CN113435000B - Boundary grid construction and battle situation judgment method based on geometric isomerism 2-to-1 game problem - Google Patents
Boundary grid construction and battle situation judgment method based on geometric isomerism 2-to-1 game problem Download PDFInfo
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Abstract
The invention discloses a boundary grid construction and battle condition judgment method based on a geometric isomerism 2-to-1 game problem. The obtained boundary grating is in an analytic form, so that whether a certain attack party is in a winning area of a certain or a certain group of defenders can be rapidly judged, whether a corresponding defender plays games with the attack party is further judged, and therefore overall task allocation and strategy formulation are completed without losing real-time performance; meanwhile, the optimal strategy of both attack and defense is considered in the boundary gate calculation process, so that the results obtained by the method are obtained based on objective conditions, and the problem of excessively strong subjectivity is avoided. The invention can well overcome the defects of the traditional battlefield form analysis and weapon equipment strategy formulation method, and supplement the prior blank in the field of battlefield real-time situation analysis and mission planning.
Description
Technical Field
The invention belongs to the field of differential game. Relates to a boundary grid construction and battle condition judgment method based on a geometric isomerism 2-to-1 game problem.
Background
The Reach-Avoid Game is an important problem in the field of differential Game and is widely applied to the fields of control, system attack and defense countermeasure and the like.
In the Reach-Avoid Game, the space is divided into a plurality of subareas, which are called target areas, game participants are divided into two groups which are mutually opposite, namely an attack party and a defender, the purpose of the attack party is to Reach the target area and Avoid being captured by the defender, the purpose of the defender is to intercept the attack party to prevent the attack party from reaching the purpose, and when one of the two parties reaches the target, the Game is terminated. The capturing condition is that the distance between the attack party and the defender is smaller than a given capturing radius at a certain moment. The goal of Reach-Avoid Game solving is to determine the final result of the overall Game problem by using the initial states of the Game participants and the specific form of the Game space, wherein the specific form is to divide the state space of the overall participants into an attack winning region (ADR) and a defender winning region (DDR), a certain strategy is always present for the corresponding party in each region to ensure own winning, and the hypersurface used for dividing the regions is called a boundary grid.
The general method of computing the world grid is as follows: firstly, constructing a Hamilton function of a game problem, and obtaining an optimal control strategy form of a participant through a maximum and minimum principle. Dividing the boundary of the target area into an available area (namely, a part of boundary that an attack party can enter the target area) and an unavailable area (namely, a part of boundary that a defender can capture the attack party), so as to obtain the available part boundary dividing the two part areas, and obtaining a boundary grating by using the available part boundary as an integration starting point and utilizing the optimal control strategy form obtained by a Hamilton function and reversing the state equation of the integration system. However, in the case of 2-to-1 game, the game problem has various termination conditions, so that the boundary of the target area has various conditions and is difficult to describe, and therefore, the specific form of the point of origin of the point is difficult to determine and difficult to use by the traditional method of reverse integration. In addition, the traditional back-step integration method needs to carry out integration calculation, has large calculation amount of game problems for three or more participants, is difficult to meet the real-time requirements of battlefield, and cannot provide references for making attack and defense strategies in the battlefield environment. Meanwhile, the world is focused on cooperative combat among heterogeneous weapon systems, and the traditional Reach-Avoid Game only considers a boundary grid construction method under the isomorphic state of a defender, so that the research on heterogeneous conditions is deficient.
The traditional battlefield judgment and military equipment mission planning are often carried out by adopting a simulation method or an expert scoring method, and the traditional battlefield judgment and military equipment mission planning can not be applied in a real-time battlefield environment because a large amount of modeling work and differential equation calculation are needed; the latter is based on expert scores, so subjectivity is too strong, and objective battlefield situations may not be reflected.
Disclosure of Invention
In view of the above, the invention provides a geometric-based heterogeneous 2-to-1 Avoid Game gate interface analysis construction method, and the interface gate form is obtained through a geometric method, so that the calculation load can be effectively reduced, and the real-time performance of an algorithm can be improved; the background is the game problem under heterogeneous conditions, and can be suitable for the countermeasure environment of a wider military system;
geometric-based heterogeneous 2-to-1 Avoid Game gate analysis construction method, game problem area omega is obtained, the area is a convex area, and the area is segmentedDivided into gaming areas Ω P Is in contact with the target area omega T Is provided with two defending parties D 1 ，D 2 The optimal track of the attack party A participating in two defending parties and one attack party is a linear track, and v D,1 ＞v D,2 ＞v A ；
According to two defending parties D 1 ，D 2 Obtaining an Apolloni circle according to the initial position coordinates and the speed information of the vehicle; according to Apolloni circle and line segmentAnd the definition of the Apolloni circle, the following conclusions are obtained: line segment->The point in Apolloni circle determines the game problem result by the slower party, line segment +.>Determining a game problem result by a faster party at a point outside the Apolloni circle; and then, according to the conclusion, selecting a corresponding defender, forming a new Apolloni circle with an attack party A, decomposing the 2-to-1 game problem into a 1-to-1 game problem, finding a neutral state according to the conclusion, namely, according to winning conditions of the attack party A and the defender, and solving to obtain a corresponding boundary grid.
Preferably, the Apolloni circle and line segmentThe influence conditions of the defending party include the following four types:
case one: apolloni circle and line segmentNo intersection point, D 1 Influencing game problem results, i.e. the interface grid being composed of D only 1 Determining;
and a second case: line segmentWithin the Apolloni circle, D 2 Influencing game problem results, i.e. the interface grid being composed of D only 2 Determining;
and a third case: apolloni circle and line segmentAn intersection point exists, line segment->The part in the Apolloni circle is composed of D 2 Affecting game problem results, the demarcation gate is defined by D 2 Determining; line segment->The part outside the Apolloni circle is composed of D 1 Affecting game problem results, the demarcation gate is defined by D 1 Determining;
case four: apolloni circle and line segmentTwo crossing points exist, line segment->The part in the Apolloni circle is composed of D 2 Affecting game problem results, the demarcation gate is defined by D 2 Determining; line segment->Outside of Apolloni circleDivide by D 1 Affecting game problem results, the demarcation gate is defined by D 1 And (5) determining.
Preferably, the specific method for obtaining the neutral state comprises the following steps: from the conclusion, it is derived that only line segmentsWhen there is a new Apolloni circle and only one intersection point exists, the problem results are neutral, including the following two cases:
a, when the center coordinates of the new Apolloni circle are in the line segmentIn the corresponding abscissa, only new Apolloni circle and line segment +.>When the two parts are tangent, a neutral result can be obtained;
b, when the center coordinates of the new Apolloni circle are in the line segmentWhen the corresponding abscissa is outside, only new Apolloni circle and line segment +.>When the two cross points are intersected, a neutral result can be obtained only when one cross point is intersected;
according to the two neutral conditions, a boundary gate expression is constructed.
Preferably, the specific method for obtaining the neutral state comprises the following steps: from the conclusion, it is derived that only line segmentsWhen there is a new Apolloni circle and only one intersection point exists, the problem results are neutral, including the following two cases:
a, when the center coordinates of the new Apolloni circle are in the line segmentIn the corresponding abscissa, only new Apolloni circle and line segment +.>When the two parts are tangent, a neutral result can be obtained;
b, when the center coordinates of the new Apolloni circle are in the line segmentWhen the corresponding abscissa is outside, only new Apolloni circle and line segment +.>When the two cross points are intersected, a neutral result can be obtained only when one cross point is intersected;
according to the two neutral conditions, a boundary gate expression is constructed.
Preferably, the specific construction method of the boundary gate comprises the following steps:
for one of the cases: line segmentFor the line segment from point M to point N, D is selected 1 Forming a new Apolloni circle with the attacker A, taking the abscissa which is smaller than the point M as the new Apolloni circle abscissa range, and enabling the new Apolloni circle and the line segment ∈10>Intersecting and only having one focus, and constructing an expression as a first grid; taking the abscissa corresponding to the point M to the point N as the new Apolloni circle abscissa range, and enabling a new Apolloni circle and a line segment +.>Tangent, the constructed expression is used as a second boundary grid; taking the abscissa which is larger than the corresponding abscissa of the point N as the new Apolloni circle abscissa range, and enabling new Apolloni circle and line segment +.>Intersecting and only having one focus, and constructing an expression as a third boundary gate;
for the second case: line segmentFor the line segment from point M to point N, D is selected 2 Forming a new Apolloni circle with the attacker A, taking the abscissa which is smaller than the point M as the new Apolloni circle abscissa range, and enabling the new Apolloni circle and the line segment ∈10>Intersecting and only having one focus, and constructing an expression as a first grid; taking the abscissa corresponding to the point M to the point N as the new Apolloni circle abscissa range, and enabling a new Apolloni circle and a line segment +.>Tangent, the constructed expression is used as a second boundary grid; taking the abscissa which is larger than the corresponding abscissa of the point N as the new Apolloni circle abscissa range, and enabling new Apolloni circle and line segment +.>Intersecting and only having one focus, and constructing an expression as a third boundary gate;
for the third case: line segmentFor the line segment from point M to point N, two defenders D 1 ，D 2 The Apolloni circle and the line segment thus formed +.>The intersection point of (2) is P r D is selected for 2 Forming a first Apolloni circle with an attacker A, taking an abscissa smaller than a point M as an abscissa range of the first Apolloni circle, and enabling the first Apolloni circle and a line segment +.>Intersecting and only having one focus, and constructing an expression as a first grid; in the way of point M to point P r The corresponding abscissa is used as the abscissa range of the first Apolloni circle, so that the first Apolloni circle and the line segment are enabled to be +.>Tangent, the constructed expression is used as a second boundary grid; to be greater than point P r The corresponding abscissa is used as the abscissa range of the first Apolloni circle, so that the first Apolloni circle and the line segment MP r Intersecting and only having one focus, and constructing an expression as a third boundary gate; d is selected for 1 Form a second Apolloni circle with attacker A at point P r The abscissa corresponding to the point N is used as the abscissa range of the second Apolloni circle, so that the second Apolloni circle and the line segment are in the +.>Tangent, the constructed expression is used as a fourth boundary gate; taking the abscissa which is larger than the corresponding abscissa of the point N as the abscissa range of the second Apolloni circle, and enabling the second Apolloni circle and the line segment to be +.>Intersecting and only having one focus, and constructing an expression as a fifth boundary gate; four times for the case: let line segment->For the line segment from point M to point N, two defenders D 1 ，D 2 The Apolloni circle and the line segment thus formed +.>The intersection point of (2) is P l And P r D is selected for 1 Forming a first Apolloni circle with an attacker A, taking an abscissa smaller than a point M as an abscissa range of the first Apolloni circle, and enabling the first Apolloni circle and a line segment +.>Intersecting and only having one focus, and constructing an expression as a first grid; in the way of point M to point P l The corresponding abscissa is used as the abscissa range of the first Apolloni circle, so that the first Apolloni circle and the line segment are enabled to be +.>Tangent, the constructed expression is used as a second boundary grid; to be greater than point P l The corresponding abscissa is used as the abscissa range of the first Apolloni circle, so that the first Apolloni circle and the line segment MP l Intersecting and only having one focus, and constructing an expression as a third boundary gate; d is selected for 2 Form a second Apolloni circle with attacker A at point P l To point P r The corresponding abscissa is used as the abscissa range of the second Apolloni circle, so that the second Apolloni circle and the line segment are +.>Tangent, the constructed expression is used as a fourth boundary gate; to be greater than point P r The corresponding abscissa is used as the abscissa range of the second Apolloni circle, so that the second Apolloni circle and the line segment P l P r Intersecting and only having one focus, and constructing an expression as a fifth boundary gate; d is selected for 1 Form a third Apolloni circle with attacker A at point P r The abscissa corresponding to the point N is used as the third Apolloni circle abscissa range, so that the third Apolloni circle and the line segment are in the +.>Tangent, the constructed expression is used as a sixth boundary gate; taking the abscissa corresponding to the point N as the third Apolloni circle abscissa range, and enabling the third Apolloni circle and the line segment to be +.>Intersecting and having only one focus, the constructed expression acts as a seventh bounding gate.
The invention also provides a battle situation judging method, wherein a battle area is taken as a game problem area omega, a corresponding boundary grid is obtained by adopting the boundary grid analysis and construction method in any one of the schemes, the game problem area is divided into two areas by utilizing the obtained boundary grid, wherein the area adjacent to a target area is an attack winning area, and the other area is a defending winning area; when the attack direction is in the attack direction winning zone, the attack direction will win and when the defender direction is in the defender winning zone, the defender will win.
The beneficial effects are that:
the invention researches a boundary gate construction and battle condition judgment method based on a geometric isomerism 2-to-1 game problem. The invention decomposes the 2-to-1 problem into the 1-to-1 problem using an Apolloni circle. Meanwhile, by constructing the boundary grid, related personnel can rapidly judge whether the attack party is in ADR or DDR, so that the party of the attack party and the defender wins, and the characteristic can be used for judging the battlefield form and planning the military equipment task in real time.
The boundary grid obtained by calculation is in an analytic form, so that whether a certain attack party is in a winning area of a certain defender or a certain group of defenders can be rapidly judged, whether the corresponding defenders and the attack party play games or not is judged, and therefore overall task allocation and strategy formulation are completed without losing real-time performance; meanwhile, the optimal strategy of both attack and defense is considered in the boundary gate calculation process, so that the results obtained by the method are obtained based on objective conditions, and the problem of excessively strong subjectivity is avoided. The invention can well overcome the defects of the traditional battlefield form analysis and weapon equipment strategy formulation method, and supplement the prior blank in the field of battlefield real-time situation analysis and mission planning. Meanwhile, in the civil field, an unmanned plane is adopted to catch evasions or to catch wild animals, and game states can be rapidly judged through the fence.
Drawings
FIG. 1 is a flow chart of the overall process of the present invention;
FIG. 2 is a diagram of a game space example according to the present invention;
FIG. 3 is an example of a 2-to-1 gate structure in accordance with the present invention;
FIGS. 4 (a), 4 (b), 4 (c) and 4 (d) are explanatory views of the problem resolution in step 1 of the present invention, the boundary of the target areaThe solid line part of (2) is defending part D 1 Corresponding part, dotted line part is defender D 2 A corresponding portion;
FIG. 5 is an example of a 2-to-1 gate structure according to the present invention;
Detailed Description
The invention will now be described in further detail with reference to the drawings and to specific examples.
The invention uses a geometric method to analyze and construct the 2-to-1 Reach-Avoid Game gate under heterogeneous conditions, and solves the problems that the available part boundary condition of the target area is complex, the overall calculation complexity is high and the heterogeneous condition is difficult to calculate in the traditional back-step integration method. Constructing a corresponding Apollonius circle by using initial position information and speed ratio of an attack party and a defending party, and boundary with a target areaAnd (5) comparing the position relations to obtain the boundary grid form under the condition of 1 to 1. Constructing a corresponding Apollonius circle by adopting initial position information and speed ratio of an isomerism defender, and forming a boundary with a target area +.>And (3) comparing the position relation, dividing the boundary of the whole target area into a plurality of subareas influenced by a certain defender, solving the 1-to-1 game grid for each subarea, and combining to obtain the original problem grid form.
The invention discloses a geometric-based heterogeneous 2-to-1 Avoid Game gate analytic construction method, which comprises the following specific steps as shown in figure 1:
acquiring a game problem area omega which is a convex area and uses the area line segmentDivided into gaming areas Ω P Is in contact with the target area omega T A coordinate system is constructed by using a line segment, wherein the left end point of the line segment is an origin, the x-axis points to the right end point (l, 0), and the y-axis is perpendicular to the x-axis. Acquiring two defenders D participating in game 1 ，D 2 With the initial position coordinate x of an attack party A D，1 ＝(x D，1 ，y D，1 )、x D，2 ＝(x D，2 ，y D，2 ) And x A ＝(x A ，y A ) And respective velocity information v D，1 、v D，2 And v A . The target of the offender is to reach the target area omega T And the defending party is prevented from capturing, and the defending party aims at intercepting the attacking party to prevent the attacking party from reaching the target, and the movements of the defending party and the attacking party both accord with the particle motion model. The capturing condition is that coordinates of an attack party and a defending party are the same at a certain moment. Obviously, the state quantity of the overall game problem is that
η＝(η 1 ，η 2 ，...，η 6 ) T ＝(x D，1 ，y D，1 ，x D，2 ，y D，2 ，x A ，y A ) T (1)
From the background of the game problem, the corresponding Hamilton function can be obtained as
Wherein the method comprises the steps ofTo be Hamilton function values for a gaming problem, the two parties in the game need to compete for the value, one party to be the largest party to be the smallest party, sin is a sine function, cos is a cosine function, v A 、υ D，1 、υ D，2 For the respective speeds of three participants phi A 、φ D，1 、φ D，2 For each instantaneous speed direction of three participants lambda i Is a common-mode variable that functions similarly to the method used in the optimal control to add the state equation as a constraint to the target functionThe number changes the overall problem into a Lagrangian multiplier with unconstrained optimization problem. Since all participants participating in the gaming problem follow the particle motion model, their control quantity u D,1 ，u D,2 ，u D,3 For the respective punctual movement direction, i.e.
Can be obtained according to the principle of maximum and minimum values
Wherein the method comprises the steps ofOptimal strategy for three game participants, < ->Representing the extremum of Hamilton function under the optimal strategy condition of game participants, and obtaining the optimal speed direction of the game participants to meet according to the formula (4)
Wherein the method comprises the steps ofOptimal direction of motion, ρ, for three game participants i Lambda is lambda i The related variables i epsilon {1,2,3,4,5,6}, and the common-state variable lambda can be known by the theory of optimal control correlation i Satisfy the equation during movement
Due to Hamilton functionThe game state eta is not revealed i Therefore, the formula (6) can be simplified into
From the following componentsLambda can be obtained i The three optimal tracks participating in the game are all linear tracks which are constant, and the optimal track of the end point in the area cannot exceed the area because the area omega is a convex area, so that the linear track is the optimal track feasible for the game participants.
Based on the background, a 2-to-1 gate is obtained, and the specific method is as follows:
step 1: the initial position coordinates and the speed information of the two defenders are utilized to obtain an Apollonius circle (Apollonius circle), and the Apollonius circle is defined as a shape formed by a group of points with fixed proportion of the distance to the two fixed points, and can be regarded as a set of all possible intersection points under the condition that the two moving objects start to move linearly from the initial points. By definition, it is assumed that there is a point P (x, y) on the corresponding Apollonius circle of the two defenders, the coordinates of which should satisfy
Is obtained by rewriting formula (8)
It can be seen that equation (9) is a circular equation in whichIs two defending parties D 1 And D 2 Gamma < 1 is the speed ratio of two defenders, under heterogeneous conditions, the speeds of the two defenders are different, under heterogeneous conditions, so the speed ratio is not 1, without losing generality, we assume D 1 At a speed greater than D 2 . Thus γ=v D，2 /υ D，1 ＜1。
As shown in the accompanying drawings 2 and 4 (a) to 4 (D), two diamond points are two defenders D 1 ，D 2 Triangle point is the position of attack A and circle is defender D 1 ，D 2 Corresponding Apollonius circle and dotIs the center of the circle. By the step 1, we can use the obtained Apollonius circle and line segment +.>Is to divide the target line segment +.>Divided into receivers D 1 Influence and reception of D 2 The two parts are affected, and the whole 2-to-1 game is decomposed into the combination of the 1-to-1 games on each part, so that the calculation complexity of the whole problem is reduced.
According to Apollonius circle definition, D can be known on the premise that the optimal path is a straight line 1 Can be prior to D 2 Reaching all points outside the circle of Apollonius, D 2 Can be prior to D 1 Reaching all points within the Apollonius circle. When the initial position of the attack party A is on the boundary grid (for example, the 3-part curve marked Barrier in the specification of figure 3 or the 7-part curve marked Barrier in the specification of figure 5), the terminal state of the game problem is always in a neutral state, namely, the terminal state A and at least one defender D i Simultaneous arrival line segmentAnd a certain point Q above. When Q is locatedWhen the Apollonius is out of circle, D 1 One point is earlier than D 2 Reaching Q, A will be equal to D 1 Reaching the game neutral terminal state, the partial boundary gate and D 2 Is irrelevant; similarly, when Q is within the Apollonius circle, the partial grating is defined by D 1 Irrespective of the fact that the first and second parts are.
The specific dividing mode is to study the circle and the line segmentWhen the circle and the line segment have no intersection point, if the line segment is out of the circle, the boundary grating is only determined by a defender with higher speed; if the line segment is within a circle, the gate is determined only by the slow defender. When the intersection point of the circle and the line segment exists, the boundary grating corresponding to the intersection part of the circle and the line segment is determined by the defender with lower speed, and the rest boundary grating is determined by the defender with higher speed. As shown in figure 4 of the description, the total possibilities of the problem are 4 groups, where the line segment +.>The solid line above is the portion where the corresponding gate is determined by the faster defender, while the dashed line is the portion where the corresponding gate is determined by the slower defender.
Step 2: and obtaining a corresponding 1-to-1 grid form by utilizing the position information and the speed information of the single defender and the speed information of the attacking party. According to the definition of the boundary grating, the boundary grating is a hypersurface in game space, so as to simplify the calculation process and obtain visual results, on the premise of not losing generality, we consider that the coordinate of a defender is fixed in the process of carrying out boundary grating calculation, the coordinate of an attack party is free, and the boundary grating is degenerated into a plane curve, and can be written into a functionAnd corresponding graphs may be drawn.
The specific boundary gate construction process is as follows:
step 2.1: obtaining an Apollonius circle by using initial position coordinates and speed information of the defending party and the attacking party:
wherein the method comprises the steps ofFor the initial position coordinates of defending and attacking parties, μ=v A /υ D The speed ratio of the attack party to the defending party is less than that of the attack party. According to the definition of the Apollonius circle and the fact that the optimal track is a straight line, the attack party can reach any point inside the Apollonius circle before the defender, and the defender can reach any point outside the Apollonius circle before the attack party. If the target line segment->There is more than one intersection point with the Apollonius circle, then there must be a point located within the Apollonius circle within the target area, then the point must be ensured to reach the target area before being captured as long as the direction of attack moves, otherwise if the target line segment ∈>The point in the target area is located outside the Apollonius circle, no matter the attack party moves linearly towards any direction, the defender can move towards the intersection point of the movement direction of the attack party and the Apollonius circle, and therefore the Apollonius circle is captured. Thus the target line segment in neutral state +.>There is one and only one intersection with the Apollonius circle. From the neutral characteristics of the grating, the value of the grating is found in the interval [0,l ] at the abscissa of the circle center of the Apollonius circle]When the border gate meets the highest point and +.>Tangent, corresponding phaseThe cutting conditions are as follows:
the method can be simplified into
Since the partial boundary grid needs to meet the target line segmentTangential to the Apollonius circle, so the center of the circle needs to satisfy the abscissa lying in the interval [0,l ]]. Thus, according to the target line segment->The initial x coordinate interval of the attack party corresponding to the part of the boundary grating is determined as follows:
formula (13) can be simplified to
To simplify the expression, we assume the interval boundary quantity p 1 ，p 2 Is that
The expression of the partial gate is
Step 2.2: for ApolloThe abscissa of the circle center of the nius circle is in the interval [0,l ]]Except for the cases, apollonius circleOnly one intersection point exists. Obviously, in this case the intersection point must be +.>Is defined by the endpoints of (a). The partial gate meets the following conditions:
then formulas (16) and (17) together form a single pair of single Reach-Avoid Game gate boundaries.
Step 3: according to step 1Line segment dividing method, through corresponding defending part position and speed information in each divided interval, using 1-to-1 grid construction method given in step 2 to obtain 2-to-1 Reach-Avoid Game grid analysis form under whole heterogeneous condition, assuming alpha=v A /υ D，1 ＜1、β＝υ A /υ D，2 < 1 indicates the speed ratio of the attack party to the defender party, and v D，1 ＞υ D，2 When the division form is the condition (a) in the figure 4 of the specification, the whole fence is formed by a defender D with higher speed 1 Determining that the boundary gate expression is
Wherein (x) B ，y B )∈Ω P As coordinates of points on the world wide grating, since defensive coordinates are regarded as fixed values during calculation, (x) B ，y B ) Representing coordinates of the attack party when on the gate, parametersIs that
When the division form is the condition (b) in the figure 4 of the specification, the whole fence is formed by a defensive party D with slower speed 1 Determining that the boundary gate expression is
The corresponding parameters are
When the division form is the condition (c) in the figure 4 of the specification, the whole fence is formed by two defenders D 1 、D 2 Co-determination, the boundary gate expression is
Wherein x is p Is thatIntersection point p with Apollonius circle m The abscissa of (a), parameter s i I epsilon {1,2,3,4} satisfies
When the division form is the condition (D) in the figure 4 of the specification, the whole fence is formed by two defenders D 1 、D 2 Co-determination, the boundary gate expression is
p l 、p r Representing line segmentsIntersection point with Apollonius circle in step 1, x l 、x r For point p l 、p r The abscissa of (a), parameter k i I.e {1,2,3,4,5,6} satisfies
The bounding grid resolution expressions for each case are described above, where we draw the corresponding bounding grid graph only for the most complex case, and the rest of the cases can be analogically derived. As shown in fig. 5 of the specification, the gate is composed of 7 parts and covers the entire game area omega P Dividing winning area omega for attack party ADR And defending party winning area omega DDR . Because the boundary grid is in the form of an analytic function, the coordinate of the attack party can be rapidly calculated and judged whether to be in the defending party winning area omega or not only by providing the position and the speed of the three parties participating in games DDR Within that, whether the defender participating in interception can successfully intercept the attack party. And in actual application, the obtained boundary grating is utilized to rapidly judge the winning condition of an attack party or a defender. For example, during combat, the win-or-lose condition of the offender and the defender can be rapidly pre-determined by the gate, specifically, the gate is obtained to divide the game problem area into two areas, wherein the area adjacent to the target area is the offender winning area, the other area is the defender winning area, the offender must win if the offender is located in the offender winning area, and the defender must win if the defender is located in the defender winning area. By the method, win-lose conditions of both parties of the fight can be rapidly judged, and a foundation is laid for subsequent fight tasks.
The judging method can be used for guiding the task allocation process of the defenders on line in the general many-to-many interception problem, namely, which defenders form a combination to intercept which attack party, and can meet the real-time requirements of scenes such as competition and countermeasure, so that an effective method is provided for attack and defense strategy formulation in the environments such as competition and countermeasure. Traditional game situation analysis and strategy formulation are often carried out by adopting a simulation method or an expert scoring method, and the former method can not be applied to a real-time competition resistant environment because a large amount of modeling work and differential equation calculation are needed; the latter is based on expert scores, so subjectivity is too strong, and objective game situation may not be reflected. The boundary grid obtained by calculation is in an analytic form, so that whether a certain attack party is in a winning area of a certain defender or a certain group of defenders can be rapidly judged, whether the corresponding defenders and the attack party play games or not is judged, and therefore overall task allocation and strategy formulation are completed without losing real-time performance; meanwhile, the optimal strategy of both attack and defense is considered in the boundary gate calculation process, so that the results obtained by the method are obtained based on objective conditions, and the problem of excessively strong subjectivity is avoided. Therefore, the invention can well overcome the defects of the traditional strategy making method and supplement the prior blank in the field of game situation analysis and game strategy making.
In summary, the above embodiments are only preferred embodiments of the present invention, and are not intended to limit the scope of the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (5)
1. A method for judging a battle situation is characterized by comprising the following steps:
step one, taking a combat zone as a game problem area omega, wherein the game problem area is a convex area, and the game problem area is formed by using line segmentsDivided into gaming areas Ω P Is in contact with the target area omega T ；
Step two, two defending parties D are arranged 1 ,D 2 The optimal track of the attack party A participating in two defending parties and one attack party is a linear trackTrace, and v D,1 ＞v D,2 ＞v A ；
According to two defending parties D 1 ,D 2 Obtaining an Apolloni circle according to the initial position coordinates and the speed information of the vehicle; according to Apolloni circle and line segmentAnd the definition of the Apolloni circle, the following conclusions are obtained: line segment->The point in Apolloni circle determines the game problem result by the slower party, line segment +.>The game problem result is determined by the faster party at the point outside the Apolloni circle, i.e. the line segment +.>Is divided into a plurality of sub-line segments by an Apolloni circle; then according to the conclusion, a corresponding defender is selected to form a new Apolloni circle with an attack side A, the 2-to-1 game problem is decomposed into the 1-to-1 game problem, and according to the conclusion, namely according to the winning conditions of the attack side A and the defender, a neutral state of the attack side A and the defender is found, wherein when a sub-line segment corresponding to the defender has an intersection point with the new Apolloni circle and only has one intersection point, the system state is the neutral state; according to the geometric relationship between the sub-line segment in the neutral state and the new Apolloni circle, the boundary grid of the corresponding defending party and the attack party A can be obtained;
dividing a game problem area into two areas by using the obtained boundary grid, wherein the area adjacent to the target area is an attack winning area, and the other area is a defending winning area; when the attack direction is in the attack direction winning zone, the attack direction will win and when the defender direction is in the defender winning zone, the defender will win.
2. The battle situation judging method according to claim 1, wherein the specific obtaining method of the neutral state includes the following two cases:
when the center coordinates of the new Apolloni circle are in the abscissa corresponding to the sub-line segment, a neutral result can be obtained only when the new Apolloni circle is tangent to the sub-line segment;
and B, when the center coordinates of the new Apolloni circle are outside the abscissa corresponding to the sub-line segment, a neutral result can be obtained only when the new Apolloni circle is intersected with the sub-line segment and only one intersection point exists.
3. The battle situation judging method according to claim 1, wherein the Apolloni circle and line segment between the two defendersThe influence conditions of the defending party include the following four types:
case one: apolloni circle and line segmentNo intersection point, D 1 Influencing game problem results, i.e. the interface grid being composed of D only 1 Determining;
and a second case: line segmentWithin the Apolloni circle, D 2 Influencing game problem results, i.e. the interface grid being composed of D only 2 Determining;
and a third case: apolloni circle and line segmentAn intersection point exists, line segment->The part in the Apolloni circle is composed of D 2 Affecting game problem results, the demarcation gate is defined by D 2 Determining; line segment->The part outside the Apolloni circle is composed of D 1 Affecting game problem results, the demarcation gate is defined by D 1 Determining;
case four: apolloni circle and line segmentTwo crossing points exist, line segment->The part in the Apolloni circle is composed of D 2 Affecting game problem results, the demarcation gate is defined by D 2 Determining; line segment->The part outside the Apolloni circle is composed of D 1 Affecting game problem results, the demarcation gate is defined by D 1 And (5) determining.
4. The battle situation judging method according to claim 3, wherein the specific acquisition of the neutral state includes two cases:
when the center coordinates of the new Apolloni circle are in the abscissa corresponding to the sub-line segment, a neutral result can be obtained only when the new Apolloni circle is tangent to the sub-line segment;
and B, when the center coordinates of the new Apolloni circle are outside the abscissa corresponding to the sub-line segment, a neutral result can be obtained only when the new Apolloni circle is intersected with the sub-line segment and only one intersection point exists.
5. The battle situation judging method of claim 4, wherein the specific construction method of the boundary grating comprises:
for one of the cases: line segmentFor the line segment from point M to point N, D is selected 1 Forming a new Apolloni circle with the attacker A, taking the abscissa which is smaller than the point M as the new Apolloni circle abscissa range, and enabling the new Apolloni circle and the line segment ∈10>Intersecting and only having one focus, and constructing an expression as a first grid; taking the abscissa corresponding to the point M to the point N as the new Apolloni circle abscissa range, and enabling a new Apolloni circle and a line segment +.>Tangent, the constructed expression is used as a second boundary grid; taking the abscissa which is larger than the corresponding abscissa of the point N as the new Apolloni circle abscissa range, and enabling new Apolloni circle and line segment +.>Intersecting and only having one focus, and constructing an expression as a third boundary gate;
for the second case: line segmentFor the line segment from point M to point N, D is selected 2 Forming a new Apolloni circle with the attacker A, taking the abscissa which is smaller than the point M as the new Apolloni circle abscissa range, and enabling the new Apolloni circle and the line segment ∈10>Intersecting and only having one focus, and constructing an expression as a first grid; taking the abscissa corresponding to the point M to the point N as the new Apolloni circle abscissa range, and enabling a new Apolloni circle and a line segment +.>Tangent, the constructed expression is used as a second boundary grid; taking the abscissa which is larger than the corresponding abscissa of the point N as the new Apolloni circle abscissa range, and enabling new Apolloni circle and line segment +.>Intersecting and only having one focus, and constructing an expression as a third boundary gate;
for the third case: line segmentFor the line segment from point M to point N, two defenders D 1 ,D 2 The Apolloni circle and the line segment thus formed +.>The intersection point of (2) is P r D is selected for 2 Forming a first Apolloni circle with an attacker A, taking an abscissa smaller than a point M as an abscissa range of the first Apolloni circle, and enabling the first Apolloni circle and a line segment +.>Intersecting and only having one focus, and constructing an expression as a first grid; in the way of point M to point P r The corresponding abscissa is used as the abscissa range of the first Apolloni circle, so that the first Apolloni circle and the line segment are enabled to be +.>Tangent, the constructed expression is used as a second boundary grid; to be greater than point P r The corresponding abscissa is used as the abscissa range of the first Apolloni circle, so that the first Apolloni circle and the line segment MP r Intersecting and only having one focus, and constructing an expression as a third boundary gate; d is selected for 1 Form a second Apolloni circle with attacker A at point P r The abscissa corresponding to the point N is used as the abscissa range of the second Apolloni circle, so that the second Apolloni circle and the line segment are in the +.>Tangent, the constructed expression is used as a fourth boundary gate; taking the abscissa which is larger than the corresponding abscissa of the point N as the abscissa range of the second Apolloni circle, and enabling the second Apolloni circle and the line segment to be +.>Intersecting and only having one focus, and constructing an expression as a fifth boundary gate;
four times for the case: line segmentFor the line segment from point M to point N, two defenders D 1 ,D 2 The Apolloni circle and the line segment thus formed +.>The intersection point of (2) is P l And P r D is selected for 1 Forming a first Apolloni circle with an attacker A, taking an abscissa smaller than a point M as an abscissa range of the first Apolloni circle, and enabling the first Apolloni circle and a line segment +.>Intersecting and only having one focus, and constructing an expression as a first grid; in the way of point M to point P l The corresponding abscissa is used as the abscissa range of the first Apolloni circle, so that the first Apolloni circle and the line segment are enabled to be +.>Tangent, the constructed expression is used as a second boundary grid; to be greater than point P l The corresponding abscissa is used as the abscissa range of the first Apolloni circle, so that the first Apolloni circle and the line segment MP l Intersecting and only having one focus, and constructing an expression as a third boundary gate; d is selected for 2 Form a second Apolloni circle with attacker A at point P l To point P r The corresponding abscissa is used as the abscissa range of the second Apolloni circle, so that the second Apolloni circle and the line segment are +.>Tangent, the constructed expression is used as a fourth boundary gate; to be greater than point P r The corresponding abscissa is used as the abscissa range of the second Apolloni circle, so that the second Apolloni circle and the line segment P l P r Intersecting and only having one focus, and constructing an expression as a fifth boundary gate; d is selected for 1 Form a third Apolloni circle with attacker A at point P r The abscissa corresponding to the point N is used as the third Apolloni circle abscissa range, so that the third Apolloni circle and the line segment are in the +.>Tangent, the constructed expression is used as a sixth boundary gate; taking the abscissa corresponding to the point N as the third Apolloni circle abscissa range, and enabling the third Apolloni circle and the line segment to be +.>Intersecting and having only one focus, the constructed expression acts as a seventh bounding gate.
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