CN114911167A - Analysis solving method and system for spacecraft finite time pursuit escape game control - Google Patents
Analysis solving method and system for spacecraft finite time pursuit escape game control Download PDFInfo
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Abstract
The invention discloses an analytic solving method and system for spacecraft limited time pursuit escape game control. The method comprises the following steps: in an LVLH coordinate system, zero control miss distance is used as a state variable, and a fixed time differential countermeasure model for spacecraft orbit pursuit escape is established; and solving the optimal control law of the analytic forms of the two pursuing parties and the corresponding optimal game path based on the established fixed time differential countermeasure model. The method can quickly solve the saddle points of the spacecraft escape pursuit game fixed time differential strategy, the escape pursuit bilateral optimal control law corresponding to the saddle points can plan game paths for the spacecraft and avoid unknown targets such as space debris, and the like, and the method has the advantages of correct and reasonable design method, quick and effective calculation process, good applicability to actual tasks, and the like.
Description
Technical Field
The invention belongs to the technical field of spacecraft pursuit and escape game path planning, and particularly relates to an analytic solving method and system for spacecraft limited-time pursuit and escape game control.
Background
With the continuous development of space rendezvous and docking technology and on-orbit service technology, a control method for a cooperative target tends to be mature, but a rendezvous or interception control method (pursuit game control technology) with maneuvering capabilities of both parties is considered to be urgently developed, so that the method has important significance for avoiding unknown targets such as space debris and the like and national space security defense. Accordingly, the problem of escape pursuit game control between two spacecrafts also becomes a hotspot problem of the current leading-edge technical theory. The problem is essentially a continuous time dynamic programming problem under bilateral active control, and two parties in a game have mutually conflicting targets: the tracking spacecraft is intended to approach the escaping spacecraft, which then needs to strive to get rid of the approach. Therefore, compared with the rendezvous control problem of cooperative targets, the spacecraft escape pursuit game control problem has more antagonism and uncertainty, and the problem dimension and the solving difficulty are doubled.
The differential strategy is a theory for researching continuous time infinite dynamic game by adopting a differential method, and is very suitable for modeling and analyzing the problems. Differential countermeasures can be divided into two types, namely qualitative differential countermeasures and quantitative differential countermeasures according to the existence of a payment function; the fixed time differential strategy is one of quantitative differential strategies, and the solution of the saddle point of the differential strategy under the premise of setting the escape game time is mainly researched. The saddle points of the differential countermeasures are Nash equilibrium points of the pursuit evasion parties, the solution of the differential countermeasures at the saddle points corresponds to a bilateral optimal pursuit game path and control law, effective reference can be provided for the design of the pursuit evasion party spacecraft game strategies, and the method has important significance for solving the space safety problem in China. For a detailed description of the theory of differential countermeasures, reference is made to the book "differential countermeasures and their applications" written by leen peak.
When the saddle point of the spacecraft fixed time differential strategy is solved, a spacecraft orbit pursuit differential strategy model is generally established to be a linear quadratic zero sum differential strategy. The solving method of the linear quadratic form zero and differential strategy can be divided into a numerical method and an analytic method, and the numerical method is basically adopted at present, and the specific solving thought is as follows: describing the relative motion state of two spacecrafts under a Local Vertical Local Horizontal (LVLH) coordinate system, establishing a corresponding fixed time differential countermeasure model, constructing a Hamilton-Jacobi-Bellman (Hamilton-Jacobi-Bellman, HJB) partial differential equation of a countermeasure (or a Hamilton function of the countermeasure according to the Pontryagin maximum value principle) according to a dynamic programming principle, converting the Hamilton function into a matrix Riccati differential equation, and solving the matrix Riccati differential equation to obtain the closed loop optimal linear feedback control law of the two spacecrafts. In consideration of the fact that the solving of the numerical method usually requires integration of the matrix Riccati differential equation, the solving time is long, and the method is difficult to be applied to engineering practice, an analytic solving method for spacecraft finite-time pursuit game control is needed to be developed, and the analytic solving method is used for realizing the rapid and real-time solving of the spacecraft path planning problem and similar derivative problems thereof on the premise of giving pursuit game time.
Disclosure of Invention
The invention particularly provides an analytic solving method and system for spacecraft finite time catch-up and escape game control, which overcome the defects of complex solving process, long solving time and the like in the prior art, reduce the solving complexity, greatly shorten the solving time and meet the requirements of quick and real-time solving in engineering practice.
In order to achieve the purpose, the invention provides an analytic solving method for spacecraft limited-time pursuit escape game control, which comprises the following steps:
s1: in an LVLH coordinate system, zero control miss distance is used as a state variable, and a fixed time differential countermeasure model for spacecraft orbit pursuit escape is established;
s2: and respectively solving the optimal control law in the analysis form of the pursuing-escaping two-party spacecraft and the corresponding optimal game path based on the fixed time differential countermeasure model established in the S1.
In order to achieve the above object, the present invention further provides an analysis solving system for spacecraft limited time catching up and escaping game control, which includes a memory and a processor, wherein the memory stores an analysis solving program for spacecraft limited time catching up and escaping game control, and the processor executes the following steps when running the analysis solving program for spacecraft limited time catching up and escaping game control:
s0: acquiring initial parameters of a spacecraft orbit pursuit game of both sides under a fixed time differential countermeasure;
s1: in an LVLH coordinate system, zero control miss distance is used as a state variable, and a fixed time differential countermeasure model for spacecraft orbit pursuit escape is established;
s2: and respectively solving the optimal control law in the analysis form of the pursuing-escaping two-party spacecraft and the corresponding optimal game path based on the fixed time differential countermeasure model established in the S1.
To achieve the above object, the present invention also provides a computer-readable storage medium having a computer program stored thereon, the computer program, when executed by a processor, implementing the steps of:
s0: acquiring initial parameters of a spacecraft orbit pursuit game of both sides under a fixed time differential countermeasure;
s1: in an LVLH coordinate system, zero control miss distance is used as a state variable, and a fixed time differential countermeasure model for spacecraft orbit pursuit escape is established;
s2: and respectively solving the optimal control law in the analysis form of the pursuing-escaping two-party spacecraft and the corresponding optimal game path based on the fixed time differential countermeasure model established in the S1.
The spacecraft finite time pursuit escaping game control analysis solving method and system provided by the invention have the following advantages:
1. the optimal game control law obtained by the analytic method is completely consistent with the result of the numerical method, but the solution is simpler, more convenient and quicker. Considering that the pursuit game of the spacecraft is more likely to be carried out in real time in an on-orbit mode, and the computing power of the on-board computer is relatively weak; therefore, the method for analyzing and solving the spacecraft fixed time differential strategy is more beneficial to the satellite-borne computer to quickly and accurately solve the game maneuvering strategy, and has better applicability in actual engineering tasks.
2. Compared with the solving process of the optimal game control law in the numerical method, the method is more intuitive and specific for the derivation and application of the optimal game control law in the analytic form, and the general rule of the pursuit of the spacecraft by both sides to the game on the premise of giving the game time is more easily reflected.
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FIG. 1 is a technical scheme diagram of a spacecraft limited time pursuit escape game control analysis solving method according to the invention;
fig. 2 is a schematic flow chart of the processor executing the solving program in the memory in the finite-time catch-up game control analysis solving system for the spacecraft of the invention;
FIG. 3 is a diagram of an optimal path for two-party spacecraft pursuit-escape gaming in the first preferred embodiment;
fig. 4 is a diagram illustrating a change rule of the position and distance of the first-party spacecraft with respect to the reference point with time in the first preferred embodiment, wherein: (a) the change rule graph of the relative position component x along with time, (b) the change rule graph of the relative position component y along with time, (c) the change rule graph of the relative position component z along with time, and (d) the change rule graph of the relative distance s along with time;
fig. 5 is a diagram illustrating a change rule of the optimal thrust acceleration of the two pursuing evasive spacecraft with time according to the first preferred embodiment, wherein: (a) for an optimum thrust acceleration component a x A time-dependent change law diagram, (b) is an optimal thrust acceleration component a y A time-dependent change law diagram, (c) is an optimal thrust acceleration component a z A change rule graph along with time, (d) is a change rule graph along with time of the optimal thrust acceleration magnitude a;
FIG. 6 is a diagram of the optimal path for the two-party spacecraft pursuit game in the second preferred embodiment;
fig. 7 is a diagram illustrating a change rule of the position and distance of the two pursuing evasive spacecraft relative to the reference point with time according to the second preferred embodiment, wherein: (a) the change rule graph of the relative position component x along with time, (b) the change rule graph of the relative position component y along with time, (c) the change rule graph of the relative position component z along with time, and (d) the change rule graph of the relative distance s along with time;
fig. 8 is a diagram illustrating a change rule of the velocity of the two pursuing evasive spacecraft relative to the reference point with time according to the second preferred embodiment, wherein: (a) is a relative velocity component v x Law of change with timeFIG. b is a relative velocity component v y A time-dependent change law diagram, wherein (c) is a relative velocity component v z A change rule graph along with time, (d) is a change rule graph along with time of the relative speed v;
fig. 9 is a diagram illustrating a change rule of the optimal thrust acceleration of the two pursuing evasive spacecraft with time according to the second preferred embodiment, wherein: (a) for an optimum thrust acceleration component a x A time-dependent change law diagram, and (b) an optimal thrust acceleration component a y A time-dependent change law diagram, (c) is an optimal thrust acceleration component a z A change rule graph along with time, (d) is a change rule graph along with time of the optimal thrust acceleration magnitude a;
fig. 10 is an internal structure diagram of a computer device used in the analysis and solution system for spacecraft limited-time catch-up game control according to the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present application more apparent, the present application is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the present application and are not intended to limit the present application.
The specific implementation mode of the invention is as follows:
in one embodiment, as shown in fig. 1, there is provided an analytic solution method for spacecraft limited-time catch-up game control, comprising the following steps:
s1: and under an LVLH coordinate system, zero control miss distance is adopted as a state variable, and a fixed time differential countermeasure model for spacecraft orbit pursuit is established.
Further, the S1 includes the following steps:
s101: establishing a fixed time differential countermeasure initial model based on a spacecraft relative motion C-W equation under an LVLH coordinate system;
1) establishing an LVLH coordinate system as a reference coordinate system, and describing the relative motion of the spacecrafts of the two pursuits by adopting a C-W equation;
and establishing an LVLH coordinate system by taking the escaping spacecraft as a reference spacecraft and taking the virtual circular orbit corresponding to the position of the escaping spacecraft as a reference spacecraft orbit. At this time, the LVLH coordinate system is established as follows: the origin of the coordinate system is located at the centroid O of the reference spacecraft, the Ox axis is along the sagittal direction of the reference spacecraft, the Oz axis is along the normal direction of the virtual circular orbit, and the Oy axis is along the tangential direction of the motion trajectory of the reference spacecraft and forms a right-hand coordinate system with the Ox axis and the Oz axis.
In an LVLH coordinate system, the C-W equation shown in the formula (1) is adopted to describe the relative motion of the two pursuing parties:
wherein x, y, z,
The position component and the velocity component of the spacecraft along the radial direction, the track direction and the normal direction relative to the reference coordinate system respectively, omega is the angular velocity of the spacecraft moving on the virtual circular orbit, a x 、a y 、a z Thrust acceleration components of the spacecraft in the radial direction, the tracking direction and the normal direction are respectively.
The relative motion state of the spacecraft is set as
Thrust acceleration vector is u ═ a x ,a y ,a z ] T Then the C-W equation can be written in the form of a state space
Wherein,
formula (2) is a linear non-homogeneous coefficient system equation; given an initial condition x (t) 0 )＝x 0 Under the premise ofThe solution of the equation is
The state transition matrix Φ (t, t) in equation (3) 0 ) Is composed of
Wherein s is sin ω Δ t, c is cos ω Δ t, Δ t is t-t 0 。
2) Establishing a differential countermeasure state equation of a spacecraft pursuit escape game;
the relative motion states of the spacecrafts of the two pursuing parties are respectively set as
And both of them meet the use condition of the C-W equation; order to
State variables of the differential countermeasures for spacecraft pursuit, a P,x 、a P,y 、a P,z 、a E,x 、a E,y 、a E,z The thrust and acceleration components of the two pursuing space vehicles along the radial direction, the tracking direction and the normal direction are respectively, then the formula (1) can be further written as
Both the pursuing and escaping spacecrafts adopt continuous thrust control, and the maximum thrust acceleration is T P 、T E Thrust acceleration directions are respectively by yaw angle alpha (T) i The angle of the projection on the xOy plane with the Ox axis) andpitch angle beta (T) i Angle with the xOy plane) indicates that alpha is 0,2 pi]，
The thrust acceleration has components in the radial, tracking and normal directions of
Where i is P, E. The equation (6) is substituted for the equation (5), and the differential countermeasure state equation of the spacecraft pursuit game can be obtained as
3) Establishing a fixed time differential countermeasure initial model of a spacecraft pursuit escape game;
setting the given pursuit game time as T ═ T f -t 0 The control quantities of the two pursuing and escaping spacecrafts are respectively thrust acceleration vectors
And
and satisfy | u P |≤T P 、|u E |≤T E The differential countermeasure state equation of the spacecraft escape pursuit game obtained by the equations (2), (6) and (7) is
Considering that the tracking spacecraft hopes to reach the end time t of the pursuit escape game f The closer the distance between the two chasing-away spacecrafts, the better, the less fuel consumed in the chasing-away game process and the opposite of the purpose of escaping the spacecrafts; the payment function of the two pursuing and fleeing parties can be respectively defined as
Wherein,
the method is characterized in that a semi-positive definite symmetric matrix represents the proportion of the distance between two spacecrafts in the escape pursuit game ending time payment function; r P ＝r P I 3 And R E ＝r E I 3 The proportion of fuel consumed by the tracking spacecraft and the escaping spacecraft in the payment function is respectively represented for positive definite symmetric matrixes. In particular, Q is the time when the objective of tracking spacecraft in a fixed time derivative strategy is rendezvous rather than interception f ＝q f I 6 。
And (4) establishing a fixed time differential game initial model of the spacecraft pursuit escape game by combining the vertical type (8) and the formula (9).
S102: defining zero control miss amount as a new state variable, and deducing and establishing a fixed time differential countermeasure model for spacecraft orbit pursuit on the basis of the original initial model.
Defining the zero-control miss amount y (t) as a new state variable, i.e.
y(t)＝DΦ(t f ,t)x(t) (10)
Wherein D ═ I 3 O 3 ]，
In particular, when the objective of tracking the spacecraft in the fixed time derivative strategy is rendezvous and not interception, D ═ I 6 。
Order to
G(t f ,t)＝DΦ(t f ,t)B (11)
The new state variable y (t) is derived from the equation (10) and substituted into the equations (8) and (11), and a new differential countermeasure state equation is obtained as
Correspondingly, the payment function of the two pursuing parties can be respectively defined as
Wherein,
the space vehicle tracking method is a semi-positive definite symmetric matrix, and represents the proportion of the zero control miss amount of a tracking space vehicle relative to an escaping space vehicle in a payment function at the ending moment of the pursuing and escaping game; in particular, when the objective of tracking the spacecraft in the fixed time derivative strategy is rendezvous rather than interception,
and (3) combining the vertical type (12) and the formula (13), so that a spacecraft pursuit game fixed time differential countermeasure model taking the zero control miss amount as a state variable can be established.
So far, a fixed time differential countermeasure model for spacecraft orbit pursuit is deduced and established again on the basis of the original initial model.
S2: and respectively solving the optimal control law in the analysis form of the pursuing-escaping two-party spacecraft and the corresponding optimal game path based on the fixed time differential countermeasure model established in the S1.
Further, the S2 includes the following steps:
s201: based on the fixed time differential countermeasure model established in S1, solving the optimal game control law of the two pursuit-fleeing spacecraft by a variational method;
based on a spacecraft orbit pursuit fixed time differential game model formed by the formulas (12) and (13), firstly, a Hamilton function of a game is constructed according to the Pontryagin maximum principle, and then the optimal game control law of the pursuit spacecraft and the spacecraft is solved by adopting a variational method. Specific methods and procedures can be found in the references [ Y.C.Ho, A.E.Bryson Jr., S.Baron.Differencential gaps and optimal throughput-evolution protocols [ J ]. IEEE Transactions on Automatic Control, AC-10(4),385- & 389,1965 ].
Decomposing by a variational method to obtain:
wherein,
and (3) solving the formula (11), the formula (16), the formula (15) and the formula (14) in sequence to obtain the optimal game control law of the two space vehicles.
S202: and resolving the optimal game control law of the two space vehicles and obtaining the corresponding optimal game path.
From the equation (4), there are some fixed time differential countermeasure models for spacecraft orbit tracking newly established in S1
Where s is sin ω τ, c is cos ω τ, τ is t f -t。
By substituting formula (17) for formula (11), the compound is obtained
Formula (18) is substituted for formula (16) and formula (15) in this order to obtain
Wherein,
the optimal game control laws of the analysis form of the spacecraft of the pursuing party and the evading party can be respectively represented by the following formula (19) and the formula (10) to replace the formula (14)
Wherein, K -1 (t f And t) can be obtained by adopting an LU decomposition inverse matrix method.
As can be seen from equation (20): when the catching up game is finished t f Optimal game control law u for pursuing both-party spacecrafts at given time P 、u E Only the current time t and the relative motion state x of both spacecrafts at the current time are related.
In particular, when the objective of tracking the spacecraft in the fixed time derivative strategy is rendezvous rather than interception,
wherein τ is t f -t，
By this time, the analytic solution of the spacecraft fixed time differential strategy is completed.
Consider two parties pursuing and escapingThe spacecraft adopts continuous thrust control, and the maximum thrust acceleration is T P 、T E (ii) a For the current time instant t e [ t ∈ [ [ t ] 0 ,t f ]When the optimum thrust acceleration obtained by the analytic method satisfies | u | P |＞T P Or | u E |＞T E When the thrust saturation is limited to
Wherein i ═ P, E. When u P |≤T P And | u E |≤T E Time, found optimum thrust acceleration u P 、u E Remain unchanged.
Optimal thrust acceleration u of both pursuing and fleeing spacecraft P 、u E And converting the game path into a geocentric inertial coordinate system, substituting the game path into a spacecraft orbit dynamics equation and solving the spacecraft orbit dynamics equation to obtain the optimal game path of the two pursuing and escaping spacecrafts in the geocentric inertial coordinate system.
The first preferred embodiment is an orbit evasion game between two spacecraft under the fixed time differential strategy. In the first preferred embodiment, the purpose of tracking the spacecraft is to intercept the escaping spacecraft at the game ending time (i.e. the relative distance between the two spacecrafts evading at the game ending time is 0), and the purpose of escaping the spacecraft is opposite to the purpose of intercepting the escaping spacecraft.
The analysis solving system for the limited-time spacecraft pursuit and escape game control comprises a memory and a processor, wherein the memory stores an analysis solving program for the limited-time spacecraft pursuit and escape game control, and the processor executes the following steps when running the analysis solving program for the limited-time spacecraft pursuit and escape game control:
s0: acquiring initial parameters of a spacecraft orbit pursuit game of both sides under a fixed time differential countermeasure;
table 1 example an initial parameter configuration
According to the initial parameters shown in Table 1, the differential countermeasure state equation related physical quantity of the two-party spacecraft orbit pursuit escape game can be obtained as
Wherein,
initial value of state variable, x, for differential countermeasures P，0 、x E，0 Pursuing the motion state of the two spacecrafts relative to the virtual circular orbit reference point at the initial moment 0 Referring to the angular velocity of the virtual circular orbit for the initial time, mu is the gravitational constant, r ref,0 ＝[x ref,0 ,y ref,0 ,z ref,0 ] T Is the position vector of the spacecraft at the reference point under the geocentric inertial coordinate system.
In addition, the payment function related parameters of the two pursuing-escaping spacecrafts are set as follows: q. q.s f ＝1×10 10 ，r P ＝8×10 7 ，r E ＝1×10 8 ，t 0 ＝0，t f 3600 s; so far, the initial parameter acquisition of the two-party spacecraft orbit pursuit game under the fixed time differential countermeasure has been completed.
S1: in an LVLH coordinate system, zero control miss distance is used as a state variable, and a fixed time differential countermeasure model for spacecraft orbit pursuit escape is established;
r is calculated according to the parameters given in Table 1 ref,0 And substitution of formula (24) to give omega 0 (ii) a Will omega 0 And t f Substitution of formula (18) to give G (t) f T) analytical expression; g (t) f And t) substituting the analytical expression of the formula (12) and substituting the payment function related parameters of the formula (13) to obtain the spacecraft orbit pursuit fixed time differential countermeasure model taking y (t) as the state variable.
S2: and respectively solving the optimal control law of the analysis form of the two pursuing-escaping spacecrafts and the corresponding optimal game path based on the fixed time differential countermeasure model established in the S1.
Will omega 0 And t f Substitution of formula (17) to give Φ (t) f T) analytical expression; q is to be f 、r P 、r E 、ω 0 And t f Substituting formula (19) and decomposing the inverse matrix through LU to obtain K -1 (t f T) analytical expression; g (t) f ,t)、K -1 (t f ,t)、Φ(t f T) analytical expressions and r P 、r E And (5) substituting an equation (20) to obtain an analytical expression of the optimal control law of the two pursuing and escaping spacecrafts. Calculating x according to the parameters given in Table 1 P,0 、x E,0 And is substituted for formula (24) to yield x 0 (ii) a X is to be 0 Substituting into the analytic expression of the optimal control law of the two parties of pursuing and fleeing to obtain t 0 Optimal control law u of tracking spacecraft and escaping spacecraft corresponding to moment P,0 、u E,0 。
Judgment u P,0 And u E,0 Is the thrust saturation limit processing condition satisfied? If so, will u P,0 、u E,0 Sequentially replacing the formula (23) and the formula (6), and updating the current optimal control laws of the tracked spacecraft and the escaped spacecraft; otherwise, the current optimal control laws of the tracked spacecraft and the escaped spacecraft are kept unchanged.
U handle P,0 、u E,0 And converting the motion state of the spacecraft into the geocentric inertial coordinate system, substituting the motion state into the spacecraft two-body orbit motion equation, and solving to obtain the motion state of the two spacecrafts pursuing to escape at the next moment in the geocentric inertial coordinate system. Calculating the motion state of the corresponding virtual circular orbit reference point according to the motion state of the escaping spacecraft at the next moment; repeating the steps S1 and S2 until the escape game ending time t f And the optimal game path for pursuing the spacecraft of both sides in the given game time can be obtained.
In a first preferred embodiment, a flow diagram of the processor executing the finite-time spacecraft evasion game control resolving program in the memory is shown in fig. 2. Specifically, the optimal game path diagram of the first embodiment of the first escape flier is shown in fig. 3, and the first escape flierThe diagram of the relative position components x, y and z and the relative distance s of the spacecraft changing with time is shown in 4, and the optimal thrust acceleration component a of the spacecraft of the two parties can be tracked x 、a y 、a z And the optimum thrust acceleration magnitude a with time are shown in fig. 5. By contrast, when the first preferred embodiment is solved by adopting a traditional numerical method, the change rule of each corresponding physical quantity along with time is consistent with that of the first preferred embodiment shown in fig. 4 and 5, and the optimal game path of the spacecrafts of the two pursuits of evasion parties is consistent with that shown in fig. 3.
The steps and initial parameters of the second preferred embodiment are the same as those of the first preferred embodiment, and the main differences are as follows: the purpose of tracking the spacecraft is to meet the escaping spacecraft at the game ending time (namely, the relative distance and the relative speed between the two spacecrafts are 0 at the game ending time), and the purpose of escaping the spacecraft is opposite to the purpose of tracking the escaping spacecraft.
A second preferred embodiment provides an analysis solving system for spacecraft limited-time pursuit game control, which includes a memory and a processor, wherein the memory stores an analysis solving program for spacecraft limited-time pursuit game control, and the processor executes the following steps when running the analysis solving program for spacecraft limited-time pursuit game control:
s0: acquiring initial parameters of a spacecraft orbit pursuit game of both sides under a fixed time differential countermeasure;
the initial parameters obtained are the same as the initial parameters in the first preferred embodiment.
S1: in an LVLH coordinate system, zero control miss distance is used as a state variable, and a fixed time differential countermeasure model for spacecraft orbit pursuit escape is established;
r is calculated according to the parameters given in Table 1 ref,0 And substitution of formula (24) to give omega 0 (ii) a Will omega 0 And t f Substitution of formula (21) to give G (t) f T) analytical expression; g (t) f And t) substituting the analytical expression of the formula (12) and substituting the payment function related parameters of the formula (13) to obtain the spacecraft orbit pursuit fixed time differential countermeasure model taking y (t) as the state variable.
S2: and respectively solving the optimal control law of the analysis form of the two pursuing-escaping spacecrafts and the corresponding optimal game path based on the fixed time differential countermeasure model established in the S1.
Will omega 0 And t f Substitution of formula (17) to give Φ (t) f T) analytical expression; q is to be f 、r P 、r E 、ω 0 And t f Substituting formula (22) and decomposing the inverse matrix through LU to obtain K -1 (t f T) analytical expression; g (t) f ,t)、K -1 (t f ,t)、Φ(t f T) analytical expressions and r P 、r E And (5) substituting an equation (20) to obtain an analytical expression of the optimal control law of the two pursuing and escaping spacecrafts. Calculating x according to the parameters given in Table 1 P,0 、x E,0 And is substituted for formula (24) to yield x 0 (ii) a X is to be 0 Substituting the analytic expressions of the optimal control laws of the two pursuit parties to obtain t 0 Optimal control law u of tracking spacecraft and escaping spacecraft corresponding to moment P,0 、u E,0 。
Judgment u P,0 And u E,0 Is the thrust saturation limit processing condition satisfied? If so, will u P,0 、u E,0 Sequentially replacing the formula (23) and the formula (6), and updating the current optimal control laws of the tracked spacecraft and the escaped spacecraft; otherwise, the current optimal control laws of the tracked spacecraft and the escaped spacecraft are kept unchanged.
U handle P,0 、u E,0 And converting the motion state of the spacecraft into the geocentric inertial coordinate system, substituting the motion state into the spacecraft two-body orbit motion equation, and solving to obtain the motion state of the two spacecrafts pursuing to escape at the next moment in the geocentric inertial coordinate system. Calculating the motion state of the corresponding virtual circular orbit reference point according to the motion state of the escaping spacecraft at the next moment; repeating the steps S1 and S2 until the escape game ending time t f And the optimal game path for pursuing the spacecraft of both sides in the given game time can be obtained.
In a second preferred embodiment, a flow diagram of the processor executing the finite-time spacecraft evasion game control resolving program in the memory is shown in fig. 2. Particularly, the optimal game path of the two-party-evasion spacecraft of the second preferred embodimentAs shown in FIG. 6, the time-dependent change of the relative position components x, y, z and the relative distance s of the two pursuing and escaping spacecrafts is shown in FIG. 7, and the relative velocity component v of the two pursuing and escaping spacecrafts is shown in FIG. 6 x 、v y 、v z And the graph of the relative velocity v changing with time is shown as 8, and the optimal thrust acceleration component a of the two pursuing escaping spacecrafts x 、a y 、a z And the optimum thrust acceleration magnitude a with time is shown in fig. 9. By contrast, the second preferred embodiment is solved by adopting a traditional numerical method, the change rule of each corresponding physical quantity along with time is consistent with that of the second preferred embodiment shown in the figure 7, the figure 8 and the figure 9, and the optimal game path of the two space vehicles of the pursuit evasion party is consistent with that shown in the figure 6.
Comparing the specific steps executed by the processor in the above preferred embodiment when running the program with the flowchart (as shown in fig. 2), it can be found that: when the processor runs the program, the actual flow is not sequentially realized from front to back according to the sequence described by the specific execution steps. It should be understood that the processor, when executing the program, is not limited to the exact order in which the steps are performed, and that the steps may be performed in other orders, unless explicitly stated otherwise. Moreover, at least a portion of a step includes multiple sub-steps or multiple stages, which are not necessarily performed at the same time, but may be performed at different times, and the order of performing the sub-steps or stages is not necessarily sequential, but may be performed alternately or in alternation with other steps or at least a portion of the sub-steps or stages of other steps.
In another embodiment, a computer device used by an analytic solution system for spacecraft limited-time catch-up gambling control is provided, the computer device can be a terminal, and the internal structure diagram can be shown in fig. 10. The computer device includes a processor, a memory, a network interface, a display screen, and an input device connected by a system bus. Wherein the processor of the computer device is configured to provide computing and control capabilities, and the memory of the computer device comprises a non-volatile storage medium and an internal memory. The non-volatile storage medium stores an operating system and a computer program. The internal memory provides an environment for the operating system and the computer program to run on the non-volatile storage medium. The network interface of the computer device is used for communicating with an external terminal through a network connection. The computer program is executed by a processor to realize an analytic solving method for spacecraft finite-time catch-up game control. The display screen of the computer equipment can be a liquid crystal display screen or an electronic ink display screen, and the input device of the computer equipment can be a touch layer covered on the display screen, a key, a track ball or a touch pad arranged on the shell of the computer equipment, an external keyboard, a touch pad or a mouse and the like.
Those skilled in the art will appreciate that the architecture presented in FIG. 10 is merely a block diagram of some of the structures associated with the disclosed aspects and is not intended to limit the computing devices to which the disclosed aspects may be applied, and that a particular computing device may include more or less components than those shown, or may combine certain components, or have a different arrangement of components.
In another embodiment, a computer-readable storage medium is provided, on which a computer program is stored, which computer program, when being executed by a processor, carries out the steps of the above-mentioned preferred embodiments.
It will be understood by those skilled in the art that all or part of the processes of the above preferred embodiments may be implemented by a computer program, which may be stored in a non-volatile computer-readable storage medium, and which, when executed, may comprise processes of the above method embodiments. Any reference to memory, storage, database, or other medium used in the embodiments provided herein may include non-volatile and/or volatile memory, among others. Non-volatile memory can include read-only memory (ROM), Programmable ROM (PROM), Electrically Programmable ROM (EPROM), Electrically Erasable Programmable ROM (EEPROM), or flash memory. Volatile memory can include Random Access Memory (RAM) or external cache memory. By way of illustration and not limitation, RAM is available in a variety of forms such as Static RAM (SRAM), Dynamic RAM (DRAM), Synchronous DRAM (SDRAM), Double Data Rate SDRAM (DDRSDRAM), Enhanced SDRAM (ESDRAM), Synchronous Link DRAM (SLDRAM), Rambus Direct RAM (RDRAM), direct bus dynamic RAM (DRDRAM), and memory bus dynamic RAM (RDRAM).
The technical features of the above embodiments can be arbitrarily combined, and for the sake of brevity, all possible combinations of the technical features in the above embodiments are not described, however, as long as there is no contradiction between the combinations of the technical features, the scope of the present description should be considered as being described in the present specification.
The embodiments described above only express several embodiments of the present invention, and the description thereof is specific and detailed, but the scope of protection of the present invention is not limited to the embodiments described above, and all technical solutions belonging to the idea of the present invention belong to the scope of protection of the present invention. It should be noted that several variations and modifications without departing from the inventive concept will occur to those skilled in the art, and such variations and modifications should also be considered as within the scope of the present invention. Therefore, the protection scope of the present invention should be subject to the appended claims.
Claims (10)
1. An analytic solving method for spacecraft finite time pursuit escape game control is characterized by comprising the following steps:
s1: in an LVLH coordinate system, zero control miss distance is used as a state variable, and a fixed time differential countermeasure model for spacecraft orbit pursuit escape is established;
the S1 includes the steps of:
s101: establishing a fixed time differential countermeasure initial model based on a spacecraft relative motion C-W equation under an LVLH coordinate system;
s102: and defining zero control miss distance as a new state variable, and deducing and establishing a fixed time differential countermeasure model for spacecraft orbit pursuit on the basis of the fixed time differential countermeasure initial model.
S2: and respectively solving the optimal control law in the analysis form of the pursuing and escaping two spacecrafts and the corresponding optimal game path based on the fixed time differential countermeasure model established in the S1.
The S2 includes the following steps:
s201: based on the fixed time differential countermeasure model established in S1, solving the optimal game control law of the two pursuit-fleeing spacecraft by a variational method;
s202: and resolving the optimal game control law of the spacecrafts of the two pursuits and the escaped spacecrafts in an analyzing manner and obtaining a corresponding optimal game path.
2. The method for analyzing, controlling and solving the finite-time catching-up game of the spacecraft of claim 1, wherein in the step S101, the reference spacecraft is used as the escaping spacecraft, and the virtual circular orbit corresponding to the position of the escaping spacecraft is used as the reference spacecraft orbit, so as to establish the LVLH coordinate system.
3. The method for analyzing, controlling and resolving the limited-time spacecraft escape pursuit game according to claim 1, wherein in S101, an initial model of a fixed-time differential game for spacecraft escape pursuit game is established as follows:
the differential countermeasure state equation for establishing the spacecraft pursuit escape game is
Wherein,
u is the relative motion state of the two spacecrafts pursuing and escaping under the LVLH coordinate system P ＝[a P,x ,a P,y ,a P,z ] T And u E ＝[a E,x ,a E,y ,a E,z ] T Thrust acceleration vectors u of the tracked spacecraft and the escaped spacecraft respectively P 、u E Satisfy | u P |≤T P 、|u E |≤T E ，T P 、T E Maximum thrust plus escape spacecraft in continuous thrust control mode for tracking spacecraft and escape spacecraft respectivelyThe speed of the motor is controlled by the speed of the motor,
and
respectively is a system matrix and a control matrix of the C-W equation in a state space description form, and omega is the motion angular velocity of the reference spacecraft in the C-W equation.
Setting the given pursuit game time as T ═ T f -t 0 The payment functions of the two pursuing and fleeing parties are respectively
Wherein t is the current time, t 0 To catch up to the start of the game, t f The game end time is the pursuit of escape;
the method is characterized in that a semi-positive definite symmetric matrix represents the proportion of the distance between two spacecrafts in the escape pursuit game ending time payment function; r P ＝r P I 3 And R E ＝r E I 3 The proportion of fuel consumed by the tracking spacecraft and the escaping spacecraft in the payment function is respectively represented for positive definite symmetric matrixes.
And (3) combining the vertical type (1) and the formula (2), so that a fixed time differential game initial model of the spacecraft pursuit escape game can be established.
4. The finite-time catching game control analysis solving method for spacecraft according to claim 3, wherein in the step S102, the zero-control miss distance is used as a new state variable to establish a fixed-time differential countermeasure model for spacecraft orbit catching as follows:
defining the zero control miss amount y (t) as a new state variable, i.e.
y(t)＝DΦ(t f ,t)x(t) (3)
Wherein D ═ I 3 O 3 ]In the form of a matrix of coefficients,
the state transition matrix corresponding to the system matrix in the equation, where s is sin ω τ, c is cos ω τ, and τ is t f And t is the remaining time of the escape game corresponding to the current moment.
Let coefficient matrix
G(t f ,t)＝DΦ(t f ,t)B (4)
The new state variable y (t) is derived according to the formula (3) and substituted into the formulas (1) and (4), and the new differential countermeasure state equation is obtained
Correspondingly, the payment functions of the two pursuing parties are respectively
Wherein,
the space vehicle tracking method is a semi-positive definite symmetric matrix, and represents the proportion of the zero control miss amount of a tracking space vehicle relative to an escaping space vehicle in a payment function at the ending moment of the pursuing and escaping game; r P ＝r P I 3 And R E ＝r E I 3 The proportion of fuel consumed by the tracking spacecraft and the escaping spacecraft in the payment function is respectively represented for positive definite symmetric matrixes.
And (5) and the formula (6) are combined, so that a spacecraft pursuit game fixed time differential countermeasure model taking the zero control miss amount as a state variable can be established.
5. The spacecraft limited-time catch-up game control analysis solving method according to claim 4, wherein in S101, Q is used when the purpose of tracking spacecraft is rendezvous but not interception f ＝q f I 6 (ii) a In S102, D ═ I 6 ，
6. The method for analyzing, controlling and solving finite-time catching game of spacecraft according to claim 4, wherein in the step S201, the optimal game control law for catching both spacecraft by using a variational method is as follows:
wherein, G (t) f ,t)、K(t f T) are coefficient matrices;
further, there are
Wherein M is P (t f ,t)、M E (t f And t) are coefficient matrices.
And solving the formula (4), the formula (9), the formula (8) and the formula (7) in sequence to obtain the optimal game control law of the two-party pursuit spacecraft.
7. The method for solving the finite-time catching-up game control analysis of the spacecraft of claim 6, wherein in the step S202, the method for solving the optimal game control law in the form of catching-up both-side spacecraft analysis comprises the following steps:
wherein,
by substituting the formula (10), the formula (11) and the formula (3) for the formula (7), the optimal game control laws in the analysis form of the spacecraft of the two parties can be obtained
Wherein, K -1 (t f And t) can be obtained by adopting an LU decomposition inverse matrix method.
8. The method for solving the finite-time catching game control of spacecraft of claim 7, wherein in step S202, when the purpose of tracking spacecraft is rendezvous and not interception,
wherein,
9. an analytic solving system for spacecraft limited time pursuit and escape game control is characterized by comprising a processor and a memory: the memory stores an analytic solver of spacecraft limited-time pursuit game control, and the processor executes the steps of the method according to any one of claims 1-8 when running the analytic solver of spacecraft limited-time pursuit game control.
10. A computer-readable storage medium, on which a computer program is stored, which, when being executed by a processor, carries out the steps of the method according to any one of claims 1 to 8.
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