Optimal strategy generation method, device and equipment for spacecraft pulse game

Abstract

The invention discloses an optimal strategy generation method, device and equipment for a spacecraft pulse game. Comprising a boundary condition of a Hamiltonian function based on a tracker, a boundary condition of a Hamiltonian function of an escaper, a boundary condition of a costate variable in the Hamiltonian function and a sequence pulse differential game model. The Hamiltonian function and interior point function sequence of the tracker and the Hamiltonian function and interior point function sequence of the escaper construct a multi-point boundary value configuration expression; and solving an initial game strategy corresponding to the tracker and the escaper, and carrying out iterative solution on the initial game strategy based on an iterative design variable constructed by the initial game strategy, the multi-point boundary value configuration expression, the pulse interval, the pulse amplitude and the situation awareness time delay to obtain an optimal pulse maneuvering strategy. During orbital transfer adjustment, when strategy design is carried out on a spacecraft according to a pulse maneuvering mode, an optimal pulse game interception strategy suitable for discontinuous maneuvering of the spacecraft is provided.

Description

Technical Field

[0001] The present invention relates to the field of spacecraft technology, and in particular to a method, device and equipment for generating an optimal strategy for a spacecraft impulse game. Background Art

[0002] When studying political, economic, military and other multi-faceted game problems, differential game methods are often used to model them as bilateral or multilateral optimal control problems. All participants together form a multi-player differential system, which contains concepts such as constraints, objective functions and equilibrium solutions, in order to ultimately solve the optimal or balanced strategy outcomes for multiple parties. However, in the past, differential game models were often continuously differentiable game models, but many existing spacecraft are only equipped with chemical fuel propulsion systems. When adjusting orbits, strategies need to be designed according to pulse maneuvers. The use of pulse maneuvers will make it difficult to model the game process of spacecraft, mainly manifested in the following aspects: 1) The state of the spacecraft changes suddenly, resulting in the dynamic equation being non-differentiable at the discontinuity point of the state, which does not meet the continuous differentiability conditions of differential game theory; 2) Under the assumption of continuous maneuvers, the spacecraft only needs to control the direction of the constant thrust of the engine, but under the assumption of pulse maneuvers, the number, time, size, direction, etc. of pulses will become decision variables of the spacecraft, and the decision space will increase significantly; 3) When both parties use pulse maneuvers, the order in which the pulse sequences of both parties are applied will also affect the results of the pursuit and escape game. Therefore, the differential game model under the pulse thrust mode has not yet been formed, and the optimal pulse maneuver strategy under the pulse thrust mode cannot be obtained. Summary of the invention

[0003] The main technical problem solved by the present invention is to propose an optimal pulse game interception strategy suitable for discontinuous maneuvers of spacecraft when designing a strategy for the spacecraft according to a pulse maneuvering method during orbit change adjustment.

[0004] According to the first aspect, an embodiment provides an optimal strategy generation method for a spacecraft impulse game, including:

[0005] Obtain the pulse interval, pulse amplitude and situational awareness delay of the pursuer in the pursuit-escape pulse maneuver game scenario, as well as the pulse interval, pulse amplitude and situational awareness delay of the escaper; wherein the pulse interval refers to the time interval between two adjacent pulses;

[0006] Constructing a sequence pulse differential game model, and constructing the Hamiltonian function and interior point function sequence of the tracker and the Hamiltonian function and interior point function sequence of the escaper; wherein the sequence pulse differential game model includes constraint equations constructed according to the pulse interval and the pulse amplitude;

[0007] Constructing a multi-point boundary value configuration expression based on the boundary conditions of the Hamiltonian function of the tracker, the boundary conditions of the Hamiltonian function of the escaper, the boundary conditions of the co-state variables in the Hamiltonian function, the sequential pulse differential game model, the Hamiltonian function and interior point function sequence of the tracker, and the Hamiltonian function and interior point function sequence of the escaper;

[0008] The initial game strategies corresponding to the tracker and the escaper are solved, and corresponding iterative design variables are constructed according to the initial game strategies. The initial game strategies are iteratively solved based on the iterative design variables, the multi-point boundary value configuration expression, the pulse interval, the pulse amplitude and the situation awareness delay to obtain the optimal pulse maneuvering strategy.

[0009] In some embodiments, solving the initial game strategies corresponding to the tracker and the escaper includes:

[0010] Starting from the first pulse phase of a preset multiple pulse phases:

[0011] Acquire the initial orbital positions of the tracker and the escaper in the current pulse phase, wherein each pulse phase is preset with the initial orbital position of the tracker and the initial orbital position of the escaper;

[0012] When there is a pulse thrust on the tracker and the escaper, the pulse maneuver strategy of the tracker and the pulse maneuver strategy of the escaper are solved based on the initial orbital position of the tracker and the initial orbital position of the escaper, wherein the pulse maneuver strategy of the tracker includes a plurality of tracking strategies, and the pulse maneuver strategy of the escaper includes a plurality of escape strategies;

[0013] Selecting the best tracking strategy among the pulse maneuver strategies of the tracker, and selecting the best escape strategy among the pulse maneuver strategies of the escaper;

[0014] If the tracker satisfies the preset game conditions when using the optimal tracking strategy and the escaper satisfies the preset game conditions when using the optimal escape strategy, or the number of pulse stages that have been experienced is greater than the preset pulse stage threshold, the optimal tracking strategy and the optimal escape strategy are used as the initial game strategies;

[0015] Otherwise, enter the next pulse stage to calculate the updated optimal tracking strategy and optimal escape strategy until the tracker satisfies the preset game condition when using the optimal tracking strategy and the escaper uses the optimal escape strategy to play the pursuit and escape game, or the number of pulse stages that have been experienced is greater than the preset game stage threshold, and the updated optimal tracking strategy and optimal escape strategy are used as the initial game strategy; wherein, the game condition is that the remaining interception time is less than the pulse interval of the tracker or less than the pulse interval of the escaper, and the remaining interception time represents the time required for the tracker to intercept the escaper.

[0016] In some embodiments, the initial game strategy includes the pulse maneuver sequence of the tracker, the pulse maneuver sequence of the escaper and the game deadline, and the pulse maneuver sequence of the tracker includes Δv P1 ,...,Δv PN , where Δv P1 ,...,Δv PN represent the speed of the pursuer under the action of the 1st... to the Nth pulse respectively, and the pulse maneuver sequence of the escaper includes Δv E1 ,...,Δv EN , Δv E1 ,...,Δv EN represent the speed of the escaper under the action of the 1st... to Nth pulses respectively, P represents the pursuer, E represents the escaper, and N represents the number of pulses;

[0017] The iterative design variables include:

[0018] X=[Δv P1 ,...,Δv PN ,Δv E1 ,...,Δv EN ,κ P1 ,...,κ PN ,κ E1 ,...,κ EN ,λ Pr0, λ Er0 ,λ Pv0, t f ] T

[0019] Where X represents the iterative design variable, t f represents the game deadline in the initial game strategy, i.e., the interception time, κ P1 ,...,κ PN Denotes the Lagrange multiplier corresponding to different trackers, κ E1 ,...,k ENDenotes the Lagrange multiplier corresponding to the different escapers, λ Pr0, λ Er0, λ Pv0 They respectively represent the position co-state of the tracker at the initial moment, the position co-state of the escaper at the initial moment, and the velocity co-state of the tracker at the initial moment.

[0020] In some embodiments, the interior point function sequence of the tracer and the interior point function sequence of the escaper include:

[0021] Φ ik =γ ik χ ik +κ ik σ ik ,i=P,E

[0022] Among them, Φ ik represents the interior point function sequence of the spacecraft, γ ik and κ ik represents the Lagrange multiplier of the spacecraft, χ ik represents the position difference of the spacecraft, σ ik represents the velocity impulse inequality of the spacecraft, P represents the tracker, E represents the escaper, k represents the time node, k=1,2,...,N, N represents the number of pulses, wherein when i=P, the spacecraft refers to the tracker, and when i=E, the spacecraft refers to the escaper.

[0023] In some embodiments, the Hamiltonian function of the tracker and the Hamiltonian function of the escaper include:

[0024]

[0025] Among them, H i represents the Hamiltonian function of the spacecraft, P represents the pursuer, E represents the escaper, T represents the transposed symbol, represents the position co-state of the spacecraft, v i is the speed of the spacecraft, represents the velocity co-state of the spacecraft, g i represents the gravitational acceleration of the spacecraft, wherein when i=P, the spacecraft is a pursuer, and when i=E, the spacecraft is an escaper.

[0026] In some embodiments, the constraint equation includes:

[0027]

[0028] N f :=r P (t f )-r E (t f)=0

[0029] Wherein, P represents the tracker, E represents the escaper, represents the instant after the pulse acts on the spacecraft, represents the instant before the pulse acts on the spacecraft, χ ik represents the position difference of the spacecraft, represents the position of the spacecraft immediately after the pulse, represents the position of the spacecraft immediately before the pulse, σ ik The velocity impulse inequality for the spacecraft, Δv ik represents the velocity difference of the spacecraft, represents the speed of the spacecraft immediately after the pulse, The velocity of the spacecraft immediately before the pulse, ΔV i max represents the maximum pulse amplitude of the spacecraft, k represents the time node, N i represents the number of pulses of the spacecraft, N f represents the intercept constraint at the terminal position, t f represents the interception time, r P (t f ) represents the position of the tracker at the time of interception, r E (t f ) represents the position of the escapee at the moment of interception, wherein when i=P, the spacecraft refers to the pursuer, and when i=E, the spacecraft refers to the escapee.

[0030] In some embodiments, the sequential pulse differential game model further includes a state equation and a payoff function, wherein the state equation is constructed based on the orbital positions and velocities of the pursuer and the escaper.

[0031] In some embodiments, the iteratively solving the initial game strategy based on the iterative design variables, the multi-point boundary value configuration expression, the pulse interval, the pulse amplitude and the situation awareness delay to obtain the optimal pulse maneuver strategy includes:

[0032] Constructing the trajectory of the tracker at different pulse phases and the trajectory of the escaper at different pulse phases;

[0033] Generate a trajectory end error expression according to the trajectory of the tracker at different pulse phases and the trajectory of the escaper at different pulse phases;

[0034] The target shooting method is used as the iterative algorithm, the iterative design variable is used as the iterative correction value, the multi-point boundary value configuration expression is used as the constraint condition, the pulse interval, the pulse amplitude, and the situation awareness delay are used as input parameters, and the initial game strategy is iteratively solved according to the trajectory end error expression to obtain the optimal pulse maneuvering strategy.

[0035] According to the second aspect, an embodiment provides an optimal strategy generation device for a spacecraft impulse game, comprising:

[0036] A data preparation module is used to obtain the pulse interval, pulse amplitude and situational awareness delay of the pursuer in the pursuit-escape pulse maneuver game scenario, as well as the pulse interval, pulse amplitude and situational awareness delay of the escaper; wherein the pulse interval refers to the time interval between two adjacent pulses;

[0037] A model building module, used to build a sequence pulse differential game model, and to build the Hamiltonian function and interior point function sequence of the tracker and the Hamiltonian function and interior point function sequence of the escaper; wherein the sequence pulse differential game model includes a constraint equation constructed according to the pulse interval and the pulse amplitude;

[0038] A multi-point boundary value conversion module, for constructing a multi-point boundary value configuration expression based on the boundary conditions of the Hamiltonian function of the tracker, the boundary conditions of the Hamiltonian function of the escaper, the boundary conditions of the co-state variables in the Hamiltonian function, the sequential pulse differential game model, the Hamiltonian function and interior point function sequence of the tracker, and the Hamiltonian function and interior point function sequence of the escaper;

[0039] A game strategy generation module is used to solve the initial game strategies corresponding to the tracker and the escaper, and construct corresponding iterative design variables according to the initial game strategies, and iteratively solve the initial game strategies based on the iterative design variables, the multi-point boundary value configuration expression, the pulse interval, the pulse amplitude and the situation awareness delay to obtain the optimal pulse maneuver strategy.

[0040] According to the third aspect, an embodiment provides an optimal strategy generation device for a spacecraft impulse game, including:

[0041] Memory, used to store programs;

[0042] A processor is used to implement a game strategy generation method by executing a program stored in the memory.

[0043] According to the optimal strategy generation method, device and equipment of the spacecraft pulse game of the above-mentioned embodiment, since a sequential pulse differential game model is constructed, it can be applied to the case of discontinuous maneuvers of the spacecraft when subjected to pulse thrust, and a multi-point boundary value configuration expression is constructed based on the sequential pulse differential game model, the Hamiltonian function and interior point function sequence of the constructed tracker, and the Hamiltonian function and interior point function sequence of the escaper. The interior point function sequence of the tracker and the interior point function sequence of the escaper take into account the sequence situation when the pulse maneuver is used as a discontinuous point, and are converted into a multi-point boundary value problem. In the process of solving the pulse maneuver strategy, iterative design variables are constructed according to the solved initial game strategy, and iterative solutions are performed to improve the accuracy of the strategy solution, so that an accurate optimal pulse maneuver strategy can be obtained. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 A flowchart of a method for generating an optimal strategy for a spacecraft impulse game according to an embodiment of the present application;

[0045] Figure 2 A flowchart of solving the initial game strategy corresponding to the tracker and the escaper according to an embodiment;

[0046] Figure 3 A flowchart of an embodiment of the invention is to iteratively solve the initial game strategy based on iterative design variables, multi-point boundary value configuration expressions, pulse intervals, pulse amplitudes and situation awareness delays to obtain the optimal pulse maneuver strategy;

[0047] Figure 4a , Figure 4b , Figure 4c and Figure 4d are respectively the game results in one embodiment;

[0048] Figure 5a , Figure 5b , Figure 5c and Figure 5d are respectively the game results in one embodiment;

[0049] Figure 6 A schematic diagram of the structure of an optimal strategy generation device for a spacecraft impulse game according to an embodiment. DETAILED DESCRIPTION

[0050] The present invention is further described in detail below by specific embodiments in conjunction with the accompanying drawings. Wherein similar elements in different embodiments adopt associated similar element numbers. In the following embodiments, many detailed descriptions are for making the present application better understood. However, those skilled in the art can easily recognize that some features can be omitted in different situations, or can be replaced by other elements, materials, methods. In some cases, some operations related to the present application are not shown or described in the specification, this is to avoid the core part of the present application being overwhelmed by too much description, and for those skilled in the art, it is not necessary to describe these related operations in detail, and they can fully understand the related operations according to the description in the specification and the general technical knowledge in the art.

[0051] In addition, the features, operations or characteristics described in the specification can be combined in any appropriate manner to form various implementations. At the same time, the steps or actions in the method description can also be interchanged or adjusted in a manner that is obvious to those skilled in the art. Therefore, the various sequences in the specification and the drawings are only for the purpose of clearly describing a certain embodiment and are not meant to be a required sequence, unless otherwise specified that a certain sequence must be followed.

[0052] The serial numbers of the components in this document, such as "first", "second", etc., are only used to distinguish the objects described and do not have any order or technical meaning. The "connection" and "coupling" mentioned in this application, unless otherwise specified, include direct and indirect connections (couplings).

[0053] In the game, it is assumed that both the pursuer and the evader have perfect state observation. The pursuer will only maneuver after observing the evader's escape pulse maneuver. The amplitude of the pulse maneuvers of both parties has an upper limit but the direction is free. The purpose of each game is interception / counter-interception, and the time of the game is free. If the pursuer can complete the interception within one escape-pursuit pulse maneuver round, this game is a single pulse interception game. If the pursuer fails to successfully intercept the evader within one pulse maneuver round, and both parties can apply multiple pulse maneuvers at equal intervals, then the game becomes a multi-pulse interception game. Since both parties can observe the latest situation of the other party before making a decision in the next maneuver, this game forms a multi-pulse sequence decision game. The use of pulse maneuvers will make it difficult to model the game process of spacecraft, mainly manifested in the following aspects: 1) The state of the spacecraft changes suddenly, resulting in the dynamic equation being non-differentiable at the discontinuity point of the state, which does not meet the continuous differentiability conditions of differential game theory; 2) Under the assumption of continuous maneuvers, the spacecraft only needs to control the direction of the constant thrust of the engine, but under the assumption of pulse maneuvers, the number, time, size, direction, etc. of pulses will become decision variables of the spacecraft, and the decision space will increase significantly; 3) When both parties use pulse maneuvers, the order in which the pulse sequences of both parties are applied will also affect the results of the pursuit and escape game. Therefore, the differential game model under the pulse thrust mode has not yet been formed, and the optimal pulse maneuver strategy under the pulse thrust mode cannot be obtained.

[0054] Please refer to Figure 1 In an embodiment of the present invention, a method for generating an optimal strategy for a spacecraft impulse game is proposed, including steps S1 to S4, which are described in detail below.

[0055] Step S1: Obtain the pulse interval, pulse amplitude and situational awareness delay of the pursuer in the pursuit-escape pulse maneuver game scenario, as well as the pulse interval, pulse amplitude and situational awareness delay of the escaper.

[0056] In some embodiments, in the pursuit-escape pulse maneuver game scenario, the tracker and the escaper are both initially operating in the low earth orbit or near the geostationary orbit. The orbital position of the tracker or the escaper is expressed by the classical orbital elements, semi-major axis, eccentricity, orbital inclination, right ascension of the ascending node, and latitude argument. Therefore, the orbital position of the tracker can be expressed as [a P ,e P ,i P ,Ω P ,θ uP ]=[6778.137km,0,0°,0°,0°], the orbital position of the escape vehicle can be expressed as [a E ,e E ,i E ,Ω E ,θ uE] = [6878.137km, 0, 0°, 0°, 1.5°], where a P represents the semi-major axis of the tracker, e P represents the eccentricity of the tracker, i P represents the tracker's orbital inclination, Ω P represents the right ascension of the ascending node of the tracker, θ uP Indicates the latitude argument of the tracker. The pulse interval of the tracker is T P is 1000s, the pulse interval of the escaper is T E is 1000s, and the pulse amplitude of the tracker is ΔV P max =150m / s, the pulse amplitude of the escaper is ΔV E max =50m / s, the tracker's situational awareness delay τ P is 0s, the situational awareness delay of the escape device is τ E The pulse interval of the tracker and the pulse interval of the escaper refer to the time interval between two adjacent pulses.

[0057] Step S2: Construct a sequential pulse differential game model, and construct the Hamiltonian function and interior point function sequence of the tracker and the Hamiltonian function and interior point function sequence of the escaper.

[0058] In some embodiments, the sequential pulse differential game model includes constraint equations constructed based on pulse intervals and pulse amplitudes. The sequential pulse differential game model also includes state equations and payoff functions, wherein the state equations are constructed based on the orbital positions and velocities of the pursuer and the escaper.

[0059] In some embodiments, the constraint equations include:

[0060]

[0061] N f :=r P (t f )-r E (t f )=0

[0062] Among them, P represents the tracker, E represents the escaper, represents the instant after the pulse acts on the spacecraft, represents the instant before the pulse acts on the spacecraft, χ ik represents the position difference of the spacecraft, represents the position of the spacecraft immediately after the pulse, represents the position of the spacecraft immediately before the pulse, σ ik The velocity impulse inequality for the spacecraft, Δv ik represents the velocity difference of the spacecraft, represents the speed of the spacecraft immediately after the pulse, The velocity of the spacecraft immediately before the pulse, ΔV i max represents the maximum pulse amplitude of the spacecraft, k represents the time node, N i Represents the number of pulses of the spacecraft, N f represents the intercept constraint at the terminal position, t f represents the interception time, r P (t f ) represents the position of the tracker at the time of interception, r E (t f ) represents the position of the escape vehicle at the moment of interception, where when i=P, the spacecraft refers to the pursuer, and when i=E, the spacecraft refers to the escape vehicle.

[0063] In some embodiments, if the tracker can complete the interception as early as possible, the interception time satisfies t f ≤T E , then the game ends in a single pulse phase. If the escaper maximizes the interception time so that t f >T E , then the escaper will be able to apply a second pulse, and the game will enter a multi-stage multi-pulse game. E represents the pulse interval of the escaper. Assume that the number of pulses of the chaser and escaper is N P and N E , the pulse maneuver sequence of the tracker is Δv P1 ,Δv P2 ,..., The impulse maneuver sequence of the escape vehicle is Δv E1 ,Δv E2 ,..., The time of the first pulse of both parties is t P1 and t E1 , then the tracker takes effect at the k+1th pulse at t Pk+1 =t E1 +τ P +k·T P ,k=1,2,3,...,N P -1, the time of action of the escaper at the k+1th pulse is t Ek+1 =t E1 +k·T E ,k=1,2,3,...,N E -1, where τ P Represents the situational awareness latency of the tracker.

[0064] In some embodiments, the sequential pulse differential game model further includes a state equation, wherein the state equation is constructed based on the orbital position and velocity of the pursuer and the escaper. The position vectors of the two spacecraft in the geocentric inertial system are r P and r E , when the perturbation force is ignored, the motion of the two obeys Kepler orbital dynamics, or in other words, presents the state equation in the game:

[0065]

[0066] Among them, μ represents the gravitational constant of the central celestial body, r P and r E They represent the position vectors of the pursuer and escaper in the geocentric inertial system, r P0 represents the track position of the tracker at the initial moment, v P0 represents the speed of the tracker at the initial moment, r E0 represents the orbital position of the escaper at the initial moment, v E0 represents the velocity of the escaper at the initial moment, t 0 Indicates the initial moment of the game.

[0067] In some embodiments, the sequential pulse differential game model includes a payoff function, where each pulse of the escaper is not only a response to the preceding chaser pulse maneuver, but also takes into account the future chaser maneuvers to continuously maximize the interception time in the case of a multi-pulse game:

[0068]

[0069] Where J represents the payment function, N P and N E represents the number of pulses of the chaser and escaper, t f Indicates the interception time.

[0070] In some embodiments, the solution that satisfies the above formula is called a multi-stage Steinberg equilibrium. The game behavior of each decision-making step of both parties is reflected in that not only the local optimality of the action for the current time is considered, but also the global optimality for subsequent actions is considered prospectively.

[0071] In some embodiments, since the pulse interruption time of the tracker and the escaper are inconsistent, the Hamiltonian function of the tracker and the Hamiltonian function of the escaper are constructed respectively, including:

[0072]

[0073] Among them, H i represents the Hamiltonian function of the spacecraft, P represents the pursuer, E represents the escaper, T represents the transposed symbol, represents the position co-state of the spacecraft, vi is the speed of the spacecraft, represents the velocity co-state of the spacecraft, g i represents the gravitational acceleration of the spacecraft, where when i=P, the spacecraft refers to the pursuer, and when i=E, the spacecraft refers to the escaper.

[0074] In some embodiments, at the discontinuous points of the pulse action, that is, the interior points, the interior point function sequences of the tracker and the interior point function sequences of the escaper are respectively constructed, including:

[0075] Φ ik =γ ik χ ik +κ ik σ ik ,i=P,E

[0076] Among them, Φ ik represents the interior point function sequence of the spacecraft, γ ik and κ ik represents the Lagrange multiplier of the spacecraft, χ ik represents the position difference of the spacecraft, σ ik represents the velocity impulse inequality of the spacecraft, P represents the pursuer, E represents the escaper, k represents the time node, k=1,2,...,N, N represents the number of pulses, among which, when i=P, the spacecraft refers to the pursuer, and when i=E, the spacecraft refers to the escaper.

[0077] In some embodiments, a terminal function Φ is constructed at the terminal time f The constraints on the terminal moment and the items related to the terminal moment in the payment function are included, expressed as: Φ f =t f .

[0078] Step S3: Construct a multi-point boundary value configuration expression based on the boundary conditions of the Hamiltonian function of the tracker, the boundary conditions of the Hamiltonian function of the escaper, the boundary conditions of the co-state variables in the Hamiltonian function, the sequential pulse differential game model, the Hamiltonian function and interior point function sequence of the tracker, and the Hamiltonian function and interior point function sequence of the escaper.

[0079] In some embodiments, according to the transversal condition of the Hamiltonian function, the boundary condition of the Hamiltonian function of the tracker and the boundary condition of the Hamiltonian function of the escaper can be obtained as follows: i (t f )=-1.

[0080] In some embodiments, the boundary conditions of the co-state variables in the Hamiltonian function can be obtained according to the transversal conditions of the co-state variables:

[0081]

[0082]

[0083] in, represents the position co-state of the tracker at the instant before the pulse, represents the position co-state of the tracker at the instant after the pulse acts, Φ k represents the interior point function sequence of the spacecraft, Indicates the position of the tracker at the moment before the pulse acts. represents the position co-state of the escaper at the instant before the pulse acts, represents the position co-state of the escaper at the instant after the pulse acts on it, It indicates the position of the escaper at the instant before the pulse acts. represents the velocity co-state at the instant after the pulse acts on the tracker, represents the velocity of the tracker at the instant after the pulse acts on it, κ Pk represents the Lagrange multiplier of the tracker at time k, Δv Pk represents the speed of the tracker under the action of the kth pulse.

[0084] In some embodiments, Δr(t f )=r P (t f )-r E (t f ) represents the terminal position vector difference between the pursuer and the escaper.

[0085] In some embodiments, since the escaper's impulse maneuver acts at the initial moment, the escaper's trajectory will be divided into N segments by N interior points, and the tracker's trajectory will be divided into N+1 segments by N interior points. Therefore, the multi-point boundary value configuration expression is constructed as follows:

[0086]

[0087] in, represents the position vector of the escaper at the initial moment of the game, r E0 represents the orbital position of the escaper at the initial moment, represents the velocity co-state of the escaper at the initial moment of the game, κ E1 represents the Lagrange multiplier of the escaper when k = 1, Δv E1 represents the speed of the escaper under the action of the first pulse, represents the position co-state of the escaper at the instant before the pulse acts, represents the position co-state of the escaper at the instant after the pulse acts on it, represents the velocity co-state at the instant before the pulse acts on the escaper, represents the velocity co-state at the instant after the pulse acts on the escaper, It represents the position of the escaper at the instant after the pulse acts on it. represents the position of the escaper before the pulse acts, r E (t f ) represents the position vector of the escape device at the time of interception, r P (t f ) represents the position vector of the tracker at the interception moment, represents the velocity co-state of the escape vehicle at the moment of interception, represents the position co-state of the tracker at the instant before the pulse, represents the position co-state of the tracker at the instant after the pulse acts, κ Pk represents the Lagrange multiplier of the tracker at time k, Δv Pk represents the speed of the tracker under the kth pulse, It indicates the position of the tracker at the moment of the pulse. Indicates the position of the tracker at the moment before the pulse acts. Represents the velocity co-state at the instant after the pulse is applied to the tracker.

[0088] In some embodiments, the additional N+1 unknown parameters of the tracker correspond to N+1 equations, expressed as:

[0089]

[0090] Among them, κ i1 ,...κ iN represents the Lagrange multiplier of the spacecraft at time nodes k = 1, ..., N, σ i1 ,...σ iN represents the velocity impulse inequality of the spacecraft at the time nodes k = 1, ..., N, H(t f )=-1 represents the boundary condition of the Hamiltonian function of the spacecraft.

[0091] Step S4: Solve the initial game strategies corresponding to the pursuer and the evader, and construct the corresponding iterative design variables according to the initial game strategies. The initial game strategies are iteratively solved based on the iterative design variables, multi-point boundary value configuration expressions, pulse intervals, pulse amplitudes, and situation awareness delays to obtain the optimal pulse maneuver strategy.

[0092] Please refer to Figure 2 In some embodiments, step S4 solves the initial game strategies corresponding to the tracker and the escaper, including steps S41 to S45, wherein steps S41 to S45 are executed from the first pulse phase of the preset multiple pulse phases, as described in detail below.

[0093] Step S41: Obtain the initial orbital positions of the tracker and the escaper in the current pulse phase.

[0094] In some embodiments, each pulse phase is preset with an initial orbital position of the tracker and an initial orbital position of the escaper.

[0095] Step S42: When there is pulse thrust on the tracker and the escaper, the pulse maneuver strategy of the tracker and the pulse maneuver strategy of the escaper are solved based on the initial orbital position of the tracker and the initial orbital position of the escaper.

[0096] In some embodiments, the pulse maneuver strategy of the pursuer includes multiple pursuit strategies, and the pulse maneuver strategy of the escaper includes multiple escape strategies.

[0097] Step S43: Select the optimal tracking strategy among the pulse maneuvering strategies of the tracker, and select the optimal escape strategy among the pulse maneuvering strategies of the escaper.

[0098] In some embodiments, the optimal tracking strategy in the pulse maneuver strategy of the tracker is solved The strategy that minimizes the interception time is the optimal tracking strategy. Similarly, the optimal escape strategy in the escape device's pulse maneuver strategy is

[0099] Step S44: If the tracker satisfies the preset game conditions when using the optimal tracking strategy and the evader uses the optimal escaping strategy for the pursuit and escape game, or the number of pulse stages that have been experienced is greater than the preset pulse stage threshold, the optimal tracking strategy and the optimal escaping strategy are used as the initial game strategies.

[0100] In some embodiments, the preset pulse phase threshold is 10.

[0101] Step S45: Otherwise, enter the next pulse stage to calculate the updated optimal tracking strategy and optimal escape strategy until the tracker uses the optimal tracking strategy and the escaper uses the optimal escape strategy to play the pursuit and escape game and the preset game conditions are met, or the number of pulse stages that have been experienced is greater than the preset game stage threshold, and the updated optimal tracking strategy and optimal escape strategy are used as the initial game strategy.

[0102] In some embodiments, the game condition is that the remaining interception time is less than the pulse interval of the tracker or less than the pulse interval of the escaper, and the remaining interception time represents the time required for the tracker to intercept the escaper.

[0103] In some embodiments, since the spacecraft is assumed to have equally spaced pulses and the waiting time is not considered, the number of remaining pulses depends entirely on the remaining interception time of the game and satisfies the following formula:

[0104]

[0105] Among them, N Est Indicates the number of remaining pulses, floor indicates the rounding down function, Indicates the estimated remaining intercept time.

[0106] In some embodiments, for the initial game strategy that has been obtained, the iterative process from the initial solution to the exact solution can be completed by the shooting method, wherein the iterative design variables are constructed according to the pulse maneuver sequence of the tracker, the pulse maneuver sequence of the escaper and the game deadline in the initial game strategy. The pulse maneuver sequence of the tracker includes Δv P1 ,...,Δv PN , where Δv P1 ,...,Δv PN represent the speed of the pursuer under the action of the 1st... to the Nth pulse respectively. The pulse maneuver sequence of the escaper includes Δv E1 ,...,Δv EN , Δv E1 ,...,Δv EN They represent the speed of the escaper under the action of the 1st... to the Nth pulse, P represents the pursuer, E represents the escaper, and N represents the number of pulses. Therefore, the iterative design variables include:

[0107] X=[Δv P1 ,...,Δv PN ,Δv E1 ,...,Δv EN ,κ P1 ,...,κ PN ,κ E1 ,...,κ EN ,λ Pr0 ,λ Er0 ,λ Pv0 ,t f ] T

[0108] Where X represents the iterative design variable, t f represents the game deadline in the initial game strategy, i.e., the interception time, κ P1 ,...,κ PN represents the Lagrange multiplier corresponding to different trackers, κ E1 ,...,κ EN represents the Lagrange multiplier corresponding to different escapers, λPr0, λ Er0, λ Pv0, They respectively represent the position co-state of the tracker at the initial moment, the position co-state of the escaper at the initial moment, and the velocity co-state of the tracker at the initial moment.

[0109] Please refer to Figure 3 In some embodiments, step S4 iteratively solves the initial game strategy based on iterative design variables, multi-point boundary value configuration expressions, pulse intervals, pulse amplitudes, and situation awareness delays to obtain the optimal pulse maneuver strategy, including steps S47 to S49, which are described in detail below.

[0110] Step S47: construct the trajectory of the tracker at different pulse stages, and the trajectory of the escaper at different pulse stages.

[0111] In some embodiments, since the timing of the tracker pulse maneuver is not consistent with the timing of the escaper pulse maneuver, for the tracker, The first segment of the trajectory can be obtained by integrating the following formula:

[0112]

[0113] in, represents the position of the spacecraft in the first trajectory, represents the velocity of the spacecraft in the first trajectory, represents the position co-state of the spacecraft, represents the velocity co-state of the spacecraft, v i represents the speed of the tracker, g i is the gravitational acceleration of the spacecraft, represents the position of the tracker immediately after the initial pulse. It represents the speed of the tracker immediately after the initial pulse. It represents the position co-state of the tracker immediately after the initial pulse. represents the velocity co-state of the tracker at the instant after the initial pulse, r P0 represents the position of the tracker at the initial moment, v P0 represents the speed of the tracker at the initial moment, λ Pr0 represents the position co-state of the tracker at the initial moment, λ Pv0 Represents the velocity covariance of the tracker at the initial moment.

[0114] In some embodiments, for the escaper, the first trajectory can be obtained by integrating the following formula:

[0115]

[0116] In some embodiments, if the number of pulses is greater than or equal to two, the k-th trajectory of the chaser and escaper can be obtained by integrating the following equation:

[0117]

[0118] Step S48: Generate a trajectory end error expression based on the trajectory of the tracker at different pulse phases and the trajectory of the escaper at different pulse phases.

[0119]

[0120] Step S49: Using the shooting method as the iterative algorithm, the iterative design variable as the iterative correction value, the multi-point boundary value configuration expression as the constraint condition, the pulse interval, pulse amplitude, and situation awareness delay as input parameters, the initial game strategy is iteratively solved according to the trajectory end error expression to obtain the optimal pulse maneuvering strategy.

[0121] In some embodiments, the trajectory end error expression can be used as a constraint limit on the error of the end position in the multi-segment trajectory of the tracker and the escaper, and the shooting method is used as the iterative algorithm, the iterative design variables are used as the iterative correction values, the multi-point boundary value configuration expression is used as the constraint condition, and the pulse interval, pulse amplitude, and situation awareness delay are used as input parameters to iteratively solve the initial game strategy and obtain the optimal pulse maneuvering strategy.

[0122] Please refer to Figure 4a , Figure 4b , Figure 4c and Figure 4d It can be seen that in some embodiments, in the pursuit-escape pulse maneuver game scenario, the track position of the tracker is [a P ,e P ,i P ,Ω P ,θ uP ]=[6778.137km,0,0°,0°,0°], the orbital position of the escape vehicle is [a E ,e E ,i E ,Ω E ,θ uE ] = [6878.137km, 0, 0°, 0°, 1.5°], where a P represents the semi-major axis of the tracker, e P represents the eccentricity of the tracker, i P represents the tracker's orbital inclination, Ω P represents the right ascension of the ascending node of the tracker, θ up Indicates the latitude argument of the tracker. The pulse interval of the tracker is T P is 1000s, the pulse interval of the escaper is TE is 1000s, and the pulse amplitude of the tracker is ΔV P max =150m / s, the pulse amplitude of the escaper is ΔV E max =50m / s, the tracker's situational awareness delay τ P is 0s, the situational awareness delay of the escape device is τ E 0s. The tracker and the escaper differ in orbital altitude of 100km and phase angle of 1.5°. Both parties perform pulse maneuvers in sequence, with the escaper taking the initiative and the tracker taking the initiative. Assuming that the tracker has no situational awareness delay, that is, the tracker can perform pulse maneuvers immediately after the escaper applies a pulse, then this game will be completed within one pulse cycle, and the interception time corresponding to the Steinberg equilibrium solution where both parties adopt the optimal pulse maneuver strategy is 961s. Reference Figure 4a , Figure 4b , Figure 4c and Figure 4d It can be seen that the optimal pulse maneuvering directions of both the pursuer and the escaper are almost the same. From a temporal perspective, the pulse moments of the tracker and the escaper are the same, which is very close to the simultaneous decision-making game, but in fact the two are completely different. The fundamental difference is that the tracker has a wider set of situation information than the escaper, because the tracker has observed the latest situation of the escaper. From the concept of the solution, this solution is not a Nash equilibrium solution either. Because if the tracker's strategy is fixed, the escaper will not actively maintain an interception relationship with the tracker when freely choosing the pulse direction, so the interception may not be guaranteed in the end.

[0123] Please refer to Figure 5a , Figure 5b , Figure 5c and Figure 5d It can be seen that in some embodiments, in the pursuit-escape pulse maneuver game scenario, the track position of the tracker is [a P ,e P ,i P ,Ω P ,θ uP ]=[6778.137km,0,0°,0°,0°], the orbital position of the escape vehicle is [a E ,e E ,i E ,Ω E ,θ uE ] = [6878.137km, 0, 0°, 0°, 4°]. Tracker pulse interval T P is 1000s, the pulse interval of the escaper is T E is 1000s, and the pulse amplitude of the tracker is ΔV P max =150m / s, the pulse amplitude of the escaper is ΔV E max =50m / s, the tracker's situational awareness delay τP is 100s, the situational awareness delay of the escape device is τ E The phase angle between the pursuer and the escaper is 4°, and considering the 100-second situational awareness delay, the game cannot be completed within the single pulse interval of 1000 seconds, and it will evolve into a multi-pulse interception game. Since the pursuer finally intercepted the escaper at 1862.8 seconds, the game is ultimately a two-pulse game. The absolute trajectory, relative trajectory, distance change, and pulse direction of the game are shown in Figure 1. Figure 5a , Figure 5b , Figure 5c and Figure 5d As shown in Figure 2, it can be found that the optimal pulse directions of the pursuer and escaper are still approximately parallel, and the optimal pulse direction is not always along the tangent direction of the trajectory.

[0124] Please refer to Figure 6 In some embodiments, a device for generating an optimal strategy for a spacecraft pulse game includes a data preparation module 10, a model building module 20, a multi-point edge value conversion module 30 and a game strategy generation module 40.

[0125] The data preparation module 10 is used to obtain the pulse interval, pulse amplitude and situational awareness delay of the tracker in the pursuit-escape pulse maneuver game scenario, as well as the pulse interval, pulse amplitude and situational awareness delay of the escaper; wherein the pulse interval refers to the time interval between two adjacent pulses.

[0126] The model building module 20 is used to construct a sequential pulse differential game model, and to construct the Hamiltonian function and interior point function sequence of the tracker and the Hamiltonian function and interior point function sequence of the escaper; wherein the sequential pulse differential game model includes constraint equations constructed according to the pulse interval and pulse amplitude.

[0127] The multi-point boundary value conversion module 30 is used to construct a multi-point boundary value configuration expression based on the boundary conditions of the Hamiltonian function of the tracker, the boundary conditions of the Hamiltonian function of the escaper, the boundary conditions of the co-state variables in the Hamiltonian function, the sequential pulse differential game model, the Hamiltonian function and interior point function sequence of the tracker, and the Hamiltonian function and interior point function sequence of the escaper.

[0128] The game strategy generation module 40 is used to solve the initial game strategies corresponding to the tracker and the evader, and construct the corresponding iterative design variables according to the initial game strategies. The initial game strategies are iteratively solved based on the iterative design variables, multi-point boundary value configuration expressions, pulse intervals, pulse amplitudes and situation awareness delays to obtain the optimal pulse maneuver strategy.

[0129] In some embodiments, a device for generating an optimal strategy for a spacecraft pulse game is characterized by comprising: a memory for storing programs; and a processor for implementing a game strategy generation method by executing the programs stored in the memory.

[0130] Those skilled in the art will appreciate that all or part of the functions of the various methods in the above-mentioned embodiments can be implemented by hardware or by computer programs. When all or part of the functions in the above-mentioned embodiments are implemented by computer programs, the program can be stored in a computer-readable storage medium, and the storage medium can include: read-only memory, random access memory, disk, optical disk, hard disk, etc., and the program is executed by a computer to implement the above-mentioned functions. For example, the program is stored in the memory of the device, and when the program in the memory is executed by the processor, all or part of the above-mentioned functions can be implemented. In addition, when all or part of the functions in the above-mentioned embodiments are implemented by computer programs, the program can also be stored in a storage medium such as a server, another computer, disk, optical disk, flash disk or mobile hard disk, and can be downloaded or copied and saved in the memory of the local device, or the system of the local device is updated, and when the program in the memory is executed by the processor, all or part of the functions in the above-mentioned embodiments can be implemented.

[0131] The above specific examples are used to illustrate the present invention, which is only used to help understand the present invention and is not intended to limit the present invention. For those skilled in the art, according to the concept of the present invention, some simple deductions, modifications or substitutions can be made.

Claims

1. A method for generating an optimal strategy for a spacecraft impulse game, characterized in that: include: Obtain the pulse interval, pulse amplitude and situational awareness delay of the pursuer in the pursuit-escape pulse maneuver game scenario, as well as the pulse interval, pulse amplitude and situational awareness delay of the escaper; wherein the pulse interval refers to the time interval between two adjacent pulses; Constructing a sequence pulse differential game model, and constructing the Hamiltonian function and interior point function sequence of the tracker and the Hamiltonian function and interior point function sequence of the escaper; wherein the sequence pulse differential game model includes constraint equations constructed according to the pulse interval and the pulse amplitude; Constructing a multi-point boundary value configuration expression based on the boundary conditions of the Hamiltonian function of the tracker, the boundary conditions of the Hamiltonian function of the escaper, the boundary conditions of the co-state variables in the Hamiltonian function, the sequential pulse differential game model, the Hamiltonian function and interior point function sequence of the tracker, and the Hamiltonian function and interior point function sequence of the escaper; The initial game strategies corresponding to the tracker and the escaper are solved, and corresponding iterative design variables are constructed according to the initial game strategies. The initial game strategies are iteratively solved based on the iterative design variables, the multi-point boundary value configuration expression, the pulse interval, the pulse amplitude and the situation awareness delay to obtain the optimal pulse maneuvering strategy.

2. The optimal strategy generation method according to claim 1, characterized in that: The solving of the initial game strategies corresponding to the tracker and the escaper includes: Starting from the first pulse phase of a preset number of pulse phases: Acquire the initial orbital positions of the tracker and the escaper in the current pulse phase, wherein each pulse phase is preset with the initial orbital position of the tracker and the initial orbital position of the escaper; When there is a pulse thrust on the tracker and the escaper, the pulse maneuver strategy of the tracker and the pulse maneuver strategy of the escaper are solved based on the initial orbital position of the tracker and the initial orbital position of the escaper, wherein the pulse maneuver strategy of the tracker includes a plurality of tracking strategies, and the pulse maneuver strategy of the escaper includes a plurality of escape strategies; Selecting the best tracking strategy among the pulse maneuver strategies of the tracker, and selecting the best escape strategy among the pulse maneuver strategies of the escaper; If the tracker satisfies the preset game conditions when using the optimal tracking strategy and the escaper satisfies the preset game conditions when using the optimal escape strategy, or the number of pulse stages that have been experienced is greater than the preset pulse stage threshold, the optimal tracking strategy and the optimal escape strategy are used as the initial game strategies; Otherwise, enter the next pulse stage to calculate the updated optimal tracking strategy and optimal escape strategy until the tracker satisfies the preset game condition when using the optimal tracking strategy and the escaper uses the optimal escape strategy to play the pursuit and escape game, or the number of pulse stages that have been experienced is greater than the preset game stage threshold, and the updated optimal tracking strategy and optimal escape strategy are used as the initial game strategy; wherein, the game condition is that the remaining interception time is less than the pulse interval of the tracker or less than the pulse interval of the escaper, and the remaining interception time represents the time required for the tracker to intercept the escaper.

3. The optimal strategy generation method according to claim 1, characterized in that: The initial game strategy includes the pulse maneuver sequence of the tracker, the pulse maneuver sequence of the escaper and the game deadline. The pulse maneuver sequence of the tracker includes Δv P1 ,...,Δv PN , where Δv P1 ,...,Δv PN represent the speed of the pursuer under the action of the 1st... to the Nth pulse respectively, and the pulse maneuver sequence of the escaper includes Δv E1 ,...,Δv EN , Δv E1 ,...,Δv EN represent the speed of the escaper under the action of the 1st... to Nth pulses respectively, P represents the pursuer, E represents the escaper, and N represents the number of pulses; The iterative design variables include: X=[Δv P1 ,...,Δv PN ,Δv E1 ,...,Δv EN ,k P1 ,...,k PN ,k E1 ,...,k EN ,l Pr0 ,l Er0 ,l Pv0 ,t f ] T Where X represents the iterative design variable, t f represents the game deadline in the initial game strategy, i.e., the interception time, κ P1 ,...,κ PN Denotes the Lagrange multiplier corresponding to different trackers, κ E1 ,...,κ EN represents the Lagrange multiplier corresponding to the different escapers, λ Pr0, λ Er0, λ Pv0 They respectively represent the position co-state of the tracker at the initial moment, the position co-state of the escaper at the initial moment, and the velocity co-state of the tracker at the initial moment.

4. The optimal strategy generation method according to claim 1, characterized in that: The interior point function sequence of the tracker and the interior point function sequence of the escaper include: F ik =c ik x ik +k ik s ik ,i=P,E Among them, Φ ik represents the interior point function sequence of the spacecraft, γ ik and κ ik represents the Lagrange multiplier of the spacecraft, χ ik represents the position difference of the spacecraft, σ ik represents the velocity impulse inequality of the spacecraft, P represents the tracker, E represents the escaper, k represents the time node, k=1,2,...,N, N represents the number of pulses, wherein when i=P, the spacecraft refers to the tracker, and when i=E, the spacecraft refers to the escaper.

5. The optimal strategy generation method according to claim 1, characterized in that: The Hamiltonian function of the tracker and the Hamiltonian function of the escaper include: Among them, H i represents the Hamiltonian function of the spacecraft, P represents the pursuer, E represents the escaper, T represents the transposed symbol, represents the position co-state of the spacecraft, v i is the speed of the spacecraft, represents the velocity co-state of the spacecraft, g i represents the gravitational acceleration of the spacecraft, wherein when i=P, the spacecraft is a pursuer, and when i=E, the spacecraft is an escaper.

6. The optimal strategy generation method according to claim 1, characterized in that: The constraint equations include: N f :=r P (t f )-r E (t f )=0 Wherein, P represents the tracker, E represents the escaper, represents the instant after the pulse acts on the spacecraft, represents the instant before the pulse acts on the spacecraft, χ ik represents the position difference of the spacecraft, represents the position of the spacecraft immediately after the pulse, represents the position of the spacecraft immediately before the pulse, σ ik The velocity impulse inequality for the spacecraft, Δv ik represents the velocity difference of the spacecraft, is the velocity of the spacecraft immediately after the pulse, The velocity of the spacecraft immediately before the pulse, ΔV imax represents the maximum pulse amplitude of the spacecraft, k represents the time node, N i represents the number of pulses of the spacecraft, N f represents the intercept constraint at the terminal position, t f represents the interception time, r P (t f ) represents the position of the tracker at the time of interception, r E (t f ) represents the position of the escapee at the moment of interception, wherein when i=P, the spacecraft refers to the pursuer, and when i=E, the spacecraft refers to the escapee.

7. The optimal strategy generation method according to claim 1, characterized in that: The sequential pulse differential game model also includes a state equation and a payoff function, wherein the state equation is constructed according to the orbital positions and velocities of the pursuer and the escaper.

8. The optimal strategy generation method according to claim 1, characterized in that: The iteratively solving the initial game strategy based on the iterative design variables, the multi-point boundary value configuration expression, the pulse interval, the pulse amplitude and the situation awareness delay to obtain the optimal pulse maneuver strategy includes: Constructing the trajectory of the tracker at different pulse phases and the trajectory of the escaper at different pulse phases; Generate a trajectory end error expression according to the trajectory of the tracker at different pulse phases and the trajectory of the escaper at different pulse phases; The target shooting method is used as the iterative algorithm, the iterative design variable is used as the iterative correction value, the multi-point boundary value configuration expression is used as the constraint condition, the pulse interval, the pulse amplitude, and the situation awareness delay are used as input parameters, and the initial game strategy is iteratively solved according to the trajectory end error expression to obtain the optimal pulse maneuvering strategy.

9. An optimal strategy generation device for spacecraft impulse game, characterized in that: include: A data preparation module is used to obtain the pulse interval, pulse amplitude and situational awareness delay of the pursuer in the pursuit-escape pulse maneuver game scenario, as well as the pulse interval, pulse amplitude and situational awareness delay of the escaper; wherein the pulse interval refers to the time interval between two adjacent pulses; A model building module, used to build a sequence pulse differential game model, and to build the Hamiltonian function and interior point function sequence of the tracker and the Hamiltonian function and interior point function sequence of the escaper; wherein the sequence pulse differential game model includes a constraint equation constructed according to the pulse interval and the pulse amplitude; A multi-point boundary value conversion module, for constructing a multi-point boundary value configuration expression based on the boundary conditions of the Hamiltonian function of the tracker, the boundary conditions of the Hamiltonian function of the escaper, the boundary conditions of the co-state variables in the Hamiltonian function, the sequential pulse differential game model, the Hamiltonian function and interior point function sequence of the tracker, and the Hamiltonian function and interior point function sequence of the escaper; A game strategy generation module is used to solve the initial game strategies corresponding to the tracker and the escaper, and construct corresponding iterative design variables according to the initial game strategies, and iteratively solve the initial game strategies based on the iterative design variables, the multi-point boundary value configuration expression, the pulse interval, the pulse amplitude and the situation awareness delay to obtain the optimal pulse maneuver strategy.

10. An optimal strategy generation device for spacecraft impulse game, characterized in that: include: Memory, used to store programs; A processor, configured to implement the optimal strategy generation method according to any one of claims 1 to 8 by executing the program stored in the memory.

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