A differential game method for three-dimensional target defense based on motion camouflage

Abstract

The present invention discloses a three-dimensional target defense differential game method based on motion camouflage, comprising the following steps: establishing a three-dimensional coordinate system and a three-dimensional dynamic model of the target, pursuer, and defender; decomposing the target-pursuer-defender three-party game into two sub-games; for each sub-game, constructing a Hamiltonian containing the relative dynamic equation, applying the Pontryagin maximum / minimum principle to solve the co-state equation, deriving the guidance strategy of each party, and updating the strategy in real time through a discretized time series; based on the initial conditions and the guidance strategy, solving the saddle point solution for each time period in a simulation environment, and generating dynamic guidance instructions to complete the attacker's capture mission. This method, for the target-pursuer-defender (TAD) three-party game, achieves efficient interception of the pursuer by the defender and dynamic protection of the target by the target by the defender by decomposing the game problem, designing an equilibrium strategy in three-dimensional space, and combining the motion camouflage constraints of biomimicry.

Description

Technical Field

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

[0002] Pursuit-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 real-world 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-escape game: players use three-degree-of-freedom control. A more practical application of the pursuit-escape game is the target-pursuer-defender differential game (TAD), in which the pursuer's goal is to capture the target, the defender's goal 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 various visually perceptive insects to capture prey, first proposed by Srini Vasan et al. It generally refers to the situation where, from the perspective of the avoider, the pursuer always stays in a straight line between a reference point and the avoider, and maintains a constant azimuth angle.

[0004] In existing technologies, linear quadratic differential games (LQDGs) are widely used in multi-agent adversarial problems, but they are often limited to two-dimensional space and fail to effectively integrate dynamic constraints with real-time policy updates. For example, the three-party game model proposed by Rusnak et al. does not address three-dimensional maneuvering strategies, and the state feedback optimal policy proposed by Garcia is computationally complex and difficult to apply in real time. Regarding biomimetic strategies, the motion camouflage proposed by Srinivasan only targets single-tracking scenarios and has not been extended to three-party games.

[0005] Existing technologies have the following defects: 1. Dimensional limitations: Traditional methods are mostly limited to two-dimensional space and cannot be directly applied to actual three-dimensional scenarios; 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, resulting in high maneuvering energy consumption and limited interception success rates; 4. High computational complexity: Directly solving the three-party game is difficult and lacks an effective decomposition method. Summary of the Invention

[0006] To address 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, by decomposing the game problem, designing an equilibrium strategy in three-dimensional space, and combining the motion camouflage constraints of biomimicry, the defender can efficiently intercept the pursuer and dynamically protect the target.

[0007] In order to achieve the above object, the specific technical solutions adopted by the present invention are 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 conditions

[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 three-party guidance strategy is derived. The strategy is updated in real time through the discretized time series.

[0012] Step D: Based on 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 subgames: 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, the game constraints are ensured to be satisfied. The guidance strategy can be predicted based on the current game state. We provide specific initial conditions and verify the effectiveness of the strategy through simulation. This paper provides some theoretical support and algorithmic framework for intelligent agent confrontation tasks in complex three-dimensional environments (such as air combat interception and anti-drone defense).

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

[0015] Using this approach, we propose a novel differential game strategy for target defense based on biomimetic guidance, utilizing differential game theory to solve saddle point problems. Simulation results demonstrate that our proposed method enables the defender to achieve motion camouflage conditions and capture the pursuer. This provides insights for non-suicidal escort missions or interception missions with speed restrictions. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] To more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present invention. Those skilled in the art can also derive other drawings based on these drawings without inventive effort.

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

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

[0027] Figure 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 orientations or positional relationships based on the orientations 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", etc. are only used for descriptive purposes and cannot be understood as indicating or implying relative importance or implicitly indicating the number of the indicated technical features. Therefore, features defined as "first", "second", etc. 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 expressly specified or limited, the terms "mounted," "connected," and "connected" should be understood in a broad sense. For example, they may refer to fixed connections, detachable connections, or integral connections; mechanical connections or electrical connections; direct connections or indirect connections through an intermediate medium; and internal communication between two components. Those skilled in the art will understand the specific meanings of the above terms in the present invention based on 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: Establish a three-dimensional coordinate system and a three-dimensional dynamic model of the target, pursuer, and defender. The three-dimensional dynamic model includes the position coordinates, heading angle, pitch angle, and corresponding control parameters of each 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 As shown in 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 as V i =[V xi ,V yi ,V zi ] T , where i∈{P,E,D}, the corresponding velocity 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] Let R PD ,R PE ,R DE 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 termination time t of the 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 the 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 cover 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 its minimum value of 0 only when the defender is located on the PE segment. This just meets the requirement that the defender is located 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] Question 1: Consider the P, E game and the defender's MC strategy, find the strategy of 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 subgame, construct a Hamiltonian containing the relative dynamics equation, apply the Pontryagin maximum / minimum principle to solve the co-state equation, derive the guidance strategy of each party, and update the strategy in real time through the discretized time series;

[0057] Specifically, this example decomposes the TAD problem into two subgames. The first subgame is a pursuit-escape game involving a target and a pursuer. The second subgame is a defense game between a defender and a pursuer. From this, each player's maneuver strategy 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 position information of the target is obtained by its state and terminal time prediction, a Pψ ,a Pθ ,a Eψ ,a Eθ They represent the angular velocity control quantities of the pursuer and target in the heading and pitch directions, Sgn() is the sign function, and v P ,v E represent the speed 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 ERepresent the sine and cosine components of the target's heading and pitch angles respectively.

[0070] Since the Hamiltonian and dynamic equations of formula (9) are in the control variable a P and a E is decoupled. Therefore, the Isaac condition holds, and each player's optimal strategy satisfies the Nash solution. According to Pontryagin's 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 governing 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, since t f – The non-negativity of t and the proximity in the line of sight allow t to be updated in real time during the game. 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, regarding the defender's strategy:

[0084] Here, in this example, we derive the defender's strategy. 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 a saddle point solution in the two-player game between the defender and the pursuer:

[0085] like Figure 3 As shown, the motion camouflage (MC) strategy not only needs to capture the pursuer, but also requires that some constraints be satisfied during the game. To avoid the highly nonlinear and high-dimensional two-point boundary value problem in the objective function, this study discretizes the game time and adopts a saddle point solution within 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, the defender has two objectives: one is to maintain the geometric constraints of motion camouflage, and the other is to capture the pursuer. Geometrically, PD + ED ≥ PE, and this inequality is minimized when the defender lies on the PE segment. In other words, when the MC constraints are satisfied, the objective function should be minimized to 0.

[0090] At the same time, V P >V E = ∑ i = 1 i ∈ ...

[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 the solution of the covariate variables. Therefore, the main focus is on explaining how to deal with the covariate 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 quantity in the heading and pitch directions, Sgn() is the sign function, 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 Indicates 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, with the main differences being the objective function and the terminal time.

[0101] The governing 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 formula (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, based on 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 Equation (19).

[0109] By analyzing the objective function (17), it is clear that the pursuer's Nash strategy here is not to capture the escaper as in Section (10). Instead, its strategy is to destroy the MC condition established by the defender. Therefore, the pursuer's strategy, as discussed later in this invention, is achieved through a capture strategy, as derived from 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 attacker's capture task.

[0111] Example 2

[0112] The difference between this embodiment and embodiment 1 is that in this embodiment, the steering 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 Figure 4As shown in Figure 2, the chaser will win at 16.9 seconds, while the defender will capture the chaser at 14.7 seconds. Figure 5 As shown, the above theorem is verified. The change of distance between players and motion camouflage index is as follows Figure 6 、 7 shown.

[0116] Example 3

[0117] The difference between this embodiment and embodiment 1 is that in this embodiment, the steering 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 Figure 8 As shown in Figure 2, the chaser will win at 15.9 seconds, while the defender will capture the chaser at 8.5 seconds. The angle parameters are as follows: Figure 9 As shown, the above theorem is verified. The change of distance between players and motion camouflage index is as follows Figure 10 、 11 shown.

[0121] This paper 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 evader, and the other between the pursuer and the defender, with the defender operating under motion-camouflage constraints. The final open-loop strategy is derived using Isaacs' differential game theory and based on real-time state prediction over the duration of the game.

[0122] The heading and pitch angle control proposed in this paper are derived from the maneuvering 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 speed-limited 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. It will be apparent to those skilled in the art that various changes, modifications, substitutions, and variations of these embodiments, including components, without departing from the principles and spirit of the present invention are still within the scope of protection of the present invention.

Claims

1. A three-dimensional target defense differential game method based on motion camouflage, characterized in that: The steps include: Step A: Establish a three-dimensional coordinate system and a three-dimensional dynamic model of the target, pursuer, and defender. The three-dimensional dynamic model includes the position coordinates, heading angle, pitch angle, and corresponding control parameters of each agent. Step B: decomposing the target-pursuer-defender three-party game into two sub-games, wherein the two sub-games are 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 subgame, construct a Hamiltonian containing the relative dynamics equation, apply the Pontryagin maximum / minimum principle to solve the co-state equation, derive the guidance strategy of each party, and update the strategy in real time through the discretized time series; Step D: Based on the initial conditions and the guidance strategy, solving the saddle point solution for each time period through a simulation environment, and generating dynamic guidance instructions to complete the attacker's capture task; 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: in, 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; 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 Indicates the terminal position information of the pursuer, x Ef 、y Ef 、z Ef represents the terminal location information of the evader, which is predicted by its state and terminal time, a Pψ , a Pθ , a Eψ , a Eθ They represent the angular velocity control quantities of the pursuer and target in the heading and pitch directions, Sgn() is the sign function, and v P ,v E represent the speed 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 Represent the sine and cosine components of the target's heading and pitch angles respectively; 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 the interception path is optimized 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,w Ψ ,w θ ] Among them, w Ψ , w 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, 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 are the distances between the defender and the pursuer, the defender and the target, and the pursuer and the target, respectively.

4. The differential game method for three-dimensional target defense based on motion camouflage according to claim 1, 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 quantity in the heading and pitch directions, Sgn() is the sign function, 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 Indicates the location information of the evader after each time interval.

5. 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 is obtained by discretizing the time series, predicting the terminal state in each time interval, and updating the guidance strategy in real time based on the co-state equation and the terminal co-state condition.

6. The differential game method for three-dimensional target defense based on motion camouflage according to claim 1, characterized in that: In step D, a simulation environment is built using MATLAB, and ODE functions are used to solve differential equations and verify the effectiveness of the guidance strategy, thereby generating dynamic trajectories of the pursuer's interception time and the defender's capture time.

7. The differential game method for three-dimensional target defense based on motion camouflage according to claim 1, characterized in that: The attacker's maneuverability is superior to the target's, and the defender's angular velocity control capability is superior to the attacker's, specifically meeting the following requirements: Among them, V P 、V E and V D are the speeds of the pursuer, target, and defender, respectively, a Pm 、a Em and a Dm are the maximum maneuvering speeds of the pursuer, target, and defender, respectively.

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