CN114996841A - Optimal coplanarity transfer search method based on tangent initial value - Google Patents
Optimal coplanarity transfer search method based on tangent initial value Download PDFInfo
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Abstract
The invention discloses an optimal coplanarity transfer search method based on a tangent initial value, and relates to an optimal coplanarity transfer search method based on a tangent initial value. The invention aims to solve the problems that the existing method is limited by optimization variables, has long calculation time and low solving efficiency and is difficult to meet the large-scale track transfer requirement. The process is as follows: firstly, finishing initial condition setting of pulse optimization; secondly, limiting the thrust direction in the speed tangential direction, setting an optimization index as fuel optimum, and performing global search on 2n-3 variables needing to be optimized for the tangential pulse fuel optimum solution by using a genetic algorithm to obtain the global fuel optimum solution under the tangential pulse condition; thirdly, completing initialization of the fmincon algorithm; and fourthly, limiting the pulse direction in the track plane, setting an optimization index as the optimal fuel, and carrying out local optimization on the 3n-4 optimization variables by utilizing an fmincon algorithm to obtain the optimal local fuel solution. The invention is used in the field of spacecraft orbit transfer.
Description
Technical Field
The invention relates to an optimal coplanarity transfer search method based on a tangent initial value.
Background
The orbital dynamics of a spacecraft is the fundamental science of aerospace engineering. The design of the orbit of the spacecraft is the key for ensuring the smooth execution of the working task of the spacecraft. Since the time that mankind entered space, the theory of orbit design developed rapidly, and relevant methods and techniques were verified by a large number of practical projects. The orbit transfer is a basic problem of orbit design, and means that a spacecraft is actively transferred from an original orbit to a target orbit through a maneuvering process, and design indexes such as fuel consumption, transfer time, control accuracy and the like are generally considered.
At present, the number of on-orbit satellites is increasing, and the requirement of orbit design for transferring tasks is increased. This puts higher demands on the solution efficiency of the transfer task. When solving the fuel optimal transfer orbit, a global optimal pulse transfer scheme is usually obtained through a global search algorithm such as a genetic algorithm according to the number of pulses required by a task. However, the traditional method is limited by the optimization variables (3n-4 optimization variables), has long calculation time and low solving efficiency, and is difficult to meet the large-scale track transfer requirement.
Disclosure of Invention
The invention aims to provide an optimal coplanar transfer search method based on tangent initial values, which aims to solve the problems that the existing method is limited by optimization variables (3n-4 optimization variables), the calculation time is long, the solving efficiency is low, and the large-scale track transfer requirements are difficult to deal with.
The optimal coplanarity transfer search method based on the tangent initial value comprises the following specific processes:
step one, finishing initial condition setting of pulse optimization:
setting iteration times and population quantity parameters of a genetic algorithm according to initial orbit parameters, target orbit parameters and pulse number n required by a task of the spacecraft;
step two, solving a tangent pulse global optimal solution by a genetic algorithm:
based on the first step, limiting the thrust direction in the speed tangent direction, setting an optimization index as the optimal fuel, and performing global search on 2n-3 variables needing to be optimized for the optimal solution of the tangent pulse fuel by utilizing a genetic algorithm to obtain the optimal solution of the global fuel under the tangent pulse condition;
step three, finishing initialization of the fmincon algorithm:
setting the radial value of the global optimal solution under the tangent pulse condition obtained by the genetic algorithm to 0, and substituting the radial value into the fmincon algorithm as an initial guess;
step four, solving the local optimal solution of the free direction pulse by using an fmincon algorithm:
and (3) limiting the pulse direction in the track plane according to the initial conditions given in the step one, setting an optimization index as the optimal fuel, and carrying out local optimization on 3n-4 optimization variables by using an fmincon algorithm to obtain the optimal local fuel solution under the pulse condition in the free direction.
The invention has the beneficial effects that:
the invention provides a coplanar orbit transfer fuel optimal fast search algorithm based on tangent pulse initial guess. The tangential pulse is a pulse with the collinear thrust direction and speed direction, has the characteristics of easiness in realization, high safety, convenience in solution and approximate optimal energy in engineering, and the tangential direction is the thrust direction with the fastest energy for changing the track. The algorithm effectively improves the calculation efficiency by searching the global optimal solution of the tangent pulse and utilizing the characteristics of less optimization variables (2n-3 optimization variables) of the tangent pulse and approximate optimal fuel, and takes the solution as the initial guess of the local optimization algorithm to obtain the final solution by adopting the local optimization algorithm, thereby ensuring the optimal fuel and saving the calculation time.
In the invention, firstly, the pulse is directly limited to be the tangent pulse, so that the optimization variables are reduced, after a global optimal solution under the tangent condition is obtained, the solution is taken as an initial guess of a local optimization algorithm, and then the final solution is obtained by adopting the local optimization algorithm. The algorithm utilizes the characteristic of less optimization variables of the tangent pulse, effectively improves the calculation efficiency, can be converged to the same group of fuel optimal solution with the original free pulse global search method, saves the calculation time, and can be used for any number of pulses.
Drawings
FIG. 1 is a flow chart of an algorithm for coplanar orbit transferred fuel optimum search based on a tangent initial value;
FIG. 2 is a flow chart illustrating a conventional method for solving coplanar rail transfer fuel optimality;
FIG. 3 is a schematic view of a track transfer process;
FIG. 4 is a two pulse fuel comparison plot;
FIG. 5 is a three pulse fuel comparison plot.
Detailed Description
The first specific implementation way is as follows: the method for searching for the optimal coplanarity transfer based on the tangent initial value comprises the following specific processes:
step one, finishing initial condition setting of pulse optimization:
setting parameters such as iteration times, population quantity and the like of a genetic algorithm according to initial orbit parameters, target orbit parameters and pulse number n required by a task of the spacecraft;
step two, solving a tangent pulse global optimal solution by a genetic algorithm:
based on the first step, limiting the thrust direction in the speed tangent direction, setting an optimization index as the optimal fuel, and performing global search on 2n-3 variables needing to be optimized for the optimal solution of the tangent pulse fuel by utilizing a genetic algorithm to obtain the optimal solution of the global fuel under the tangent pulse condition;
step three, completing initialization of the fmincon algorithm:
according to initial conditions such as pulse number and the like, parameters such as the maximum iteration times and the convergence error limit of the fmincon algorithm are reasonably set so as to ensure that the convergence precision reaches the simulation requirement, the radial value of each pulse of the global optimal solution obtained by the genetic algorithm under the tangent pulse condition is set to be 0, and then the radial value is substituted into the fmincon algorithm to serve as an initial guess; the maximum iteration times of the fmincon algorithm can be increased, and a convergence condition is reasonably set to obtain an optimal solution meeting the convergence condition;
step four, solving the local optimal solution of the free direction pulse by using an fmincon algorithm:
and (3) limiting the pulse direction in the track plane according to the initial conditions given in the step one, setting an optimization index as the optimal fuel, and carrying out local optimization on 3n-4 optimization variables by using an fmincon algorithm to obtain the optimal local fuel solution under the pulse condition in the free direction.
The second embodiment is as follows: the difference between this embodiment and the first embodiment is that the initial condition setting for completing the pulse optimization in the first step is as follows:
setting parameters such as iteration times, population quantity and the like of a genetic algorithm according to initial orbit parameters, target orbit parameters and pulse number n required by a task of the spacecraft;
the specific process is as follows:
the orbit parameters of the spacecraft at the initial moment are given under a J2000 geocentric inertial coordinate system, and the initial orbit parameters of the spacecraft comprise: semi-major axis a 0 Eccentricity e 0 Track inclination i 0 And the right ascension omega 0 Angle distance omega from the proximal point 0 ；
The target orbit parameters include: semi-major axis a f Eccentricity e f Track inclination i f And the right ascension omega f Angle distance omega from the proximal point f (ii) a Where the orbital inclination and ascent intersection declination should be the same as for the spacecraft at the initial moment, i 0 ＝i f ，Ω 0 ＝Ω f Namely, the initial orbit and the target orbit of the spacecraft meet the coplanar condition;
when the eccentricity of an initial orbit and a target orbit is large in an initial condition, the situation that the difference between the global optimal solution of the tangent pulse and the global optimal solution of the free direction pulse is large can occur, namely the pulse is applied to the global optimal solution of the tangent pulse and is not close to the pulse application place of the global optimal solution of the direction free pulse, and a local optimal solution can be obtained by using the method provided by the invention with certain possibility; the population number and the iteration times of the genetic algorithm are reasonably set to ensure the convergence and the convergence speed;
the dynamic models adopted in the whole process of the invention are two-body models, namely perturbation factors such as earth non-spherical perturbation, sunlight pressure, atmospheric resistance and the like are not considered;
wherein mu represents the gravitational constant of the earth, r and v respectively represent the position vector and the velocity vector of the spacecraft under the J2000 earth-centered inertial coordinate system, and | r | represents the size of the corresponding position vector;
the first derivative of r is represented as,
the first derivative of v is indicated.
Other steps and parameters are the same as those in the first embodiment.
The third concrete implementation mode: the difference between the present embodiment and the first or second embodiment is that, in the second step, the genetic algorithm solves the tangent pulse global fuel optimal solution:
based on the first step, the thrust direction is limited in the tangential direction, optimization indexes are set as fuel optimum, 2n-3 variables needing to be optimized for the tangential pulse fuel optimum solution are searched globally by utilizing a genetic algorithm, and the global fuel optimum solution under the tangential pulse condition is obtained;
the specific process is as follows:
in step two, the calculation and optimization process of the fuel consumption is as follows:
step two, assuming that the pulse number is n, the number of state variables needing to be optimized for obtaining the fuel optimal solution is at least 2n-3, and selecting the state variable to be optimized as X ═ theta ═ to 1 ,θ 2 ,...,θ n-1 ,Δv 1 ,Δv 2 ,...,Δv n-2 ]；
Wherein, theta i Represents the position of the spacecraft in the ith pulse and the rise point of the initial orbit in the radial direction OXThe value range of the included angle is [0,2 pi ]]；Δv i The pulse of the ith time is not excessively large considering the velocity of the elliptic orbit of the earth satellite, and the value range is set to be [ -10,10 ]]The direction same as the original speed direction is taken as positive, and the unit is km/s;
step two, according to the initial orbit parameter [ a ] 0 ,e 0 ,i 0 ,Ω 0 ,ω 0 ]And the state variable to be optimized can be subjected to orbit recursion, and by taking the first pulse process as an example, the orbit recursion process is represented as follows:
from initial orbit parameters and theta 1 It can be obtained that the spacecraft is at theta before the first pulse is applied 1 Six number of tracks [ a ] 1 ,e 1 ,i 1 ,Ω 1 ,ω 1 ,f 1 ]Wherein the first five orbital parameters are the same as the initial orbital parameters, i.e. satisfy a 1 ＝a 0 、e 1 ＝e 0 、i 1 ＝i 0 、Ω 1 ＝Ω 0 、ω 1 ＝ω 0 True near point angle f 1 ＝θ 1 -ω 1 (ii) a By passing at theta 1 Six tracks before pulse application and pulse size Δ v 1 Solving for theta after the first applied pulse 1 Six number of tracks [ a ] 1 ′,e 1 ′,i 1 ′,Ω 1 ′,ω 1 ′,f 1 ′]；
Step two and step three, the first five items of six track numbers are known easily, namely a 1 ′＝a 2 、e 1 ′＝e 2 、i 1 ′＝i 2 、Ω 1 ′＝Ω 2 、ω 1 ′＝ω 2 (ii) a Through theta 2 Can find f 2 ＝θ 2 -ω 2 (ii) a From which the second pulse av can be applied 2 Six number of front tracks [ a 2 ,e 2 ,i 2 ,Ω 2 ,ω 2 ,f 2 ]By at θ 2 Six number of tracks and pulse Δ v before applying pulse 2 Solving for second applied pulse at θ 2 Six number of tracks [ a ] 2 ′,e 2 ′,i 2 ′,Ω 2 ′,ω 2 ′,f 2 ′]；
The orbit recursion is carried out in this way until the value obtained is theta n-2 To apply a pulse Δ v n-2 Rear orbital parameter [ a ] n-2 ′,e n-2 ′,i n-2 ′,Ω n-2 ′,ω n-2 ′,f n-2 ′]By knowing the orbit parameters of the target orbit [ a ] f ,e f ,i f ,Ω f ,ω f ]And theta in the state variable to be optimized n-1 The magnitude Deltav of the last two tangent pulses can be obtained by the existing method n-1 ，Δv n 。
Therefore, the total number of state variables to be optimized in the whole solving process is 2n-3, namely, by optimizing X ═ theta 1 ,θ 2 ,...,θ n-1 ,Δv 1 ,Δv 2 ,...,Δv n-2 ]The optimal fuel transfer orbit under the tangent pulse condition can be obtained through solving; and (3) carrying out global search on 2n-3 variables needing to be optimized for obtaining the fuel optimal transfer orbit under the tangent pulse condition through a genetic algorithm to obtain a global fuel optimal solution under the tangent pulse condition.
Other steps and parameters are the same as those in the first or second embodiment.
The fourth concrete implementation mode: the difference between this embodiment and the first to the third embodiment is that the second step is based on the initial orbit parameter [ a ] 0 ,e 0 ,i 0 ,Ω 0 ,ω 0 ]And the state variable to be optimized can be subjected to orbit recursion, and by taking the first pulse process as an example, the orbit recursion process is represented as follows:
from initial orbit parameters and theta 1 It can be obtained that the spacecraft is at theta before the first pulse is applied 1 Six number of tracks [ a ] 1 ,e 1 ,i 1 ,Ω 1 ,ω 1 ,f 1 ]Wherein the first five orbital parameters are the same as the initial orbital parameters, i.e. satisfy a 1 ＝a 0 、e 1 ＝e 0 、i 1 ＝i 0 、Ω 1 ＝Ω 0 、ω 1 ＝ω 0 True proximal angle f 1 ＝θ 1 -ω 1 (ii) a By passing at theta 1 Six tracks before applying pulse and pulse size Deltav 1 Solving for theta after the first applied pulse 1 Six number of tracks [ a ] 1 ′,e 1 ′,i 1 ′,Ω 1 ′,ω 1 ′,f 1 ′]；
The specific process is as follows:
position and velocity vectors r of the spacecraft on the first pulse application 1 ,v 1 ]Can be expressed as
r 1 ＝C 3 (Ω 1 )C 1 (i 1 )C 3 (ω 1 )[|r 1 |sinf 1 ；|r 1 |cosf 1 ；0]
In the formula, C 1 、C 3 Representing a rotation matrix of the form:
wherein α represents Ω 1 、i 1 Or ω 1 ；
At theta 1 To apply a pulse Δ v 1 Then, the position vector is not changed, the speed vector is changed, and the position vector and the speed vector [ r ] of the spacecraft are changed at the moment 1 ′,v 1 ′]Can be expressed as
r 1 ′＝r 1
So that the position velocity vector r after the pulse is applied can be used 1 ′，v 1 ' solve toAt theta 1 To apply a pulse Δ v 1 Six number of rear tracks [ a ] 1 ′,e 1 ′,i 1 ′,Ω 1 ′,ω 1 ′,f 1 ′]。
Other steps and parameters are the same as those in one of the first to third embodiments.
The fifth concrete implementation mode is as follows: the difference between this embodiment and one of the first to fourth embodiments is that in the third step, it is easy to know that a is a for the first five items of six tracks 1 ′＝a 2 、e 1 ′＝e 2 、i 1 ′＝i 2 、Ω 1 ′＝Ω 2 、ω 1 ′＝ω 2 (ii) a Through theta 2 Can find f 2 ＝θ 2 -ω 2 (ii) a From which the second pulse av can be applied 2 Six number of front track [ a ] 2 ,e 2 ,i 2 ,Ω 2 ,ω 2 ,f 2 ]By at θ 2 Six number of tracks and pulse Δ v before applying pulse 2 Solving for second applied pulse at θ 2 Six number of tracks [ a ] 2 ′,e 2 ′,i 2 ′,Ω 2 ′,ω 2 ′,f 2 ′]；
The orbit recursion is carried out in this way until the value obtained is theta n-2 To apply a pulse Δ v n-2 Rear orbital parameter [ a ] n-2 ′,e n-2 ′,i n-2 ′,Ω n-2 ′,ω n-2 ′,f n-2 ′]By knowing the orbital parameters of the target orbit [ a ] f ,e f ,i f ,Ω f ,ω f ]And theta in the state variable to be optimized n-1 The magnitude Deltav of the last two tangent pulses can be obtained by the existing method n-1 ，Δv n ；
The process is as follows:
θ＝θ n -θ n-1
in which theta represents theta n Angle of included angle and theta n-1 The difference of the included angles; λ represents an intermediate variable;
the relationship between the flight direction angle gamma and the true approach point angle can be used to obtain
In the formula of gamma j Indicating the application of the j-th pulse av j Front flight direction angle, | r j L denotes the application of the jth pulse Δ v j Position vector magnitude of time, p j Representing the radius of the track;
multiplication of both sides by p n sinθ，
Can be obtained by finishing
Order to
Then
In the formula k 1 、k 2 The intermediate variable is represented by a number of variables,
represents an intermediate variable;
the transfer orbit half-diameter and the angular momentum in the process are respectively
The magnitude Deltav of the last two pulses can thus be determined n-1 ，Δv n
The total fuel consumption can be expressed as
Other steps and parameters are the same as in one of the first to fourth embodiments.
The sixth specific implementation mode: the difference between this embodiment and one of the first to fifth embodiments is that the fmincon algorithm in step four solves the local fuel optimum solution of the free direction pulse:
limiting the pulse direction in the plane of the track according to the initial conditions given in the step one, setting optimization indexes as fuel optimization, and carrying out local optimization on 3n-4 optimization variables by using an fmincon algorithm to obtain a local fuel optimal solution under the pulse condition in the free direction;
in step four, when the pulse directions are all pulses in free directions, the calculation and optimization process of the fuel consumption is as follows:
assuming that the number of pulses is n, what is needed to obtain the optimal solution isThe number of state variables is at least 3n-4, and the state variable is selected to be X ═ theta 1 ,θ 2 ,...,θ n ,Δv i1 ,Δv i2 ,...,Δv i(n-2) ,Δv j1 ,Δv j2 ,...,Δv j(n-2) ]Wherein, theta k The included angle between the position of the spacecraft and the rise point radial direction OX of the initial orbit at the kth pulse is shown, and the value range is [0,2 pi ]]；Δv ik Is the radial component of the kth pulse and has the value range of [ -10,10 [)]，Δv jk Is the size of the circumferential component of the kth pulse and has the value range of [ -10,10 [)]The direction same as the original speed direction is taken as positive, and the unit is km/s;
similar to the tangential orbit recursion process, the first pulse process can be passed through the initial orbit parameters and θ 1 ，Δv i1 ，Δv j1 Recursion is carried out to obtain a position velocity vector r after pulse is applied 1 ′，v 1 ' to find out six tracks after pulse application, and then passing through theta 2 To obtain f 2 By passing at theta 2 Six number of tracks and pulse Δ v before applying pulse i2 ,Δv j2 Solving for second applied pulse at θ 2 Six number of tracks [ a ] 2 ′,e 2 ′,i 2 ′,Ω 2 ′,ω 2 ′,f 2 ′]Then the next step of track recursion can be carried out;
finally, solving to obtain six tracks after the pulse is applied for the (n-2) th time, knowing the track parameters of the target track, and considering the variable theta to be optimized n-1 ，θ n The problem is converted into the traditional optimal double-pulse transfer problem, and the optimal double-pulse transfer can be used for obtaining delta v i(n-1) ，Δv j(n-1) ，Δv in ，Δv jn So that the whole process requires 3n-4 state variables to be optimized, i.e. the total fuel consumption of the whole transfer process can be obtained
Other steps and parameters are the same as in one of the first to fifth embodiments.
The following examples were used to demonstrate the beneficial effects of the present invention:
the first embodiment is as follows:
two-pulse example
The following is a set of two-pulse specific examples, the orbit parameters of which are shown in table 1:
TABLE 1 orbital parameters of example one
The genetic algorithm sets the PopulationSize to 150 and the Generations to 100. The fmincon algorithm sets maxfenevals to 6000, maxter to 5000, and TolCon to 1 e-6.
The simulation results are shown in table 2:
TABLE 2 EXAMPLES simulation results
Therefore, the method of the invention and the traditional method converge to the same group of solutions and have great advantages in computational efficiency.
Three pulse example
The following is a specific example of a set of three pulses, the trajectory parameters of which are shown in table 3:
TABLE 3 orbital parameters of example two
The parameter settings of the algorithm are the same as in the first embodiment. The simulation results are shown in table 4:
TABLE 4 simulation results of example two
Therefore, the method provided by the invention converges to the same group of solutions as the traditional method, and has great advantages in computational efficiency.
Overall effect of the method
A large number of examples were simulated for reliable conclusions:
and selecting a random number of 300-2000 km for the near-to-ground height of the initial track, wherein the near-to-ground height of the target track is 0.3-3 times of the near-to-ground height of the initial track, and the multiple is the random number. Initial orbital eccentricity e 0 Target orbit eccentricity e f Is a random number of 0 to 1. Omega 0 、ω f Is a random number of 0-2 pi. The random numbers are uniformly distributed, 600 groups of two pulses are used for simulation, and 250 groups of three pulses are used for simulation. The parameter settings of the algorithm are as above.
The comparison of the obtained conventional method with the calculation time is shown in table 5.
TABLE 5 calculation time comparison Table
The fuel optimum solution obtained by the method used in the present invention was compared with the fuel of the fuel optimum solution obtained by the conventional method, and the comparison results are shown in fig. 4 and 5. The percentage of the legend is the percentage of fuel consumed by the method compared with the traditional method, and because the fuel consumed by mutual transfer between the two tracks is the same, the method is convenient for drawing and defaults to e 0 Less than e f . The height of the vertical axis represents the coincidence of e 0 、e f Number of classified simulation groups.
From the results of fig. 4 and 5, it can be seen that the fuel index obtained by the method of the present invention and the conventional method have almost no difference in the case of a small eccentricity. The method used in the present invention is universal, as earth orbiting satellites generally do not design large eccentricity orbits.
As can be seen, a large number of examples show that the method provided by the invention can be converged to the same group of solutions as the traditional method, and has the advantage of 1-2 orders of magnitude in calculation time.
The present invention is capable of other embodiments and its several details are capable of modifications in various obvious respects, all without departing from the spirit and scope of the present invention.
Claims (6)
1. An optimal coplanarity transfer search method based on tangent initial values is characterized in that: the method comprises the following specific processes:
step one, finishing initial condition setting of pulse optimization:
setting iteration times and population quantity parameters of a genetic algorithm according to initial orbit parameters, target orbit parameters and pulse number n required by a task of the spacecraft;
step two, solving a tangent pulse global optimal solution by a genetic algorithm:
based on the first step, limiting the thrust direction in the speed tangent direction, setting an optimization index as the optimal fuel, and performing global search on 2n-3 variables needing to be optimized for the optimal solution of the tangent pulse fuel by utilizing a genetic algorithm to obtain the optimal solution of the global fuel under the tangent pulse condition;
step three, finishing initialization of the fmincon algorithm:
setting the radial value of the global optimal solution under the tangent pulse condition obtained by the genetic algorithm to 0, and substituting the radial value into the fmincon algorithm as an initial guess;
step four, solving the local optimal solution of the free direction pulse by using an fmincon algorithm:
and (3) limiting the pulse direction in the track plane according to the initial conditions given in the step one, setting an optimization index as the optimal fuel, and carrying out local optimization on 3n-4 optimization variables by using an fmincon algorithm to obtain the optimal local fuel solution under the pulse condition in the free direction.
2. The optimal coplanar transition search method based on tangent initial value as claimed in claim 1, wherein: the initial condition setting for completing the pulse optimization in the first step is as follows:
setting iteration times and population quantity parameters of a genetic algorithm according to initial orbit parameters, target orbit parameters and pulse number n required by a task of the spacecraft; the specific process is as follows:
the orbit parameters of the spacecraft at the initial moment are given under a J2000 geocentric inertial coordinate system, and the initial orbit parameters of the spacecraft comprise: semi-major axis a 0 Eccentricity e 0 Track inclination i 0 And the right ascension omega 0 Angle distance omega from the near center point 0 ；
The target orbit parameters include: semi-major axis a f Eccentricity e f Track inclination i f The right ascension channel omega f Angle distance omega from the near center point f (ii) a Where the orbital inclination and ascent intersection declination should be the same as for the spacecraft at the initial moment, i 0 ＝i f ，Ω 0 ＝Ω f Namely, the initial orbit and the target orbit of the spacecraft meet the coplanar condition;
the adopted dynamic model is a two-body model, namely factors such as earth non-spherical perturbation, sunlight pressure and atmospheric resistance perturbation are not considered;
wherein mu represents the gravitational constant of the earth, r and v represent the position vector and the velocity vector of the spacecraft under the J2000 geocentric inertial coordinate system, and | r | represents the size of the position vector;
the first derivative of r is represented as,
the first derivative of v is indicated.
3. The optimal coplanar transition search method based on tangent initial value as claimed in claim 2, wherein: solving the tangent pulse global fuel optimal solution by the genetic algorithm in the step two:
based on the first step, the thrust direction is limited in the tangential direction, optimization indexes are set as fuel optimum, 2n-3 variables needing to be optimized for the tangential pulse fuel optimum solution are searched globally by utilizing a genetic algorithm, and the global fuel optimum solution under the tangential pulse condition is obtained; the specific process is as follows:
step two, assuming that the pulse number is n, the number of state variables needing to be optimized for obtaining the fuel optimal solution is at least 2n-3, and selecting the state variable to be optimized as X ═ theta ═ 1 ,θ 2 ,...,θ n-1 ,Δv 1 ,Δv 2 ,...,Δv n-2 ]；
Wherein, theta i The included angle between the position of the spacecraft and the rise-crossing point of the initial orbit in the radial direction OX is shown in the ith pulse, and the value range is [0,2 pi ]]；Δv i The value range of the ith pulse is set as [ -10,10 [ ]]Taking the direction same as the original speed direction as positive and the unit as km/s;
step two, according to the initial orbit parameter [ a ] 0 ,e 0 ,i 0 ,Ω 0 ,ω 0 ]And carrying out track recursion on the state variable to be optimized, wherein the track recursion process is represented as follows:
from initial orbit parameters and theta 1 The first time before pulse is applied can be obtained that the spacecraft is at theta 1 Six number of tracks [ a ] 1 ,e 1 ,i 1 ,Ω 1 ,ω 1 ,f 1 ]Wherein the first five orbital parameters are the same as the initial orbital parameters, i.e. satisfy a 1 ＝a 0 、e 1 ＝e 0 、i 1 ＝i 0 、Ω 1 ＝Ω 0 、ω 1 ＝ω 0 True proximal angle f 1 ＝θ 1 -ω 1 (ii) a By passing at theta 1 Six tracks before applying pulse and pulse size Deltav 1 Solving for theta after the first applied pulse 1 Six number of tracks [ a ] 1 ′,e 1 ′,i 1 ′,Ω 1 ′,ω 1 ′,f 1 ′]；
Step two and step three, for the first five items of six track numbers, there is a 1 ′＝a 2 、e 1 ′＝e 2 、i 1 ′＝i 2 、Ω 1 ′＝Ω 2 、ω 1 ′＝ω 2 (ii) a Through theta 2 Can find f 2 ＝θ 2 -ω 2 (ii) a From which the second pulse av can be applied 2 Six number of front track [ a ] 2 ,e 2 ,i 2 ,Ω 2 ,ω 2 ,f 2 ]By passing at theta 2 Six number of tracks and pulse Δ v before applying pulse 2 Solving for second applied pulse at θ 2 Six number of tracks [ a ] 2 ′,e 2 ′,i 2 ′,Ω 2 ′,ω 2 ′,f 2 ′]；
Until obtaining at theta n-2 Where a pulse Δ v is applied n-2 Rear orbital parameter [ a ] n-2 ′,e n-2 ′,i n-2 ′,Ω n-2 ′,ω n-2 ′,f n-2 ′]By knowing the orbit parameters of the target orbit [ a ] f ,e f ,i f ,Ω f ,ω f ]And theta in the state variable to be optimized n-1 The magnitude Deltav of the last two tangent pulses is obtained n-1 ，Δv n 。
4. The optimal coplanar transition search method based on tangent initial value as claimed in claim 3, wherein: in the second step, the initial orbit parameter [ a ] is used as the basis 0 ,e 0 ,i 0 ,Ω 0 ,ω 0 ]And the state variables to be optimized can be subjected to orbit recursion, and the orbit recursion process is expressed as follows:
from initial orbit parameters and theta 1 It can be obtained that the spacecraft is at theta before the first pulse is applied 1 Six number of tracks [ a ] 1 ,e 1 ,i 1 ,Ω 1 ,ω 1 ,f 1 ]Wherein the first five orbital parameters are the same as the initial orbital parameters, i.e. satisfy a 1 ＝a 0 、e 1 ＝e 0 、i 1 ＝i 0 、Ω 1 ＝Ω 0 、ω 1 ＝ω 0 True proximal angle f 1 ＝θ 1 -ω 1 (ii) a By passing at theta 1 Six tracks before applying pulse and pulse size Deltav 1 Solving for theta after the first applied pulse 1 Six number of tracks [ a ] 1 ′,e 1 ′,i 1 ′,Ω 1 ′,ω 1 ′,f 1 ′]；
The specific process is as follows:
position and velocity vectors r of the spacecraft on the first pulse application 1 ,v 1 ]Can be expressed as
r 1 ＝C 3 (Ω 1 )C 1 (i 1 )C 3 (ω 1 )[|r 1 |sinf 1 ；|r 1 |cosf 1 ；0]
In the formula, C 1 、C 3 Representing a rotation matrix of the form:
wherein α represents Ω 1 、i 1 Or ω 1 ；
At theta 1 To apply a pulse Δ v 1 Then, the position vector is not changed, the speed vector is changed, and the position vector and the speed vector [ r ] of the spacecraft are changed at the moment 1 ′,v 1 ′]Can be expressed as
So that the position and velocity vector r after the pulse is applied can be used 1 ′，v 1 Solving at θ 1 To apply a pulse Δ v 1 Six number of rear tracks [ a ] 1 ′,e 1 ′,i 1 ′,Ω 1 ′,ω 1 ′,f 1 ′]。
5. The optimal coplanar transition search method based on tangent initial value as claimed in claim 4, wherein: in the second step and the third step, the first five items of six orbits are known to have a 1 ′＝a 2 、e 1 ′＝e 2 、i 1 ′＝i 2 、Ω 1 ′＝Ω 2 、ω 1 ′＝ω 2 (ii) a Through theta 2 Can find f 2 ＝θ 2 -ω 2 (ii) a From which the second pulse av can be applied 2 Six number of front track [ a ] 2 ,e 2 ,i 2 ,Ω 2 ,ω 2 ,f 2 ]By passing at theta 2 Six number of tracks and pulse Δ v before applying pulse 2 Solving for second applied pulse at θ 2 Six number of tracks [ a ] 2 ′,e 2 ′,i 2 ′,Ω 2 ′,ω 2 ′,f 2 ′]；
Until a value at θ is obtained n-2 To apply a pulse Δ v n-2 Rear orbital parameter [ a ] n-2 ′,e n-2 ′,i n-2 ′,Ω n-2 ′,ω n-2 ′,f n-2 ′]By knowing the orbital parameters of the target orbit [ a ] f ,e f ,i f ,Ω f ,ω f ]And theta in the state variable to be optimized n-1 The magnitude Deltav of the last two tangent pulses is obtained n-1 ，Δv n ；
The process is as follows:
θ＝θ n -θ n-1
in which theta represents theta n Angle of included angle and theta n-1 The difference of the included angles; λ represents an intermediate variable;
the relationship between the flight direction angle gamma and the true approach point angle can be used to obtain
In the formula of gamma j Indicating the application of the j-th pulse av j Front flight direction angle, | r j L denotes the application of the jth pulse Δ v j Position vector magnitude of time, p j Representing the half-diameter of the track;
multiplication of both sides by p n sinθ，
Can be obtained by finishing
Order to
Then
In the formula k 1 、k 2 It is indicated that the intermediate variable(s),
representing an intermediate variable;
the transfer orbit has a radius and an angular momentum of
The magnitude Deltav of the last two pulses can thus be determined n-1 ，Δv n
The total fuel consumption can be expressed as
6. The optimal coplanar transition search method based on tangent initial value as claimed in claim 5, wherein: solving the local fuel optimal solution of the free direction pulse by using an fmincon algorithm in the fourth step:
limiting the pulse direction in the plane of the track according to the initial conditions given in the step one, setting optimization indexes as fuel optimization, and carrying out local optimization on 3n-4 optimization variables by using an fmincon algorithm to obtain a local fuel optimal solution under the pulse condition in the free direction; the process is as follows:
assuming that the number of pulses is n, a state variable X ═ θ is selected 1 ,θ 2 ,...,θ n ,Δv i1 ,Δv i2 ,...,Δv i(n-2) ,Δv j1 ,Δv j2 ,...,Δv j(n-2) ]Wherein, θ k The included angle between the position of the spacecraft and the rise point of the initial orbit in the radial direction OX is shown in the k-th pulse, and the value range is [0,2 pi ]]；Δv ik Is the radial component of the kth pulse and has the value range of [ -10,10 [)]，Δv jk Is the size of the circumferential component of the kth pulse and has the value range of [ -10,10 [)]The direction same as the original speed direction is taken as positive, and the unit is km/s;
by initial orbit parameters and theta 1 ，Δv i1 ，Δv j1 Recursion is carried out to obtain a position velocity vector r after pulse is applied 1 ′，v 1 ' to find out six tracks after pulse application, and then passing through theta 2 To obtain f 2 By passing at theta 2 Six number of tracks and pulse Δ v before applying pulse i2 ,Δv j2 Solving for θ after second pulse application 2 Six number of tracks [ a ] 2 ′,e 2 ′,i 2 ′,Ω 2 ′,ω 2 ′,f 2 ′]；
Finally, solving to obtain six tracks after the pulse is applied for the (n-2) th time, knowing the track parameters of the target track, and considering the variable theta to be optimized n-1 ，θ n Obtaining Δ v using optimal dipulse transfer i(n-1) ，Δv j(n-1) ，Δv in ，Δv jn So that the whole process needs 3n-4 state variables to be optimized, i.e. the total fuel consumption of the whole transfer process can be obtained
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