CN113221365B - Processing method for burnup constraint in spacecraft flight game - Google Patents
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Abstract
The invention relates to a method for processing fuel consumption constraints in a spacecraft flight game, in particular to a method for processing the fuel consumption constraints of two spacecrafts when the two spacecrafts play the game, and belongs to the field of aerospace. The implementation method of the invention comprises the following steps: the method comprises the steps of establishing a spacecraft differential game model under the condition of considering spacecraft fuel consumption constraint, deducing corresponding optimal conditions based on an optimal control theory, solving by using a target shooting method, approaching the condition of considering the fuel consumption constraint from the condition of not considering the spacecraft fuel consumption constraint, finally obtaining a solution of the spacecraft differential game problem considering the fuel consumption constraint, and adopting a corresponding control processing strategy according to a spacecraft game problem solving result so as to solve related technical problems in the field of spacecraft game. The method has the advantages of good convergence and strong applicability.
Description
Technical Field
The invention relates to a method for processing fuel consumption constraints in a spacecraft flight game, in particular to a method for processing the fuel consumption constraints of two spacecrafts when the two spacecrafts play the game, and belongs to the field of aerospace.
Background
In the future space game countermeasure mission, the problem of spacecraft escape plays an important role. The spacecraft game problem is characterized in that the spacecrafts participating in the game have maneuvering capabilities, one is a chaser, and the other is an evacuee. The chaser finishes chasing the evacuee in the shortest time as much as possible, the evacuee lengthens the game time as much as possible, and the chaser and the evacuee do not cooperate. The solution to the game problem, namely how the chaser can complete the capture of the evacuee in the shortest time, is an important component of the overall planning of the space confrontation task. The method aims at solving the problem of poor convergence caused by difficulty in processing the fuel consumption constraint in the spacecraft game problem, and develops research on the problem of processing the fuel consumption constraint in the spacecraft game problem. The invention provides a processing method for burn-up constraint in a spacecraft flight game by combining a given spacecraft pursuit, spacecraft escape initial state (position, speed and mass), burn-up constraint, thrust amplitude constraint and the like, and can effectively solve the problem of poor convergence caused by the burn-up constraint in the spacecraft game.
In the developed research of spacecraft differential Game problem, in the prior art [1] (see Pontani M, conway B A. Numerical Solution of the Three-Dimensional Orbital burst-evolution Game [ J ]. Journal of Guidance Control & Dynamics,2009,32 (2): 474-487), by using a semi-direct point-matching nonlinear programming method and taking the acceleration as a Control variable, the optimal trajectory of pursuing spacecraft and avoiding spacecraft is obtained. However, the method cannot ensure the optimality of the trajectories of the two spacecrafts, and the quality change of the spacecrafts is not considered, so that the problem of the differential game of the spacecrafts considering the fuel consumption constraint cannot be solved.
In the prior art [2] (see Shen H-X, casalino L.Revisit of the Three-Dimensional Orbital Pursuit-evolution Game [ J ]. Journal of guiding, control, and Dynamics,2018,41 (8): 1823-1831), an indirect method is adopted for optimization, and the problem of Three-Dimensional orbit Pursuit escape considering mass consumption is researched, but after the spacecraft fuel consumption constraint is added, the problem convergence domain is reduced, the convergence is deteriorated, and therefore the problem of the spacecraft differential Game considering the fuel consumption constraint cannot be solved.
Disclosure of Invention
The invention discloses a method for processing fuel consumption constraint in spacecraft flight game, which aims to solve the technical problems that: under the condition of spacecraft fuel consumption constraint, solving the spacecraft game problem based on an optimal control theory, and adopting a corresponding control processing strategy according to the spacecraft game problem solving result so as to solve the related technical problem in the field of spacecraft game. The method has the advantages of good convergence and strong applicability.
The purpose of the invention is realized by the following technical scheme.
The invention discloses a processing method for fuel consumption constraint in spacecraft flight game, which is characterized in that a spacecraft differential game model is established under the condition of considering the spacecraft fuel consumption constraint, corresponding optimal conditions are deduced on the basis of an optimal control theory, a target shooting method is utilized for solving, meanwhile, the fuel consumption constraint condition is approached to the fuel consumption constraint condition under the condition of not considering the spacecraft fuel consumption constraint condition, and finally, the solution of the spacecraft differential game problem under the condition of considering the fuel consumption constraint is obtained.
The invention discloses a processing method for fuel consumption constraint in a spacecraft flight game, which is characterized in that under the condition of considering the fuel consumption constraint of a spacecraft, a differential game model of the spacecraft is established, corresponding optimal conditions are deduced based on an optimal control theory, a target shooting method is used for solving, meanwhile, the condition of not considering the fuel consumption constraint of the spacecraft is approached to the condition of considering the fuel consumption constraint, and finally, the solution of the differential game problem of the spacecraft considering the fuel consumption constraint is obtained.
The invention discloses a method for processing fuel consumption constraint in a spacecraft flight game, which comprises the following steps:
the method comprises the following steps: establishing a spacecraft differential game model in the earth mass center inertial system, and giving initial conditions x of chasing spacecraft p0 Initial condition x of evacuee e0 Minimum mass m of chaser pf Minimum mass m of evacuee ef 。
After the pursuit spacecraft has been launched out of the atmosphere, the pursuit spacecraft starts the game with the evasive spacecraft. Orbital perturbations are ignored because the entire process is short in duration. The dynamic equation of the spacecraft is as follows
Wherein r and v are position and velocity loss of the spacecraft respectively, m is spacecraft mass, and T and I are sp Thrust and specific impulse of the engine, mu is the gravitational constant of the central celestial body, g 0 Is sea level gravitational acceleration. The corner mark i = p, e represents a pursuit spacecraft or an escape spacecraft.
During the spacecraft gaming process, the initial states of the two spacecrafts are known
x p (t 0 )＝x p0 ,x e (t 0 )＝x e0 (2)
Wherein t is 0 Is an initial time, x p To chase the spacecraft state, x e To escape the spacecraft regime.
The game ending condition is that the positions of the two spacecrafts are superposed
r p (t f )＝r e (t f ) (3)
Wherein t is f Is the end time.
The burnup of each spacecraft at the end of the game cannot be greater than the maximum burnup
m p (t f )≥m pf ,m e (t f )≥m ef (4)
Wherein m is pf To chase the minimum mass, m, of the spacecraft ef To avoid minimum mass of the spacecraft;
in addition, the spacecraft has an upper limit on engine thrust throughout the game play, i.e.
||T i ||≤c i (5)
Wherein c is i Is the upper thrust limit of the engine.
In the game countermeasure problem of the spacecrafts, the pursuit spacecrafts and the escape spacecrafts have maneuverability, the pursuit spacecrafts use corresponding strategies to complete the game as soon as possible, and the escape spacecrafts are not captured by the pursuit spacecrafts as far as possible. Here, a performance index is defined as shown in formula (6), and the tracking of the spacecraft is minimized while the evasion of the spacecraft is maximized.
Step two: and the fuel consumption constraint of the pursuing spacecraft and the evading spacecraft is ignored, the problem of the spacecraft game without the fuel consumption constraint is solved, the convergence of solving the problem of the spacecraft game without the fuel consumption constraint is enhanced, and the solution of the spacecraft game problem without the fuel consumption constraint is obtained.
Due to the accuracy and optimality of the indirect method, the spacecraft game countermeasure problem is solved by the indirect method. Applying an optimal control theory, establishing a Hamiltonian H as follows:
wherein λ ri ,λ vi ,λ mi Are the covariates of the corresponding spacecraft.
By applying the Pontryagin maximum principle, the chaser should minimize the Hamilton function and the evacuee should maximize, and the corresponding optimal control directions of the two spacecrafts are determined
By substituting the above equation into the Hamiltonian, then
Wherein κ p ＝-1,κ e ＝1
Setting a switching function
And under the condition that no singular arc section exists, the thrust of the two spacecrafts is bang-bang controlled:
in addition, the covariate equation is
The terminal cross-sectional conditions obtained from the terminal conditions (3) are
λ rp (t f )+λ re (t f )＝0 (13)
Since the tip speed and burn-up are unconstrained, the method
λ vp (t f )＝λ ve (t f )＝0 (14)
λ mp (t f )＝λ me (t f )＝0 (15)
The combination of the vertical type (10), (12) and (15) can be known as lambda mp ＞0,λ me If < 0 is always true in the whole process, the engines of the two spacecrafts are always started in game countermeasure, and the variable lambda can be removed mi And λ mi Constraints are concerned and engine on-off judgment is not required.
Since the end time is free, there is a static condition corresponding to the end time
Because the initial value of the covariate is unbounded, the covariate is normalized, and the performance index and the constant value-covariate factor lambda are normalized 0 Multiplication, without changing the solution of the problem, then
The static condition becomes
Other constraints are not changed, but initial constraints of the covariates are added
‖λ|＝1 (19)
Wherein
In this way, the optimal control problem becomes a two-point boundary value problem, and the target shooting method is used for solving the problem. In the solving process, the convergence is reduced along with the increase of the total time of the game, and two methods are provided for enabling the game problem with longer time to be converged quickly.
Method one, transition from one-sided problem to two-sided problem. The unilateral problem is that only the maneuver of the pursuing spacecraft is considered, but the maneuver capability of the evading spacecraft is not considered, and only the maneuver capability of the evading spacecraft is considered
As an unknown quantity, the problem is greatly simplified. The unilateral problem has better convergence, and the solution of the unilateral problem is obtained
Then, it is used as the chaser co-modal variable and constant co-modal factor lambda in the bilateral problem 0 The convergence of the bilateral problem is greatly improved by the guessed value of (2).
And the second method adopts a step-by-step adjustment method. Selection of thrust T pm ＞c p And T em ＜c e Define the pursuit and evasion thrusts as
Wherein epsilon [0,1] is a parameter, and when epsilon =0, the problem is the original problem.
The game confrontation problem of short time ending is easier to converge, and the game confrontation time is shorter than the original problem when the epsilon =1, so the solution is easy. And after a corresponding bilateral problem solution lambda is obtained under the condition that epsilon =1, epsilon is gradually reduced, and lambda is used as an initial value of the new game problem. And iterating until epsilon =0 to obtain a solution of the original game problem.
Step three: judging whether the fuel consumption constraint of the spacecraft is met, if so, obtaining a result which is the solution of the final spacecraft game problem; if the fuel consumption constraint does not meet the requirement, the fuel consumption constraint of the spacecraft is added, and the weak fuel consumption constraint is gradually strengthened to obtain the final solution of the spacecraft game problem considering the fuel consumption constraint.
And judging after obtaining the optimal solution without considering the condition of the fuel consumption constraint, and if the obtained result fuel consumption does not meet the preset constraint, adding the corresponding terminal fuel consumption constraint into the model.
m p (t f )＝m pf ,m e (t f )＝m ef (21)
Different from the situation without the burnup constraint, the situation with the burnup constraint is more complicated, and the shutdown situation exists in the game process. Therefore, under the condition of fuel consumption constraint, the co-modal quality lambda of the spacecraft with the fuel consumption constraint mi And its associated constraints are not negligible.
For the optimality condition, the co-state equation (12), the switching function (10), the thrust control (11), the cross-section conditions (13) - (14) and the normalization condition (19) are unchanged, the mass-related terminal state is replaced by (21) from (15), and the terminal mass co-state constraint is absent. For the end static condition, since the end co-modal quality is not zero, it becomes:
under the condition that the intersection can still be completed when no fuel consumption constraint exists, in order to increase the problem convergence, a gradual adjustment method is still adopted when the problem of spacecraft game countermeasure with fuel consumption constraint is solved. First, if the evacuee consumed fuel is greater than the given fuel, the evacuee is treated first, and at this time, the chaser mass is not constrained, i.e., the chaser end mass is free. The terminal mass constraint is shown in equations (23) and (24):
m e (t f )＝m es (23)
m es ＝m ef +ε(m em -m ef ) (24)
wherein m is em For evacuee end quality without burnup constraint, ε ∈ [0,1]The constraint is a parameter, and when epsilon =0, the constraint is corresponding to the preset terminal quality constraint; when epsilon =1, the constraint is a condition corresponding to the burnup-free constraint. Under the gradual adjustment method, the target practice equation can be gradually transited to the target condition, and finally the solution of the required equation is obtained.
In the same way, after the solution corresponding to the fuel consumption constraint of the evacuee is obtained, a gradual adjustment method is applied to the chaser. The terminal mass constraint is shown in equations (25) and (26):
m p (t f )＝m ps (25)
m ps ＝m pf +ε(m pm -m pf ) (26)
wherein m is pm To consider only the chaser end quality when evacuee end burnup constraints, ε ∈ [0,1]]The constraint is a parameter, and when the epsilon =0, the constraint is corresponding to the preset terminal quality constraint; when ε =1, the constraint is a condition corresponding to when only the evacuee end burnup constraint is considered. Under the gradual adjustment method, the shooting equation is gradually transited to the target condition, and finally the solution of the spacecraft game countermeasure problem under the fuel consumption constraint is obtained.
Step four: and (4) according to the solution of the spacecraft game problem which is obtained in the step three and takes the fuel consumption constraint into consideration, adopting a corresponding control processing strategy, and further solving the related technical problem in the field of spacecraft game.
Has the advantages that:
1. the invention discloses a processing method for burn-up constraint in spacecraft flight game, which adopts a method of gradually adjusting transition and thrust from a unilateral problem to a bilateral problem, so that the time length of the game problem is gradually prolonged from short, the convergence of the problem is enhanced, and the method can be suitable for solving the game problem for a long time.
2. The invention discloses a processing method for burnup constraint in spacecraft flight game, which considers the condition of no burnup constraint, adds the fuel up constraint of an escape spacecraft after obtaining the solution of the condition of no burnup constraint, and gradually strengthens the fuel up constraint of the escape spacecraft; similarly, the fuel consumption constraint of the chasing spacecraft is added later, the fuel consumption constraint of the chasing spacecraft is gradually strengthened, the problem of no fuel consumption constraint is gradually transited to the problem of the game of considering the fuel consumption constraint, the convergence of the problem is strengthened, the game problem of the spacecraft considering the fuel consumption constraint can be successfully and quickly solved, and the efficiency of solving the related technical problems in the field of spacecraft games is improved.
Drawings
FIG. 1 is a flowchart of a processing method for burnup constraints in a spacecraft flight game, which is disclosed by the invention.
FIG. 2 is a game result diagram of a no-burnup constraint situation of the processing method for burnup constraint in spacecraft flight game disclosed by the invention.
FIG. 3 is a thrust directional diagram of two spacecrafts without fuel consumption constraint condition for a fuel consumption constraint processing method in a spacecraft flight game disclosed by the invention. Wherein a is a pursuit spacecraft thrust directional diagram under the condition of no fuel consumption constraint, and b is an evasion spacecraft thrust directional diagram under the condition of no fuel consumption constraint.
Fig. 4 is a game result diagram only considering the situation of evading the fuel consumption constraint of the spacecraft, which is disclosed by the invention, for the processing method of the fuel consumption constraint in the spacecraft flight game.
FIG. 5 is a thrust direction diagram of two spacecrafts, which only considers the situation of avoiding the burnup constraint of the spacecrafts, in the processing method for the burnup constraint in the flight game of the spacecrafts disclosed by the invention. Wherein, a is a pursuit spacecraft thrust directional diagram only considering the condition of avoiding the spacecraft fuel consumption constraint, and b is an avoiding spacecraft thrust directional diagram only considering the condition of avoiding the spacecraft fuel consumption constraint.
Fig. 6 is a game result diagram of the processing method for the burnup constraint in the spacecraft flight game, which is disclosed by the invention, and which considers the situation of evading the spacecraft and chasing the burnup constraint of the spacecraft.
FIG. 7 is a thrust direction diagram of two spacecrafts considering the situations of avoiding the spacecraft and tracing the fuel consumption constraint of the spacecraft in the processing method for the fuel consumption constraint in the spacecraft flight game disclosed by the invention. Wherein a is a pursuit spacecraft thrust directional diagram considering the fuel consumption constraint condition of the two spacecrafts, and b is an evasion spacecraft thrust directional diagram considering the fuel consumption constraint condition of the two spacecrafts.
Detailed Description
For a better understanding of the objects and advantages of the present invention, reference should be made to the following detailed description taken in conjunction with the accompanying drawings and examples.
Example 1:
in order to verify the feasibility of the method, the game problem of the spacecraft and the synchronous orbit satellite near the earth is selected and solved by the method. The pursuing spacecraft engine is set to provide 16N of thrust, the evading spacecraft thrust is set to be 8N, the specific impulse of the two spacecraft engines is 350s, and the mass of the two spacecraft engines is 200kg. Wherein the available burnup of the escape spacecraft is 10kg, and the available burnup of the chase spacecraft is 50kg. The initial state parameters of the two spacecrafts are shown in the table 1.
TABLE 1 two spacecraft orbital parameters
As shown in fig. 1, the processing method for the burnup constraint in the flight game of the spacecraft disclosed in this embodiment specifically includes the following steps:
the method comprises the following steps: and establishing a spacecraft differential game model in the earth mass center inertial system.
The pursuing spacecraft has launched beyond the atmosphere and begins the game with the evasive spacecraft. The spacecraft dynamics equation is as follows:
wherein r and v are position and velocity loss of the spacecraft respectively, m is spacecraft mass, and T and I are sp Thrust, specific impulse, g of the engine 0 Is the sea level gravitational acceleration. The corner mark i = p, e represents a pursuit spacecraft or an evasive spacecraft.
During the spacecraft gaming process, the initial states of the two spacecrafts are known
x p (t 0 )＝x p0 ,x e (t 0 )＝x e0
The game ending condition is that the positions of the two spacecrafts are superposed
r p (t f )＝r e (t f )
The mass of each spacecraft at the end of the game must not be below its minimum mass
m p (t f )≥m pf ,m e (t f )≥m ef
In addition, the spacecraft has an upper limit on engine thrust throughout the game play, i.e.
||T i ||≤c i
In the problem of game confrontation of the spacecrafts, the pursuing spacecrafts and the evasion spacecrafts have maneuvering capabilities, the pursuing spacecrafts use corresponding strategies to complete the game as soon as possible, and the evasion spacecrafts are not captured by the pursuing spacecrafts as far as possible. Here, the performance index is defined as follows, and the following formula is used to minimize the pursuit of the spacecraft and maximize the evasion of the spacecraft.
Step two: and the fuel consumption constraint of the pursuing spacecraft and the evading spacecraft is ignored, the problem of the spacecraft game without the fuel consumption constraint is solved, the convergence of solving the problem of the spacecraft game without the fuel consumption constraint is enhanced, and the solution of the spacecraft game problem without the fuel consumption constraint is obtained.
Firstly, the tail end fuel consumption constraints of the two spacecrafts are not considered, the method is used for solving, finally, the two spacecrafts are in a track diagram as shown in figure 2, and it can be seen that the chaser is located near the earth at the initial moment, the evacuee is in a synchronous orbit, and the positions of the two spacecrafts are completely overlapped when the two spacecrafts complete game. The thrust direction changes in the two-spacecraft game process are shown in fig. 3, and the thrust direction changes continuously. The two-space vehicle state at the time of the intersection is shown in table 2.
TABLE 2 two spacecraft orbital parameters
The final betting time is 5.494h. The fuel consumption of the pursuing spacecraft is 92.198kg, and the fuel consumption of the evading spacecraft is 46.099kg. It can be seen that the burnup exceeds a predetermined value for either pursuit or evasion of the spacecraft, and therefore, the end burnup needs to be constrained.
Step three: judging whether the fuel consumption constraint of the spacecraft is met, if so, obtaining a result which is the solution of the final spacecraft game problem; if the fuel consumption constraint does not meet the requirement, the fuel consumption constraint of the spacecraft is added, and the weak fuel consumption constraint is gradually strengthened to obtain the final solution of the spacecraft game problem considering the fuel consumption constraint.
And judging that the fuel consumption of the pursuing spacecraft and the evading spacecraft exceeds a preset value. And (4) restraining the end quality of the escape spacecraft by using a step-by-step adjustment method. The mass constraint of the escape spacecraft tail end is as follows:
m e (t f )＝m es
m es ＝m ef +ε(m em -m ef )
wherein m is em ＝153.901kg,m ef ＝190kg。
And (4) transiting the parameter epsilon from 1 to 0, taking the result of each problem as the initial value of the next problem, and solving the game problem only considering escaping from the fuel consumption constraint of the spacecraft. The result of the constraint on only evasive spacecraft burnup is shown in figure 4, where the two spacecraft eventually meet. The thrust direction changes in the two-spacecraft game process are shown in fig. 5, and the thrust direction changes continuously. The two-space vehicle state at the time of the intersection is shown in table 3.
TABLE 3 two spacecraft orbital parameters
The final game time is 4.783h, which is shorter than the case without the burn-up constraint, but the burn-up constraint does not have a large effect on the game time due to the large initial orbit difference. The chase spacecraft burnup was 80.269kg, which still exceeded the predetermined value, thus requiring a constraint on the tip burnup. The escape spacecraft is burned by 10kg, is continuously started at the initial stage of the game, keeps in a sliding state after being shut down for 1.192h, and finally meets the pursuit spacecraft.
And because the combustion consumption of the chasing spacecraft is too large, the tail end mass of the chasing spacecraft is restrained by continuously applying a gradual adjustment method. The terminal mass constraint of the chasing spacecraft is as follows:
m p (t f )＝m ps
m ps ＝m pf +ε(m pm -m pf )
wherein m is pm ＝119.731kg,m pf ＝150kg
And (4) transiting the parameter epsilon from 1 to 0, taking the result of each problem as the initial value of the next problem, and solving the game problem only considering escaping from the fuel consumption constraint of the spacecraft. The resulting diagram is shown in fig. 6, where the two spacecraft meet eventually. The thrust direction changes in the two-spacecraft game process are shown in fig. 7, and the thrust direction changes continuously. The two-space vehicle state at the time of the intersection is shown in table 4.
TABLE 4 two spacecraft orbital parameters
The method can successfully solve the problem of spacecraft game considering the fuel consumption constraint, and the final game time is 5.364h, which is close to the situation without the fuel consumption constraint. The fuel consumption of the pursuing spacecraft is 50kg, and the fuel consumption of the evading spacecraft is 10kg. Therefore, the method provided by the invention successfully solves the spacecraft flight game problem in consideration of the fuel consumption constraint.
Step four: and (4) adopting a corresponding control processing strategy according to the solution of the spacecraft game problem which is obtained in the step three and takes the fuel consumption constraint into consideration, thereby solving the related technical problem in the field of spacecraft game.
The above detailed description is intended to illustrate the objects, aspects and advantages of the present invention, and it should be understood that the above detailed description is only exemplary of the present invention and is not intended to limit the scope of the present invention, and any modifications, equivalents, improvements and the like made within the spirit and principle of the present invention should be included in the scope of the present invention.
Claims (3)
1. A processing method for burnup constraint in spacecraft flight game is characterized by comprising the following steps: comprises the following steps of (a) preparing a solution,
the method comprises the following steps: establishing a spacecraft differential game model in the earth mass center inertial system, and giving initial conditions x of chasing spacecraft p0 Initial condition x of evacuee e0 The smallest mass m of the chaser pf Minimum mass m of evacuee ef ；
The first implementation method of the method is that,
after the pursuit spacecraft is launched out of the atmosphere, starting the game with the escape spacecraft; neglecting orbital perturbations due to the short duration of the whole process; the dynamic equation of the spacecraft is as follows
Wherein r and v are position and velocity loss of the spacecraft respectively, m is spacecraft mass, and T and I are sp Thrust, specific impulse, g of the engine 0 Is sea level gravitational acceleration; the corner mark i = p, e represents a pursuit spacecraft or an evasive spacecraft;
during the spacecraft gaming process, the initial states of the two spacecrafts are known
x p (t 0 )＝x p0 ,x e (t 0 )＝x e0 (2)
Wherein t is 0 Is an initial time, x p To chase a spacecraft state, x e Escape spacecraft state;
the game ending condition is that the positions of the two spacecrafts are superposed
r p (t f )＝r e (t f ) (3)
Wherein t is f Is the end time;
the burnup of each spacecraft at the end of the game cannot be greater than the maximum burnup
m p (t f )≥m pf ,m e (t f )≥m ef (4)
In addition, the spacecraft has an upper limit on engine thrust throughout the game play confrontation, i.e.
||T i ||≤c i (5)
Defining a performance index as shown in a formula (6), and tracking the spacecraft to the minimum and avoiding the spacecraft to the maximum;
step two: neglecting the fuel consumption constraint of the pursuing spacecraft and the evading spacecraft, solving the problem of the spacecraft game without the fuel consumption constraint, enhancing the convergence of solving the problem of the spacecraft game without the fuel consumption constraint, and further obtaining the solution of the spacecraft game neglecting the fuel consumption constraint;
the second step is realized by the method that,
due to the accuracy and optimality of the indirect method, the problem of the spacecraft game countermeasure is solved by adopting the indirect method; applying an optimal control theory, establishing a Hamiltonian H as follows:
wherein λ is ri ,λ vi ,λ mi The covariates of the corresponding spacecraft;
by applying the Pontryagin maximum principle, the chase should minimize the Hamiltonian and the evasion should maximize, and the corresponding optimal control directions of the two spacecrafts are determined
By substituting the above equation into the Hamiltonian, then
Wherein κ p ＝-1,κ e ＝1
Setting a switching function
And under the condition that no singular arc section exists, the thrust of the two spacecrafts is bang-bang controlled:
in addition, the covariate equation is
The terminal cross-sectional conditions obtained from the terminal conditions (3) are
λ rp (t f )+λ re (t f )＝0 (13)
Since the tip speed and burn-up are unconstrained, therefore
λ vp (t f )＝λ ve (t f )＝0 (14)
λ mp (t f )＝λ me (t f )＝0 (15)
The simultaneous type (10), (12) and (15) can be known as lambda mp ＞0,λ me If < 0 is always true in the whole process, the engines of the two spacecrafts are always started in the game countermeasure, and the variable lambda can be removed mi And λ mi Constraint is concerned, and engine on-off judgment is not required;
since the end time is free, there is a static condition corresponding to the end time
Because the initial value of the covariate is unbounded, the covariate is normalized, the performance index is multiplied by the constant value-covariate factor, and the solution of the problem is not changed, then
The static condition becomes
Other constraints are not changed, but initial constraints of covariates are added
||λ||＝1 (19)
Wherein
In the method, the optimal control problem is changed into a two-point boundary value problem, and a target shooting method is used for solving the problem; in the solving process, the convergence is reduced along with the lengthening of the total time of the game, and two methods are provided for rapidly converging the long-time game problem;
the method I is characterized in that the single-side problem is transited to the double-side problem; the unilateral problem is that only the maneuver of the pursuing spacecraft is considered, but the maneuver capability of the evading spacecraft is not considered, and only the maneuver capability of the evading spacecraft is considered
As an unknown quantity, the problem is greatly simplified; the unilateral problem has good convergence, and the solution of the unilateral problem is obtained
Then, it is used as the chaser co-modal variable and constant co-modal factor lambda in the bilateral problem 0 The guessed value of (2) greatly improves the convergence of the bilateral problem;
the second method adopts the homotopy method; selection of thrust T pm ＞c p And T em ＜c e Define the pursuit and evasion thrusts as
Wherein epsilon belongs to [0,1] is a parameter, and when epsilon =0, the problem is the original problem;
the game confrontation problem ending in a short time is easy to converge, and the game confrontation time is shortened in comparison with the original problem when the epsilon =1, so that the solution is easy; obtaining a corresponding bilateral problem solution lambda under the condition that epsilon =1 * Then, let ε decrease gradually, use λ * The game is an initial value of a new game problem; iterating until epsilon =0 to obtain a solution of the original game problem;
step three: judging whether the fuel consumption constraint of the spacecraft is met, if so, obtaining a result which is the solution of the final spacecraft game problem; if not, adding the spacecraft fuel consumption constraint, gradually strengthening and then constraining to obtain the final solution of the spacecraft game problem considering the fuel consumption constraint.
2. The processing method for the burnup constraint in the aircraft flight game as recited in claim 1, wherein: and step four, adopting a corresponding control processing strategy according to the solution of the spacecraft game problem which is obtained in the step three and takes the fuel consumption constraint into consideration, and further solving the related technical problem in the spacecraft game field.
3. The processing method for the fuel consumption constraint in the flight game of the spacecraft as recited in claim 1, wherein: the third step is to realize the method as follows,
judging after obtaining the optimal solution of the condition without considering the fuel consumption constraint, if the obtained result fuel consumption does not meet the preset constraint, adding the corresponding terminal fuel consumption constraint into the model;
m p (t f )＝m pf ,m e (t f )＝m ef (21)
unlike the case where there is no burn-up constraint,the condition of the fuel consumption constraint is more complex, and the shutdown condition exists in the game process; therefore, under the condition of fuel consumption constraint, the co-modal quality lambda of the spacecraft with the fuel consumption constraint mi And its associated constraints are not negligible;
for the optimality condition, the co-state equation (12), the switching function (10), the thrust control (11), the cross-section conditions (13) - (14) and the normalization condition (19) are unchanged, the terminal state related to the quality is replaced by (15) to (21), and the terminal quality co-state constraint is not generated; for the end static condition, since the end co-modal quality is not zero, it becomes:
under the condition that the intersection can still be completed without the burn-up constraint, in order to increase the problem convergence, the homotopy method is still adopted when the problem of spacecraft game countermeasure with the burn-up constraint is solved; firstly, if the fuel consumed by the evacuee is larger than the given fuel, the evacuee is processed firstly, and the mass of the chaser is not restricted at the moment, namely the tail end mass of the chaser is free; the terminal mass constraint is shown in equations (23) and (24):
m e (t f )＝m es (23)
m es ＝m ef +ε(m em -m ef ) (24)
wherein m is em For evacuee end quality without burnup constraint, ε ∈ [0,1]The constraint is a parameter, and when the epsilon =0, the constraint is corresponding to the preset terminal quality constraint; when the epsilon =1, the constraint is the corresponding condition of the condition of no burnup constraint; under the homotopy method, the target practice equation gradually transits to the target condition, and finally the solution of the required equation is obtained;
in the same way, after the solution corresponding to the fuel consumption constraint of the evacuee is obtained, the homotopy method is applied to the chaser; the terminal mass constraint is shown in equations (25) and (26):
m p (t f )＝m ps (25)
m ps ＝m pf +ε(m pm -m pf ) (26)
wherein m is pm To consider only the chaser end quality when evacuee end burnup constraints, ε ∈ [0,1]]The constraint is a parameter, and when epsilon =0, the constraint is corresponding to the preset terminal quality constraint; when epsilon =1, the constraint is a corresponding condition when only the fuel consumption constraint of the end of the evacuee is considered; under the homotopy method, the target shooting equation is gradually transited to the target condition, and finally the solution of the spacecraft game countermeasure problem under the fuel consumption constraint is obtained.
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