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Вестник РУДН. Серия: Инженерные исследования RUDN Journal of Engineering Research
2024;25(1):7-20
ISSN 2312-8143 (Print); ISSN 2312-8151 (Online) journals.rudn.ru/engineering-researches
DOI: 10.22363/2312-8143-2024-25-1-7-20 UDC 629.78 EDN: FDPCQT
Research article / Научная статья
Non-Coplanar Rendezvous in Near-Circular Orbit with the Use a Low Thrust Engine
Andrey A. Baranov1 , Adilson P. Oliviob H
a Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, Russia b RUDN University, Moscow, Russia
H pedrokekule@mail.ru
Article history
Received: November 5, 2023 Revised: January 9, 2024 Accepted: January 14, 2024
Conflicts of interest
The authors declare that there is no conflict of interest.
Authors' contribution
Undivided co-authorship.
Abstract. Presented method allows one to calculate the of maneuvers performed on several turns using a low-thrust engine. These maneuvers ensure the flight of an active spacecraft within a given area of the target space object. The flight is carried out in the vicinity of a circular orbit. Simplified mathematical models of motion are used to solve this problem. The influence of the non-centrality of the gravitational field and atmosphere is not taken into account in the calculations. The process of determining the parameters of the maneuvers is divided into several stages: in the first and third stages, the parameters of the impulse transfer and the transfer carried out by the low-thrust engine are calculated using analytical methods. In the second stage, the distribution of maneuvering between turns, ensuring a successful solution to the meeting problem, is determined by changing one variable. This method is characterized by its simplicity and high reliability in determining the parameters of maneuvers, which makes it applicable on board a spacecraft. As part of the study, an analysis of the dependence of the total characteristic velocity of solving the meeting problem on the amount of engine thrust was also carried out. The maneuver parameters can be refined using an iterative procedure to take into account the main disturbances.
Keywords: spacecraft, near-circular orbit, velocity impulse, calculation of maneuver parameters, space object, low thrust engine
For citation
Baranov AA, Olivio AP. Non-coplanar rendezvous in near-circular orbit with the use a low thrust engine. RUDN Journal of Engineering Research 2024;25(1):7-20. http://doi.org/10.22363/2312-8143-2024-25-1-7-20
© Baranov A.A., Olivio A.P., 2024
(D© I This work is licensed under a Creative Commons Attribution 4.0 International License KzM^K^H https://creativec0mm0ns.0rg/licenses/by-nc/4.Q/legalc0de
Некомпланарная встреча на околокруговой орбите с помощью двигателя малой тяги
A.A. Баранов3 , А.П. Оливиоь а
a Институт прикладной математики имени М.В. Келдыша РАН, Москва, Россия b Российский университет дружбы народов, Москва, Россия
н pedrokekule@mail.ru
История статьи
Поступила в редакцию: 5 ноября 2023 г. Доработана: 9 января 2024 г. Принята к публикации: 14 января 2024 г.
Заявление о конфликте интересов
Авторы заявляют об отсутствии конфликта интересов.
Вклад авторов
Нераздельное соавторство.
Аннотация. Представлен метод, позволяющий вычислить параметры маневров, выполняемых на нескольких витках с применением двигателя малой тяги. Эти маневры обеспечивают перелет активного космического аппарата в пределы заданной области целевого космического объекта. Перелет осуществляется в окрестности круговой орбиты. Для решения данной задачи применяются упрощенные математические модели движения. Влияние нецентральности гравитационного поля и атмосферы в расчетах не учитывается. Процесс определения параметров маневров разбит на несколько этапов: на первом и третьем этапах параметры импульсного перехода и перехода, осуществляемого двигателем малой тяги, вычисляются с использованием аналитических методов. На втором этапе распределение маневрирования между витками, обеспечивающее успешное решение задачи встречи, определяется путем изменения одной переменной. Данный метод отличается простотой и высокой надежностью в определении параметров маневров, что делает его применимым на борту космических аппаратов. В рамках исследования также проведен анализ зависимости суммарной характеристической скорости решения задачи встречи от величины тяги двигателя. Параметры маневров могут быть уточнены с помощью итерационной процедуры, чтобы учесть основные возмущения.
Ключевые слова: космический аппарат, околокруговая орбита, импульс скорости, расчет параметров маневров, космический объект, малая тяга
Для цитирования
Baranov A.A., Olivio A.P. Non-coplanar rendezvous in near-circular orbit with the use a low thrust engine // Вестник Российского университета дружбы народов. Серия: Инженерные исследования. 2024. Т. 25. № 1. С. 7-20. http://doi.org/ 10.22363/2312-8143-2024-25-1-7-20
Introduction
The problem of meeting in a near-circular orbit using low-thrust engines is important in the practice of spacecraft (SC) flights. Typical examples are the problem of rendezvous and docking of spacecraft, the implementation of a group flight of several spacecraft, the formation of a given configuration of satellite systems, during removal of space debris, during servicing of spacecraft and other astro-nautical missions involving more than one spacecraft.
Due to the great complexity of solving problems of spacecraft meeting with greater accuracy, over the past few years many authors have developed algorithms for solving the problem of spacecraft meeting [1-2].
Currently, three main approaches are widely used in solving complex problems of multi-impulse maneuvering of spacecraft. In the first case, the problems of maneuvering in the orbital plane and the problems of rotating the orbital plane are solved independently. This scheme was used, for example, when approaching the Shuttle spacecraft with an
orbital station, to control the movement of geostationary satellites [3], satellites within satellite systems [4-6], and so on. The advantage of this scheme lies in its simplicity and reliability, and the disadvantage is the excessive cost of characteristic velocity for maneuvering.
In the second case, numerical methods are used to find optimal solutions for the most complex multi-impulse problems, taking into account a wide range of restrictions [7-8]. The simplex method is most often used [9-10].
In the third method, at the initial stage, the solution to the Lambert problem is used to determine the parameters of the two-impulse solutions to the meeting problem. Then the behavior of the hodograph of the basis vector corresponding to the found solution is analyzed, and, if necessary, additional velocity impulses are added to obtain the optimal solution. This approach was first used in the works of Lion and Handelsman [11], Jezewski and Rozendaal [12].
There are also methods that are at the intersection of different approaches. For example, in [13-14] numerical and analytical methods for solving the multi-impulse meeting problem are proposed, combining the advantages of the first and second of the previously listed approaches. They make it possible to use the results obtained in the early papers of T. Edelbaum [15], J.P. Marec [2], when solving modern practical problems.
Since the 1960s, the process of using electric rocket engines (ERM) on spacecraft began. Thanks to their high specific impulse, electric propulsion engines can significantly reduce fuel costs for orbital maneuvering. However, the low (compared to traditional liquid rocket engines) thrust of electric propulsion engines leads to the need to take into account their long-term operation.
Problems of this type take a special place a special place among the problems of optimal maneuvering of a spacecraft. A significant number of articles have been devoted to them, and several very interesting monographs have been published [16; 17]. Particularly noteworthy are the papers of V.G. Petukhov [18-20]. Due to the complexity of the problems in which it is assumed that maneuvering is carried out using a propulsion system (PS), they have traditionally been solved numerically and by methods using the Pontryagin maximum principle or the continuation method. In recent years, Yu.P. Ulybyshev has successfully used
the interior point method to solve problems with long maneuvers [21].
In the method considered in this paper, the non-coplanar meeting problem is solved both in the impulse formulation and taking into account the long-term operation of the low-thrust engine [22-24].
To analyze the relative motion of a spacecraft in the vicinity of circular orbits, it is necessary to use special mathematical models of motion. The most popular mathematical model of the relative motion of a spacecraft in the vicinity of circular orbits is the Hill-Clohessy-Wiltshire (HCW) model. Linearized differential equations for the relative motion of a spacecraft in the vicinity of a circular orbit for the problem of rendezvous and docking were obtained by Clohessy-Wiltshire in 1960 [25], but back in the 19th century similar equations were used by Hill in his theory of lunar motion [26].
In this mathematical model, to obtain the equations of relative motion, a rotating (orbital) coordinate system and linearization of the differential equations of relative motion are used, based on the assumption that the distance between the considered spacecraft is small compared to the average orbital radius. This work uses linearized equations obtained by P.E. Elyasberg [27]. They were obtained using a cylindrical coordinate system and are significantly more convenient for solving the problem of long-duration encounters, in which there are significant deviations along the orbit.
Due to the increase in the number of maneuvering spacecraft and the increase in the efficiency of solving problems, there is currently a tendency to transfer the process of calculating maneuvers on board the spacecraft. This leads to the need to simplify the process of calculating maneuver parameters and increase the reliability of this process. The algorithm considered in this paper has precisely these properties.
1. Formulation of the meeting problem
The problem of calculating the parameters of transfer maneuvers between close near-circular orbits is solved in an approximate impulse formulation, within the framework of unperturbed Keplerian motion.
The conditions for transferring with the help of N velocity impulses in a fixed time from the initial orbit to a given point of the final non-coplanar orbit (meeting problem) in a linear approximation can be written in the form Ilyin and Kuzmak [22]:
Z (A V>n9i +2A Vti cos 9y )=Ae,, (1)
i=1
N
Z (-A Vri cos 9i+2A Ft,.sin9i )=Ae,, (2)
i=1
Z 2 A Vti = A a, (3)
i=1
N
Z (2A v. (1 - cos 9i)+A Vti (-39i+4sin9i))=At, (4)
If=1-AKz¿sin9¿ = Az,
Ef=1 AVzi cos9¿ =AKZ,
(5)
(6)
where
Aex = e/cos©/ - eocos©o, Aey= e/sin©/ - eosin©o, Aa=(a/-ao)/ro, At=ho(t/-to),
Az= zo/ro, AVz=AVzo/Vo,
AVti = a Vt* /Vo, AVri = A Vr* /Vo, AV^AV^Wo.
Here «/», «o» — the indices corresponding to the final and initial orbits, e/, eo — the eccentricities of the orbits; a/, ao — semi-axes major of orbits; ©/, ©o — angles between the direction to the pericenter of the corresponding orbit and the direction to a point specified on the final orbit (the Ox axis is directed to this point); t/ — the required time of arrival at a given point, to — the time at which, when moving along the initial orbit, the projection of the radius vector onto the plane of the final orbit hits the ray passing through the given meeting point; zo — the deviation of the spacecraft in the initial orbit from the plane of the final orbit at time to; Vzo — lateral relative velocity at this moment; Vo, ho — orbital and angular velocities of movement along the reference circular orbit of radius ro (ro = a/); N — number of velocity impulses; 9¿ — the angle of application of the i-th velocity impulse, measured from the direction to a given meeting point in the direction of the SC motion; a Vt* , A Vr* , AVZ* — transversal, radial
and lateral components of the i-th velocity
impulse, respectively. It is necessary to take into account that the angles 9i — negative, because it was assumed that at a given point 9/= 0.
The problem of searching for parameters of optimal maneuvers can be formulated as follows: it is necessary to determine AV„-, AVi, AVzi, 9i (i = 1, ..., N), at which the total characteristic velocity of maneuvers AV is minimal.
AV
¿=1 ¿=1
AVr2¿ + A72 + A7Z2¿,
under restrictions (1)-(6).
In this paper problem of the meeting is solved in several stages. At the first stage, the problem of impulses transfer between non-coplanar orbits is solved (Section 2). The velocity impulses for solving the transfer problem are then distributed among the turns allowed for maneuvering to ensure that equation (4) is satisfied (Section 3). In sections (4) and (5), a solution to the low-thrust transfer problem is sought.
The maneuver parameters can be refined using an iterative procedure to take into account all disturbances (the influence of compression of the Earth, atmosphere, etc.).
2. Algorithm for solving the transfer problem
When solving the problem of transfer between non-coplanar orbits, five equations of the system (1)-(6) are used.
The angle ^ (the angle of application of the first velocity impulse) is searched and for each successive value of the angle the values of the velocity impulses and the angle are found:
Ae2 - Aa2
AVm = —f-T,
4(AeySin^1^ + Aexcos<p^f - A a)
AVt2=f-AVtl,
-f- - AVtl sin
tan^2 = a£-
— - AVtlcos<plf
(7)
(8) (9)
and then from equations (5)-(6) the values of the lateral components of the velocity impulses are determined:
AV,
zl
= -(■
Az cos + AVZ sin
(1o)
i=1
AV,
z2
Az COS ф! + àVz sin ф-L З1п(ф1-ф2)
(11)
From the entire set of solutions found, the one that provides the minimum total characteristic velocity is selected. Further, the parameters of this solution are indicated by the index «m» SVt\m,
AVz\m, 91m, AVt2m, AVz2m, ty2m.
3. Algorithm for solving the meeting problem
When solving the meeting problem, the values of the velocity impulses AVt1, AV2, determined when solving the transfer problem, are distributed among N turns allowed for maneuvering:
Here AVm, AVun h AV2t1, AV2m are the magnitudes of the velocity impulses on the first and last permitted turns of maneuvering, which are a part of the first and second velocity impulses of solving the transfer problem.
Substituting the values of velocity impulses calculated using formulas (15), (16) into (12) and (13) we obtain:
Wltm = 2?=1Wlti = 0.5N(Wltl + AVltN); (17) W2tm = Tf=1AV2ti = 0.5N(AV2tl + W2tN). (18)
Using (17) and (18), we obtain formulas for determining a v a v :
0 " r 1tN r 2 tN
N
AVltm=Z avw ; i=1
N
W2m =ZAV2ti .
(12)
(13)
i=1
Here N is the number of turns on which maneuvering is allowed.
The lateral components are distributed in proportion to the transversal
|ДМ
and
AV2zi=^AK2zm
|AVs
2t1
(14)
The further goal is to select such a distribution of the magnitudes of the velocity impulses along the turns so that equation (4) is satisfied.
To significantly simplify the solution of the problem, we assume that the magnitudes of the velocity impulses along the turns change linearly:
A VUl = Л Vm +
AV2tN =
0.5 N
■w2tl.
(19)
(20)
Substituting the found values a v A V into
0 ""UN'" r2 tN
formulas (15) and (16), we obtain:
AVlti =
= 2AV1(i - 1)/N(N -1) + AVltl [l - ^g2], (21)
W2ti =
= 2AV2(i - 1)/N(N -1) + AV2tl [l -(22)
Thus, we found the values of all velocity impulses, expressed only in terms of AVm and AV2ti. Substituting them into equation (3), we obtain a linear equation with two unknowns a V111, A V211.
The coefficients for velocity impulses are known, since their angles of application are known:
Ф 1i = Ф 1 m + 2 n( N i - N ), Ф 2 i = Ф 2 m + 2 n ( N i - N ) .
(23)
(24)
By sorting through the value of the variable +( A V1tN -A V1t1)( i - 1)/( N - 1), (15) AVm, within the specified limits, for each value
from equation (3) we find the value of the variable
Л V2 й = Л V2 +
AVM.
+(A V2 N -A V2 n)( i - 1)/( N - 1). (16)
Then, using (23) and (24), we find the values of all velocity impulses. Adding the modules of all
velocity impulses, we find the total characteristic velocity of the next solution. The solution whose total characteristic velocity is minimal is accepted as a solution to the meeting problem. If the total characteristic velocity of the found solution coincides with the total characteristic velocity of the solution to the transfer problem, then a solution with the minimum possible total characteristic velocity was found.
If the duration of the largest velocity impulse does not exceed 20°, then the solution is close to an impulse one and we consider that the problem has already been solved. Taking into account all disturbances (non-centrality of the gravitational field, the influence of the atmosphere, etc.), the operation of a real propulsion system can be carried out using the iterative procedure described in Section 5. If the duration of the maneuvers is significant, then we proceed to solving the problem taking into account low thrust PS.
4. Solving the problem with «low thrust»
It is assumed that the orientation of the propulsion system during the execution of the maneuver is fixed in the orbital coordinate system.
For each turn, we find what changes in eccentricity and semi-major axis produce the found velocity impulses determined at this turn
Lelix=2AVlti cos + 2AV2ti cos q>2i, (25)
Aeliy =2AVlti sin + 2AF2ti sin (26)
Aat =2A7lti + 2AV2ti. (27)
Similarly for changing the lateral parameters on a turn
AViz=AFlzi cos ^ + AV2Z( cos (28)
AZi = AV1Z( sin + AV2ti sin (29)
Then we determine the required duration of low-thrust maneuvers that will produce the same change in these elements [20]:
A^! = 2arcsin^^, (30)
A^2 = 2arcsin^-^. (31)
Thus, turn by turn we find the duration of all maneuvers. The low thrust problem has been correctly solved. If the arcsine argument is greater than 1, then there is no solution (with the existing thrust and mass of the spacecraft, it is impossible to solve the meeting problem for a given number of turns).
The found solution with "low thrust" gives the same change in the eccentricity vector and orientation of the orbital plane as the original impulse solution, because the midpoints of long maneuvers coincide with the moments of application of velocity impulses.
However, the difficulty is that the change in the semi-major axis becomes larger than necessary, since it changes with orbital orientation more effectively than eccentricity. Therefore, as a result of the maneuvers, an error remains in the formation of the required value of the semi-major axis, and to eliminate this error, you can use the iterative procedure described in [20].
Let us assume that the initial deviation of the semi-major axis was Aa0 — Q,^ — Uq (for example, Aa0 > 0), and the deviations Aa0, Aex0, Aey0, Ai0, A-80 (the angle between the line of intersection of the orbital planes and the line of apses relative orbits) were used in determining the parameters of the maneuvers.
As a result of performing the calculation maneuvers, the semi-major axis at will be formed (a1 > Of). In the next iteration, deviations Aax = Aa0 + a,f — a1; Aex0, Aey0, Ai0, A-90 will be used, at the next iteration Aa2 = Aax + a^ — a2, etc., until the semi-major axis is formed with the given accuracy.
Since at each turn the same change in the semimajor axis will be made as in the impulse solution, the meeting problem will be solved with the same accuracy.
5. Iterative procedure
In the formulated meeting problem, linearized equations of motion are used, the non-centrality of the gravitational field, the influence of the atmosphere, etc. are not taken into account. This leads to the fact that the actual accuracy of fulfilling the terminal conditions in system (1)-(6) will be insufficient. To solve a problem with a given accuracy, you can use an iterative scheme [7-8], which consists of the following stages:
1. In the beginning of the next iteration, an "approximate" problem is solved: under the previously accepted simplifying assumptions, the parameters of maneuvers that ensure the formation of a "target" orbit are determined (at the first iteration, the "target" orbit coincides with the final orbit).
2. Then, taking into account the calculated maneuvers, using models of all necessary disturbances, an "accurate" prediction of the spacecraft motion is carried out and the parameters of the formed orbit are found.
3. The deviations of the parameters of the formed orbit from the corresponding parameters of the final orbit are calculated.
4. If the deviations exceed the permissible ones, then the parameters of the "target" orbit are changed by the value of the calculated deviations, and the next iteration is carried out.
5. The procedure ends when the terminal conditions are met with the specified accuracy.
6. For "accurate" forecasting, as a rule, numerical and/or high-precision numerical-analytical integration are used. It is possible to use different forecast methods at different iterations, but the accuracy of the forecast should increase with the number of the current iteration.
7. During numerical integration, the influence of the non-centrality of the gravitational field, atmosphere, light pressure, etc. is taken into account, the operation of the spacecraft engines is carefully modeled, therefore, despite the fact that the maneuver parameters are found at each iteration using the simplest motion model, but as a result of an iterative procedure, they ensure access to the final orbit with the required accuracy.
6. Example of solving the non-coplanar meeting problem
Let us consider the motion of a spacecraft (SC) relative to point O, moving in an undisturbed near circular orbit with a radius of 6871 km. Let us take the gravitational parameter of the Earth equal to 3.9860044-1014 m3/s2. Let us consider the flight problem using N velocity impulses in a fixed time from the initial orbit to a given point in the final orbit from a point in phase space r0 = (10, 100, -5) km, v0 = (1, -10, 3) m /c to the origin, that is, to the point rf = (0, 0, 0) km, with a velocity vf = (0, 0, 0) m/s. For the problem, we will take the initial mass of the spacecraft equal to 1000 kg, the specific
impulse of the spacecraft propulsion system is 220 seconds (2157.463 m/s), and the thrust (T) will be varied in the range from 1 to 100 N. The flight is carried out in N = 15 turns.
Solution of the two-impulse transfer problem
Table 1 shows the results of calculations of the parameters of the optimal two-impulse transfer between non-coplanar orbits, that is, the values of the transversal and lateral components of the velocity impulses, the angles of application of the first and second impulses are given as well. The angle of application of the first velocity impulse was varied from 0 to 360° with a step of 0.75°. It can be seen that the minimum value of the characteristic velocity that a spacecraft (SC) must have for the transfer maneuver is 10.308 m/s.
Multi-impulse solution to the meeting problem
To obtain an impulse solution to the meeting problem, the velocity impulses of the two-impulse solution are distributed between 15 turns so that condition (4) is satisfied. For this purpose, the algorithm described in Section 3. The value of the first velocity impulse is varied within the range from -3.452 m/s to 0.5 m/s with a step of 0.023 m/s.
Table 2 shows parameters of the distributed impulse solution
Table 3 shows the deviations of orbital elements for each turn corresponding to the influence of distributed velocity impulses.
This impulse solution can be transformed to take into account the real thrust of the engine.
The process of obtaining a solution for 1N thrust is shown below.
At the first stage, the durations of the maneuvers are calculated, which for a real low thrust (1N) provide the changes in the orbital elements shown in Table 3 at each orbit (except for the semi-major axis). These durations are shown in Table 4.
Then the change in the semi-major axis produced for a given duration of maneuvers is calculated and a new target value of the semi-major axis is formed for the next iteration. These data are shown in Table 5.
The next iteration is performed and the parameters of the new impulse solution, the duration of the maneuver and changes made of the semi-major axis under the influence of low thrust and errors in the correction of the semi-major axis are shown in Tables 6, 7 and 8.
Table 1
Results of the calculation the parameters of the optimal non-coplanar impulse transfer problem
° AVtl,m/s AVt2, m/s AVzl ,m/s AVz2,m/s AVt,m/s AVz,m/s AKj , m/s bM2 , m/s bM,m/s
155 55.851 -3.452 2.367 -0.637 -6.372 5.819 7.01 3.51 6.798 10.308
Table 2
Distribution of the two-impulse solution by turns
N AVtl ,m/s AVt2 ,m/s AVzl ,m/s AVz2 ,m/s AVt,m/s AVZ, m/s kV1, m/s bM2,m/s IV ,m/s
1 -0.022 0.314 -0.004 -0.844 0.336 0.848 0.023 0.9 0.923
2 -0.052 0.291 -0.01 -0.784 0.343 0.794 0.053 0.837 0.89
3 -0.082 0.269 -0.015 -0.724 0.351 0.739 0.083 0.773 0.856
4 -0.111 0.247 -0.021 -0.664 0.358 0.685 0.113 0.709 0.822
5 -0.141 0.224 -0.026 -0.604 0.366 0.63 0.143 0.645 0.788
6 -0.171 0.202 -0.031 -0.545 0.373 0.576 0.174 0.581 0.755
7 -0.2 0.18 -0.037 -0.485 0.38 0.522 0.204 0.517 0.721
8 -0.23 0.158 -0.043 -0.425 0.388 0.467 0.234 0.453 0.687
9 -0.26 0.136 -0.048 -0.365 0.395 0.413 0.264 0.389 0.653
10 -0.29 0.113 -0.053 -0.305 0.403 0.358 0.294 0.325 0.619
11 -0.319 0.091 -0.059 -0.245 0.41 0.304 0.325 0.261 0.586
12 -0.349 0.069 -0.064 -0.185 0.418 0.249 0.349 0.198 0.547
13 -0.379 0.047 -0.07 -0.125 0.425 0.195 0.385 0.134 0.519
14 -0.408 0.024 -0.075 -0.066 0.433 0.141 0.415 0.07 0.485
15 -0.438 0.002 -0.081 -0.006 0.44 0.087 0.446 0.006 0.452
S -3.452 2.367 -0.637 -6.372 5.819 7.01 3.51 6.798 10.308
Tabee 3
Results of deviations of orbital elements by turns
N Aeliz x 10"4 Aeliy x 10 4 Aet x 10"4 Vei ° Aaoi x 10"4 AVzi x 10"4 Azt x 10"4
1 -0.41 -0.71 0.816 59.877 0.765 0.627 -0.915 55.578
2 -0.306 -0.691 0.755 66.096 0.629 0.589 -0.847 55.163
3 -0.203 -0.675 0.705 73.302 0.492 0.552 -0.779 54.68
4 -0.099 -0.66 0.667 81.466 0.356 0.514 -0.711 54.112
5 0.005 -0.644 0.644 -89.598 0.219 0.476 -0.642 53.435
6 0.108 -0.629 0.638 -80.254 0.083 0.439 -0.574 52.612
7 0.212 -0.614 0.649 -70.979 -0.053 0.401 -0.506 51.594
8 0.315 -0.598 0.676 -62.228 -0.19 0.364 -0.438 50.302
9 0.419 -0.583 0.718 -54.318 -0.326 0.326 -0.37 48.609
10 0.522 -0.568 0.771 -47.388 -0.463 0.288 -0.302 46.299
11 0.626 -0.552 0.835 -41.432 -0.599 0.251 -0.234 42.976
12 0.729 -0.537 0.906 -36.362 -0.736 0.213 -0.166 37.832
13 0.833 -0.521 0.983 -32.057 -0.872 0.176 -0.097 29.028
14 0.936 -0.506 1.064 -28.395 -1.009 0.138 -0.029 11.991
15 1.04 -0.491 1.15 -25.267 -1.145 0.1 0.039 -21.159
Table 4
Duration of the maneuver for N = 15
N Афц ° AVzi° Лф, °
1 1.427 59.874 61.301
2 3.347 55.244 58.591
3 5.268 50.71 55.978
4 7.19 46.259 53.449
5 9.115 41.881 50.996
6 11.041 37.567 48.608
7 12.971 33.306 46.277
8 14.905 29.093 43.998
9 16.843 24.92 41.763
10 18.786 20.78 39.566
11 20.734 16.667 37.401
12 22.689 12.575 35.264
13 24.65 8.5 33.15
14 26.618 4.436 31.054
15 28.595 0.377 28.972
Table 5
Changes made of the semi-major axis under the influence of low thrust and errors in the correction of the semi-major axis
N Aaoi x 10"4 Aat x 10"4 Sat x 10"4 Aalf x 10"4
1 0.765 0.804 -0.039 0.727
2 0.629 0.659 -0.0304 0.598
3 0.492 0.516 -0.0235 0.469
4 0.356 0.374 -0.018 0.338
5 0.219 0.232 -0.013 0.2065
6 0.083 0.092 -0.0089 0.074
7 -0.053 -0.048 -0.0056 -0.059
8 -0.19 -0.187 -0.00278 -0.193
9 -0.326 -0.326 -0.00036 -0.327
10 -0.463 -0.464 -0.0018 -0.461
11 -0.599 -0.603 0.00375 -0.596
12 -0.736 -0.741 0.00565 -0.73
13 -0.872 -0.88 0.0076 -0.865
14 -1.009 -1.018 0.0097 -0.999
15 -1.145 -1.157 0.01203 -1.133
Table 6
Parameters of the new impulse solution for N = 15
N AVtl ,m/s AVt2 , m/s AVzl ,m/s AVz2,m/s AVt,m/s AVZ, m/s AV1, m/s AV2 , m/s AV,m/s
1 -0.04 0.317 0.045 0.843 0.357 0.888 0.060 0.901 0.961
2 -0.066 0.294 0.029 0.791 0.36 0.82 0.072 0.844 0.916
3 -0.092 0.271 0.014 0.73 0.363 0.744 0.093 0.779 0.872
13 -0.375 0.046 -0.079 0.124 0.421 0.203 0.383 0.132 0.515
14 -0.404 0.024 -0.086 0.065 0.428 0.151 0.413 0.069 0.482
15 -0.435 0.003 -0.089 -0.009 0.438 0.098 0.444 0.009 0.453
S -3.501 2.378 -0.501 5.402 5.879 7.073 3.585 6.82 10.405
It can be seen that the accuracy of the semimajor axis formation has increased.
It took four iterations to solve the problem. The information about the fourth iteration is given below (in Table 9).
Fourth iteration. In the next iteration, an impulse solution is first sought for the deviations of the orbital elements at each turn.
Then, the duration of the maneuvers is determined and shown in Table 10.
The change made in the semi-major axis is determined and shown in Table 11.
The good accuracy of the semi-major axis formation was obtained, so the iterative procedure is completed.
The duration of maneuvers is converted into impulse values. These results are shown in Table 12.
Maneuvers are calculated in a similar way for various thrust values from a given range.
The results are shown in the summary Table 13.
Table 7
Duration of the maneuver for N= 15
N A<Pu ° A^2i°
1 3.82 60.588 64.408
2 4.57 55.777 60.347
3 5.944 51.103 57.047
13 24.551 8.425 32.976
14 26.499 4.379 30.878
15 28.498 0.636 29.134
Table 8
Changes made of the semi-major axis under the influence of low thrust and errors in the correction of the semi-major axis
N Aaoi x 10"4 Aat x 10"4 Sat x 10"4 Aali x 10"4
1 0.765 0.717 0.00483 0.775
2 0.629 0.617 0.0116 0.61
3 0.492 0.494 -0.0017 0.467
4 0.356 0.3614 -0.0055 0.333
5 0.219 0.225 -0.00569 0.201
6 0.083 0.0876 -0.00461 0.0694
7 -0.053 -0.0503 -0.00314 -0.0622
8 -0.19 -0.188 -0.00164 -0.194
9 -0.326 -0.326 -0.00022 -0.327
10 -0.463 -0.454 0.0011 -0.46
11 -0.599 -0.602 0.00233 -0.593
12 -0.736 -0.723 -0.0124 -0.7425
13 -0.872 -0.869 0.00466 -0.867
14 -1.009 -1.014 0.00566 -0.993
15 -1.145 -1.15 0.00437 -1.129
Table 9
Parameters of the next impulse solution for N= 15
N AVtl ,m/s AVt2,m/s AVzl ,m/s AVz2,m/s AVt,m/s AVz,m/s kV1,m/s bM2, m/s IV, m/s
1 -0.031 0.315 0.019 0.848 0.346 0.867 0.036 0.905 0.941
2 -0.061 0.293 0.015 0.788 0.354 0.803 0.063 0.841 0.904
3 -0.093 0.271 0.016 0.73 0.364 0.746 0.094 0.779 0.873
13 -0.373 0.046 -0.084 0.124 0.419 0.208 0.382 0.132 0.514
14 -0.401 0.024 -0.096 0.065 0.425 0.161 0.412 0.069 0.481
15 -0.434 0.005 -0.093 -0.012 0.439 0.105 0.444 0.013 0.457
S -3.496 2.381 -0.517 6.382 5.877 7.039 3.562 6.834 10.396
Table 10
Duration of the maneuver for N = 15
N Афи° Лф2, °
1 2.273 60.18 62.453
2 3.991 55.573 59.564
3 5.999 51.128 57.127
13 24.499 8.401 32.9
14 26.406 4.437 30.843
15 28.463 0.832 29.295
Table 11
Changes made of the semi-major axis under the influence of low thrust and errors in the correction of the semi-major axis
N Aaoi x 10"4 Aat x 10"4 Sat x 10"4 Aali x 10"4
1 0.765 0.774 -0.00872 0.738
2 0.629 0.638 -0.009 0.6
3 0.492 0.492 0.000198 0.468
4 0.356 0.356 0.0000162 0.333
5 0.219 0.219 -0.0000054 0.2
6 0.083 0.0831 -0.000121 0.0672
7 -0.053 -0.0532 -0.000263 -0.0646
8 -0.19 -0.19 -0.0002505 -0.385
9 -0.326 -0.326 -0.0000469 -0.327
10 -0.463 -0.463 0.000289 -0.458
11 -0.599 -0.6 0.000481 -0.589
12 -0.736 -0.734 -0.00196 -0.745
13 -0.872 -0.875 0.0029 -0.857
14 -1.009 -1.01 0.001031 -0.987
15 -1.145 -1.145 0.000124 -1.127
Tabee 12
Parameters of the solution with low thrust for N= 15
N AVtl,m/s AVt2, m/s AVzl ,m/s A Vz2,m/s A Vt,m/s A Vz,m/s ДК-l ,m/s W2 ,m/s IV,m/s
1 0.035 0.33 0.006 -0.888 0.365 0.894 0.036 0.947 0.983
2 0.062 0.305 0.011 -0.82 0.367 0.831 0.063 0.875 0.938
3 0.093 0.28 0.017 -0.755 0.373 0.772 0.095 0.805 0.900
4 0.12 0.256 0.022 -0.688 0.376 0.71 0.122 0.734 0.856
5 0.148 0.231 0.027 -0.623 0.379 0.650 0.150 0.664 0.815
6 0.176 0.207 0.032 -0.558 0.383 0.59 0.179 0.595 0.774
7 0.204 0.184 0.038 -0.494 0.388 0.532 0.208 0.527 0.735
8 0.232 0.16 0.043 -0.431 0.392 0.474 0.236 0.460 0.696
9 0.261 0.137 0.048 -0.368 0.398 0.416 0.265 0.393 0.658
10 0.29 0.114 0.054 -0.306 0.404 0.36 0.295 0.327 0.622
11 0.319 0.091 0.059 -0.245 0.41 0.304 0.324 0.261 0.586
12 0.349 0.07 0.064 -0.187 0.419 0.251 0.355 0.200 0.554
13 0.379 0.046 0.07 -0.124 0.425 0.194 0.385 0.132 0.518
14 0.409 0.024 0.075 -0.065 0.433 0.14 0.416 0.069 0.485
15 0.441 0.005 0.081 -0.012 0.446 0.093 0.448 0.013 0.461
S 3.518 2.44 0.647 -6.564 5.958 7.211 3.577 7.003 10.580
Table 13
Parameters of the solution with respect to maximal thrust magnitude
T, N IV, m/s М, kg
1 10.580 4.892
2 10.377 4.798
5 10.32 4.772
10 10.318 4.771
100 10.308 4.766
Conclusion
The paper describes an algorithm for calculating the parameters of the multi-turn, multi-impulse meeting. The main advantage of the proposed algorithm is its simplicity and reliability, which allows it to be used not only in ground control centers, but also on board a spacecraft. In the same time, this algorithm makes it possible to obtain an optimal solution to the problem in the case when the initial phase belongs to the optimal phase range and the total characteristic velocity of solving the meeting problem coincides with the total character-ristic velocity of the optimal solution to the transfer problem. The algorithm makes it possible to obtain a solution even in the case when maneuvers are performed by low-thrust engines. Each stage of the algorithm is transparent for understanding and control. The examples given in the article confirm the performance of this algorithm and the high quality of the resulting solution.
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About the authors
Andrey A. Baranov, Candidate of Physical and Mathematical Sciences, Leading Researcher, Institute for Applied Mathematics, Russian Academy of Sciences, Moscow, Russia; eLIBRARY SPIN-code: 6606-3690; ORCID: 0000-00031823-9354; E-mail: andrey_baranov@list.ru
Adilson P. Olivio, Postgraduate, Department of Mechanics and Control Processes, Academy of Engineering, RUDN University, Moscow, Russia; eLIBRARY SPIN-code: 7628-6084; ORCID: 0000-0001-5632-3747; E-mail: pedrokekule@ mail.ru
Сведения об авторах
Баранов Андрей Анатольевич, кандидат физико-математических наук, ведущий научный сотрудник, Институт прикладной математики имени М.В. Келдыша, Российская академия наук, Москва, Россия; eLIBRARY SPIN-код: 6606-3690; ORCID: 0000-0003-1823-9354; E-mail: andrey_baranov@list.ru
Оливио Адилсон Педро, аспирант, департамент механики и процессов управления, инженерная академия, Российский университет дружбы народов, Москва, Россия; eLIBRARY SPIN-код: 6606-3690; ORCID: 0000-0001-56323747; E-mail: pedrokekule@mail.ru
