capture time, motion camouflage index and three-party game path.

9. The differential game method for three-dimensional target defense based on motion camouflage according to claim 1 is characterized in that: The attacker's mobility is better than the target's, and the defender's angular velocity control ability is better than the attacker's, specifically satisfying: Among them, V P 、V E and V D are the speeds of the pursuer, target and defender respectively, ω Pm ,ω Em and ω Dm are the maximum turning angular velocities of the pursuer, target, and defender, respectively.

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