A pursuit-evasion game method with elliptic motion constraint

Abstract

This invention relates to the field of aviation control technology, specifically to a pursuit-escape game method with elliptical motion constraints, used for pursuit-escape missions of unmanned systems with motion constraints. Based on the abstraction of the motion region as an ellipse (circle), the pursuit party's motion region is modeled as a circle with a given radius, and the escape party's motion region is modeled as an ellipse with given semi-major and semi-minor axes. This establishes an elliptical motion-constrained pursuit-escape game model, and a distance function is established within it. By transforming the bilateral optimization problem into an extremum problem, the equilibrium solutions for both the pursuer and escape parties are obtained using the extremum principle based on the distance function. Furthermore, the optimal pursuit strategy for the pursuer and the optimal escape strategy for the escape party are analytically obtained, effectively solving the pursuit-escape problem of unmanned systems within a constrained motion region (ellipse).

Description

Technical Field

[0001] This invention relates to the field of aviation control technology, specifically to a pursuit and escape game method with elliptical motion constraints, for pursuit and escape missions of unmanned systems with motion constraints. Background Technology

[0002] Unmanned system pursuit and escape missions are a game theory problem with wide-ranging applications, serving as a crucial step in typical scenarios such as security, counter-terrorism, and patrol. However, most previous research on pursuit and escape game theory has primarily focused on problem modeling and analysis within free space. Free space refers to the unrestricted movement of the participating agents (pursuers and escapees) in two- or three-dimensional space. Air combat and missile interception missions can be considered typical examples of free-space pursuit and escape missions, where the trajectories of aircraft and missiles are unrestricted in three-dimensional airspace. However, in some mission scenarios, the movement areas of both the pursuer and escapee are not free and unlimited, but rather constrained and restricted. For example, drones used for high-value target protection can only pursue or expel threatening targets within a certain protected area, while the threatening targets, due to fuel or communication limitations, can only escape or move within a certain range. This constrained movement area can typically be simply represented as an ellipse of a certain size, resulting in the pursuer and escapee only being able to move freely within different but defined ellipses. However, there is still a gap in existing research on pursuit and escape game problems with elliptical motion constraints, which greatly limits the development of pursuit and escape missions of unmanned systems. Summary of the Invention

[0003] In view of the current background technology, the present invention provides a pursuit and escape game method with elliptical motion constraints, which solves the technical problem that unmanned systems cannot complete pursuit and escape tasks within an elliptical motion area.

[0004] This invention is achieved through the following technical solution:

[0005] A pursuit-escape game method with elliptical motion constraints includes the following steps:

[0006] Establish an elliptical motion-constrained pursuit and escape game model, establish a distance function based on the elliptical motion-constrained pursuit and escape game model, and input the specific parameters of the elliptical motion-constrained pursuit and escape game model into the distance function;

[0007] Within the distance function, the escape phase angle of the fleeing party is kept constant. Based on the principle of extreme values, the optimal pursuit phase angle function with the escape phase angle as the independent variable is obtained when the pursuit distance is minimized, which serves as its pursuit strategy.

[0008] Substitute the best pursuit phase angle function of the pursuer into the distance function, and solve the best escape phase angle of the escaper when the pursuit distance is maximized based on the principle of extrema, which can be used as its escape strategy.

[0009] Input the optimal escape phase angle into the optimal pursuit phase angle function to obtain the optimal pursuit phase angle;

[0010] Outputting the optimal pursuit phase angle of the pursuer yields the optimal pursuit strategy of the pursuer, and outputting the optimal escape phase angle of the escaper yields the optimal escape strategy of the escaper, thus completing the elliptical motion-constrained pursuit-escape game.

[0011] The preferred calculation formula for the pursuit-escape game model is as follows:

[0012]

[0013] Where P represents the pursuing party; E represents the escaping party; x P The x-coordinate represents the pursuing party's position on the x-axis; the y-coordinate represents the pursuing party's position P Indicates the y-coordinate of the pursuing party; x E Represents the x-coordinate of the fleeing party on the x-axis; y E α represents the y-coordinate of the fleeing party; β represents the phase angle of the pursuing party on the circle; r represents the radius of the circle containing the pursuing party; a represents the semi-major axis of the ellipse containing the fleeing party; b represents the semi-minor axis of the ellipse containing the fleeing party; L represents the length between the center of the circle containing the pursuing party and the center of the ellipse containing the fleeing party; d represents the distance between the two parties.

[0014] Preferably, a distance function is established based on the elliptical motion-constrained pursuit-escape game model, wherein the formula for calculating the distance function is as follows:

[0015]

[0016] Where α represents the phase angle of the pursuer on the circle; β represents the phase angle of the escaper on the ellipse; d represents the distance between the pursuer and the escaper; r represents the radius of the circle containing the pursuer; a represents the semi-major axis of the ellipse containing the escaper; b represents the semi-minor axis of the ellipse containing the escaper; and L represents the length between the center of the circle containing the pursuer and the center of the ellipse containing the escaper.

[0017] Furthermore, the specific parameters of the elliptical motion-constrained pursuit-escape game model include the radius r of the circle containing the pursuer, the major semi-axis a of the ellipse containing the escaper, the minor semi-axis b of the ellipse containing the escaper, and the length L between the center of the circle containing the pursuer and the center of the ellipse containing the escaper.

[0018] Furthermore, by inputting the specific parameters of the elliptical motion-constrained pursuit-escape game model into the distance function, the steps for calculating the optimal strategy for both the pursuer and the escaper are as follows:

[0019] With the phase angle β of the fleeing party on the ellipse fixed, calculate the first partial derivative of the distance d between the pursuer and the fleeing party with respect to the phase angle α of the pursuer on the circle. Set this partial derivative to zero to obtain the possible values ​​α1 and α2 of the phase angle α of the pursuer on the circle when the distance d between the pursuer and the fleeing party reaches its extreme value; where, α2 = π + α1;

[0020] Within the distance function, the phase angle β of the fleeing party on the ellipse is kept constant. The second partial derivative of the distance d between the pursuer and the fleeing party with respect to the phase angle α of the pursuer on the circle is calculated. The possible values ​​α1 and α2 are input into the second partial derivative to calculate the extreme points. Among them, the possible extreme points are greater than α1, which is the minimum value and α2, which is the maximum value.

[0021] Let the minimum value α1 be the solution for the pursuer's optimal strategy in the chase game, denoted as α. * (β), and α * (β) Inputting the distance function, the formula for the distance function between the pursuer and the fleeing party when the pursuer adopts the optimal strategy is as follows:

[0022]

[0023] Where, α * (β) represents the solution of the optimal pursuit phase angle for the pursuer in the pursuit-escape game; L represents the length between the center of the circle where the pursuer is located and the center of the ellipse where the escaper is located; a represents the major semi-axis of the ellipse where the escaper is located; b represents the minor semi-axis of the ellipse where the escaper is located; β represents the phase angle of the escaper on the ellipse; r represents the radius of the circle where the pursuer is located.

[0024] Furthermore, the calculation process for the escapee's optimal strategy is as follows:

[0025] When the optimal strategy is adopted for the pursuer, the partial derivative of the distance function between the pursuer and the fleeing party with respect to the phase angle β of the fleeing party on the ellipse is calculated, and the phase angle β of the fleeing party on the ellipse is set to zero. This yields the possible extreme points of β when the distance reaches an extreme value. The specific formula is as follows:

[0026]

[0027] Among them, when At that time, β has only 2 extreme points;

[0028] when At that time, β has 4 extreme points;

[0029] Where L represents the length between the center of the circle containing the pursuer and the center of the ellipse containing the escaper; a represents the major semi-axis of the ellipse containing the escaper; b represents the minor semi-axis of the ellipse containing the escaper; and β represents the phase angle of the escaper on the ellipse.

[0030] Furthermore, by considering the possible extreme points of β when the distance reaches its extreme value, and based on the value of the length L between the center of the circle containing the pursuer and the center of the ellipse containing the escaper, the optimal escape phase angle β for the escaper can be determined from the possible extreme points. * , among which, if Then the optimal escape phase angle β for the escaper * =0, at this point the equilibrium solution of the distance between the pursuer and the pursuer is:

[0031] d(α * ,β * )=L+br

[0032] Wherein: the optimal pursuit phase angle α of the pursuing side * =0;

[0033] if There are two optimal strategies for the escapee: or At this point, the equilibrium solution for the distance between the pursuing and fleeing parties is:

[0034]

[0035] in: or

[0036] Where, α * The optimal pursuit phase angle for the pursuing side; β * denoted as the optimal escape phase angle for the escaping side; L represents the length between the center of the circle containing the pursuing side and the center of the ellipse containing the escaping side; a represents the major semi-axis of the ellipse containing the escaping side; b represents the minor semi-axis of the ellipse containing the escaping side; and r represents the radius of the circle containing the pursuing side.

[0037] Preferably, the elliptical motion constraint is that the motions of both the pursuer and the escaper lie on an ellipse in a two-dimensional plane.

[0038] Compared with the prior art, the present invention has the following beneficial technical effects:

[0039] This invention provides a pursuit-escape game method with elliptical motion constraints. Based on the abstraction of the motion region as an ellipse (circle), the pursuit party's motion region is modeled as a circle with a given radius, and the escape party's motion region is modeled as an ellipse with given semi-major and semi-minor axes. This establishes an elliptical motion-constrained pursuit-escape game model, and a distance function is established within this model. By transforming the bilateral optimization problem into an extremum problem, equilibrium solutions for both the pursuer and escape party are obtained using the extremum principle based on the distance function. Furthermore, the optimal pursuit strategy for the pursuer and the optimal escape strategy for the escape party are analytically obtained. By simplifying the feasible motion regions of both parties by abstracting them into elliptical (circular) motion constraints, this invention effectively solves the problem of pursuit-escape missions in unmanned systems operating within constrained motion regions (ellipses). This invention aligns with the actual motion characteristics of drones, unmanned vehicles, and unmanned ships, which have fuel consumption or communication distance limitations, and therefore has wide applicability. Attached Figure Description

[0040] Figure 1 This is a flowchart of the pursuit and escape game method in this invention;

[0041] Figure 2 This refers to a pursuit scenario with elliptical motion constraints that is applicable to this invention.

[0042] Figure 3 This is a graph showing the variation of the distance between the pursuing and fleeing parties as a function of angles α and β under the parameter conditions of Example 1 of this invention;

[0043] Figure 4 This is a graph showing the change in distance between the pursuing and fleeing parties as a function of angles α and β under the parameter conditions of Example 2 of this invention;

[0044] Figure 5 This is a family of curves showing the variation of the distance between the pursuing and fleeing parties under different angles β under the parameter conditions of Example 2 of this invention;

[0045] Figure 6 This is the curve showing the change of the distance between the pursuing and fleeing parties under the parameter conditions of Example 2 of the present invention as a function of α when β = 1; Detailed Implementation

[0046] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0047] It should be noted that the terms "first," "second," etc., in the specification, claims, and accompanying drawings of this invention are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments of the invention described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover a non-exclusive inclusion; for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0048] The present invention will now be described in further detail with reference to the accompanying drawings:

[0049] This invention provides a pursuit-escape game theory method with elliptical motion constraints. The pursuer's movement region is modeled as a circle with a given radius, and the escaper's movement region is modeled as an ellipse with given semi-major and semi-minor axes. A pursuit-escape game model based on the "minimum-maximum distance" principle is then established, and the equilibrium solution to the aforementioned game problem is obtained using the extremum principle. This method is relatively simple in form and has clear physical meaning, providing an effective solution for pursuit-escape tasks in unmanned systems with motion constraints.

[0050] Specifically, according to Figure 1 As shown, this pursuit and escape game method includes the following steps:

[0051] S1. Establish an elliptical motion-constrained pursuit and escape game model, establish a distance function based on the elliptical motion-constrained pursuit and escape game model, and input the specific parameters of the elliptical motion-constrained pursuit and escape game model into the distance function;

[0052] S2. Within the distance function, keep the escape phase angle of the fleeing party constant, and solve the optimal pursuit phase angle function with the escape phase angle as the independent variable when the pursuit distance is minimized based on the principle of extreme values. Use this function as the pursuit strategy.

[0053] S3. Substitute the best pursuit phase angle function of the pursuer into the distance function, and solve the best escape phase angle of the escaper when the pursuit distance is the maximum based on the principle of extrema, and use it as its escape strategy.

[0054] S4. Input the optimal escape phase angle into the optimal pursuit phase angle function to obtain the optimal pursuit phase angle;

[0055] S5. Output the best pursuit phase angle of the pursuer to obtain the best pursuit strategy of the pursuer, and output the best escape phase angle of the escaper to obtain the best escape strategy of the escaper, thus completing the elliptical motion constrained pursuit and escape game.

[0056] Specifically, the calculation formula for the pursuit-escape game model is as follows:

[0057]

[0058] Where P represents the pursuing party; E represents the escaping party; x P The x-coordinate represents the pursuing party's position on the x-axis; the y-coordinate represents the pursuing party's position P Indicates the y-coordinate of the pursuing party; x E Represents the x-coordinate of the fleeing party on the x-axis; y E α represents the y-coordinate of the fleeing party; β represents the phase angle of the pursuing party on the circle; r represents the radius of the circle containing the pursuing party; a represents the semi-major axis of the ellipse containing the fleeing party; b represents the semi-minor axis of the ellipse containing the fleeing party; L represents the length between the center of the circle containing the pursuing party and the center of the ellipse containing the fleeing party; d represents the distance between the two parties.

[0059] The distance function is established based on the elliptical motion-constrained pursuit-escape game model, and the formula for calculating the distance function is as follows:

[0060]

[0061] Where α represents the phase angle of the pursuer on the circle; β represents the phase angle of the escaper on the ellipse; d represents the distance between the pursuer and the escaper; r represents the radius of the circle containing the pursuer; a represents the semi-major axis of the ellipse containing the escaper; b represents the semi-minor axis of the ellipse containing the escaper; and L represents the length between the center of the circle containing the pursuer and the center of the ellipse containing the escaper.

[0062] Specifically, the parameters of the elliptical motion-constrained pursuit-escape game model include the radius r of the circle where the pursuer is located, the major semi-axis a of the ellipse where the escaper is located, the minor semi-axis b of the ellipse where the escaper is located, and the length L between the center of the circle where the pursuer is located and the center of the ellipse where the escaper is located.

[0063] Specifically, the parameters of the elliptical motion-constrained pursuit-escape game model are input into the distance function, and the calculation steps within the distance function are as follows:

[0064] With the phase angle β of the fleeing party on the ellipse fixed, calculate the first partial derivative of the distance d between the pursuer and the fleeing party with respect to the phase angle α of the pursuer on the circle. Set this partial derivative to zero to obtain the possible values ​​α1 and α2 of the phase angle α of the pursuer on the circle when the distance d between the pursuer and the fleeing party reaches its extreme value; where, α2 = π + α1.

[0065] Within the distance function, the phase angle β of the fleeing party on the ellipse is kept constant. The second partial derivative of the distance d between the pursuer and the fleeing party with respect to the phase angle α of the pursuer on the circle is calculated. The possible values ​​α1 and α2 are input into the second partial derivative to calculate the extreme points. Among them, the possible extreme points are greater than α1, which is the minimum value and α2, which is the maximum value.

[0066] Let the minimum value α1 be the solution for the pursuer's optimal strategy in the chase game, denoted as α. * (β), and α * (β) Input to the distance function to obtain the distance function between the pursuer and the fleeing party when the pursuer adopts the optimal strategy;

[0067] The formula for the distance function between the pursuer and the fleeing party when the pursuer adopts the optimal strategy is as follows:

[0068]

[0069] Where, α * (β) represents the solution of the best strategy of the pursuer in the pursuit-escape game; L represents the length between the center of the circle where the pursuer is located and the center of the ellipse where the escaper is located; a represents the semi-major axis of the ellipse where the escaper is located; b represents the semi-minor axis of the ellipse where the escaper is located; β represents the phase angle of the escaper on the ellipse; r represents the radius of the circle where the pursuer is located.

[0070] Specifically, the process of determining the optimal strategy for the fleeing party from the possible extreme points obtained by calculating the distance function between the pursuer and the fleeing party when the pursuer adopts the optimal strategy is as follows:

[0071] When the optimal strategy is adopted for the pursuer, the partial derivative of the distance function between the pursuer and the fleeing party with respect to the phase angle β of the fleeing party on the ellipse is calculated, and the phase angle β of the fleeing party on the ellipse is set to zero. This yields the possible extreme points of β when the distance reaches an extreme value. The specific formula is as follows:

[0072]

[0073] Among them, when At that time, β has only 2 extreme points;

[0074] when At that time, β has 4 extreme points;

[0075] Where L represents the length between the center of the circle containing the pursuer and the center of the ellipse containing the escaper; a represents the major semi-axis of the ellipse containing the escaper; b represents the minor semi-axis of the ellipse containing the escaper; and β represents the phase angle of the escaper on the ellipse.

[0076] Specifically, by considering the possible extreme points of β when the distance reaches its extreme value, and based on the value of the length L between the center of the circle containing the pursuer and the center of the ellipse containing the escaper, the optimal strategy β for the escaper is determined from the possible extreme points. * , among which, if Then the best strategy β for the escapee * =0, at this point the equilibrium solution of the distance between the pursuer and the pursuer is:

[0077] d(α * ,β * )=L+br

[0078] Among them: the best strategy of the pursuing side α * =0;

[0079] if There are two optimal strategies for the escapee: or At this point, the equilibrium solution for the distance between the pursuing and fleeing parties is:

[0080]

[0081] in: or

[0082] Where, α * The best strategy for the pursuer; β * The optimal strategy for the fleeing side is defined by L, which represents the length between the center of the circle containing the pursuer and the center of the ellipse containing the fleeing side; a represents the major semi-axis of the ellipse containing the fleeing side; b represents the minor semi-axis of the ellipse containing the fleeing side; and r represents the radius of the circle containing the pursuer.

[0083] Example

[0084] A pursuit-escape game is conducted based on elliptical motion constraints, where the movements of both the pursuer and the escapee lie on an ellipse in a two-dimensional plane, such as... Figure 2 As shown;

[0085] Step 1: Establish an elliptical motion-constrained pursuit-escape game model;

[0086] Its model is as follows:

[0087]

[0088] The relevant symbols in the formula are defined as follows:

[0089] P – indicates the pursuing party;

[0090] E – Indicates the fleeing party;

[0091] x P —This represents the coordinates of the pursuing party on the x-axis;

[0092] y P —This represents the y-coordinate of the pursuing party;

[0093] x E —This represents the coordinates of the fleeing party on the x-axis;

[0094] y E —This represents the y-coordinate of the fleeing party;

[0095] α — represents the phase angle of the pursuing side on the circle;

[0096] β — represents the phase angle of the escape side on the ellipse;

[0097] r represents the radius of the circle containing the pursuing player;

[0098] a — represents the semi-major axis of the ellipse containing the escaping side;

[0099] b — represents the minor semi-axis of the ellipse containing the escaping party;

[0100] L represents the length between the center of the circle containing the pursuing player and the center of the ellipse containing the escaping player.

[0101] d — indicates the distance between the pursuing and the fleeing parties.

[0102] The values ​​of each parameter in the above model are shown in Table 1.

[0103] Step 2: Input the specific parameters of the pursuit-escape game model, including: r, a, b, L, and establish the distance function between distance d and parameters α and β according to formula (1):

[0104]

[0105] Step 3: Keeping β constant, calculate the first partial derivative of distance d with respect to parameter α, and set this partial derivative to zero. This gives us two possible values ​​of α, α1 and α2, when distance d reaches its extreme value.

[0106]

[0107] Step 4: Keep β constant and calculate the second partial derivative of distance d with respect to parameter α. Substitute the two possible extrema obtained in S2 into the second partial derivative. It can be found that: no matter what value β takes, the second partial derivative corresponding to α1 is greater than zero, while the second partial derivative corresponding to α2 is less than zero. Therefore, α1 is a minimum value and α2 is a maximum value.

[0108] Step 5: Take α1 as the optimal strategy solution for the pursuer in the chase game, denoted as α * Substituting (β) into the distance function, i.e., formula (2), we obtain the distance function between the pursuer and the fleeing party when the pursuer adopts the optimal strategy:

[0109]

[0110] Step 6: Take the partial derivative of the distance function given by formula (4) with respect to β, and set it equal to zero to obtain the possible extreme points of β when the distance reaches an extreme value:

[0111]

[0112] That is: when When β has only 2 extreme points; when At this time, β has 4 extreme points.

[0113] Step 7: Based on the value of L, determine the optimal escape strategy β from the possible extreme points obtained in S5. * .Right now:

[0114] ①If Then the best strategy β for the escapee * =0, at this point the equilibrium solution of the distance between the pursuer and the pursuer is:

[0115] d(α * ,β * )=L+br (6)

[0116] Among them: the best strategy of the pursuing side α * =0.

[0117] ②If There are two optimal strategies for the escapee: or At this point, the equilibrium solution for the distance between the pursuing and fleeing parties is:

[0118]

[0119] in: or

[0120] Step 8: Based on the values ​​of r, a, b, and L, output the result of the elliptical motion constraint pursuit-escape game: α * β * d(α) * ,β * );

[0121] The above structure is listed in Table 1. Table 1 shows that the present invention calculates the equilibrium solution of the pursuit-escape game with elliptical motion constraints, and the number of solutions matches the theoretical expectation. To further illustrate the correctness of the solution, graphs showing the variation of distance d with angles α and β under two typical cases are plotted, as follows: Figure 3 and Figure 4 As shown.

[0122]

[0123] Table 1. Parameter settings and corresponding equilibrium solutions for the pursuit and escape game model;

[0124] according to Figure 3 and Figure 4 As shown, the two game equilibrium solutions corresponding to Example 1 are two saddle points of the game problem under this condition; the game equilibrium solution corresponding to Example 3 is the unique saddle point of the game problem under this condition. Because... Figure 3 and Figure 4 Since the curves are three-dimensional, it is not convenient to show more details and patterns on paper. Therefore, the results of Example 2 are plotted as a family of two-dimensional planar curves, such as... Figure 5 and Figure 6 As shown.

[0125] in, Figure 5 The graph shows how distance *d* varies with α for different values ​​of β. It also plots the minimum value for each curve, indicated by a red asterisk. As can be seen from the graph, the minimum value differs for different β values. By filtering the minimum values, the maximum value *max min d* can be obtained. Comparison shows that this solution is precisely the equilibrium solution obtained by the method of this invention. Figure 6 This is a detailed display of a specific curve in the family of curves. As can be seen from the figure, each specific β angle corresponds to a curve that evolves with α, but there is indeed a minimum value.

[0126] In this embodiment, the elliptical motion constraint is that the motions of both the pursuer and the escaper lie on an ellipse in a two-dimensional plane, specifically:

[0127] The pursuing party's motion lies on a circle with center C. P The coordinates are (-L, 0), and the radius is r;

[0128] The motion of the escaper lies on an ellipse, with its center point C. E The coordinates of the object are (0,0), its major semi-axis is a, and its minor semi-axis is b. Obviously, a≥b.

[0129] The line containing the semi-major axis of the ellipse intersects with C. P C E If the lines are perpendicular, then equivalently, the line containing the minor semi-axis of the ellipse intersects with line C. P C E The lines overlap;

[0130] To ensure that the circle containing the pursuing party does not intersect the ellipse containing the escaping party, L≥r+b is required.

[0131] In summary, a pursuit-escape game model based on the min-max form was first established, including optimization indices, optimization variables, and motion constraints. The optimization indices are described by the distance between the pursuer and the escaper, the optimization variables are the phase angle of the pursuer on the circle and the phase angle of the escaper on the ellipse, and the motion constraints are the radius of the pursuer's circle, the semi-major axis and semi-minor axis of the escaper's ellipse, and the distance between the center of the circle and the center of the ellipse. Then, a solution method for the aforementioned min-max optimization problem was established. This invention abstracts the feasible movement regions of both the pursuer and the escaper into elliptical (circular) shapes, simplifying the motion constraints in many practical problems, but conforming to the actual movement characteristics of drones, unmanned vehicles, and unmanned ships with fuel consumption or communication distance limitations, thus having wide applicability. The pursuit-escape problem solution method provided by this patent transforms the bilateral optimization problem into an extremum problem, thereby analytically obtaining the optimal strategies of the pursuer and the escaper under the Nash equilibrium. The application of this model and method can provide an effective solution for pursuit-escape problems with elliptical motion constraints, thus laying a theoretical foundation for the application of unmanned systems in security, counter-terrorism, and patrol tasks.

[0132] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the specific implementation of the present invention. Any modifications or equivalent substitutions that do not depart from the spirit and scope of the present invention should be covered within the scope of protection of the claims of the present invention.

Claims

1. A pursuit-escape game method with elliptical motion constraints, characterized in that, Includes the following steps: Establish an elliptical motion-constrained pursuit and escape game model, establish a distance function based on the elliptical motion-constrained pursuit and escape game model, and input the specific parameters of the elliptical motion-constrained pursuit and escape game model into the distance function; Within the distance function, the escape phase angle of the fleeing party is kept constant. Based on the principle of extreme values, the optimal pursuit phase angle function with the escape phase angle as the independent variable is obtained when the pursuit distance is minimized, which serves as its pursuit strategy. Substitute the best pursuit phase angle function of the pursuer into the distance function, and solve the best escape phase angle of the escaper when the pursuit distance is maximized based on the principle of extrema, which can be used as its escape strategy. Input the optimal escape phase angle into the optimal pursuit phase angle function to obtain the optimal pursuit phase angle; Outputting the best pursuit phase angle of the pursuer yields the best pursuit strategy of the pursuer, and outputting the best escape phase angle of the escaper yields the best escape strategy of the escaper, thus completing the elliptical motion constrained pursuit-escape game. The calculation formula for the pursuit-escape game model is as follows: in, Indicates the pursuing party; Indicates the party escaping; Indicates the pursuing side is x Coordinates on the axis; Indicates the pursuing side is y Coordinates on the axis; Indicates the escaped party is x Coordinates on the axis; Indicates the escaped party is y Coordinates on the axis; This indicates the phase angle of the pursuing party on the circle; This represents the phase angle of the escaping side on the ellipse; Indicates the radius of the circle containing the pursuing player; Represents the semi-major axis of the ellipse containing the escaping party; Represents the minor semi-axis of the ellipse containing the escaping party; It represents the length between the center of the circle containing the pursuing party and the center of the ellipse containing the escaping party; Indicates the distance between the pursuing and the fugitive.

2. The pursuit-escape game method with elliptical motion constraints according to claim 1, characterized in that, A distance function is established based on the elliptical motion-constrained pursuit-escape game model, and the formula for calculating the distance function is as follows: in, This indicates the phase angle of the pursuing party on the circle; This represents the phase angle of the escaping side on the ellipse. Indicates the distance between the pursuing and fugitive parties; Indicates the radius of the circle containing the pursuing player; Represents the semi-major axis of the ellipse containing the escaping party; Represents the minor semi-axis of the ellipse containing the escaping party; It represents the length between the center of the circle containing the pursuing player and the center of the ellipse containing the escaping player.

3. The pursuit-escape game method with elliptical motion constraints according to claim 2, characterized in that, The specific parameters of the elliptical motion-constrained pursuit game model include the radius of the circle containing the pursuer. r The semi-major axis of the ellipse containing the escape direction a The minor semi-axis of the ellipse containing the escape direction b The length between the center of the circle containing the pursuing party and the center of the ellipse containing the escaping party. L .

4. The pursuit-escape game method with elliptical motion constraints according to claim 2, characterized in that, The specific parameters of the elliptical motion-constrained pursuit-escape game model are input into the distance function, and the steps for calculating the optimal strategy for both the pursuer and the escaper are as follows: The phase angle of the fixed escape side on the ellipse With the value remaining unchanged, calculate the distance between the pursuing and fleeing parties. Regarding the phase angle of the pursuing side on the circle The first-order partial derivative of the equation is taken, and then set to zero to obtain the distance between the pursuing and fleeing parties. When taking the extreme value, the phase angle of the pursuing side on the circle Possible values and possible values ;in, ; Within the distance function, the phase angle of the escape side on the ellipse is fixed. With the value remaining unchanged, calculate the distance between the pursuing and fleeing parties. Regarding the phase angle of the pursuing side on the circle The second-order partial derivatives, and the possible numerical values and possible values The extreme points are calculated by inputting the values ​​into the second-order partial derivatives respectively. It is possible that the extreme point is greater than the given value. It is the minimum value. It is the maximum value; Minimum value The solution to the best strategy for the pursuer in a chase game is denoted as... and will The formula for the distance function obtained by inputting the distance data into the distance function, when the pursuer adopts the optimal strategy, is as follows: ； in, This represents the solution for the optimal pursuit phase angle during a pursuit-escape game. It represents the length between the center of the circle containing the pursuing party and the center of the ellipse containing the escaping party; Represents the semi-major axis of the ellipse containing the escaping party; Represents the minor semi-axis of the ellipse containing the escaping party; This represents the phase angle of the escaping side on the ellipse; This represents the radius of the circle containing the pursuing party.

5. A pursuit-escape game method with elliptical motion constraints according to claim 4, characterized in that, The calculation process for the optimal strategy of the escaping party is as follows: The phase angle of the distance function between the pursuer and the fleeing party on the ellipse when the optimal strategy is adopted for the pursuer. Find the partial derivative and let the phase angle of the escape side on the ellipse be... When the distance is equal to zero, the extreme value of the distance is obtained. The possible extreme points are shown in the following formula: Among them, when hour, There are only 2 extreme points; when hour, There are 4 extreme points; in, It represents the length between the center of the circle containing the pursuing party and the center of the ellipse containing the escaping party; Represents the semi-major axis of the ellipse containing the escaping party; Represents the minor semi-axis of the ellipse containing the escaping party; This represents the phase angle of the escaping side on the ellipse.

6. A pursuit-escape game method with elliptical motion constraints according to claim 5, characterized in that, When the distance reaches its extreme value Possible extreme point scenarios depend on the length between the center of the circle containing the pursuing player and the center of the ellipse containing the escaping player. The optimal escape phase angle is determined from the possible extreme points by taking the value of . , among which, if Then the optimal escape phase angle for the escaping party At this point, the equilibrium solution between the pursuing and fleeing parties is: Among them: the optimal pursuit phase angle of the pursuing side ; if Then the best strategy for the escapee is twofold: or At this point, the equilibrium solution for the distance between the pursuer and the pursuer is: in: or ; in, The optimal pursuit phase angle for the pursuing side; The optimal escape phase angle for the escaping party; It represents the length between the center of the circle containing the pursuing party and the center of the ellipse containing the escaping party; Represents the semi-major axis of the ellipse containing the escaping party; Represents the minor semi-axis of the ellipse containing the escaping party; This represents the radius of the circle containing the pursuing party.

7. A pursuit-escape game method with elliptical motion constraints according to claim 1, characterized in that, The elliptical motion constraint means that the motions of both the pursuer and the escaper lie on an ellipse in a two-dimensional plane.

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