CN110032768B - Four-pulse orbit intersection optimization method using accurate dynamic model - Google Patents
Four-pulse orbit intersection optimization method using accurate dynamic model Download PDFInfo
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Abstract
The invention discloses a four-pulse orbit intersection optimization method using an accurate dynamics model, which is suitable for multi-circle multi-pulse accurate dynamics orbit intersection optimization calculation with long-term drift. The pulse-tracking strategy for a given initial and end state and fixed transition time optimizes the calculation. Because the magnitude of other perturbation terms is far smaller than that of the J2 perturbation term, the analytic optimal solution of the J2 perturbation term is considered to be closer to the optimal solution of the numerical total-perturbation model, and the numerical solution can be approximated through the analytic solution. The invention designs a reasonable approximation method, and finds out the deviation between the analytic optimal solution and the numerical optimal solution, so that the terminal state meets the intersection constraint. The invention solves the problems of large calculation amount and low optimization speed of the pulse transfer optimization method in the prior art.
Description
Technical Field
The invention belongs to the technical field of space control, and particularly relates to a four-pulse orbit intersection optimization method using an accurate dynamic model.
Background
For the orbital crossing control of the medium and low earth orbits for tens of days, if the natural drift caused by perturbation is reasonably utilized, the fuel consumption is greatly reduced. At present, most pulse transfer optimization methods directly establish a numerical optimization model and use an accurate dynamic model to carry out integration to solve. Because the number of turns is large, a large number of local optimal solutions exist, the calculation amount is large, and the optimization speed is low. Because the magnitude of other perturbation terms is far smaller than that of the J2 perturbation term, the analytic optimal solution of the J2 perturbation can be considered to be closer to the optimal solution of the numerical full-scale-up model, and the numerical solution can be approximated through the analytic solution. However, even a slight model difference may cause the integrated state to completely deviate from the target because the orbit transition time is long. Therefore, a reasonable approximation method needs to be designed, and the deviation between the analytic optimal solution and the numerical optimal solution is found out, so that the terminal state meets the intersection constraint.
Disclosure of Invention
The invention aims to provide a four-pulse orbit intersection optimization method using an accurate dynamic model, and solves the problems of large calculation amount and low optimization speed of a pulse transfer optimization method in the prior art.
The invention adopts the technical scheme that a four-pulse orbit intersection optimization method using an accurate dynamic model is implemented according to the following steps:
step 1, establishing a Lambert transfer calculation analysis model containing J2 perturbation and represented by six elements of track number, wherein the Lambert transfer calculation analysis model is used for calculating the speed increment between every two orbital transfer pulses;
step 2, establishing a pulse optimization analytic model containing J2 perturbation and solving;
step 3, establishing a full perturbation lambert transfer numerical calculation model;
and 4, establishing a conversion model from the analytic model to the full-shot model and solving to obtain the optimal transfer.
The present invention is also characterized in that,
the step 1 is implemented according to the following steps:
step 1.1, an analytic kinetic equation containing J2 perturbation expressed by six elements [ a, e, i, omega, M ] of the orbital number is as follows:
wherein, J 2 Is a second order term of the earth's gravity, re is the earth's radius,
is the average angular velocity of the track;
step 1.2, firstly, solving the problem of Lambert transfer calculation with fixed time and positions at two ends under an analytic dynamic model, and defining LambertJ2_ M to represent a calculation process:
[v in ,v out ]＝lambertJ2_M(r in ,r out ,t) (2)
wherein r is in ,r out Respectively, the input start-stop position vector, t is the transition time, v in ,v out Respectively for the start-stop velocity vectors to be output, i.e. solving for [ v in ,v out ]So that the state [ r in ,v in ]Extrapolation according to equation (1) to r out ,v out ]；
Step 1.3, calculating the lambert transfer initial end and end speeds only considering the earth two-body gravity model:
then v is measured in As solution variables, use
As initial value, solve so that [ r 0 ，v in ]Extrapolating the position vector r after t time according to equation (1) f Initial velocity v of in Namely:
[r,v]＝f(r 0 ,v in ,t)
J＝|r-r f |＝0 (4)
wherein f represents the analytic extrapolation process according to formula (1) to obtain v in And v out 。
In step 2, the initial state is set as [ A ] 0 ,e 0 ,i 0 ,Ω 0 ,ω 0 ,M 0 ]Corresponding to a position velocity of [ r ] 0 ,v 0 ]The target state is [ A ] f ,e f ,i f ,Ω f ,ω f ,M f ]Corresponding to position velocityIs [ r ] f ,v f ]The starting time is fixed to T 0 End time T f The method comprises the following steps of (1) fixing, wherein the number of pulses is fixed to be 4, calculating the orbit after applying the pulses by adopting an equation (1), namely firstly pushing the orbit of the spacecraft to a first pulse time, pushing the orbit of the spacecraft to a second pulse time after applying the pulses, pushing the orbit of the spacecraft to a third pulse time after applying the pulses, and then calculating the speed of the spacecraft after applying the pulses at the third pulse time and the speed of the spacecraft before applying the pulses at the terminal time by a double-pulse lambert transfer algorithm of a J2 analytical model according to the position and terminal position constraints of the spacecraft at the moment, so that the sizes of the third pulse and the fourth pulse can be calculated, and the method is implemented specifically according to the following steps:
step 2.1, solving variable x has 9 dimensions:
the duration of the first pulse relative to the start time, i.e. the gliding section, is as follows:
t 1 ＝(T f -T 0 )x[0] (5)
the first pulse has the following magnitude and direction:
the duration of the second pulse relative to the first pulse is defined as less than 1 track cycle:
t 2 ＝(T f -T 0 )(1-x[0])x[4] (7)
the second pulse has the following magnitude and direction:
the interval between the third and fourth pulses is defined as less than 1 track cycle, the fourth pulse being the terminal crossing time:
t 3 ＝(T f -T 0 )(1-x[0])(1-x[4])x[8] (9)
step 2.2, set [ r 3 ,v 3 ]For extrapolating to a third pulse from the trajectory after the second pulseThe state obtained by the time of impact is according to the double-pulse lambert transfer algorithm of the J2 analytic model, [ v [ ] in ,v out ]＝lambertJ2_M(r in ,r out T) function, calculating the speed increment of the third and fourth pulses, i.e.:
[v 3 in ,v 4 out ]＝lambertJ2_M(r 3 ,r f ,t 3 ) (10)
in the formula, v 3 in ,v 4 out The velocity after the third pulse and before the fourth pulse, respectively, is then:
step 2.3, the total optimization index is to minimize the 4-pulse total velocity increment:
J＝|dv 1 |+|dv 2 |+|dv 3 |+|dv 4 | (12)
step 2.4, solving by using a differential evolution algorithm, wherein the solving step comprises initialization; then carrying out mutation, crossing and selection operations on the population to update the population; and the updating process is repeated until the convergence condition is satisfied.
Step 3 is implemented specifically according to the following steps:
step 3.1, the numerical kinetic equation containing the perturbation term is as follows:
wherein: r is the satellite position vector, v is the velocity vector, μ is the Earth's gravitational constant, a p Is perturbation acceleration;
step 3.2, defining lambert j2_ H to represent the calculation process:
[v in ,v out ]＝lambertJ2_H(r in ,r out ,t) (14)
wherein r is in ,r out Start and stop positions of respective inputsLet vector, t be transfer time, v in ,v out Respectively for the start-stop velocity vectors to be output, i.e. solving for [ v in ,v out ]So that the state [ r in ,v in ]Extrapolation according to equation (13) to r out ,v out ]；
Step 3.3, the velocity obtained in the step 2 is taken as an initial value, and then v is taken in As solution variables, namely:
[r,v]＝g(r 0 ,v in ,t)
J＝|r-r f |＝0 (15)
wherein g represents the numerical integration extrapolation process according to formula (13) to obtain v in And v out 。
Step 4 is implemented according to the following steps:
step 4.1, define normalized optimization variable x = [ x ] 0 ,x 1 ,x 2 ,x 3 ,x 4 ,x 5 ]Having 6 dimensions, the positions r of the second and third pulse instants 2 ,r 3 Amount of adjustment of r 1 ,r f Keeping unchanged:
in the formula, r 2 '、r 3 ' are the positions of the second and third pulse moments, respectively, under a numerical dynamical model;
step 4.2, according to the difference of the flat transient transformation, if k =10km, the speeds of the spacecrafts before and after 4 pulses are applied are calculated by position vectors according to multi-circle lambert transfer:
in the formula (I), the compound is shown in the specification,
and
respectively representing the slave r of a spacecraft 1 Is transferred to r 2 The initial velocity and the terminal velocity at',
and
respectively representing the slave r of a spacecraft 2 ' transfer to r 3 The initial velocity and the terminal velocity at',
and
respectively representing the slave r of a spacecraft 3 ' transfer to r f The initial and terminal velocities;
step 4.3, the velocity increment dv of four pulses i I =1,2,3,4 are each
Step 4.4, the optimization index and the constraint are expressed as:
J＝|dv 1 |+|dv 2 |+|dv 3 |+|dv 4 | (19)
and (4) solving by using a differential evolution algorithm to obtain the four-pulse optimal transfer.
The invention has the beneficial effects that the four-pulse orbit intersection optimization method using the accurate dynamic model is suitable for multi-circle multi-pulse accurate dynamic orbit intersection optimization calculation with long-term drift. The pulse tracking strategy for given initial and end states and fixed transition times optimizes the calculation. Because the magnitude of other perturbation terms is far smaller than that of the J2 perturbation term, the analytic optimal solution of the J2 perturbation can be considered to be closer to the optimal solution of the numerical full-scale-up model, and the numerical solution can be approximated through the analytic solution. The invention designs a reasonable approximation method, and finds out the deviation between the analytic optimal solution and the numerical value optimal solution, so that the terminal state meets the intersection constraint.
Drawings
FIG. 1 is a flow chart of a four-pulse orbital intersection optimization method of the present invention using an accurate kinetic model.
FIG. 2 is a schematic diagram of an analytic perturbation optimization model of a four-pulse orbital intersection optimization method using an accurate kinetic model.
FIG. 3 is a conversion model of analytic initial values to numerically optimal solutions for a four-pulse orbital intersection optimization method using an accurate kinetic model according to the present invention.
Detailed Description
The invention is described in detail below with reference to the drawings and the detailed description.
The invention discloses a four-pulse orbit intersection optimization method using an accurate dynamic model, which is implemented by the following steps as shown in a flow chart shown in figure 1:
step 1, establishing a Lambert transfer calculation analysis model containing J2 perturbation and represented by six elements of track number, wherein the Lambert transfer calculation analysis model is used for calculating the speed increment between every two orbital transfer pulses and is implemented according to the following steps:
step 1.1, an analytic kinetic equation containing J2 perturbation expressed by six elements [ a, e, i, omega, M ] of the orbital number is as follows:
wherein, J 2 Is a second order term of the earth's gravity, re is the earth's radius,
is the average angular velocity of the track;
step 1.2, firstly, the problem of Lambert transfer calculation with fixed time and two end positions under an analytic dynamic model is solved, and LambertJ2_ M is defined to represent a calculation process:
[v in ,v out ]＝lambertJ2_M(r in ,r out ,t) (2)
wherein r is in ,r out Respectively, the input start-stop position vector, t is the transition time, v in ,v out Respectively for the start-stop velocity vectors to be output, i.e. solving for [ v in ,v out ]So that the state [ r in ,v in ]Extrapolation according to equation (1) to r out ,v out ]；
Step 1.3, calculating the lambert transfer initial end and end speeds only considering the earth two-body gravity model:
then v is measured in As solution variables, use
As initial value, solve so that [ r 0 ，v in ]Extrapolating the position vector r after t time according to equation (1) f Initial velocity v of in Namely:
[r,v]＝f(r 0 ,v in ,t)
J＝|r-r f |＝0 (4)
wherein f represents the analytical extrapolation process according to formula (1) to obtain v in And v out 。
Step 2, establishing a pulse optimization analytic model containing J2 perturbation and solving:
in step 2, the initial state is set as [ A ] 0 ,e 0 ,i 0 ,Ω 0 ,ω 0 ,M 0 ]Corresponding to a position velocity of [ r ] 0 ,v 0 ]Target state is [ A ] f ,e f ,i f ,Ω f ,ω f ,M f ]Corresponding to a position velocity of [ r ] f ,v f ]The starting time is fixed to T 0 End time T f Fixing, the number of pulses is fixed to 4, calculating the orbit after applying the pulses by adopting an equation (1), namely firstly pushing the orbit of the spacecraft to a first pulse time, then pushing to a second pulse time after applying the pulses, pushing to a third pulse time after applying the pulses, and then carrying out the calculationAccording to the position and the end position constraint of the spacecraft at the moment, the velocity of the spacecraft after the pulse is applied at the third pulse moment and the velocity of the spacecraft before the pulse is applied at the terminal moment are inversely calculated by a double-pulse lambert transfer algorithm of a J2 analytic model, so that the magnitudes of the third pulse and the fourth pulse can be calculated, and the method is implemented according to the following steps:
step 2.1, solving variable x has 9 dimensions, as shown in fig. 2:
the duration of the first pulse relative to the start time, i.e. the glide phase, is as follows:
t 1 ＝(T f -T 0 )x[0] (5)
the first pulse has the following magnitude and direction:
the duration of the second pulse relative to the first pulse is defined as less than 1 track cycle:
t 2 ＝(T f -T 0 )(1-x[0])x[4] (7)
the second pulse has the following magnitude and direction:
|dv 2 |＝dv max x[5]
dv 2x ＝|dv 2 |cos(2πx[6])
dv 2y ＝|dv 2 |sin(2πx[6])cos(2πx[7])
dv 2z ＝|dv 2 |sin(2πx[6])sin(2πx[7]) (8)
the interval between the third and fourth pulses is defined as less than 1 track cycle, the fourth pulse instant being the terminal crossing instant:
t 3 ＝(T f -T 0 )(1-x[0])(1-x[4])x[8] (9)
step 2.2, set [ r ] 3 ,v 3 ]For the state obtained from the extrapolation of the track after the second pulse to the third pulse instant, the double-pulse lambert transfer algorithm according to the J2 analytical model, [ v ] v in ,v out ]＝lambertJ2_M(r in ,r out T) function, calculating the speed increment of the third and fourth pulses, i.e.:
[v 3 in ,v 4 out ]＝lambertJ2_M(r 3 ,r f ,t 3 ) (10)
in the formula, v 3 in ,v 4 out The velocity after the third pulse and before the fourth pulse, respectively, is then:
step 2.3, the total optimization index is to minimize the 4-pulse total velocity increment:
J＝|dv 1 |+|dv 2 |+|dv 3 |+|dv 4 | (12)
step 2.4, solving by using a differential evolution algorithm, wherein the solving step comprises initialization; then carrying out mutation, crossover and selection operations on the population to update the population; and the updating process is repeated until the convergence condition is satisfied.
Step 3, establishing a full perturbation lambert transfer numerical calculation model, and specifically implementing the steps as follows:
step 3.1, the numerical kinetic equation containing the perturbation term is as follows:
wherein: r is the satellite position vector, v is the velocity vector, μ is the earth's gravitational constant, a p Is perturbation acceleration; (mainly comprises earth non-spherical perturbation, atmospheric resistance, sunlight pressure, sun-moon attraction and the like which are functions of satellite positions and speeds, and mature calculation methods are available).
Step 3.2, define lamberti j2_ H to represent the calculation process:
[v in ,v out ]＝lambertJ2_H(r in ,r out ,t) (14)
wherein r is in ,r out Respectively, input start-stop position vector, t transfer time, v in ,v out Respectively for the start-stop velocity vectors to be output, i.e. solving for [ v in ,v out ]So that the state [ r in ,v in ]Extrapolation according to equation (13) to r out ,v out ]；
Step 3.3, the speed obtained in the step 2 is used as an initial value, and then a Minpack toolkit is used
V is to be in As solution variables, namely:
[r,v]＝g(r 0 ,v in ,t)
J＝|r-r f |＝0 (15)
wherein g represents the numerical integral extrapolation according to equation (13) to obtain v in And v out 。
Step 4, establishing a conversion model from the analytic model to the full-shot model and solving to obtain the optimal transfer, which is implemented according to the following steps:
and 4.1, because the magnitude of other perturbation terms is smaller than that of the J2 perturbation term, the analytic optimal solution of the J2 perturbation is considered to be closer to the optimal solution of the numerical total-perturbation model. It can be set that the transition time between pulses is constant when comparing the numerical optimal solution with the analytic optimal solution, and the position of the middle two pulse application time has a fine adjustment amount, see fig. 3, and a normalized optimization variable x = [ x ] is defined 0 ,x 1 ,x 2 ,x 3 ,x 4 ,x 5 ]Having 6 dimensions, the positions r of the second and third pulse instants 2 ,r 3 Amount of adjustment of r 1 ,r f Keeping the steps unchanged:
in the formula, r 2 '、r 3 ' the positions of the second and third pulse moments under the numerical dynamics model, respectively;
step 4.2, according to the difference of the flat transient transformation, if k =10km, the speeds of the spacecraft before and after the application of the 4 pulses are calculated by the position vector according to the multi-turn lambert transfer:
in the formula (I), the compound is shown in the specification,
and
respectively representing the slave r of a spacecraft 1 Is transferred to r 2 The initial velocity and the terminal velocity at,
and
respectively representing the slave r of a spacecraft 2 ' transfer to r 3 The initial velocity and the terminal velocity at,
and
respectively representing the slave r of a spacecraft 3 ' transfer to r f The initial and terminal velocities;
step 4.3, the velocity increment dv of four pulses i I =1,2,3,4 are each
Step 4.4, the optimization index and the constraint are expressed as:
J＝|dv 1 |+|dv 2 |+|dv 3 |+|dv 4 | (19)
and (4) solving by using a differential evolution algorithm to obtain the four-pulse optimal transfer.
Examples
In a specific application example, the adopted tracking star orbit and the target star orbit are from data files of GTOC9 (ninth International orbit optimization Competition).
The known orbits of the tracking star and the target star are shown in Table 1, respectively, the start time (unit MJD2000, T) f Same) is T 0 =24111.63, the meeting time is T f =24114.51, transfer duration T =2.88 days.
TABLE 1 number of tracking stars and target star orbits
The initial position velocity and the target position velocity are first calculated.
[r 0 ,v 0 ]＝[522053.631,-1114330.156,-6901285.848,-4424.105,-6071.874,638.533]
[r f ,v f ]＝[4740455.489,4037622.516,-3516701.059,-1932.103,-3298.295,-6419.572]
According to the attached figure 2, an optimization model under the 4-pulse analytic kinetic equation is established. The process is that the number of pulses is fixed to be 4, the calculation of the orbit after the pulses are applied adopts an equation (1), namely, firstly, the orbit of the spacecraft is pushed to the first pulse time, then the orbit of the spacecraft is pushed to the second pulse time after the pulses are applied, the orbit of the spacecraft is pushed to the third pulse time after the pulses are applied, then the speed of the spacecraft after the pulses are applied at the third pulse time and the speed of the spacecraft before the pulses are applied at the terminal time are inversely calculated by a double-pulse lambert transfer algorithm of a J2 analytical model according to the position and terminal position constraints of the spacecraft at the moment, and the sizes of the third pulse and the fourth pulse can be calculated accordingly. Therefore, the solving variable x has 9 dimensions, and the total optimization index is that the total speed increment of 4 pulses is minimized.
And (3) calculating to obtain an analytic optimal solution by using a differential evolution algorithm:
the dimension of the differential evolution algorithm is set to be 9, and the number of the populations is set to be 100. Firstly, initializing, then carrying out mutation, intersection and selection operations on the population to update the population, repeating the updating process until a convergence condition is met, wherein the optimal solution is as follows:
X opt [9]＝[0.0081254,0.7172128,-0.409457,0.0197869,0.4598666,0.00027222897,-0.708910,0.543359,0.00739929]. The corresponding speed increment and time are shown in Table 2, and the total speed increment is 102.4m/s:
TABLE 2 optimal speed increment
Establishing an analysis model to an optimization model of a full-shooting model:
according to the equations (5) to (11), the position vector of the analytic optimal transfer orbit at the application positions of the 1 st, 2 nd and 3 rd pulses is calculated as:
r 1 ＝[4969597.813050483,3916697.428043888,-3013178.929439275]m
r 2 ＝[4888736.738673554,3782212.216964675,-3302647.922404093]m
r 3 ＝[-4169891.907243956,-3208239.320903509,4667574.196774956]m
setting the value of the optimum transfer trajectory according to equation (16)
Where k =1 000m.
x 6 is 6-dimensional optimization quantity of the conversion model, and the optimization index is formula (19).
And (3) calculating to obtain a numerical optimal solution by using a differential evolution algorithm:
setting the dimension of the differential evolution algorithm to be 3, the population number to be 20, and solving the optimal solution as follows:
x = [0.1309456, -0.106957, -0.10677567, -0.86676, -0.540467,0.89727]. The corresponding optimum transfer trajectory can be calculated according to equations (16) to (18), see table 3. The total velocity increment was 104.3m/s.
TABLE 3 numerical solution of optimal speed increment
Compared with the result of the jet propulsion laboratory JPL in the ninth international orbit optimization competition, the optimization result given by the jet propulsion laboratory JPL is 105.2m/s under the condition of the same calculation condition, and the optimization effect of the jet propulsion laboratory JPL is proved to be better.
Claims (2)
1. A four-pulse orbital intersection optimization method using an accurate dynamic model is characterized by comprising the following steps:
step 1, establishing a Lambert transfer calculation analysis model containing J2 perturbation and represented by six elements of track number, wherein the Lambert transfer calculation analysis model is used for calculating the speed increment between every two orbital transfer pulses;
the step 1 is specifically implemented according to the following steps:
step 1.1, an analytic kinetic equation containing J2 perturbation expressed by six elements [ a, e, i, omega, M ] of the track number is as follows:
wherein, J 2 Is a second order term of the earth's gravity, re is the radius of the earth,
is the average angular velocity of the track;
step 1.2, firstly, solving the problem of Lambert transfer calculation with fixed time and positions at two ends under an analytic dynamic model, and defining LambertJ2_ M to represent a calculation process:
[v in ,v out ]＝lambertJ2_M(r in ,r out ,t) (2)
wherein r is in ,r out Respectively, the input start-stop position vector, t is the transition time, v in ,v out Respectively for the start-stop velocity vectors to be output, i.e. solving for [ v in ,v out ]So that the state [ r in ,v in ]Extrapolation according to formula (1) equal to [ r ] after t time out ,v out ]；
Step 1.3, calculating the lambert transfer initial end and end speeds only considering the earth two-body gravity model:
then v is measured in As solution variables, use
As initial value, solve so that [ r 0 ,v in ]Extrapolating the position vector r after t time according to equation (1) f Initial velocity v of in Namely:
[r,v]＝f(r 0 ,v in ,t)
J＝|r-r f |＝0 (4)
wherein f represents the analytical extrapolation process according to formula (1) to obtain v in And v out ；
Step 2, establishing a pulse optimization analytic model containing J2 perturbation and solving;
the initial state is set as [ A ] in the step 2 0 ,e 0 ,i 0 ,Ω 0 ,ω 0 ,M 0 ]Corresponding to a position velocity of [ r ] 0 ,v 0 ]The target state is [ A ] f ,e f ,i f ,Ω f ,ω f ,M f ]Corresponding to a position velocity of [ r ] f ,v f ]The starting time is fixed to T 0 End time T f The method comprises the following steps of (1) fixing, wherein the number of pulses is fixed to be 4, calculating the orbit after applying the pulses by adopting an equation (1), namely firstly pushing the orbit of the spacecraft to a first pulse time, pushing the orbit of the spacecraft to a second pulse time after applying the pulses, pushing the orbit of the spacecraft to a third pulse time after applying the pulses, and then calculating the speed of the spacecraft after applying the pulses at the third pulse time and the speed of the spacecraft before applying the pulses at the terminal time by a double-pulse lambert transfer algorithm of a J2 analytical model according to the position and terminal position constraints of the spacecraft at the moment, so that the sizes of the third pulse and the fourth pulse can be calculated, and the method is implemented specifically according to the following steps:
step 2.1, solving the variable x and having 9 dimensions:
the duration of the first pulse relative to the start time, i.e. the glide phase, is as follows:
t 1 ＝(T f -T 0 )x[0] (5)
the first pulse has the following magnitude and direction:
the duration of the second pulse relative to the first pulse is defined as less than 1 track cycle:
t 2 ＝(T f -T 0 )(1-x[0])x[4] (7)
the second pulse has the following magnitude and direction:
|dv 2 |＝dv max x[5]
dv 2x ＝|dv 2 |cos(2πx[6])
dv 2y ＝|dv 2 |sin(2πx[6])cos(2πx[7])
dv 2z ＝|dv 2 |sin(2πx[6])sin(2πx[7]) (8)
the interval between the third and fourth pulses is defined as less than 1 track cycle, the fourth pulse being the terminal crossing time:
t 3 ＝(T f -T 0 )(1-x[0])(1-x[4])x[8] (9)
step 2.2, set [ r ] 3 ,v 3 ]For the state obtained by extrapolation from the track after the second pulse to the third pulse instant, according to the double pulse lambert transfer algorithm of the J2 analytical model, [ v ] v in ,v out ]＝lambertJ2_M(r in ,r out T) function, calculating the speed increment of the third and fourth pulses, i.e.:
[v 3 in ,v 4 out ]＝lambertJ2_M(r 3 ,r f ,t 3 ) (10)
in the formula, v 3 in ,v 4 out The velocity after the third pulse and before the fourth pulse, respectively, is then:
step 2.3, the total optimization index is to minimize the 4-pulse total velocity increment:
J＝|dv 1 |+|dv 2 |+|dv 3 |+|dv 4 | (12)
step 2.4, solving by using a differential evolution algorithm, wherein the solving step comprises initialization; then carrying out mutation, crossover and selection operations on the population to update the population; and repeating the updating process until a convergence condition is satisfied;
step 3, establishing a full perturbation lambert transfer numerical calculation model;
the step 3 is specifically implemented according to the following steps:
step 3.1, the numerical kinetic equation containing the perturbation term is as follows:
wherein: r is the satellite position vector, v is the velocity vector, μ is the Earth's gravitational constant, a p Is perturbation acceleration;
step 3.2, defining lambert j2_ H to represent the calculation process:
[v in ,v out ]＝lambertJ2_H(r in ,r out ,t) (14)
wherein r is in ,r out Respectively, the input start-stop position vector, t is the transition time, v in ,v out Respectively for the start-stop velocity vectors to be output, i.e. solving for [ v in ,v out ]So that the state [ r ] in ,v in ]Extrapolation according to equation (13) to r out ,v out ]；
Step 3.3, the velocity obtained in the step 2 is taken as an initial value, and then v is taken in As solution variables, namely:
[r,v]＝g(r 0 ,v in ,t)
J＝|r-r f |＝0 (15)
wherein g represents the numerical integration extrapolation process according to formula (13) to obtain v in And v out ；
And 4, establishing a conversion model from the analytic model to the full-shot model and solving to obtain the optimal transfer.
2. The method for optimizing the four-pulse orbital intersection by using the precise kinetic model according to claim 1, wherein the step 4 is implemented by the following steps:
step 4.1, define normalized optimization variable x = [ x = 0 ,x 1 ,x 2 ,x 3 ,x 4 ,x 5 ]Having 6 dimensions, the positions r of the second and third pulse instants 2 ,r 3 Amount of adjustment of r 1 ,r f Keeping unchanged:
in the formula, r 2 '、r 3 ' the positions of the second and third pulse moments under the numerical dynamics model, respectively;
step 4.2, according to the difference of the flat transient transformation, if k =10km, the speeds of the spacecraft before and after the application of the 4 pulses are calculated by the position vector according to the multi-turn lambert transfer:
in the formula (I), the compound is shown in the specification,
and
respectively representing the slave r of a spacecraft 1 Is transferred to r 2 The initial velocity and the terminal velocity at',
and
respectively representing the slave r of a spacecraft 2 ' transfer to r 3 The initial velocity and the terminal velocity at',
and
respectively representing the spacecraft slave r 3 ' transfer to r f The initial velocity and the terminal velocity;
step 4.3, the velocity increment dv of four pulses i I =1,2,3,4 are each
Step 4.4, the optimization index and the constraint are expressed as:
J＝|dv 1 |+|dv 2 |+|dv 3 |+|dv 4 | (19)
and solving by using a differential evolution algorithm to obtain the four-pulse optimal transfer.
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