CN116541968A - Determination method of ground-moon DRO optimal transfer orbit - Google Patents

Determination method of ground-moon DRO optimal transfer orbit Download PDF

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CN116541968A
CN116541968A CN202310774499.0A CN202310774499A CN116541968A CN 116541968 A CN116541968 A CN 116541968A CN 202310774499 A CN202310774499 A CN 202310774499A CN 116541968 A CN116541968 A CN 116541968A
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杨洪伟
秦理民
李爽
徐聪
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Nanjing University of Aeronautics and Astronautics
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Abstract

The invention discloses a method for determining an optimal transfer orbit of a lunar DRO, which comprises the following steps: calculating a DRO track family through a differential correction method and a continuation method; the earth-moon transfer orbit design is carried out by combining the single lunar attraction assistance, and the transfer orbit segments define search parameters and search ranges; performing track recursion by utilizing a reverse integral of a circular limiting three-body problem dynamics model to obtain track parameters meeting the conditions; the optimization variables and performance indexes are designed, and the initial orbit parameters obtained by searching are segmented, so that the parameter sensitivity can be reduced to improve the solving efficiency of an optimization algorithm; and optimizing and solving the fuel optimal transfer track meeting the constraint condition by utilizing the segmented search results through a sequence secondary optimization algorithm.

Description

Determination method of ground-moon DRO optimal transfer orbit
Technical Field
The invention belongs to the technical field of aerospace, and particularly relates to a method for determining an optimal transfer orbit of a ground-month DRO.
Background
With the beginning of the united states, the european union, etc. to deploy lunar space stations (LOP-G) capable of covering lunar space, lunar space has become an important area of space development in the next stage. The construction of the near-month space station can greatly improve the long-term exploration capability of human beings for deep space exploration, and comprises the exploration tasks of other planets and asteroid of the solar system. DRO (Distant Retrograde Orbit, remote retrograde orbit) is first mentioned as a planar solution to CRTBP, this orbit being the orbit by henon when studying simple periodic symmetric orbits under Hill model, which is made up of orbits around the secondary main celestial body in retrograde fashion in the CRTBP rotational coordinate system. In previous studies, DROs were shown to be capable of exhibiting long-term stability. At the same time, however, stability results in inflexibility of the track mission design in terms of low cost transfer. Because DROs lack a constant flow type structure, conventional low energy transition trajectory designs cannot freely approach the DRO itself with constant flow. However, the stability of the DROs is critical for the structure of the new earth-moon space station. Therefore, it is necessary to study the low fuel consumption track transferred to the DRO in the earth-month space, design an earth-month DRO fuel optimal track calculation method, analyze the speed consumption and transfer time required by transfer, and obtain an optimal transfer track diagram.
The existing design method needs to build a huge database in advance, is large in calculation amount, and does not comprise an optimization solving process. In addition, existing design approaches are directed to only one particular DRO and not the entire track family.
Disclosure of Invention
In view of the above-mentioned shortcomings of the prior art, an object of the present invention is to provide a method for determining an optimal transfer trajectory of a DRO in the earth and moon, so as to design a search strategy that can calculate a transfer better than in the existing research and obtain a corresponding optimization result.
A method for determining an optimal transfer orbit of a lunar DRO comprises the following steps:
step 1: calculating a target DRO and a target insertion point;
step 2: determining search parameters and search ranges;
step 3: track recursion is performed by utilizing a reverse integral of a circular limiting three-body problem dynamics model, and a cut-off condition is set to obtain track design parameters meeting the condition;
step 4: designing optimization variables and performance indexes to construct a nonlinear optimization problem, and segmenting the initial track parameters obtained by searching to obtain indexing initialization conditions;
step 5: and (3) optimizing and solving the optimal fuel transfer track meeting the constraint condition by using a sequence secondary optimization algorithm based on the segmented initial condition obtained in the step (4).
As a preferable technical solution, the step 1 specifically includes:
under a rotating coordinate system that the motion of the spacecraft follows the motion of the two main celestial bodies, the state quantity of the spacecraft is thatThe kinetic equation is:
(1)
wherein r represents a position vector, v represents a velocity vector, and x, y and z represent components of the velocity v vector respectively;representation ofxIs to ask for the help of (1)>Representing the quadratic derivative of x, +.>Representation ofyIs to ask for the help of (1)>Representing the quadratic derivative of y, +.>Represents a derivative of z, +.>Representing a second derivative of z;
Urepresenting a fake of a systemPotential energy function, defined as:
(2)
wherein, the liquid crystal display device comprises a liquid crystal display device,for the mass ratio of the system, +.>M is lunar mass, M is earth mass;
the distance between the spacecraft and the main star and the secondary star are respectivelyr 1 Andr 2 ：
(3)
the jacobian constant j is expressed as:
(4)
correction of initial velocity and time of flight by multiple differential correction processesTThe terminal position is enabled to meet the condition to obtain DRO, and then the complete DRO family is obtained through the natural parameter prolongation process;
the target insertion point is separated from the initial point by definition on the DROX 0 Time of flightt dro The target position after that.
As a preferable technical solution, the step 2 specifically includes:
determining search parameters and search ranges, the search parameters includingxShaft speed incrementSpeed increment of y axisTransfer time->；
Taking all initial values which can reach the height of the target lunar borrowing point to perform subsequent searching;
the initial value search of the lunar attraction auxiliary is carried out at the lunar borrowing point, and the initial value search comprises the speed incrementSpeed increment angleaAnd transfer time->。
As a preferable technical solution, the step 3 specifically includes:
applying insertion maneuver at a target position point, obtaining a track capable of reaching the height of a target lunar borrowing force point in a reverse integration mode, applying lunar fly maneuver at the lunar borrowing force point, obtaining a transfer track from the lunar to a low earth orbit in a reverse integration mode, and stopping integration if the track collides with the lunar;
all initial values which can reach the height of the target near-place and can not collide with the moon are taken.
As a preferable technical solution, the step 4 specifically includes:
the lunar gravity point meets the height of the near moonh M And relative moon track angleConstraint that the near-site meets the near-site heighth E Relative to earth track angle->Constraint;
the track angle being the angle between the velocity vector and the normal perpendicular to the position vector, i.eWherein->A position vector for the spacecraft relative to the earth or moon;
for lunar DRO track segments, free variablesC 1 Comprising speed incrementsWith lunar to DRO time of flightThe method comprises the steps of carrying out a first treatment on the surface of the Free variable for near-site and lunar force-borrowing point track sectionC 2 Comprising the speed increment of the force-aid point>Transfer time with earth month->The free variable set is:
(5)
the equation constraint equation is:
(6)
in the method, in the process of the invention,representing a selected near moon altitude, +.>Indicating the selected relative lunar track angle, +.>Indicating the selected near-ground altitude, +.>Representing a selected relative earth track angle;
to prevent the transfer trajectory from colliding with the moon or earth, an inequality constraint equation is defined as:
(7)
in the middle ofUDIn order to normalize the length units,R m andR e radius for moon and earth;
the objective function is:
(8)。
as a preferable technical solution, the step 5 specifically includes: and (3) solving the optimal transfer problem in a segmented mode according to the segmentation initialization conditions obtained in the step (4) by combining a sequence quadratic programming algorithm, and finally obtaining the transfer track with the minimum speed increment consumption.
The invention has the beneficial effects that:
the method is suitable for the design of the optimal transfer track of the DRO fuel under the circular restriction three-body model, a double grid search strategy is adopted to obtain detailed initial value guess of the track design, and the optimal transfer track of the fuel is obtained by segmenting the initial value of the track design obtained by searching and combining a sequence secondary optimization algorithm. The calculated transfer trajectory is less fuel-consuming than in previous studies.
Drawings
Fig. 1 is a flow chart of a method for determining an optimal transfer orbit of a DRO in a ground month according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of a final DRO family according to an embodiment of the present invention;
FIGS. 3 and 4 are graphs of the change in speed delta and time of flight versus Jacobian constant for a lunar DRO transfer according to one embodiment of the invention;
fig. 5 is a schematic diagram of a geodetic DRO optimal transfer track according to an embodiment of the present invention.
Detailed Description
The invention will be further described with reference to examples and drawings, to which reference is made, but which are not intended to limit the scope of the invention.
Referring to fig. 1, the embodiment provides a method for determining an optimal transfer track of a DRO in the earth and moon, which comprises the following steps:
1) Calculating a target DRO and a target insertion point;
2) Defining search parameters and search ranges;
3) Track recursion is performed by utilizing a reverse integral of a circular limiting three-body problem dynamics model, and a cut-off condition is set to obtain track design parameters meeting the condition;
4) Designing optimization variables and performance indexes to construct a nonlinear optimization problem, and segmenting initial track parameters obtained by searching;
5) And (3) optimizing and solving the optimal fuel transfer track meeting the constraint condition by using a sequence secondary optimization algorithm based on the obtained segmentation initial condition in the step (4).
Taking a ground month DRO fuel optimal transfer track solution with a target insertion point as a remote point as an example:
step 1, in the planar circular limiting trisomy problem, in the planar circular limiting trisomy application, the distance between celestial bodies, the motion period of two celestial bodies and the mass of the system are typically normalized to 1, 2 pi and 1, respectively. Mass ratio of the systemWhere M is lunar mass and M is earth mass. The movement of the spacecraft is defined in a rotational coordinate system following the movement of the two main celestial bodies, the state quantity of the spacecraft is +.>The kinetic equation is:
for a DRO with transverse amplitude of 80000km, it is assumed that=0.52, track integration was performed by the kinematic equation defined above. Correction of the initial speed by several differential correction processes>And time of flightTThe end position is made to meet the conditionAnd obtaining the complete DRO family through a natural parameter continuation process after DRO is reached. The final DRO family is shown in fig. 2. All tracks were integrated with an ode45 integrator in MATLAB, with absolute and relative errors set to 1X 10, respectively −14 And 1X 10 −12 。
The target insertion point is separated from the initial point by definition on the DROX 0 Flyingt dro =T p The target position after the time/2,T p is the orbital period of the DRO.
Step 2, determining search parameters and search ranges, wherein the search parameters comprisexShaft speed incrementY-axis speed increment->Transfer time->；
Taking all initial values which can reach the height of the target lunar borrowing point to perform subsequent searching;
the initial value search of the lunar attraction auxiliary is carried out at the lunar borrowing point, and the initial value search comprises the speed incrementSpeed increment angleaAnd transfer time->The initial search range is shown in table 1:
TABLE 1 initial search Range for Fuel optimal parameters
Parameters (parameters) Minimum value Maximum value Spacing of
xShaft speed increment -0.1 0.1 0.002
xShaft speed increment -0.1 0.1 0.002
Moon to DRO time of flight 0 -4 ——
Speed increment size 0.1 0.2 0.01
Speed increment angle 0 2π π/36
Transfer time 0 -2 ——
All parameters in the table are expressed in normalized units.
And step 3, applying insertion maneuver at the target position point, obtaining a track capable of reaching the height of the target lunar aid force point in a reverse integration mode, applying lunar fly maneuver at the lunar aid force point, obtaining a transfer track from the lunar to the low earth orbit in a reverse integration mode, and stopping integration if the track collides with the lunar. And taking all initial values which can reach the height of the target near-site and cannot collide with the moon to carry out subsequent design.
Step 4, in order to construct the fuel optimal orbit, the lunar gravity-assisted point meets the height of the near-moon pointh M And relative moon track angleConstraint that the near-site meets the near-site heighth E Relative to earth track angle->Constraint;
the track angle being the angle between the velocity vector and the normal perpendicular to the position vector, i.eWherein->A position vector for the spacecraft relative to the earth or moon;
for lunar DRO track segments, free variablesC 1 Comprising speed incrementsWith lunar to DRO time of flightThe method comprises the steps of carrying out a first treatment on the surface of the Free variable for near-site and lunar force-borrowing point track sectionC 2 Comprising the speed increment of the force-aid point>Transfer time with earth month->The free variable set is:
(5)
the equation constraint equation is:
(6)
in the method, in the process of the invention,representing a selected near moon altitude, +.>Indicating the selected relative lunar track angle, +.>Indicating the selected near-ground altitude, +.>Representing a selected relative earth track angle;
to prevent the transfer trajectory from colliding with the moon or earth, an inequality constraint equation is defined as:
(7)
in the middle ofUDIn order to normalize the length units,R m andR e radius for moon and earth;
the objective function is:
(8)。
according to the initial guess obtained in the step 3, which is transferred from the Earth LEO (Low Earth Orbit) to the DRO, all the initial guesses are directly put into an optimization program, so that the optimization algorithm may fail and an optimal track cannot be obtained, and the searched initial values can be divided into intervals, so that the performance and the calculation efficiency of searching the optimal solution by the optimization algorithm can be greatly improved.
And 5, selecting a near-place height of 400km, a relative earth track angle of 0 degree, a near-moon height of 200km and a relative moon track angle of 0 degree according to the segmentation initial guess obtained in the step 4. And solving the optimal transfer problem in a segmented way by combining a sequence quadratic programming algorithm, and finally obtaining the transfer track with the minimum speed increment consumption. The nonlinear optimization problem is solved by using an 'fmincon' optimization solver of MATLAB, a solving algorithm is set as 'sqp', and a constraint tolerance is set as 1 multiplied by 10 −8 . The change relation between the speed increment and the flight time required by the ground month DRO transfer and the jacobian constant is shown in fig. 3 and 4, the transfer track is shown in fig. 5, the abscissa in fig. 5 is dimensionless, and the normalized length is the ground month distance.
According to the final solving result, the method selects the target DRO with a larger jacobian constant range and finds a more optimal earth-moon transfer track, and according to experiments, compared with the transfer track in the technical scheme provided by the prior art, the technical scheme provided by the embodiment saves about 10% of fuel consumption.
The present invention has been described in terms of the preferred embodiments thereof, and it should be understood by those skilled in the art that various modifications can be made without departing from the principles of the invention, and such modifications should also be considered as being within the scope of the invention.

Claims (6)

1. The method for determining the optimal transfer orbit of the lunar DRO is characterized by comprising the following steps of:
step 1: calculating a target DRO and a target insertion point;
step 2: determining search parameters and search ranges;
step 3: track recursion is performed by utilizing a reverse integral of a circular limiting three-body problem dynamics model, and a cut-off condition is set to obtain track design parameters meeting the condition;
step 4: designing optimization variables and performance indexes to construct a nonlinear optimization problem, and segmenting the initial track parameters obtained by searching to obtain indexing initialization conditions;
step 5: and (3) optimizing and solving the optimal fuel transfer track meeting the constraint condition by using a sequence secondary optimization algorithm based on the segmented initial condition obtained in the step (4).
2. The method according to claim 1, wherein the step 1 specifically includes:
in a rotating coordinate system in which the movement of the spacecraft follows the earth and moon movements, the state quantity of the spacecraft isThe kinetic equation is:
(1)
wherein r represents a position vector, v represents a velocity vector, and x, y and z represent components of the velocity v vector respectively;representation ofxIs to ask for the help of (1)>Representing the quadratic derivative of x, +.>Representation ofyIs to ask for the help of (1)>Representing the quadratic derivative of y, +.>Represents a derivative of z, +.>Representing the second order of zSeeking a derivative;
Ua pseudo potential energy function representing a system is defined as:
(2)
wherein, the liquid crystal display device comprises a liquid crystal display device,for the mass ratio of the system, +.>M is lunar mass, M is earth mass;
the distance between the spacecraft and the main star and the secondary star are respectivelyr 1 Andr 2 ：
(3)
the jacobian constant j is expressed as:
(4)
correction of initial velocity and time of flight by multiple differential correction processesTThe terminal position is enabled to meet the condition to obtain DRO, and then the complete DRO family is obtained through the natural parameter prolongation process;
the target insertion point is separated from the initial point by definition on the DROX 0 Time of flightt dro The target position after that.
3. The method according to claim 1, wherein the step 2 specifically includes:
determining search parameters and search ranges, the search parameters includingxShaft speed incrementSpeed increment of y axisTransfer time->；
Taking all initial values which can reach the height of the target lunar borrowing point to perform subsequent searching;
the initial value search of the lunar attraction auxiliary is carried out at the lunar borrowing point, and the initial value search comprises the speed incrementSpeed increment angleaAnd transfer time->。
4. The method according to claim 1, wherein the step 3 specifically includes:
applying insertion maneuver at a target position point, obtaining a track capable of reaching the height of a target lunar borrowing force point in a reverse integration mode, applying lunar fly maneuver at the lunar borrowing force point, obtaining a transfer track from the lunar to a low earth orbit in a reverse integration mode, and stopping integration if the track collides with the lunar;
all initial values which can reach the height of the target near-place and can not collide with the moon are taken.
5. The method according to claim 1, wherein the step 4 specifically includes:
lunar force-borrowing point meeting heighth M And the relative moon flight path angleConstraint, near-site meeting heighth E Relative to earth track angle->Constraint;
the track angle being the angle between the velocity vector and the normal perpendicular to the position vector, i.eWhereinA position vector for the spacecraft relative to the earth or moon;
for lunar DRO track segments, free variablesC 1 Comprising speed incrementsAnd moon to DRO flight time +.>The method comprises the steps of carrying out a first treatment on the surface of the Free variable for near-site and lunar force-borrowing point track sectionC 2 Comprising the speed increment of the force-aid point>Transfer time with earth month->The free variable set is:
(5)
the equation constraint equation is:
(6)
representing a selected near moon altitude, +.>Indicating the selected relative lunar track angle, +.>Indicating the selected near-ground altitude, +.>Representing a selected relative earth track angle;
to prevent the transfer trajectory from colliding with the moon or earth, an inequality constraint equation is defined as:
(7)
in the middle ofUDIn order to normalize the length units,R m andR e radius for moon and earth;
the objective function is:
(8)。
6. the method according to claim 1, wherein the step 5 specifically includes: and (3) solving the optimal transfer problem in a segmented mode according to the segmentation initialization conditions obtained in the step (4) by combining a sequence quadratic programming algorithm, and finally obtaining the transfer track with the minimum speed increment consumption.
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