CN116680509A - Dynamic matching method for multi-spacecraft escape-tracking game task - Google Patents
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Abstract
The invention discloses a dynamic matching method for a multi-spacecraft escape-tracking game task, and belongs to the technical field of aerospace. The method is based on a spacecraft dynamics model, the many-to-many problem is converted into a plurality of one-to-one sub-problems, namely the many-to-many chase-escaping game problem is disassembled into top-layer optimization and bottom-layer optimization, which are respectively calculated, and in the top-layer optimization, the state information of the chase-escaping two sides is converted into a corresponding weight matrix, so that the optimal matching problem between an escape device and a tracker is solved; in the bottom optimization, solving the optimal control strategy of both sides of the pursuit. The invention can generate an effective task matching strategy by evaluating the mobility, position and speed of the two sides of the pursuit. According to the method, the determined optimal control strategy of the two escaping sides is solved, the dynamic optimal control of the two escaping sides is realized, the number of escaping devices intercepted by the tracking side is the largest, the terminal distance is smaller, and the optimal pursuit of the two escaping sides is realized. The method has the characteristics of simple control method, small calculated amount, strong real-time performance and the like.
Description
Technical Field
The invention relates to a task matching method for a multi-spacecraft chase-escaping game, in particular to a dynamic task matching method considering state change of a spacecraft, which is suitable for tasks of avoiding or tracking a plurality of satellites or space fragments, and belongs to the technical field of aerospace.
Background
In recent years, the global aerospace technology is rapidly developed, the aerospace launching frequency is continuously improved, the huge constellation of the commercial low-orbit satellite is started to blow out to develop the wave, and urgent needs are brought forward for the research of the many-to-many chase game of the spacecraft. At present, the research on the game problem of the tracking and escaping of the spacecraft is mostly limited to the categories of one tracking and one escaping and more tracking and one escaping, and the problem of more tracking and more escaping is rarely related.
The existing one-chasing-one-escaping problem is mostly based on two-person zero and game models to solve two-party strategies, a learner researches a differential game problem between a reconnaissance satellite and a non-cooperative target on an elliptical orbit and a double spacecraft chasing-escaping problem of a near-earth orbit, and the main algorithms provided include an indirect heuristic algorithm, a semi-direct control parameterization method, a hybrid algorithm based on multiple targeting, a sensitivity method and the like. Aiming at the track-pursuit game problem under incomplete information and aiming at evades in games, a learner puts forward an optimal strategy based on estimation and an improved online parameter estimation algorithm. With the continuous development of the aerospace technology and the chase-back theory, the research on the problem of the chase-back game of the spacecraft gradually deepens to the expansion of the multi-chase-back-and-back three-party game, and the main algorithms applied are an indirect heuristic algorithm, a level set method, a distributed online mission planning algorithm, a centralized mission planning algorithm and the like.
With the continuous development of the aerospace technology, the future spacecraft chase-escaping game has the characteristics of clustering, intellectualization and antagonism, the actual task requirements are difficult to meet in the existing orbit game form, and the research of the many-to-many spacecraft chase-escaping game is necessary to be developed, which is also the necessary trend of the deep development of the chase-escaping theory. In addition, the huge constellation and the abandoned spacecraft and fragments thereof which are distributed over each orbit in the space severely occupy the earth orbit resources, thereby seriously threatening the national security and the space resource security of the whole human. In order to relieve space resource shortage and ensure the safe operation of the spacecraft, the space debris removal task is necessary to be carried out, and countermeasures are also needed to be made for monitoring and interception behaviors possibly implemented by a giant constellation. For space debris removal, the cost performance of releasing a plurality of small satellites to remove a plurality of fragments through one-time launching task is high; for actions such as monitoring and interception implemented by giant constellations, the scenario often involves multiple spacecraft. This is a typical military and civil task, which all involve multi-objective optimal matching and track optimization problems, and deserve intensive research. In order to make research more universal, the invention regards fragments and giant constellations as non-cooperative targets with maneuverability, and instead, the invention researches the problem of multi-to-multi spacecraft chase-and-flee game, and provides a method for effectively solving the task matching of the multi-spacecraft.
Disclosure of Invention
Aiming at the problem of task allocation of a multi-spacecraft escape-following game, aiming at the problem that the state of the spacecraft changes along with time to influence an escape strategy, the main purpose of the invention is to provide a dynamic allocation method for the multi-spacecraft escape-following game task, which converts the multi-to-multi problem into a plurality of one-to-one sub-problems based on a spacecraft dynamics model so as to solve the optimal control strategy of the escape-following two parties, namely, the invention can give the dynamic matching scheme and the optimal control strategy of the escape-following two parties only by evaluating the maneuverability, position and speed information of the escape-following two-party spacecraft after the task time is given, and has the characteristics of simple control method, small calculated amount, strong real-time performance and the like. The method and the device can ensure that the square spacecraft can obtain the overall optimal tracking effect.
The invention aims at realizing the following technical scheme:
the invention discloses a dynamic matching method for a multi-spacecraft escape game task, which comprises the following steps:
step one: and (3) establishing a related coordinate system related to the simplified chase-and-flee game model, and selecting a point which is closer to the chase-and-flee spacecraft as a reference point, wherein the reference point runs around the earth along a Kepler orbit. And establishing a reference coordinate system Sxyz by taking the reference point as an origin, wherein the Sx axis points to the reference point from the geocenter, the Sy axis is positioned in the track plane and points to the movement direction of the reference point, the Sy axis is perpendicular to the Sx axis, and the Sz axis meets the right-hand rule.
The distance from the earth to the reference point is far greater than the relative distance between the two pursuing parties, and the following equation is used for describing the motion of the pursuing spacecraft relative to the reference point, and the form is as follows:
the subscript k takes E or P and respectively represents an escape device and a tracker; x is the state quantity of the spacecraft, comprises position and speed information, and X, y and z represent the position components of the spacecraft relative to a reference point; u is the control quantity input of the spacecraft, u x ,u y ,u z Representing the corresponding acceleration component; ω represents the orbital angular velocity of the reference point; a and B are coefficient matrices of state quantity and control quantity respectively.
Step two: and quantifying the maneuverability, the relative distance and the relative speed of both the chase and evasion sides into a weight matrix.
(1) Constructing a judgment matrix: the maneuverability, relative distance and relative speed of both sides of the chase and escape are factors that need to be considered when the tracker chooses to chase the target. The three factors were compared two by two, and a judgment matrix c= (C) was constructed according to the 1-9 scale method shown in table 1 ij ) 3×3 . Wherein c ij Representing the judgment of the factor i relative to j, c ji ＝1/c ij Judging the relative i of the representation factors j;
table 1 1-9 scale method
(2) Hierarchical single ordering: and calculating a normalized feature vector w of the maximum feature root lambda of the judgment matrix, wherein each element of the normalized feature vector w represents the relative importance weight of each factor for other factors.
(3) And (3) consistency detection: a consistency ratio cr=ci/RI and a consistency index ci= (λ -n)/(n-1) are defined. RI is a random consistency index, and the value of RI is related to the number n of factors. When CR is smaller than 0.1, the inconsistency degree of the judgment matrix is judged to be within the allowable range, and the test is passed, otherwise, the judgment matrix is needed to be reconstructed, namely, the step two (1) is repeated.
(4) Establishing factor weights: and converting the maneuverability, the relative distance and the relative speed of the two escaping sides into corresponding weight values.
The value index i and j are numbers of the escape device and the tracker; mu (mu) a ,μ d ,μ v Weight values of the maneuverability, the distance and the relative speed of the two escaping parties respectively; v is spacecraft speed; a is the acceleration amplitude; r is the relative distance between the two sides of the escape; d (D) 1 And D 2 Respectively representing the nearest distance and the farthest distance of both sides of the chase and flee, and obtaining the position information of both sides; k (k) d Calculating coefficients for the distances, and taking values according to experience; k (k) v For the velocity direction coefficient, 0.25 is taken when the number of escapers is less than the tracker, and 1 is taken in other cases. In the formula (6), the mobility weight is between 0 and 1, if the mobility of the escape device is stronger than that of the tracker, the tracking difficulty is higher, the weight value is closer to 0, and the formula (8) is similar to the above. The formula (7) accords with the principle that the tracker selects escape targets nearby, and the closer the distance between the two parties is, the closer the distance weight is to 1.
(5) Calculating a weight matrix: establishing membership function according to the maneuverability, relative distance and relative speed of the escape device and the tracker, and each element T of the weight matrix ij The calculation method is as follows:
wherein mu ij And (5) calculating the factor weight matrix obtained in the step (4).
Because the mobility, the relative distance and the relative speed of both the two sides of the flight path can influence the final flight path effect within a limited time, the importance of three factors needs to be compared. As the spacecraft is motorized in a low-thrust mode, the speed change is small in a given time, and the importance degree of the maneuvering capability is the weakest. So far, as to which of the two remaining factors is more important, the number and state information of the escape spacecraft are given, and a weight matrix and a matching scheme are respectively generated according to two importance sequences of relative speed, relative distance, maneuverability and relative distance, relative speed and maneuverability, so that the advantages and disadvantages of the escape result are compared. The results show that the importance of the three factors is ranked from high to low: the relative speed, relative distance, and mobility, thereby generating 6 different judging matrices meeting the conditions.
Step three: based on the weight matrix obtained in the second step, the multi-spacecraft escape-tracking problem is converted into an optimal matching problem of the weighted bipartite graph, and the escape devices are matched for each tracker. The multi-to-multi-spacecraft escape-following game problem is divided into three types of cases of equal number of escape-following, less escape-following and less escape-following problems to match until the complete match of equal subgraphs is found.
For the basic concept of bipartite graph: if the vertices in the graph are divided into two disjoint subsets, and the two vertices associated with each edge in the graph respectively belong to different vertex sets, the graph is a bipartite graph, and in the bipartite graph, a match is a set of edges, wherein any two edges have no common vertex, and concepts of a matching point, a matching edge, a non-matching point, a non-matching edge and the like are defined. The maximum match of the bipartite graph is the match with the largest number of matching edges among all matches in one graph. The maximum match can be solved by finding an augmented path to augment the matching edges and matching points in the match. In the bipartite graph, from an unmatched point, paths formed by alternating sequences of unmatched edges, matched edges and unmatched edges … … are called alternating paths; if the alternate path eventually reaches another unmatched point, the alternate path is referred to as an augmented path. If each side of the bipartite graph has a weight, the bipartite graph is the weighted bipartite graph. If the escape device and the tracker are regarded as left and right vertexes of the bipartite graph, the matching relation between the two escaping sides is regarded as the edges of the bipartite graph, the weight of the two escaping sides is calculated in the step two, and the optimal matching of the two escaping sides is the edge weight and the maximum complete matching of the bipartite graph with the weight.
Let weighted bipartite graph m= { P, E, L, S }, vertex set p= { P 1 ,P 2 ,…,P m The tracker is represented by the set of vertices e= { E 1 ,E 2 ,…,E n The set S provides the weight value of each edge. If tracker i selects escape j, edge l ij Existing, edge weights are defined by s ij And (3) representing. Before matching starts, each tracker can select any one escape device as a tracking target, and three situations of the many-to-many spacecraft escape game problem are represented by weighted bipartite graphs.
Three types of situations of the many-to-many spacecraft chase-and-evasion game problem: the equal number of the following escape problems, less escape problems, more escape problems and less escape problems are matched by the following methods:
(1) Initializing a top mark value; selecting one with fewer top points as a departure set and the other as a set to be matched, and setting a top mark value for each top point in the weighted bipartite graph; because the weight and the maximum complete match are to be found, the top standard values of all vertexes in the departure set are set as the maximum weight of the edge which can be connected with the top standard values, and all top standard values of the set to be matched are set as 0; for any one edge l in weighted bipartite graph ij P is to be ensured i +e j ≥s ij Is always true; the tracking party P is marked P i Top label of escape party E is E j ；
(2) A perfect match is found. Equal subgraphs, i.e., top-level and subgraphs equal to edge weights, are first defined. For each vertex in L, find the augmented path in the equal subgraph, find the path P1-E1 for P1 in the equal subgraph. Then, similarly, the augmented path is searched for P2 in the equal subgraph, the top label and the only P2-E1 equal to the edge weight are matched, but E1 is not found, and the third step (3) is performed.
(3) If no augmented path is found, the top-level value is modified. At this time, the matching of the bipartite graph is not a perfect match, and the equal subgraph needs to be expanded. The superscript modification rule is: the top standard value of the matched point in the set to be matched is increased by d, and the variable d=min { p } i +e j -s ij }. If any one dot-dash line is added to the equal subgraph, P2 can find the augmented path P2-E2 (or P2-E3 or P2-E1-P1-E2). Since the difference between the left and right top labels of each dot-dash line and the edge weight is not the same (for example, for P1-E2, the value is 10+0-7=3, the difference between P2-E2 is 1, and the difference between P2-E3 is 3), the top label should be subtracted by the minimum value in order to make the weight sum as large as possible. The minimum value is 1, so that the top standard value of the departure set P1 and P2 is reduced by 1, the top standard value of the set E1 to be matched is increased by 1, and according to the operation, P2 finds the amplification path P2-E2, and the weight of the bipartite graph is reduced least.
(4) Repeating the steps (2) and (3) until a complete match of the equal subgraphs is found.
Step four: and after the number and the state of the two escaping sides are given, determining the corresponding relation between the two escaping sides according to the second step and the third step, and further solving the optimal control strategy of the two escaping sides.
The state variables of the escape device and the tracker are respectively X-shaped E And X P The representation is u for controlling quantity input E And u P The relative state variables of both chase sides are denoted by X. The equation of state is written as
And establishing a payment function according to the terminal distance and the fuel consumption of the two escaping parties:
wherein 0 and T f Respectively representing a game starting time and a terminal time; s is a semi-positive definite symmetric matrix, representing the terminal distance weight; r is R P And R is E Are positive definite symmetric matrixes, and represent the energy weights of the tracker and the escape device respectively.
After introducing the covariate lambda (t), formula (11) is rewritten as
The cross-sectional functional is:
the equation of the synergy and the equation of state of formula (12) are as follows:
order theThe optimal control strategy of both sides of the chase is obtained
Wherein u is kmax Is the acceleration amplitude of the spacecraft.
Formula (15) is further rewritten as a boundary condition using the principle of Pontryagin minima
Wherein, in order to simplify the formula (16), let Q P ＝B(R P ) -1 B T ，Q E ＝B(R E ) -1 B T The matrix function P (t) satisfies:
and (3) determining the corresponding relation between the two escaping sides according to formulas (16) to (17), and further solving to obtain the optimal control strategy of the escaping sides.
The method also comprises the following steps: according to the optimal control strategy of the two escaping sides determined by solving in the step four, the dynamic optimal control of the two escaping sides is realized, the number of escaping devices intercepted by the tracing side is the largest, the terminal distance is smaller, and the optimal pursuit of the two escaping sides is further realized.
The beneficial effects are that:
1. the invention discloses a dynamic matching method for a multi-spacecraft chase-escaping game task, which disassembles the multi-to-multi-chase-escaping game problem into top-layer optimization and bottom-layer optimization to be respectively solved. In top-level optimization, converting state information of both escaping sides into corresponding weight matrixes, and further solving the problem of optimal matching between an escaping device and a tracker; in the bottom optimization, solving the optimal control strategy of both sides of the pursuit.
2. The dynamic matching method for the multi-spacecraft escape-following game task can generate an effective task matching strategy only by evaluating the maneuverability, position and speed of the escape-following parties, and has smaller calculated amount compared with random matching.
3. According to the dynamic matching method for the multi-spacecraft escape-tracking game task, the states of the two escape-tracking spacecraft can change along with time in the game process, and the matching strategy can be dynamically adjusted according to the state change of the spacecraft, so that the method has the characteristic of high instantaneity.
4. The dynamic matching method for the multi-spacecraft chase-escaping game task disclosed by the invention is used for respectively analyzing the multi-spacecraft chase-escaping game problems under three scenes of equal number chase-escaping, multiple chase-escaping and multiple chase-escaping, and the dynamic matching strategy generated by the invention is obviously superior to a static matching strategy, namely, the tracking targets are properly switched to help the tracking spacecraft to timely adjust the chase strategy, so that the overall optimal chase effect is realized.
5. The dynamic matching method for the multi-spacecraft escape game task can ensure that the number of escape devices intercepted by a tracker is the largest, and the terminal distance is smaller.
Drawings
FIG. 1 is a flow chart of a dynamic matching method for a multi-spacecraft chase escaping game mission disclosed in the present invention;
FIG. 2 is a reference frame of a step-space vehicle dynamics model of the present invention;
FIG. 3 is a step three, non-weighted bipartite illustration of the present invention;
FIG. 4 is a schematic diagram of a third search for an augmented path according to the present invention; where fig. 4 a) is the initial match, fig. 4 b) is the augmented path, and fig. 4 c) is the augmented match;
FIG. 5 is a bipartite graphic intent corresponding to the three types of multi-spacecraft chase escaping game problem in step three of the present invention; wherein fig. 5 a) is a bipartite graph corresponding to an equal number of chase-escaping questions, fig. 5 b) is a bipartite graph corresponding to a multiple chase-escaping question, and fig. 5 c) is a bipartite graph corresponding to a multiple chase-escaping question;
FIG. 6 is a schematic diagram of the step three of the present invention for finding an augmented path for P1, wherein FIG. 6 a) is an equal sub-graph of the bipartite graph and FIG. 6 b) is an augmented path found for P1;
FIG. 7 is a schematic diagram of the step three of finding an augmented path for P2 according to the present invention;
FIG. 8 is a simulation diagram of a static matching scheme of a moderate number of chase-back problems in the present example, wherein FIG. 8 a) is a graph of the relative distance change between the chase-back parties, and FIG. 8 b) is a graph of the motion trail change between the chase-back parties;
FIG. 9 is a simulation diagram of a dynamic matching scheme of a moderate number of chase-back problems in the present example, wherein FIG. 9 a) is a graph of the relative distance change between the chase-back parties, and FIG. 9 b) is a graph of the motion trail change between the chase-back parties;
FIG. 10 is a simulation diagram of a static matching scheme of multiple chase and multiple evasion problems in the present embodiment, wherein FIG. 10 a) is a graph of the relative distance between the chase and evasion parties, and FIG. 10 b) is a graph of the motion trail of the chase and evasion parties;
FIG. 11 is a simulation diagram of a dynamic matching scheme of multiple-chase-less escape problems in the present embodiment, wherein FIG. 11 a) is a graph of the relative distance change between the chase-escaping parties, and FIG. 11 b) is a graph of the motion trail change between the chase-escaping parties;
FIG. 12 is a simulation diagram of a static matching scheme of a few-chase multi-escape problem in the embodiment of the present invention, wherein FIG. 12 a) is a graph of the relative distance change between the chase escaping parties, and FIG. 12 b) is a graph of the motion trail change between the chase escaping parties;
fig. 13 is a simulation diagram of a dynamic matching scheme of a few-chase multi-escape problem in the embodiment of the present invention, in which fig. 13 a) is a graph of a relative distance change between the chase escaping parties, and fig. 13 b) is a graph of a motion trail change between the chase escaping parties.
Detailed Description
For a better description of the objects and advantages of the present invention, the following description will be given with reference to the accompanying drawings and examples.
The main purpose of the scene to which the present embodiment is applied is to realize monitoring, observation and identification of enemy satellites or space debris. At ordinary times, the spacecraft cluster is positioned near the target cluster, a monitoring task for the target is executed, and target information is collected. When a target is abnormal or space situation is intense, the spacecraft needs to maneuver close to the target so as to take further measures. Therefore, the invention provides a dynamic matching method for the multi-spacecraft chase-flight game task, and in the multi-spacecraft chase-flight, if the matching is carried out only once at the initial moment, the matching mode is called static matching; if the re-matching is performed at regular intervals, the matching mode is called dynamic matching. In order to verify the effectiveness of the algorithm, static and dynamic matching schemes are respectively calculated and provided for 3 scenes of equal number of chase escapes, more chase escapes and less chase escapes, and the chase escapes effects of different schemes are compared.
The reference track was set to a sun synchronous frozen track with a height of 800km, and the initial track number of the reference points is shown in table 2. The parameter settings in the payment function are shown in Table 3, where r p And r e To control the weight parameter s r Sum s v S is a terminal weight parameter 0 For the initial value of the terminal distance weight matrix, S f And the final value of the terminal distance weight matrix. The simulation step size is set to be 1s, k d Taking 0.01. The state settings of the two sides of the 3 sides of the escaping scene are shown in tables 4-6, and when the nearest distance between the two sides of the escaping scene is smaller than 1km, the tracker is considered to intercept successfully. The equal number of the chased games is 1000s, the time interval of dynamic matching is 100s, the time interval of the multiple chase and multiple chase games is 1200s, and the time interval of dynamic matching is 150s.
TABLE 2 reference Point track count
Table 3 parameter value table
Table 4 initial state of both sides under equal amount chase
TABLE 5 initial State of both sides under more-less escape
TABLE 6 initial State of both sides under less-chase and more-chase
As shown in fig. 1, the dynamic matching method for the multi-spacecraft escape game task disclosed in the embodiment includes the following steps:
step one: and (3) establishing a related coordinate system related to the simplified chase-and-flee game model, and selecting a point which is closer to the chase-and-flee spacecraft as a reference point, wherein the reference point runs around the earth along a Kepler orbit. And establishing a reference coordinate system Sxyz by taking the reference point as an origin, wherein the Sx axis points to the reference point from the geocenter, the Sy axis is positioned in the track plane and points to the movement direction of the reference point, the Sy axis is perpendicular to the Sx axis, and the Sz axis meets the right-hand rule.
The distance from the earth to the reference point is far greater than the relative distance between the two pursuing parties, and the following equation is used for describing the motion of the pursuing spacecraft relative to the reference point, and the form is as follows:
the subscript k takes E or P and respectively represents an escape device and a tracker; x is the state quantity of the spacecraft, comprises position and speed information, and X, y and z represent the position components of the spacecraft relative to a reference point; u is the control quantity input of the spacecraft, u x ,u y ,u z Representing the corresponding accelerationA degree component; ω represents the orbital angular velocity of the reference point; a and B are coefficient matrices of state quantity and control quantity respectively.
Step two: and quantifying the maneuverability, the relative distance and the relative speed of both the chase and evasion sides into a weight matrix.
(1) Constructing a judgment matrix: the maneuverability, relative distance and relative speed of both sides of the chase and escape are factors that need to be considered when the tracker chooses to chase the target. The importance ratings of the maneuverability, the relative distance and the speed of the two escaping parties are respectively 1, 3 and 7 according to the 1-9 scale method and the test result, and the judgment matrix is as follows:
(2) Hierarchical single ordering: and calculating a normalized eigenvector w= [0.081 0.18840.7306] of the maximum eigenvalue lambda= 3.065 of the judgment matrix, wherein each element represents the relative importance weight of each factor to other factors.
(3) And (3) consistency detection: a consistency ratio cr=ci/RI and a consistency index ci= (λ -n)/(n-1) are defined. RI is a random consistency index, the value of the RI is related to the number n of factors, the value of n is 3, the corresponding RI=0.525, CI=0.032, CR=0.062 <0.1, the consistency check is passed, and the judgment matrix is effective.
(4) Establishing factor weights: and converting the maneuverability, the relative distance and the relative speed of the two escaping sides into corresponding weight values.
The value index i and j are numbers of the escape device and the tracker; mu (mu) a ,μ d ,μ v Weight values of the maneuverability, the distance and the relative speed of the two escaping parties respectively; v is spacecraft speed; a is the acceleration amplitude; r is the relative distance between the two sides of the escape; d (D) 1 And D 2 Respectively representing the nearest distance and the farthest distance of both sides of the chase and flee, and obtaining the position information of both sides; k (k) d Calculating a coefficient for the distance, and taking 0.01; k (k) v For the velocity direction coefficient, 0.25 is taken when the number of escapers is less than the tracker, and 1 is taken in other cases.
(5) Calculating a weight matrix: establishing membership function according to the maneuverability, relative distance and relative speed of the escape device and the tracker, and each element T of the weight matrix ij The calculation method is as follows:
wherein mu ij And (5) calculating the factor weight matrix obtained in the step (4).
Step three: based on the optimal weight matrix obtained in the second step, converting the multi-spacecraft escape-tracking problem into an optimal matching problem of the weighted bipartite graph, and matching the escape devices for each tracker;
let weighted bipartite graph m= { P, E, L, S }, vertex set p= { P 1 ,P 2 ,…,P m The tracker is represented by the set of vertices e= { E 1 ,E 2 ,…,E n The set S provides the weight value of each edge. If tracker i selects escape j, edge l ij Existing, edge weights are defined by s ij And (3) representing. Before matching starts, each tracker can select any one escape device as a tracking target, and three situations of the many-to-many spacecraft escape game problem can be represented by weighted bipartite graphs. After the number and the initial state of the space vehicles of both sides of the chase and flee are given, the weight value of each side in the bipartite graph can be calculated by the step two, and the step of solving the optimal matching is as follows:
(1) Initializing a top mark value; selecting one with smaller top point number as a starting set, and the otherOne party is a set to be matched, and a top label value is set for each vertex in the weighted bipartite graph; because the weight and the maximum complete match are to be found, the top standard values of all vertexes in the departure set are set as the maximum weight of the edge which can be connected with the top standard values, and all top standard values of the set to be matched are set as 0; for any one edge l in weighted bipartite graph ij P is to be ensured i +e j ≥s ij Is always true; the tracking party P is marked P i Top label of escape party E is E j ；
(2) A perfect match is found. Firstly, defining equal subgraphs, namely a top mark and a subgraph equal to edge weight, searching an augmentation path in the equal subgraphs for each vertex in L, and if the augmentation path is found, turning to step (4); if the augmented path is not found, go to step (3).
(3) If no augmented path is found, the top-level value is modified. At this time, the matching of the bipartite graph is not a perfect match, and the equal subgraph needs to be expanded. The superscript modification rule is: the top standard value of the matched point in the set to be matched is increased by d, and the variable d=min { p } i +e j -s ij }。
(4) The second and third steps are repeated until a perfect match of the equal subgraphs is found.
Step four: and step two and step three, the top layer matching result of the multi-spacecraft escape-seeking game problem can be obtained, and after the matching relation is determined, the escape strategy of the bottom layer is required to be solved.
The state variables of the escape device and the tracker are respectively X-shaped E And X P The representation is u for controlling quantity input E And u P The relative state variables of both chase sides are denoted by X. The equation of state can be written as
And establishing a payment function according to the terminal distance and the fuel consumption of the two escaping parties:
wherein 0 and T f Respectively representing a game starting time and a terminal time; s is a semi-positive definite symmetric matrix, representing the terminal distance weight; r is R P And R is E Are positive definite symmetric matrixes, and represent the energy weights of the tracker and the escape device respectively.
After introducing the covariate lambda (t), formula (11) is rewritten as:
the cross-sectional functional is:
the equation of the synergy and the equation of state of formula (13) are as follows:
order theThe optimal control strategy of both sides of the chase is obtained
Wherein u is kmax Is the acceleration amplitude of the spacecraft.
The boundary condition using Pontryagin minimum principle is further rewritten as above
Wherein, for simplifying the formula, let Q P ＝B(R P ) -1 B T ，Q E ＝B(R E ) -1 B T The matrix function P (t) satisfies:
thus, the dynamic matching scheme and the numerical results generated by the invention aiming at three types of situations of equal number of chase-evasions, multiple chase-evasions and multiple chase-evasions are shown in tables 7-12.
Table 7 equivalent number chase dynamic matching scheme
Table 8 equal number chase-evasion matching numerical results
Table 9 dynamic matching scheme for multiple chase and multiple escape
Table 10 more-less-escape matching numerical results
Table 11 dynamic matching scheme with less chase and more escaped
Table 12 number results of less-chase more-escape matching
For the equal number of chase problems, 10 matches were performed in the whole task process, and as can be seen from table 7, the dynamic matches generated three matching schemes of "1-2-4-3" (P1 trace E1, P2 trace E2, P4 trace E3, P3 trace E4, and the other similar, not described in detail), and "3-4-1-2" and "4-2-3-1", wherein the scheme obtained by the first match corresponds to the static match. As can be seen from table 8, fig. 8 and fig. 9, in the static matching scheme, the tracker successfully intercepts E2 and E3, with a distance of 3.5034km from the end of E1 and 8.5169km from the end of E4; in the dynamic matching scheme, the nearest distance between the two chase escaping parties is 0, namely the tracker successfully intercepts all the escapers, which shows that the chase escaping result of dynamic matching is obviously better than that of static matching.
For the multiple-chase and multiple-chase problem, it can be seen from fig. 10 and 11 that in the static matching scheme, E2 and E3 were successfully intercepted, but P3 failed to intercept E1, and as can be seen from table 10, the end distances of both parties were 10.1275km; as shown in Table 9, in the dynamic matching scheme, two schemes of '3-2-5' and '5-2-3' appear, and finally 3 escapers are successfully intercepted, which indicates that the dynamic matching method of the invention can be used for the multi-chase and multi-escape problem. In this problem, no matter static matching or dynamic matching, only three trackers P2, P3 and P5 participate in the pursuit task, and the escapers P1 and P4 are not allocated. This is because the allocation scheme proposed herein is based on a priority order of speed, distance, mobility, and the choice of trackers with greater motion rates is preferred when allocating chase tasks.
For the less chase and more escape problem, as can be seen from tables 11 and 12, the static matching schemes select E1, E2, E3, and E5 as tracking targets, eventually intercepting 2 escapers successfully, with a final distance of 3.8852km from E1, and a final distance of 12.0286km from E3. Dynamic matching generates four matching schemes, and finally, tracking E1, E2, E3 and E4 are selected to successfully intercept 3 escapers, and the final distance between the escapers and E1 is 1.4996km. As can be seen from fig. 12 and 13, the motion track of P2 is more roundabout than that of the static matching under the dynamic matching scheme, and the motion track of P4 is more flat. This is because, after the dynamic matching generates a new matching scheme, if the position direction of the new target is greatly different from the current motion direction of the tracker, the track of the tracker will have a roundabout characteristic, otherwise, the track of the tracker will be relatively flat. Therefore, the three examples show that the method provided by the invention can be well applied to multi-spacecraft chase-escaping game.
Step five: according to the optimal control strategy of the two escaping sides determined by solving in the step four, the dynamic optimal control of the two escaping sides is realized, the number of escaping devices intercepted by the tracing side is the largest, the terminal distance is smaller, and the optimal pursuit of the two escaping sides is further realized.
While the foregoing is directed to embodiments of the present invention, other and further embodiments of the invention may be devised without departing from the basic scope thereof, and the scope thereof is determined by the claims that follow.
Claims (2)
1. The dynamic matching method for the multi-spacecraft escape game task is characterized by comprising the following steps of:
step one: establishing a related coordinate system related to a simplified chase-flee game model, and selecting a point with a relatively close distance to the chase-flee spacecraft as a reference point, wherein the reference point runs around the earth along a Kepler orbit; establishing a reference coordinate system Sxyz by taking a reference point as an origin, wherein an Sx axis points to the reference point from a geocenter, a Sy axis is positioned in a track plane and points to the movement direction of the reference point, the Sy axis is vertical to the Sx axis, and the Sz axis meets the right-hand rule;
the distance from the earth to the reference point is far greater than the relative distance between the two pursuing parties, and the following equation is used for describing the motion of the pursuing spacecraft relative to the reference point, and the form is as follows:
the subscript k takes E or P and respectively represents an escape device and a tracker; x is the state quantity of the spacecraft, comprises position and speed information, and X, y and z represent the position components of the spacecraft relative to a reference point; u is the control quantity input of the spacecraft, u x ,u y ,u z Representing the corresponding acceleration component; ω represents the orbital angular velocity of the reference point; a and B are coefficient matrixes of state quantity and control quantity respectively;
step two: quantifying the maneuverability, the relative distance and the relative speed of both the chase and evasion sides into a weight matrix;
(1) Constructing a judgment matrix: the maneuverability, relative distance and relative speed of both sides of the pursuit and escape are factors to be considered when the tracker selects pursuit targets; the three factors were compared two by two, and a judgment matrix c= (C) was constructed according to the 1-9 scale method shown in table 1 ij ) 3×3 The method comprises the steps of carrying out a first treatment on the surface of the Wherein c ij Representing the judgment of the factor i relative to j, c ji ＝1/c ij Judging the relative i of the representation factors j;
table 1 1-9 scale method
(2) Hierarchical single ordering: calculating a normalized feature vector w of the maximum feature root lambda of the judgment matrix, wherein each element of the normalized feature vector represents the relative importance weight of each factor to other factors;
(3) And (3) consistency detection: defining a consistency ratio cr=ci/RI and a consistency index ci= (λ -n)/(n-1); RI is a random consistency index, and the value of RI is related to the number n of factors; when CR is smaller than 0.1, judging that the inconsistency degree of the judgment matrix is within the allowable range, checking to pass, otherwise, reconstructing the judgment matrix, namely repeating the step two (1);
(4) Establishing factor weights: converting the maneuverability, the relative distance and the relative speed of both sides of the chase and evasion into corresponding weight values;
the value index i and j are numbers of the escape device and the tracker; mu (mu) a ,μ d ,μ v Weight values of the maneuverability, the relative distance and the relative speed of the two escaping parties respectively; v is spacecraft speed; a is the acceleration amplitude; r is the relative distance between the two sides of the escape; d (D) 1 And D 2 Respectively representing the nearest distance and the farthest distance of both sides of the chase and flee, and obtaining the position information of both sides; k (k) d Calculating coefficients for the distances, and taking values according to experience; k (k) v Is a velocity direction coefficient; in the formula (6), the mobility weight is between 0 and 1, if the mobility of the escape device is stronger than that of the tracker, the tracking difficulty is higher, the weight value is closer to 0, and the formula (8) is similar to the formula; the formula (7) accords with the principle that the tracker selects escape targets nearby, and the closer the distance between the two parties is, the closer the distance weight is to 1;
(5) Calculating a weight matrix: maneuvering according to escapes and trackersThe ability, the relative distance and the relative speed establish membership functions, and each element T of the weight matrix ij The calculation method is as follows:
wherein mu ij Calculating the factor weight matrix obtained in the step (4);
because the mobility, the relative distance and the relative speed of both the two sides of the flight path can influence the final flight path effect in a limited time, the importance degree of three factors needs to be compared; because the spacecraft is all maneuvered in a low-thrust mode, the speed change is small in a given time, and the importance degree of the maneuvering capability is the weakest; so far, as to which of the two factors is more important, the number and state information of the escape spacecraft are given, a weight matrix and a matching scheme are respectively generated according to two important sequences of relative speed, relative distance, maneuverability and relative distance, relative speed and maneuverability, and the advantages and disadvantages of the escape result are compared; the importance of three factors, from high to low, is ranked as: the relative speed, the relative distance and the maneuverability, thereby generating 6 different judging matrixes which meet the conditions, and selecting an optimal judging matrix according to the application result;
step three: based on the weight matrix obtained in the second step, converting the multi-spacecraft escape-tracking problem into an optimal matching problem of the weighted bipartite graph, and matching the escape devices for each tracker; the multi-to-multi-spacecraft escape-following game problem is divided into three types of cases of equal number of escape-following, less escape-following and less escape-following problems to match until the complete matching of equal subgraphs is found;
for the basic concept of bipartite graph: if the vertexes in the graph are divided into two disjoint subsets, and the two vertexes associated with each edge in the graph respectively belong to different vertex sets, the graph is a bipartite graph, in the bipartite graph, one match is a set of edges, any two edges do not have common vertexes, and concepts such as a matching point, a matching edge, a non-matching point, a non-matching edge and the like are defined; the maximum matching of the bipartite graph refers to the matching with the largest number of matching edges in all the matching of one graph; the matching edges and the matching points in the matching can be increased by searching the augmentation path, so that the maximum matching is solved; in the bipartite graph, from an unmatched point, paths formed by alternating sequences of unmatched edges, matched edges and unmatched edges … … are called alternating paths; if the alternate path eventually reaches another unmatched point, the alternate path is referred to as an augmented path; if each side of the bipartite graph has a weight, the bipartite graph is the bipartite graph with the weight; if the escape device and the tracker are regarded as left and right vertexes of the bipartite graph, the matching relation between the two escaping sides is regarded as the side of the bipartite graph, the weight of the two escaping sides is calculated by the weight matrix in the second step, and the optimal matching of the two escaping sides is the complete matching of the side weight and the maximum of the bipartite graph with the weight;
let weighted bipartite graph m= { P, E, L, S }, vertex set p= { P 1 ,P 2 ,...,P m The tracker is represented by the set of vertices e= { E 1 ,E 2 ,...,E n The escape is represented by the edge set L, the pairing situation is represented by the edge set L, and the set S provides a weight value of each edge; if tracker i selects escape j, edge l ij Existing, edge weights are defined by s ij A representation; before the matching starts, each tracker can select any one escape device as a tracking target, and three situations of the multi-to-multi spacecraft escape game problem are represented by weighted bipartite graphs;
three types of situations of the many-to-many spacecraft chase-and-evasion game problem: the equal number of the following escape problems, less escape problems, more escape problems and less escape problems are matched by the following methods:
(1) Initializing a top mark value; selecting one with fewer top points as a departure set and the other as a set to be matched, and setting a top mark value for each top point in the weighted bipartite graph; because the weight and the maximum complete match are to be found, the top standard values of all vertexes in the departure set are set as the maximum weight of the edge which can be connected with the top standard values, and all top standard values of the set to be matched are set as 0; for any one edge l in weighted bipartite graph ij P is to be ensured i +e j ≥s ij Is always true; the tracking party P is marked P i Escape, escape fromTop label of square E is E j ；
(2) Searching for a complete match; firstly, defining equal subgraphs, namely a top mark and subgraphs equal to edge weights; for each vertex in L, finding an augmented path in an equal subgraph, and finding paths P1-E1 for P1 in the equal subgraph; then searching an augmentation path for P2 in the equal subgraph, wherein the top label and the edge weight are only P2-E1, but E1 is matched, so that the augmentation path is not found, and the third step (3) is carried out;
(3) If the augmented path is not found, modifying the top label value; at this time, the matching of the bipartite graph is not complete matching, and the equal subgraph needs to be expanded; the superscript modification rule is: the top standard value of the matched point in the set to be matched is increased by d, and the variable d=min { p } i +e j -s ij -a }; if any one dot-dash line is added into the equal subgraph, finding an augmented path P2-E2 or P2-E3 or P2-E1-P1-E2 through P2; because the difference between the left and right top marks and the edge weight of each dot-dash line is different, in order to make the weight sum as large as possible, the top mark needs to be subtracted by the minimum value; the minimum value is 1, so that the top standard value of the starting set P1 and the top standard value of the starting set P2 are reduced by 1, the top standard value of the set E1 to be matched is added by 1, and according to the operation, the P2 finds an amplifying path P2-E2, and the weight of the bipartite graph is reduced least;
(4) Repeating the steps (2) and (3) until the complete match of the equal subgraphs is found;
step four: after the number and the state of the two escaping sides are given, determining the corresponding relation between the two escaping sides according to the second step and the third step, and further solving the optimal control strategy of the two escaping sides;
the state variables of the escape device and the tracker are respectively X-shaped E And X P The representation is u for controlling quantity input E And u P The relative state variables of the two chase sides are represented by X; the equation of state is written as
And establishing a payment function according to the terminal distance and the fuel consumption of the two escaping parties:
wherein 0 and T f Respectively representing a game starting time and a terminal time; s is a semi-positive definite symmetric matrix, representing the terminal distance weight; r is R P And R is E Are positive definite symmetric matrixes, and respectively represent the energy weights of the tracker and the escape device;
after introducing the covariate lambda (t), formula (11) is rewritten as
The cross-sectional functional is:
the equation of the synergy and the equation of state of formula (12) are as follows:
order theThe optimal control strategy of both sides of the chase is obtained
Wherein u is kmax The acceleration amplitude of the spacecraft;
formula (15) is further rewritten as a boundary condition using the principle of Pontryagin minima
Wherein, in order to simplify the formula (16), let Q P ＝B(R P ) -1 B T ，Q E ＝B(R E ) -1 B T The matrix function P (t) satisfies:
and (3) determining the corresponding relation between the two escaping sides according to formulas (16) to (17), and further solving to obtain the optimal control strategy of the escaping sides.
2. A dynamic matching method for a multi-spacecraft chase flight mission as claimed in claim 1, further comprising the steps of: according to the optimal control strategy of the two escaping sides determined by solving in the step four, the dynamic optimal control of the two escaping sides is realized, the number of escaping devices intercepted by the tracing side is the largest, the terminal distance is smaller, and the optimal pursuit of the two escaping sides is further realized.
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| CN117434968A (en) * | 2023-12-19 | 2024-01-23 | 华中科技大学 | Multi-unmanned aerial vehicle escape-tracking game method and system based on distributed A2C |
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| CN117332684A (en) * | 2023-09-25 | 2024-01-02 | 同济大学 | Optimal capturing method under multi-spacecraft chase-escaping game based on reinforcement learning |
| CN117332684B (en) * | 2023-09-25 | 2024-04-26 | 同济大学 | Optimal capturing method under multi-spacecraft chase-escaping game based on reinforcement learning |
| CN117434968A (en) * | 2023-12-19 | 2024-01-23 | 华中科技大学 | Multi-unmanned aerial vehicle escape-tracking game method and system based on distributed A2C |
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