CN113552872A - Pursuit and escape game decision method for chaser at different speeds - Google Patents
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Abstract
The invention provides a pursuit escape game decision method for chasers at different speeds, which comprises a strategy selection method for pursuit escape parties, so that game participants can make real-time intelligent decisions aiming at complex confrontation environments, and the team cooperation capability and game antagonism of a mobile robot are improved. The invention considers the escape game problem when the escaper has speed advantage and the chaser has quantity advantage, introduces the characteristics and composition conditions of a Perfect Enclosing Formation (PEF) of the chaser, analyzes the escape strategy when the escaper is trapped in the perfect enclosing formation from the property of the perfect enclosing formation, and is simple, convenient and intuitive and can ensure successful escape; the pursuit game problem when the pursuit catcher has the advantages of speed and quantity is considered, the pursuit catcher is classified into three categories of 'catcher', 'interceptor' and 'outsider' according to the influence capacity on the safety zone of the escaper, and the strategy selection of the pursuit catcher is analyzed on the basis, so that the catching time of the pursuit catcher can be optimized.
Description
Technical Field
The invention relates to a pursuit escape game problem, in particular to a pursuit escape game decision method for a chaser at different speeds.
Background
The problem of pursuit of the mobile robot is a classic problem in the control field, and the problem mainly considers that one group of pursuits catches another group of evacuees through mutual cooperation, and simulates the pursuit process in reality through the countermeasure between intelligent bodies. The research result of the problem of multi-robot pursuit has very wide application prospect in many fields. The scheduling of industrial products is researched by establishing a pursuit model in the industrial field, so that the pipeline efficiency is optimized. The pursuit escape problem can also be applied to the aerospace field, such as tasks of cleaning space rubbish, butting and intercepting between spacecrafts and the like. In the civil field, the robot based on the pursuit escape problem can perform collaborative search and rescue tasks among multiple intelligent agents, and can perform exploration, measurement, surveying and mapping and other works in an unknown environment.
The pursuit game is based on the theory of game theory to analyze the pursuit problem, and focuses on analyzing the motion states of the pursuit two parties and researching the selection of game strategies of the two parties to obtain the Nash equilibrium of the game process. The results of subsequent state changes can be directly analyzed by mathematical derivation at a given state. Meanwhile, the critical conditions capable of generating different results can be reversely deduced, so that the effect of the influence of different parameters on the results is obtained, and the optimal strategies of both game parties are further obtained.
The problem of the escape game is mainly solved by a differential countermeasure method in the early stage. The differential strategy has great similarity with the optimal control theory, and the optimal strategy is obtained by selecting proper control input based on a dynamic system to enable a performance functional to obtain an extreme value and solving a relevant Hamilton Jacobian Bellman equation. However, the optimal control problem gives the control that is optimal for one side, while the differential countermeasures give the control that is optimal for balancing. In the pursuit and escape game problem of multiple participants, the difficulty of dimension disaster is easily encountered by differential countermeasures because the solving difficulty directly depends on the number of participants and the dynamic model thereof.
Eishax proposed the use of the aporony circle to study the problem of pursuing the escape game. The method combines the geometric strategy to research the intra-group cooperation mechanism and the inter-group game countermeasure of the pursuit both sides, has the advantages of simplicity and intuition, is suitable for the decision under the complex countermeasure environment with a large number of participants, and has very important significance for researching the pursuit game problem. The Apolloni circle is a locus circle formed by the distance ratio between a moving point and two fixed points equal to the distance ratio between two unequal known line segments, wherein the two fixed points are called focal points, and the distance ratio between the two unequal line segments is called the relative fixed ratio of the Apolloni circle. Apoloney circle has two important properties, as follows:
Property 1, the angle formed by the line connecting the aporony circle center and the escaper and any tangent is independent of the position of the escaper and the chaser.
Property 2, the straight line connecting the two tangent points on the Apollonian circle passes through the chaser and is perpendicular to the straight line connecting the chaser and the fleeer.
Compared with the traditional differential strategy method, the method has the characteristics of small operand, expandability, strong real-time performance and the like, and can realize high-efficiency decision under a complex and variable pursuit confrontation environment. However, most of the existing aporony circle geometric methods are discussed from a certain angle of a chaser or a fleeer, and only one-way decision can be given, so that the strategy selection methods of both parties cannot be involved at the same time, and the methods are not complete.
In addition, the influence of the boundary on the escape problem is not considered in the conventional decision method, so that the method cannot be applied in many practical situations.
Disclosure of Invention
In view of the above, the invention provides a pursuit escape game decision method for chasers at different speeds, which comprises a strategy selection method for pursuit escape parties, so that game participants can make real-time intelligent decisions for complex confrontation environments, and the team cooperation capability and game antagonism of a mobile robot are improved.
In order to achieve the purpose, the technical scheme of the invention is as follows:
according to the pursuit game decision method for the chasers at different speeds, when the escaper has a speed advantage and the chasers have a number advantage, the escaper approaches some of the chasers by using the speed advantage of the escaper and pulls away from the remaining chasers, and the chasers far away from the escaper move along the moving direction of the escaper to maintain the catching formation;
when the moving direction of the escaper has an intersection point with the Apolloni circle of the chaser, the chaser closer to the escaper moves towards the intersection point, in the process, a gap is generated between the farther chaser and the closer chaser, and the escaper escapes by utilizing the gap;
wherein, the Apolloni circle of the chaser is the track of X obtained by the formula (1):
wherein P is the position of the chaser, E is the position of the escaper, VPFor pursuing the speed of the person, VEIs the speed of the escaper;
the center of the Apolloni circle is O:
the radius of the Apolloni circle is r:
wherein x ispTo follow the abscissa, y, of the position of the personpIs the ordinate, x, of the position of the chasereIs the abscissa, y, of the position of the escaper eIs the ordinate of the position of the escaper,
is a fixed ratio related to the Apolloni circle;
when the chasing catcher has the advantages of speed and number and the chasing-escaping game decision, drawing an Apolloni circle of the escaper relative to each chasing catcher to obtain a safe area of the escaper, and moving the escaper in the safe area; the intersection of the Apollonian circles of the escaper and all chasers is the safety zone of the escaper at the current moment;
dividing chasers into 'outsiders', 'capturers' and 'interceptors' according to the action in the chasing process, wherein the 'outsiders' have no influence on the whole chasing process and stay at the initial position; the 'capturer' plays a role in realizing final capture in the process of pursuing and moves linearly towards the position of the escaper; the interceptor provides assistance for the capturer, moves towards the center of the Apolloni circle, and continuously compresses the safe area of the escaper until the escaper is captured.
When the pursuit escape game decision is made when the pursuit catcher has the speed and number advantages, the range set of the Apollonian circles of the escaper and the ith pursuit catcher is represented as follows:
wherein x and y are respectively the horizontal and vertical coordinates of any point in the range, and x oiAnd yoiRespectively representing the abscissa and ordinate, R, of the center of the ApollonitiIs the Apolloni circle radius.
Wherein, the chasers have the advantages of speed and quantity, but when the chasers are in the chasing escape game decision making when the speed of the chasers is uncertain, the speed of the chasers is uncertain and randomly distributed at the upper limit VPmaxAnd a lower limit VPminIn between, the speed of the escaper is constant at VEThe decision making comprises the following steps:
step 31, selecting the escape strategy of the escaper under the condition that the speed of the chaser is uncertain, comprising the following substeps:
step 31.1, the escaper compares the size of the gap between the chasers and selects the largest angle;
then, moving along the direction of the angle bisector of the maximum angle;
step 31.2, if the escaper and the chaser with the nearest distance from the two sides are in the gap thetaiSatisfying equation (6), the escaper will maintain the current moving direction;
wherein, VPFor pursuing the speed of the person, VEIs the speed of the escaper;
step 31.3, if the formula (6) is not satisfied, the escaper changes the moving direction of the escaper and reselects the gap with the largest included angle as a new moving direction;
step 32, determining the cooperative strategy of the chaser, comprising the following steps:
step 32.1, average value of upper and lower speed limits of chaser is used 
As the speed of the chaser, drawing an Apollonius circle of the chaser relative to the escaper, and optimizing the cooperative strategy of the chaser by utilizing an Apollonius circle method;
step 32.2, adding a simulated annealing algorithm to search the optimal motion direction of the chaser in a geometric strategy based on an Apollonius circle method; wherein each chaser takes into account before moving from two aspects: intercepting in the moving direction of the escaper and compressing the safety area of the escaper; the formation of the chaser team is maintained taking into account the location of other chasers.
The pursuing and escaping game decision problem under the boundary condition is considered, and the pursuing and escaping game decision problem comprises the following steps:
step 1, judging whether the escaper can escape from the parallel boundary, comprising the following steps:
step 1.1, constructing Apolloni circles of both pursuits, wherein when a plurality of pursuits exist, the intersection of the Apolloni circles is a safe region of the escaper, and the safe region indicates that the escaper always arrives earlier than the pursuit catcher in the position in the region;
step 1.2, when the safety area and the boundary have an intersection, the escaper can escape successfully, and the escape position is positioned on the intersection;
when the safety area is not intersected with the boundary, the escaper cannot escape, and the escaper target is converted to be closer to the boundary as much as possible;
Step 2, selecting the optimal strategy of the pursuing both sides, comprising the following steps:
step 2.1, the performance function of the escaper is expressed by the formula (7):
wherein, E (x)E,yE) Is the position of the escaper, F (x)F,yF) Is the closest point on the boundary to the escaper; the purpose of the escaper is to minimize the performance function J;
step 2.2, when the safety area and the boundary have an intersection, the escaper directly goes to a point which is closest to the initial position on the intersection;
when the safety area and the boundary do not intersect, respectively calculating performance functions of each point in the safety area relative to the two boundaries, selecting the point with the minimum performance function as a target position A, wherein the escape strategy of the escaper is that the escaper moves towards the point in a straight line;
step 2.3, when the safety zone and the boundary have intersection, the chaser cannot capture the escaper, and in order to prolong the escaping time of the escaper, the chaser adopts a chasing strategy when the chaser has speed and quantity advantages to make a chasing game decision; when the safe area of the escaper does not intersect with the boundary, the strategy adopted by the chaser is that the escaper goes to the target position A in a straight line and is captured at the position.
When the distance between the escaper and any one of the chasers is smaller than the capture radius H of the chaser, the escaper is captured; the situation where the chasers are all on the same side of the escaper is considered the escaper's successful escape.
Advantageous effects
The invention considers the escape game problem when the escaper has speed advantage and the chaser has quantity advantage, introduces the characteristics and composition conditions of a Perfect Enclosing Formation (PEF) of the chaser, analyzes the escape strategy when the escaper is trapped in the perfect enclosing formation from the property of the perfect enclosing formation, and is simple, convenient and intuitive and can ensure successful escape; the pursuit game problem when the pursuit catcher has the advantages of speed and quantity is considered, the pursuit catcher is classified into three categories of 'catcher', 'interceptor' and 'outsider' according to the influence capacity on the safety zone of the escaper, and the strategy selection of the pursuit catcher is analyzed on the basis, so that the catching time of the pursuit catcher can be optimized.
The invention also considers the problem of the pursuit escape game when the chaser has the advantages of speed and quantity, but the speed of the chaser is uncertain, analyzes the influence of the speed uncertainty on the pursuit process, optimizes the cooperative strategy of the chaser from the angles of maintaining the formation and reducing gaps among different chasers on the basis of the Apollonity circle method based on the average speed value, reduces the influence of speed fluctuation on the pursuit process, and improves the capturing success rate.
The invention considers the influence of the boundary on the pursuit problem and analyzes the strategy selection of the pursuit two parties under the condition that a group of parallel boundaries exist. And constructing the Apollonian circles of the escaper and each chaser, wherein the intersection of the Apollonian circles is the safety region. The escaper selects the point in the safe zone closest to the boundary to achieve the goal optimum.
Drawings
Fig. 1 is a simplified chaser surround formation according to the present invention.
FIG. 2 is a schematic diagram of the first situation in which the speed of the escaper is higher than that of the chaser, and the escaper E selects the tangent point of two Apollonian circles in the perfect surrounding formation and moves towards the tangent point.
FIG. 3 shows a first situation where the speed of the escaper is higher than that of the chaser, P, according to the embodiment of the present invention1And P3Creating a schematic view of the gap therebetween.
Fig. 4 is a schematic diagram showing the escaper selecting the gap with the largest included angle in the chaser queue as the moving direction when the speed of the chaser is uncertain in the second case according to the embodiment of the invention.
Fig. 5(a) is a schematic diagram of positions of the catcher and the escaper at a certain time when the speed of the catcher is uncertain in the second case according to the embodiment of the present invention.
Fig. 5(b) shows a schematic diagram of the motion state of each agent at the next moment in time of fig. 5 (a).
FIG. 6 is a diagram illustrating the safety region of the escaper intersecting the boundary when the boundary is considered to exist according to the present invention.
FIG. 7 is a diagram illustrating the safety region of the escaper without intersecting the boundary when the boundary is considered to exist according to the present invention.
Fig. 8 is a block diagram of a simulation experiment for the escape game problem according to the present invention.
Fig. 9 is a schematic diagram of the initial state of the game in the simulation experiment result of the present invention.
Fig. 10 is a schematic diagram of the beginning of the gap of the surrounding formation of the chaser in the simulation experiment result of the present invention.
FIG. 11 shows the simulation results of the present invention, after the surrounding formation is broken, the escaper moves to the chaser P1And P3The bisector direction of the included angle moves, so that the gap is continuously enlarged.
FIG. 12 is a schematic diagram of the same side of the escaper for all the chasers in the simulation results of the present invention.
FIG. 13(a) is a graph showing the distance between the escaper and the chaser at each moment in the simulation results of the present invention.
FIG. 13(b) is a graph showing the distance between the escaper and the nearest chaser at each moment in the simulation results of the present invention.
Fig. 14 is a schematic diagram of the case of two chasers and one escaper when a boundary exists in the simulation experiment result of the present invention, and when the safety zone of the escaper E intersects with the boundary.
Fig. 15 is a schematic diagram of the situation of two chasers and one escaper when a boundary exists, and the situation when the safety zone of the escaper E does not intersect with the boundary in the simulation experiment result of the present invention.
Detailed Description
The invention is described in detail below by way of example with reference to the accompanying drawings.
The invention comprehensively considers the pursuit and evasion parties, provides a plurality of special conditions, carries out more comprehensive discussion, and obtains a strategy selection method with stronger applicability. The method comprises the following two conditions:
in the first case, consider the following scenario: the pursuit and escape game among the intelligent robots is used for simulating military operations, missile guidance and interception and the like in the implementation, and the robot in our part has speed advantages as an escaper and needs to escape from the enclosure of the pursuer with quantity advantages. The method comprises the following steps:
step 11, simplifying the perfect surrounding formation of the chaser according to the Apollonius method, which comprises the following specific steps:
step 11.1, the position of the chaser is P, the position of the escaper is E, and the speed of the chaser is VPThe speed of the escaper is VEThe locus of X obtained by the formula (1) is an Apollonit circle.
The center of the Apolloni circle is O:
The radius of the Apolloni circle is r:
wherein x ispTo follow the abscissa, y, of the position of the personpIs the ordinate, x, of the position of the chasereIs the abscissa, y, of the position of the escapereIs the ordinate of the position of the escaper,
is a fixed ratio related to the Apolloni circle.
Step 11.2, taking a perfect enclosing formation composed of four chasers as an example, the formation can be simplified into a regular quadrangle. Preserving the location of individual Agents { P1,P2,P3,P4E, remove the apollonic circle, the vertex of the regular quadrangle is the tangent point that perfectly encloses two adjacent apollonic circles in the formation, the chaser is located at the midpoint of the side line of the quadrangle, and the escaper is located at the center of the quadrangle, as shown in fig. 1.
In step 12, in order to make a gap surrounding the formation, the escaper should approach some of the chasers and pull away from the rest of the chasers by using the advantage of the speed of the escaper, and the chasers far away from the escaper have to move along the moving direction of the escaper to maintain the formation of the arresting track. According to the Apolloni circle method, when the direction of the escaper's motion has an intersection with the chaser's Apolloni circle, the chaser's optimal strategy is to move toward the intersection. Therefore, a chaser closer to the fleeer should move toward this intersection. In the process, a gap is formed between the farther chaser and the closer chaser, and the escaper can escape through the gap.
As shown in fig. 2, the escaper E selects a tangent point that perfectly surrounds two apolloni circles in the formation, moving toward the tangent point. At the same time, two chasers P corresponding to the tangent points1And P2Should also move towards the tangent point to prevent P1And P2A gap is generated therebetween. As shown in FIG. 3, P is removed from the queue due to the difference in speed between chaser and fleeer1And P2The other chasers cannot continue to maintain perfect bounding formation, at P1And P3A gap is generated between the two, and the escaper E will use the gap to get from P1And P3To accomplish escape.
In the second case, a multi-person joint pursuit is considered, in this scenario, there are one escaper and multiple pursuers, the pursuers may be robots with intelligent behavior, the escaper being slower than the pursuer. The problem may be considered a chase gaming process in which chasers with speed and number advantages catch a single escaper. In this problem, strategy selection for optimizing capture time by a chaser is discussed, comprising the following steps:
and step 21, drawing an Apolloni circle of the escaper relative to each chaser to obtain a safety region S of the escaper. When a plurality of chasers exist near the escaper in the chasing game process, the movable range of the escaper at the moment can be found by drawing the Apolloni circle of the escaper relative to each chaser, and the escaper can arrive at any position in the range in advance of the chaser. The set of Apollonius circle ranges for the escaper and the ith chaser can be expressed as:
Wherein x and y are respectively the horizontal and vertical coordinates of any point in the range, and xoiAnd yoiRespectively representing the abscissa and ordinate, R, of the centre of the Apolloni circleiIs the Apolloni circle radius.
The intersection of the evacuee's and all chasers aporony circles is then the safe zone for the evacuee at the current time, i.e. the evacuee can arrive at each location in this zone in advance of all chasers. The safety zone S may be expressed as:
and step 22, considering the distance between the chaser and the escaper and the influence on the safety zone of the chaser, dividing the chaser into three categories according to the action in the chasing process, wherein each category of chaser has a specific subtask to assist a team to complete the task of capturing the target.
The first category of chasers, which may be referred to as "outliers," have no effect on the overall pursuit process. For the initial state of the game, when the escaper E is opposite to a certain playerCatch up person PiThe tracking person P is a person who is in the safe area S of the escaperiCan be considered as "outsiders". Because it does not limit the safe zone of the escaper, its effect is negligible throughout the pursuit. Chasers other than "outliers" are helpful to the chasing process. They can be classified into two categories according to their role in the pursuit process: one is called a "capturer" and the other is called an "interceptor". As the name implies, the "capturer" plays a role in the pursuit process to achieve the final capture, and the "interceptor" provides assistance to the "capturer" to compress the safe area S of the escaper.
Step 23, after classifying the chasers, a chasing strategy can be made according to cooperation among the chasers of different classes. And each chaser carries out respective chasing tasks according to different classifications. As a chaser of the 'outsiders', the behavior of the outsiders has no influence on the chasing action and only needs to stay at the initial position; the chaser as the interceptor moves towards the circle center direction of Apolloni to continuously compress the safe area S; the chaser, which is the "catcher", moves linearly toward the position where the escaper is located.
In the second case, the speed of the chaser is uncertain due to rough terrain and reduced power of the intelligent robot, and the chaser cannot always move at a predetermined speed. Uncertainty in chaser speed is detrimental to team cooperation. Specifically, the method comprises the following steps: (1) the path planning of the chaser is based on the speed, and the uncertainty of the speed can prevent the chaser from obtaining an optimal path; (2) uncertainty in speed can disrupt chaser's enclosure formation, and some chasers may not arrive at their intended location on time, in which case the enclosure formation can be breached resulting in a loss of chaser. Some existing pursuit strategies are mostly discussed based on the condition that the speed of a chaser is constant, so the capturing success rate of the pursuit strategies is reduced for the condition that the speed of the chaser is uncertain.
In order to improve the capturing success rate under the condition that the speed of the chaser is unstable, the invention analyzes the motion states of the chaser and the chaser under the condition, further considers the problem of the chaser and the escape game under the condition that the speed of the chaser is uncertain, and optimizes the enclosure capturing strategy of the chaser as follows:
among this problem are N chasers, 1 fleeer, the chaser being targeted to catch the fleeer and the fleeer being targeted to escape from the enclosure. The speed of the chaser is uncertain and randomly distributed at the upper limit VPmaxAnd a lower limit VPminIn between, the speed of the escaper is constant at VE. The method comprises the following steps:
step 31, selecting the escape strategy of the escaper under the condition that the speed of the chaser is uncertain, comprising the following substeps:
step 31.1 the escaper will compare the gap θ between chasers1,θ2,…,θNAnd the largest angle among them is selected. Then, the movement is performed along the bisector direction of the maximum angle. Fig. 4 is a schematic diagram showing the escaper selecting the gap with the largest included angle in the chaser queue as the moving direction when the speed of the chaser is uncertain in the second case according to the embodiment of the invention.
If the escaper's direction of movement does not need to be changed, step 31.2, it will maintain the current direction of movement. Specifically, if the gap θ between the escaper and the chaser closest to both sides is formed iIf equation (6) is satisfied, the escaper will maintain the current direction of movement.
And 31.3, if the formula (6) is not satisfied, changing the moving direction of the escaper, and reselecting the gap with the largest included angle as a new moving direction.
Step 32, in the previous case, the parameter λ of the Apollonian circle is according to the velocity V of the chaserPAnd the speed V of the escaperEAnd (4) obtaining the product. In the current escape problem, the speed of the chaser is changed continuously, and the catcher cannot use V directlyPAnd VETo calculate lambda. Although the speed of the chaser is continuously increasedThe upper and lower limits of the chaser speed variation are known, respectively VPmaxAnd VPmin. From this, a cooperative strategy of the chaser can be determined, comprising the steps of:
step 32.1, average value of upper and lower speed limits of chaser is used
As the speed of the chaser, the apolloni circle of the chaser relative to the fleeer is drawn, and the cooperative strategy of the chaser is optimized by the apolloni circle method.
And 32.2, adding a simulated annealing algorithm to find the optimal motion direction of the chaser in a geometric strategy based on the Apollonian circle method. Each chaser takes into account two aspects before moving: (1) intercepting in the moving direction of the escaper and compressing the safety area of the escaper; (2) the formation of the chaser team is maintained taking into account the location of other chasers.
FIG. 5(a) shows the positions of the chaser and the escaper at a certain time, in which the solid circle P is shown1,…,P6Representing the following person P1,…,P6The solid circle E is the escaper E, and the moving direction of the escaper at the present moment is marked by an arrow in the figure. FIG. 5(b) shows the state of motion of each agent at the next time, according to the Apollonian circle method, P6And P1Go to the aporony circle tangent point and intercept the escaper E, but the rest chasers do not go to preset positions directly for interception, and the positions of other chasers are considered, so that the movement direction is adjusted, and gaps are prevented from appearing in the formation. In the figure P2And P3Does not move directly towards the location of the escaper, but rather is shifted a little outwards, since if P is present2And P3The difference in speed is at P when the rider is moving directly towards the rider to tighten the capture formation2And P3A gap is formed between the two parts, and the escaper can escape by using the gap.
In addition, the invention also considers the problem of escape pursuit game decision under the boundary condition. The method comprises the following specific steps:
in many practical cases, the area in which the escape game is played is limited and the existence of boundaries cannot be ignored. For example, in fighting a cross-border crime, the chaser must complete the capture within the bounds of the boundary. The invention considers the condition that a group of parallel boundaries exist, the chaser has the advantages of speed and quantity, and the boundary limits the moving range of the chaser, namely the escaper arrives at the boundary and is judged to be successful escape. In this problem, first, whether the escaper can escape successfully is considered, and if the escaper cannot escape, the purpose of the escaper is to be as close to the boundary as possible. The method comprises the following steps:
Step 1, judging whether the escaper can escape from the parallel boundary, comprising the following steps:
step 1.1, constructing Apolloni circles of both pursuits, wherein when a plurality of pursuits exist, the intersection of the Apolloni circles is a safe region of the escaper, and the safe region indicates that the escaper always arrives earlier than the pursuit. The safety region is obtained as S.
Step 1.2, when the safety zone intersects the boundary, this indicates that the escaper can successfully escape, and the escape position is located at this intersection. As shown in FIG. 6, the safe zone of the escaper E intersects the boundary, and the arrow marks the direction of the escaper's progress.
When the safety area does not intersect with the boundary, the escaper cannot escape, and the escaper target is converted to be closer to the boundary as much as possible. As shown in FIG. 7, F3And F4The corresponding points are respectively the optimal points on a single apollonian circle, but in the case of two chasers the escaper cannot arrive before being caught. Considering the points in the safe area, the points A and B with the shortest distance to the two borders are calculated, due to AF1<BF2A is selected as the target location for the escaper. The escaper and the chaser go to the point A along a straight line.
Step 2, selecting the optimal strategy of the pursuing both sides, comprising the following steps:
step 2.1, the performance function of the escaper can be expressed as equation (7):
wherein, E (x)E,yE) Is the position of the escaper, F (x)F,yF) Is the closest point on the boundary to the player. The purpose of the escaper is to minimize the performance function J.
And 2.2, when the safety region and the boundary have an intersection, the escaper can successfully escape, and the escaper directly goes to a point which is closest to the initial position on the intersection. And when the safety area and the boundary do not intersect, respectively calculating the performance functions of each point in the safety area relative to the two boundaries, and selecting the point with the minimum performance function as the target position A. The evacuee's escape strategy is a straight line motion toward that point.
And 2.3, when the safety zone and the boundary have intersection, the chaser cannot capture the escaper, and in order to prolong the escaping time of the escaper, the chaser adopts the chasing strategy discussed in the second condition of the invention. When the safety zone of the escaper does not intersect with the boundary, the optimum position selected by the escaper must be located at the boundary δ of the safety zone SSThis means that the escaper and the chaser can reach the location at the same time, and therefore the chaser adopts a strategy of going straight to the target location a and capturing the escaper at the location.
Fig. 8 shows a block diagram of a simulation experiment for the escape game problem according to the present invention, which verifies the feasibility of the strategy. The simulation program can be divided into three parts of numerical value initialization, control function and drawing.
The data initialization section defines functions and variables used in the entire code, and initializes values and matrix space. The simulation maximum time is 200s, the number h of participants in the game process is 4, the game process is composed of three chasers and one escaper, the game time step is T0.05 s, the motion direction of the participants passing the game for 0.05s is adjusted, and N N/T +1 represents the total step number in the game process. The participant's position function is represented by p1, p2, p3, and E, respectively, which are matrices of order 2 xn, with the first row representing the abscissa of the participant's position and the second row the ordinate. In each time step T, the game participant always moves in a straight line, so that the direction of the movement of the game participant can be represented by a slope k, where k is a 4 × N matrix, and each row represents the movement direction of one game participant. The simulation draws the Apolloni circle of the chaser relative to the escaper at each moment, the circle center position of the Apolloni circle is stored in a 6 multiplied by N matrix R, and the radius of the Apolloni circle is stored in a 1 multiplied by N matrix R.
The control function is mainly divided into three parts: catching strategy of the catcher, escaping strategy of the escaper and judging condition of game success.
The pursuit strategy of the chaser is performed according to the apolloni circle method, and the apolloni circle center coordinates and the radius are calculated by equations (2) and (3), respectively. The direction of motion k of the chaser is calculated as follows: if the moving direction of the escaper has an intersection point with the Apolloni circle of one chaser, the chaser moves towards the intersection point; if no intersection exists, the chaser will move directly towards the location of the escaper.
The escape strategy of an escaper can be divided into two steps: the first step is to make the gap of the trap formation and the second step is to escape from the gap. The control function of the escaper can calculate the tangent point between the Apoloney circles, and after the coordinate of the tangent point is obtained, the escaper moves towards the direction of the tangent point. The player then calculates the size of the notch in the capture formation and changes the direction of movement when the notch is large enough to move toward the notch. In the process of escaping from the gap, the escaper still needs to avoid the chaser, and the specific method is to calculate the slope of a connecting line between the escaper and the chaser closest to the escaper and select the direction perpendicular to the slope to move.
The judgment condition for game termination is composed of a capture condition and an escape condition. The success of the capture is judged by the distance between the chaser and the escaper in the program, and when the distance between the escaper and any one of the chasers is smaller than the capture radius H of the chaser, the escaper is captured, and the game is ended. The successful escape refers to the successful escape of the escaper from the chaser's enclosure, and in this procedure the escaper is considered to be successfully escaped if the chasers are all on the same side of the escaper.
The function of the drawing part can be divided into two steps, drawing an image and refreshing the image. The drawing function consists of a loop structure, the positions of the chaser and the escaper at the current moment are drawn in each loop, an Apolloni circle of the chaser is drawn, the images are kept for a period of time and then the images are cleared, and the next loop is started. Each cycle lasts for a short time, and the image at each moment is refreshed continuously so as to present a dynamic effect. The program can intercept images at several moments of importance, such as the home position image, the image at which the player started moving toward the gap, and the image at the end of the game.
The simulated image consists of three parts: the location of the two parties participating in the game, the Apollonian circle of the chaser, and the time at which the game is played.
FIG. 9 shows the initial state of the game, with solid circle E being player E and solid circle P1、P2、P3To catch person P1、P2、P3Each chaser corresponds to a respective apolloni circle. Catch up person P1、P2、P3Form a perfect surrounding formation, and the Apolloni circles of the three are tangent pairwise. Subsequently, the escaper moves towards one tangent point of the Apollonian circle, P1And P2Go to the intersection point of the Apolloni circle and the advancing direction of the escaper for interception.
In FIG. 10, the person P is the chaser3The speed of the escaper E cannot be followed in time, and the surrounding formation of the chaser begins to have a gap.
As shown in FIG. 11, after the surrounding formation is broken, the escaper moves to the chaser P1And P3The bisector direction of the included angle moves, so that the gap is continuously enlarged.
In fig. 12, all chasers are on the same side of the escaper and, because the chasers are at a disadvantage in speed, they are no longer able to approach the escaper and the escaper is able to achieve escape.
Fig. 13(a) shows the distance between the escaper and each chaser at each moment, and fig. 13(b) shows the distance between the escaper and the nearest chaser at each moment. It can be seen that the escaper adopting the strategy approaches to a part of chasers firstly, and when the distance between the escaper and the chasers approaches to a certain degree, the escaper moves towards the chasers with a relatively far distance, so that gaps are generated on the formation of the chasers, and finally the escapers are escaped.
Fig. 14 and 15 show two chasers and one fleeer when a boundary exists. Wherein E is the escaper and the chaser P1And P2The corresponding Apolloni circle is C1、C2. As shown in fig. 14, when the safety area of the escaper E intersects with the boundary, the escaper moves along a straight line to the nearest point a at the intersection, and finally successfully escapes. In fig. 15, when the safety area of the escaper E does not intersect with the boundary, the escaper E cannot escape smoothly, the escaper's purpose becomes the closest to the boundary when captured, the escaper calculates the distance between each point and the boundary in the safety area, and selects a as the target point, and both the escaper and the escaper move to the point a along a straight line to achieve capture.
In summary, the above description is only a preferred embodiment of the present invention, and is not intended to limit the scope of the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (5)
1. The pursuit game decision method under different speeds of chaser is characterized in that when the escaper has speed advantage and the chaser has number advantage, the escaper approaches some chasers first by using the advantage of self speed to pull away from the rest chasers, and the chasers far away at the moment move along the moving direction of the escaper to maintain the catching formation;
When the moving direction of the escaper has an intersection point with the Apolloni circle of the chaser, the chaser closer to the escaper moves towards the intersection point, in the process, a gap is generated between the farther chaser and the closer chaser, and the escaper escapes by utilizing the gap;
wherein, the Apolloni circle of the chaser is the track of X obtained by the formula (1):
wherein P is the position of the chaser, E is the position of the escaper, VPFor pursuing the speed of the person, VEIs the speed of the escaper;
the center of the Apolloni circle is O:
the radius of the Apolloni circle is r:
wherein x ispTo follow the abscissa, y, of the position of the personpIs the ordinate, x, of the position of the chasereIs the abscissa, y, of the position of the escapereIs the ordinate of the position of the escaper,
is a fixed ratio related to the Apolloni circle;
when the chasing catcher has the advantages of speed and number and the chasing-escaping game decision, drawing an Apolloni circle of the escaper relative to each chasing catcher to obtain a safe area of the escaper, and moving the escaper in the safe area; the intersection of the Apollonian circles of the escaper and all chasers is the safety zone of the escaper at the current moment;
dividing chasers into 'outsiders', 'capturers' and 'interceptors' according to the action in the chasing process, wherein the 'outsiders' have no influence on the whole chasing process and stay at the initial position; the 'capturer' plays a role in realizing final capture in the process of pursuing and moves linearly towards the position of the escaper; the interceptor provides assistance for the capturer, moves towards the center of the Apolloni circle, and continuously compresses the safe area of the escaper until the escaper is captured.
2. The pursuit evasion gambling decision method for chasers at different speeds according to claim 1, characterized in that when the chaser has speed and number advantages for the pursuit evasion gambling decision, the range set of the apollonies circle of the escaper and the ith chaser is expressed as:
wherein x and y are respectively the horizontal and vertical coordinates of any point in the range, and xoiAnd yoiRespectively representing the abscissa and ordinate, R, of the center of the ApollonitiIs the Apolloni circle radius.
3. A pursuit evasion game decision method for chasers at different speeds according to claim 1 or 2, characterized in that the chasers have speed and quantity advantages, but the speed uncertainty of the chasers is randomly distributed at the upper limit V when the pursuit evasion game decision is made when the speed of the chasers is uncertainPmaxAnd a lower limit VPminIn between, the speed of the escaper is constant at VEThe decision making comprises the following steps:
step 31, selecting the escape strategy of the escaper under the condition that the speed of the chaser is uncertain, comprising the following substeps:
step 31.1, the escaper compares the size of the gap between the chasers and selects the largest angle;
then, moving along the direction of the angle bisector of the maximum angle;
step 31.2, if the escaper and the chaser with the nearest distance from the two sides are in the gap theta iSatisfying equation (6), the escaper will maintain the current moving direction;
wherein, VPFor pursuing the speed of the person, VEIs the speed of the escaper;
step 31.3, if the formula (6) is not satisfied, the escaper changes the moving direction of the escaper and reselects the gap with the largest included angle as a new moving direction;
step 32, determining the cooperative strategy of the chaser, comprising the following steps:
step 32.1, average value of upper and lower speed limits of chaser is used
As the speed of the chaser, drawing an Apollonius circle of the chaser relative to the escaper, and optimizing the cooperative strategy of the chaser by utilizing an Apollonius circle method;
step 32.2, adding a simulated annealing algorithm to search the optimal motion direction of the chaser in a geometric strategy based on an Apollonius circle method; wherein each chaser takes into account before moving from two aspects: intercepting in the moving direction of the escaper and compressing the safety area of the escaper; the formation of the chaser team is maintained taking into account the location of other chasers.
4. A pursuit evasion gambling decision method for chasers at different speeds according to any one of claims 1-3, wherein the decision problem of the pursuit evasion gambling under the boundary condition is considered, comprising the following steps:
Step 1, judging whether the escaper can escape from the parallel boundary, comprising the following steps:
step 1.1, constructing Apolloni circles of both pursuits, wherein when a plurality of pursuits exist, the intersection of the Apolloni circles is a safe region of the escaper, and the safe region indicates that the escaper always arrives earlier than the pursuit catcher in the position in the region;
step 1.2, when the safety area and the boundary have an intersection, the escaper can escape successfully, and the escape position is positioned on the intersection;
when the safety area is not intersected with the boundary, the escaper cannot escape, and the escaper target is converted to be closer to the boundary as much as possible;
step 2, selecting the optimal strategy of the pursuing both sides, comprising the following steps:
step 2.1, the performance function of the escaper is expressed by the formula (7):
wherein, E (x)E,yE) Is the position of the escaper, F (x)F,yF) Is the closest point on the boundary to the escaper; the purpose of the escaper is to minimize the performance function J;
step 2.2, when the safety area and the boundary have an intersection, the escaper directly goes to a point which is closest to the initial position on the intersection;
when the safety area and the boundary do not intersect, respectively calculating performance functions of each point in the safety area relative to the two boundaries, selecting the point with the minimum performance function as a target position A, wherein the escape strategy of the escaper is that the escaper moves towards the point in a straight line;
Step 2.3, when the safety zone and the boundary have intersection, the chaser cannot capture the escaper, and in order to prolong the escaping time of the escaper, the chaser adopts a chasing strategy when the chaser has speed and quantity advantages to make a chasing game decision; when the safe area of the escaper does not intersect with the boundary, the strategy adopted by the chaser is that the escaper goes to the target position A in a straight line and is captured at the position.
5. A pursuit evasion gambling decision method for chasers at different speeds according to any one of claims 1-4, wherein the escaper is captured when the distance between the escaper and any one of the chasers is less than the capture radius H of the chaser; the situation where the chasers are all on the same side of the escaper is considered the escaper's successful escape.
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