On the choice of the ballistic parameters of an on-orbit service spacecraft
The aim of this paper is to assess the ballistic potential of modern and prospective OOS spacecraft and to develop a methodology for planning rational OOS routes. Various ground- and space-based OOS systems are considered and analyzed. The expediency of their use is estimated depending on the servi...
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Інститут технічної механіки НАН України і НКА України
2019
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irk-123456789-1739792020-12-29T01:26:14Z On the choice of the ballistic parameters of an on-orbit service spacecraft Alpatov, A.P. Holdsht, Yu.M. The aim of this paper is to assess the ballistic potential of modern and prospective OOS spacecraft and to develop a methodology for planning rational OOS routes. Various ground- and space-based OOS systems are considered and analyzed. The expediency of their use is estimated depending on the service tasks to be performed. The most promising OOS schemes are identified. A technique for planning a rational sequence of orbit transfers between the orbits of the spacecraft to be serviced is proposed and illustrated by the example of a test calculation. Целью статьи является оценка баллистического потенциала современных и перспективных орбитальных сервисных аппаратов и разработка методики планирования рациональных маршрутов ОСО. Рассмотрены и проанализированы различные системы ОСО наземного и космического базирования. Проведена оценка целесообразности их использования в зависимости от планируемых для выполнения сервисных задач. Выявлены наиболее перспективные схемы ОСО. Предложена методика планирования рационального маршрута последовательного обхода орбит обслуживаемых КА и приведен тестовый пример расчета. Метою статті є оцінка балістичного потенціалу сучасних і перспективних орбітальних сервісних апаратів і розробка методики планування раціональних маршрутів ОСО. Розглянуто й проаналізовано різні системи ОСО наземного та космічного базування. Проведено оцінку доцільності їх використання в залежності від плануємих для виконання сервісних завдань. Виявлено найбільш перспективні схеми ОСО. Запропоновано методику планування раціонального маршруту послідовного обходу орбіт КА, що обслуговуються, й наведено тестовий приклад розрахунку. 2019 Article On the choice of the ballistic parameters of an on-orbit service spacecraft / A.P. Alpatov, Yu.M. Holdsht // Технічна механіка. — 2019. — № 1. — С. 25-37. — Бібліогр.: 20 назв. — англ. 1561-9184 http://dspace.nbuv.gov.ua/handle/123456789/173979 629.7 en Технічна механіка Інститут технічної механіки НАН України і НКА України |
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The aim of this paper is to assess the ballistic potential of modern and prospective OOS spacecraft and to develop a methodology for planning rational OOS routes. Various ground- and space-based OOS systems are considered and analyzed. The expediency of their use is estimated depending on the service tasks to be performed. The most promising OOS schemes are identified. A technique for planning a rational sequence of orbit transfers between the orbits of the spacecraft to be serviced is proposed and illustrated by the example of a test calculation. |
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Article |
| author |
Alpatov, A.P. Holdsht, Yu.M. |
| spellingShingle |
Alpatov, A.P. Holdsht, Yu.M. On the choice of the ballistic parameters of an on-orbit service spacecraft Технічна механіка |
| author_facet |
Alpatov, A.P. Holdsht, Yu.M. |
| author_sort |
Alpatov, A.P. |
| title |
On the choice of the ballistic parameters of an on-orbit service spacecraft |
| title_short |
On the choice of the ballistic parameters of an on-orbit service spacecraft |
| title_full |
On the choice of the ballistic parameters of an on-orbit service spacecraft |
| title_fullStr |
On the choice of the ballistic parameters of an on-orbit service spacecraft |
| title_full_unstemmed |
On the choice of the ballistic parameters of an on-orbit service spacecraft |
| title_sort |
on the choice of the ballistic parameters of an on-orbit service spacecraft |
| publisher |
Інститут технічної механіки НАН України і НКА України |
| publishDate |
2019 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/173979 |
| citation_txt |
On the choice of the ballistic parameters of an on-orbit service spacecraft / A.P. Alpatov, Yu.M. Holdsht // Технічна механіка. — 2019. — № 1. — С. 25-37. — Бібліогр.: 20 назв. — англ. |
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2025-07-15T10:50:54Z |
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1837709821053239296 |
| fulltext |
25
UDC 629.7
A. P. ALPATOV, Yu. M. HOLDSHTEIN
ON THE CHOICE OF THE BALLISTIC PARAMETERS OF AN ON-ORBIT
SERVICE SPACECRAFT
Institute of Technical Mechanics
of the National Academy of Sciences of Ukraine and the State Space Agency of Ukraine,
15 Leshko-Popel St., Dnipro 49005, Ukraine; e-mail: aalpatov@ukr.net; jura_gold@meta.ua
В даний час спостерігається суттєве зростання вартості космічних апаратів (КА). У зв'язку з цим
зростають вимоги до збільшення тривалості строків їх активного існування, надійності функціонування й
зниженню експлуатаційних витрат. Багатообіцяючим шляхом задоволення цих вимог є впровадження
технології орбітального сервісного обслуговування (ОСО). OCO дозволяє вирішувати технічні та економі-
чні проблеми за рахунок виконання сервісних операцій у космосі. Впровадження ОСО сприяє збільшенню
строків активного існування КА, підвищенню надійності їх функціонування, зниженню витрат на підтри-
мку в експлуатації орбітальних супутникових систем різного призначення. Програми ОСО з усунення КА,
що завершили активне існування, або ушкоджених КА з робочих орбіт допомагають зм'якшити проблему
нагромадження космічного сміття. Склад системи ОСО залежить від плануємих до виконання сервісних
завдань і балістичних можливостей одноразових або багаторазових сервісних космічних апаратів (СКА).
Реалізуємість і характер балістичних маневрів СКА багато в чому визначаються типом і характеристика-
ми їх маршових двигунів. Метою статті є оцінка балістичного потенціалу сучасних і перспективних орбі-
тальних сервісних апаратів і розробка методики планування раціональних маршрутів ОСО. Розглянуто й
проаналізовано різні системи ОСО наземного та космічного базування. Проведено оцінку доцільності їх
використання в залежності від плануємих для виконання сервісних завдань. Виявлено найбільш перспек-
тивні схеми ОСО. Запропоновано методику планування раціонального маршруту послідовного обходу
орбіт КА, що обслуговуються, й наведено тестовий приклад розрахунку. Методика базується на розв'язан-
ні багатокритеріальної задачі комівояжера. Багатокритеріальна задача комівояжера сформульована в тер-
мінах целочисельного лінійного програмування й приведена до однокритеріальної задачі методом адитив-
ної згортки. Новизна запропонованої методики полягає у приведенні вихідної задачі до виду багатокрите-
ріальної задачі комівояжера. Отримані результати можуть знайти застосування при обґрунтуванні, плану-
ванні й здійсненні сервісних космічних операцій.
В настоящее время наблюдается существенный рост стоимости космических аппаратов (КА). В свя-
зи с этим возрастают требования к увеличению длительности сроков их активного существования, надеж-
ности функционирования и снижению эксплуатационных расходов. Многообещающим путём удовлетво-
рения этих требований является внедрение технологии орбитального сервисного обслуживания (ОСО).
OCO позволяет решать технические и экономические проблемы за счет выполнения сервисных операций
в космосе. Внедрение ОСО способствует увеличению сроков активного существования КА, повышению
надежности их функционирования, снижению затрат на поддержание в эксплуатации орбитальных спут-
никовых систем различного назначения. Программы ОСО по устранению завершивших активное сущест-
вование или поврежденных КА с рабочих орбит помогают смягчить проблему накопления космического
мусора. Состав системы ОСО зависит от планируемых для выполнения сервисных задач и баллистических
возможностей одноразовых или многоразовых сервисных космических аппаратов (СКА). Реализуемость и
характер баллистических маневров СКА во многом определяются типом и характеристиками их марше-
вых двигателей. Целью статьи является оценка баллистического потенциала современных и перспектив-
ных орбитальных сервисных аппаратов и разработка методики планирования рациональных маршрутов
ОСО. Рассмотрены и проанализированы различные системы ОСО наземного и космического базирования.
Проведена оценка целесообразности их использования в зависимости от планируемых для выполнения
сервисных задач. Выявлены наиболее перспективные схемы ОСО. Предложена методика планирования
рационального маршрута последовательного обхода орбит обслуживаемых КА и приведен тестовый при-
мер расчета. Методика базируется на решении многокритериальной задачи коммивояжера. Многокрите-
риальная задача коммивояжера сформулирована в терминах целочисленного линейного программирова-
ния и приведена к однокритериальной задаче методом аддитивной свертки. Новизна предложенной мето-
дики состоит в приведении исходной задачи к виду многокритериальной задачи коммивояжера. Получен-
ные результаты могут найти применение при обосновании, планировании и осуществлении сервисных
космических операций.
At present, a significant increase in the cost of spacecraft is observed. Due to this fact the requirements for
their active life duration, operational reliability, and operational cost reduction become more and more stringent.
A promising way to meet these requirements is the introduction of on-orbit service (OOS). OOS allows one to
solve technical and economic problems by performing service operations in space. The introduction of OOS con-
tributes to extending the active life of spacecraft, increasing their operational reliability, and reducing the service
maintenance cost of orbital satellite systems of various purposes. OOS programs on the deorbit of used or dam-
aged spacecraft help in the mitigation of space debris problem. The composition of an OOS system depends on
А. П. Алпатов, Ю. М. Гольдштейн, 2019
Техн. механіка. – 2019. – № 1.
26
the service tasks to be performed and the ballistic capabilities of disposable or reusable service spacecraft. The
realizability and character of the ballistic maneuvers of service spacecraft are largely determined by the type and
characteristics of their sustainer engines. The aim of this paper is to assess the ballistic potential of modern and
prospective OOS spacecraft and to develop a methodology for planning rational OOS routes. Various ground- and
space-based OOS systems are considered and analyzed. The expediency of their use is estimated depending on the
service tasks to be performed. The most promising OOS schemes are identified. A technique for planning a ra-
tional sequence of orbit transfers between the orbits of the spacecraft to be serviced is proposed and illustrated by
the example of a test calculation. The technique is based on the solution of a multi-criteria traveling salesman
problem, which is formulated in terms of integer linear programming and reduced to a single-criterion problem by
the additive convolution method. The novelty of the proposed technique lies in reducing the original problem to a
multi-criteria traveling salesman problem. The results obtained may be used in the justification, planning, and
implementation of service space operations.
Keywords: traveling salesman problem, spacecraft, multi-criteria optimiza-
tion, on-orbit service, route planning.
Introduction. Due to the constant increase in the cost of spacecraft operating
in orbit, the need for their servicing increases. Orbital servicing support (OSS) is
an important direction for the increment of the efficiency of space activities. It
provides the solution of technical and economic problems through the implementa-
tion of orbital service operations. The introduction of OSS increases the duration
of the spacecraft lifetime, improves the reliability of their operations and reduces
the cost for maintenance work of the orbital satellite systems for various purposes.
The OSS also contributes to the solution of the space debris problem. Programs to
eliminate damaged satellites from work orbits help mitigate this problem.
Orbital Service Schemes. The composition and design of the OSS system are
determined by the planned servicing tasks [1 3]. The OSS system may consist of
the following elements:
- disposable or reusable launch vehicle and overclocking unit;
- disposable or reusable service spacecrafts (SSC);
- serviced spacecrafts (SC);
- orbital technical centers - base stations;
- technical means of flight control and service operations.
The same SSC can be used in a disposable or reusable mode during the per-
formance of various tasks. The location of the SSC is essential for the usage of its
reusable mode.
When ground-based, after servicing the SSC on Earth, it is launched from the
spaceport to the orbit of the serviced spacecraft
After the completion of service operations on the serviced SC, the SSC can re-
locate to the basic trajectory to refill the operating supply and wait for the next
service request or terminate its existence.
Ground-based SSC can be used in the maintenance mode of orbital objects by
the pattern of a shuttle or a sequential traversal.
For a space-based stage, the SSC is delivered in its unfilled state, entirely or in a
disassembled form, to a low near-earth orbit, where it is finally assembled and filled
with fuel. All subsequent refueling and repair works are also carried out in orbit.
There are great prospects for the creation of infrastructures in space, which
include reusable SSC and orbital base stations designed for servicing the
spacecraft when complex repairs are necessary.
Reusable SSC can serve a spacecraft by a shuttle or sequential traversal pat-
tern. Shuttle pattern option assumes that, after each service operation, the SSC is
returned to the base orbit or orbital station. In the case of a sequential bypass
scheme of several serviced spacecraft, a return to the base orbit or the orbital base
27
station is carried out after the completion of the bypass. Spacecraft service opera-
tions can be performed by direct SSC operation with the serviced spacecraft, if not,
by using it for transportation of the spacecraft from its operational orbit to the or-
bital servicing base, after that the spacecraft is serviced it returns back to the orbit
by the delivery platform. The implementation of these methods can be carried out
by one, two or multiple SSC.
The following options are possible for two SSC:
- one of the SSC is used as an observer, and the second - for the direct per-
formance of the task;
- one of the SSC is used as a base, and the second - for the direct performance
of the task.
Group application of SSC is also possible in two versions: accomplishment of
the task with the use of “swarm” tactics with prompt decision-making on the
choice of a servicing SC; in the form of a functionally complete object with a rigid
distribution of functional responsibilities between the SSC
Group application of SSC is also possible in two options:
execution of a task by using the "hive" tactics with immediate decision of ser-
vicing SSC;
in the form of a functionally complete object with a rigid distribution of func-
tional responsibilities between the SSC.
The group application of the SSC seems to be more effective for the solution
of target objectives of the OSS and increment of the stability of the functioning of
orbital complexes and systems, from the efficiency point of view.
Thrust engine of a servicing spacecraft. The feasibility and the pattern of the
SSC ballistic maneuvers are largely determined by the characteristics of their thrust
engines, which are the essential system of any SSC. SSC marching engines are di-
vided into: thermochemical, nuclear and electric, due to the type of energy used. [4].
Currently, thermo-chemical liquid rocket engines (LRE) and solid rocket en-
gines (SRE) are mainly used in SSC, the electro rocket engines (ERE) are less
commonly used. The nuclear engine for rocket vehicle applications (NERVA) are
marked for further intend to use. Fig. 1 shows the classification of the SSC
propulsion engines.
Figure 1 – Classification propulsion engines SSC
One of the most important characteristics of SSC engines is the specific thrust
impulse. Table 1 shows the ranges of changes of the specific thrust impulse for
various types of SSC engines.
gg
28
Table 1 – Ranges of changes of specific thrust impulses
Engine type SRE LRE NERVA ERE
Specific thrust
impulse, km/s 2 – 3 3 – 5 7 – 25 5 – 100
Upper stages of missiles, overclocking units or space tugs are currently mainly
used to perform the service operation of transporting a spacecraft from a “base
near-earth orbit” to a “high-energy orbit of destination”. Engines with high-boiling
components are used for long flights and engines with cryogenic components are
used for short-time flights. Table 2 shows the characteristics of LRE of
overclocking units [5].
Table 2 – Characteristics of the LRE of overclocking units
Name D, DM Briz M,
KM
Briz K,
KS,
Fregat
Agena D Centaur D
Engine 11D58M 14D30 S5.92 Bell 8096 RL10A31
Thrust, kN 85 19,62 19,91 71,6 131
Specific im-
pulse, km/s 3,538 3,255 3,270 2,855 4,356
Number of
switchings 6 10 20
Combustion
time, s 720 3000
Operational
endurance, h <7 Unlim-
ited
Unlim-
ited <7
Mass, kg 310 95 76,5
Existing overclocking units are characterized by a high unit cost of transport-
ing payloads (PL) to high orbits, as well as a low mass of PL.
In this regard, reusable SSC with ERE are actively developed. The potential
benefits of SSC with ERE overused resources are shown in [6 8]. Reusable SSC
with low-power, but economically efficient ERE is ten times more economical
than traditional overclocking units.
Distinctive features of SSC with ERE are:
– significantly greater impulse and extremely low level of flight and, as a re-
sult, a very long flight time, as compared with the LRE, SRE and NERVA;
– high level of energy consumption.
Two main types of power plants are currently being considered: solar and nu-
clear power plants and, accordingly, the solar electric propulsion engine (SEPE)
and the nuclear electric propulsion engine (NEPE).
In electro-thermal ERE, electric power is used only for heating of the working
mass the combustion chamber; its acceleration is performed due to the effect of
gas-dynamic forces in the same way as in an LRE and NERVA. It is considered to
distinguish electric heating, electric arc and induction motors, due to the principles
used for heating of the working mass.
The wide use of electro-thermal ERE is constrained by a low specific thrust
impulse (compared to other types of electric propulsion), which is not much higher
than that of chemical engines (up to 9000 m/s in promising electric propulsion pro-
jects with hydrogen as a working mass).
29
In an electrostatic engine, thrust is created by the acceleration of the like-
charged particles of the working mass in an electric field. Ionic and colloidal en-
gines are distinguished by type of accelerated particles.
In an ion engine - the working mass is first ionized, by the ion engine [9], after
those positive ions are accelerated in an electrostatic field (using an electrode sys-
tem) and, create a thrust, by flowing out of the nozzle (the electrons are injected
into it to neutralize the charge of the jet stream).
In a colloidal engine, the working mass executed in the form of positively
charged microscopic particles, which are accelerated by an electrostatic field.
In an electromagnetic ERE the acceleration of the working mass is performed
by the electromagnetic force. The thrust is created by the interaction of charged
particles with a magnetic field. Depending on the method of the power supply,
electromagnetic engines are divided into stationery (non-stop action) and pulsed.
The aggregated indicators of ERE of various types are shown in Table 3.
Table 3 – The aggregated indicators of ERE
Engines
Characteristics Electro-thermal Electro-magnetic Electro-static
Thrust, N 0,01…1 0,001…2,5 0,001…0,1
Specific impulse,
km/s 1…25 10…70 30…100
Power, W Tens -
thousands Ones-thousands Tens-
hundreds
Currently, much attention is paid to stationary plasma (SPT) and ion engines
[8, 9]. Table 4 and Table 5 present the technical characteristics of SPT and ion
engines.
Table 4 – SPT characteristics
Characteristics SPT-50 SPT-100 SPT-140
Thrust, mN 20 83 80-280
Specific impulse,
km/s 12,5 25 15 - 26
Power, W 0,350 1,221 1,2 - 6,0
Operating life, h 2250 7500 10000
Mass, kg 0,8 3,5 7
Table 5 – Characteristics of ion engines
Characteristics XIPS-13 XIPS-25 RIT-10
Thrust, mN 18 165 15
Specific impulse,
km/s 23,5 35 34
Power, kW 0,45 4,2 0,59
Country USA USA EU
It should be noted that the ERE is the most promising for reusable SSC, used
for multiple flights between the base and high-energy orbits. The efficiency of the
use of ERE will increase with increasing frequency of use of the SSC. The analysis
of the field of application of SSC with ERE shows their advantage versus SSC on
thermochemical engines in problems where flight time is not a critical parameter.
SSC based on ERE is preferably used for flights that require relatively high energy
costs and are unlimited in time. For example, for transportation from a low near-
30
earth orbit to geostationary heavy oversized structures with a limited level of per-
missible overloads.
Mathematical models of interorbital flights. Due to fuel consumption, dur-
ing the planning of SSC interorbital flights with engines of high and low thrust, in
the first approximation it will be possible to use [10] a system of differential equa-
tions for the controlled undisturbed motion of the SSC in the central field of one
attracting center with the equation of mass change:
When planning SSC interorbital flights with high and low thrust engines, in
the first approximation, you can use [10] a system of differential equations for the
controlled undisturbed motion of the SSC in the central field of one attracting cen-
ter along with the mass change equation due to fuel consumption:
,
,32
2
w
Pq
dt
dm
m
ePr
rtd
rd
(1)
where r – - is the current radius vector; – the gravitational constant of the Earth;
P – rocket engine thrust; m – the current mass of SSC; q and w – fuel con-
sumption and outflow rate of the working mass; t – relay on-off function of the
engine; te – single thrust vectoring function.
The duration of the operation, the direction of the thrust vector and the num-
ber of motor starts depend on the parameters of the initial and final orbits. During
the maneuver calculation, such conditions are determined for it that minimizes fuel
consumption or other criteria of optimality, such as, the time of flight from one
orbit to another. The system of equations (1) is supplemented by the initial condi-
tions of the maneuver and the change of the thrust vector time law. Models of per-
fectly regulated and unregulated engines are considered. Ideally adjustable engines
are subject only to the most important limitations for the type of engines that are
under consideration, such as the limits on their thrust or rate of flow. Unregulated
engines can either be turned on, and then the thrust and flow rates are constant, or
off, and then the thrust and flow rates are zero. There are no restrictions on chang-
ing the direction of thrust.
The equations of the system (1) contain variables that oscillate rapidly due to
the periodicity of motion along the trajectory. This complicates the numerical
analysis of the SSC multi-turn trajectories with low-thrust engines. To eliminate
this deficiency, the equations in equinoctial elements [11] are used. They have no
peculiarities with zero eccentricity and inclination.
Along with direct numerical integration of systems of differential equations,
the approaches based on simplifying assumptions are used to simulate SSC in-
terorbital flights. Depending on the magnitude and duration of the action of the
thrust force, pulsed maneuvers and low-pressure maneuvers are considered.
These types of maneuvers, as a rule, determined by the energy capabilities and
the operating principle of the used SSC main engines, which are divided into large
or intermediate and low thrust engines. SSC with high or intermediate propulsion
31
engines have the initial thrust-to-weight ratio, ]1,03,0[0n , and those ones with
low-thrust engines have the initial thrust-to-weight ratio .03,00 n
During inter-orbital maneuvering usage of high thrust engines, the duration of
active flight segments is negligible, as compared with the duration of passive flight
segments. Therefore, when calculating interorbital flights using high-thrust en-
gines, you can use a pulsed approximation of the thrust action, which reduces to an
abrupt change in flight speed without changing the coordinates of the SSC during
engine operation. The calculation of such a maneuver involves the determination
of the number of orientation, and points of application of velocity pulses.
The usage of SSC with a low-thrust engine for a flight between near-circular
orbits, near the Earth, leads to multiple spiral paths. The time of active flight of
such SSC is very long and in many cases may coincide with the time of flight
A characteristic feature of low thrust engines consists of the fact that their
control acceleration is quite small in comparison with local gravitational accelera-
tion. This eliminates the possibility of using the pulse approach.
Equations in osculating elements are often used as equations of motion [10],
for occasions of interorbital maneuvering using low thrust engines.
The model in osculating elements is the most convenient one for simplifying
the methods of asymptotic separation of variables[12], [13]. Firstly, this caused, by
the presence of a small parameter in an explicit form – the reactive acceleration
from thrust, which is less than the gravitational one, by several orders of magni-
tude. Secondly, the presence of a cyclic variable – the angular coordinate charac-
terizing the position of the spacecraft in orbit relative to the line of nodes or the
pericenter. The used approach allows obtaining a substantially simpler averaged
system of equations for simulation of interorbital flights of SSC with low thrust
engines.
The route optimization of the sequential bypass of the orbits of the servic-
ing spacecraft. The selection of a rational route of the SSC involves the solution
of the corresponding route-trace optimization problem. The problem of optimizing
the bypass of a sequence of serviced spacecraft belongs to the type of multicriteria
route-trace discrete-continuous problems.
The general direction for the solution of the route-trace discrete-continuous
problems is their splitting by decomposing discrete and continuous parts and inde-
pendent solution of those parts. Initially, optimization of all possible individual
flights between orbits performed, and then the optimal sequence of flights selected.
Decomposition reduces the task to the sequence of the solution of two simpler
problems and allows choosing the most efficient route for the sequential service of
spacecraft group. For cases when the decomposition of the problem of independent
continuous and discrete subtasks is impossible, we should pick the method of dy-
namic programming.
In the particular case when the sequence of serviced spacecraft is predeter-
mined, the solution of the discrete part is eliminated. After that, it remains to solve
the sequence of continuous optimization problems for interorbital flights by meth-
ods of optimal control on each of the trajectory segments.
The trace part consists of the construction of the trajectory that satisfies a set
of constraints and minimizes the functionality for the chosen sequence of visits to
the serviced spacecraft. It is an optimal control problem. The complexity of such
32
problems lies in the evaluation of the functional for a large number of considered
combinations in solving the discrete part of the problem.
Optimization of the flight trajectories of spacecraft for a long time has been
the subject of intensive research [14 - 18]. There are two main types of path opti-
mization methods: indirect and direct, or combinations thereof. Indirect methods
are reduced to the solution of a two-point boundary value problem based on the
maximum principle of L. S. Pontryagin. In such a boundary value problem, the
unknown variables are conjugate variables that have high sensitivity, which com-
plicates the search for the initial approximation. Direct methods transform the
original problem to a parametric optimization problem, which is usually solved
with help of non-linear programming methods.
The possibility of application of pulse approximation for simulating the con-
trolled motion of SSC with high thrust engines allows using numerical methods of
nonlinear programming for the optimization of the trajectories of interorbital
flights, along with the methods of the theory of optimal control.
The route part of the problem is discrete in nature and enables mathematical
formalization in the form of a generalized traveling salesman problem, in which
the SSC must go around the n orbits, starting with the reference orbit, on which it
is located, and finish its route, returning to the original orbit without having been
anywhere twice. The n destination orbits and the flight cost matrix C between
any pair of them are considered as given. Expenses for the flight between any pair
of orbits can be used in form of energy (characteristic speed or motor flight time
for an SSC with a low thrust engine), time or cost. In this problem, the total cost of
the full path starting and ending in the reference orbit presented as the evaluation
function, the presence or absence of a flight between the individual orbits of the
list in question, as well as the need to visit all of these orbits, are the limitations.
The matrix C is generally accepted as asymmetric. The size of this matrix
nn , and its diagonal elements can be arbitrary, since they are not used any-
where in the solution. The remaining elements of the matrix C in the sense of the
problem are non-negative.
We can formulate the considered traveling salesman problem formulated in
terms of integer linear programming [19–20]. We denote kie as integer variables
associated with arcs. There are as many variables as there are arcs. –
1 nnm . Each of them can take only one of two possible values: 1 or 0, de-
pending on whether or not the arc is in the desired cycle
.,,1,
,10
kinki
e ki
(2)
The objective function in this problem is the total weight of the arcs included
in the cycle:
.min
,
,,
n
ki
ki
kiki ecz
1
(3)
In addition, to the variables kie (2) the following restrictions are imposed.
From each vertex there should be exactly one arc:
33
.,1
,1
1
,
ni
e
n
ik
k
ki
(4)
Each vertex must also include one arc:
.,1
,1
1
,
nk
e
n
ki
i
ki
(5)
Finally, it is necessary that the cycle was complete, that is, consisting of n
vertices and n arcs. After all, constraints (4), (5) are satisfied, for example, by a
system of several smaller cycles, which together cover all of the n vertices. To
eliminate such subcycles, unbounded real variables iv , associated with vertices
are introduced. The system of constraints that eliminates subcycles has the follow-
ing form:
.
,
nki
nenvv ikki
2
21
(6)
Equations (2) - (6) define the mixed problem of integer linear programming
corresponding to the original traveling salesman problem. The problem is mixed
because the parts of variables are an integer, and parts are unlimited real. In total,
the problem uses 1nn integer variables with kie constraints (4), (5) and n
unbounded real variables iv . They are subject to restrictions (6). Target function
(3) is minimized.
The considered traveling salesman problem lies in the NP class of complete
tasks that cannot be solved with polynomial algorithms. Its computational com-
plexity grows exponentially with the size of the problem.
To solve it, exact and heuristic algorithms are used, and exact algorithms are
applicable only for small-sized tasks. The complexity of the problem under con-
sideration leads to a wide use of heuristic algorithms that give approximate solu-
tions in less time than exact methods.
Optimization of the route part of the problem simultaneously by several crite-
ria (for example, the Delta-v and time of the orbital flight) leads to the necessary
solution of the two-criteria salesman problem. Assignment n orbits and two ma-
trices VC and TC with the Delta-v and flight times between any pair of them are
considered as given. Conditions (2) and restrictions (4) - (6) are satisfied. The fol-
lowing objective task functions are minimized:
n
ki
ki
ki
T
ki
T
n
ki
ki
ki
V
ki
V
ecz
ecz
1
1
,
,,
,
,,
min
,min
(7)
34
The solution of the two-criterion minimization problem can be performed by
the methods of additive convolution, successive concessions or the minimum de-
viation from the ideal point. Let’s consider in more detail the method of additive
convolution.
The method of additive convolution allows to bring up the two-criteria opti-
mization problem (7) to a single-criterion one. If we set some values of weight co-
efficients 1 and 2 objective functions that satisfy the conditions
1021 ,, and 121 , then the additive convolution of two criteria
into one is performed according to the formula:
TV zzz 21 . (8)
Different in a physical sense, the elements of matrices VC and TC can vary
greatly in absolute values. It is necessary to standardize them, to eliminate this dis-
tinction and perform the convolution of equivalent criteria by formula (8). Stan-
dardization allows to bring elements of matrices VC and TC to the same area of
their change. Elements V
kic
,
~ and T
kic ,
~ of linearly standardized matrices VC~
and
TC~ are calculated by the formulas (9)
.minmax
min~,
minmax
min~
,,
,,
,
,,
,,
, T
ki
T
ki
T
ki
T
kiT
kiV
ki
V
ki
V
ki
V
kiV
ki cc
cc
c
cc
cc
c
(9)
Thus, the initial minimization problem (7) is reduced to the one-dimensional
minimization problem of the objective function (10)
n
ki
ki
ki
T
ki
V
ki eccz
1,
,,2,1 min~~ . (10)
An example of optimizing the route of a sequential bypass of orbits ser-
viced by spacecraft. Let’s present an example of optimizing the route of a sequen-
tial bypass of five orbits of spacecraft serviced, to demonstrate the proposed meth-
odology for choosing a rational orbital service route. The orbit parameters are
shown in table 6.
Table 6 – Parameters of the orbits of the servicing spacecraft
Number of
the orbit
Major semi-axis,
km Ellipticity Inclination of
orbit, deg
1 7303,80 0,010 67,84
2 7725,86 0,052 68,75
3 8566,31 0,106 67,66
4 7885,64 0,084 68,97
5 7061,74 0,056 67,80
Let us consider the orbital flight between a pair of non-coplanar elliptical or-
bits in the form of a sequence of five single-pulse flights. The first impulse is nec-
essary for an orbital flight between the initial elliptical orbit and the circular with a
radius equal to the radius of the apogee of the initial orbit in cases when the radius
of the apogee of the initial orbit is greater than the radius of the apogee of the final
35
orbit. The magnitude of the accelerating pulse applied at the apogee 1V is calcu-
lated by the formula
аннп
нп
на rr
r
r
V
2
11 , (11)
where - is the gravitational constant of the Earth, наr and нпr - the radii of the
apogee and perigee of the initial orbit.
The second velocity impulse implements the rotation of the plane of the initial
orbit by the angle of non-coplanarity until it coincides with the plane of the fi-
nal orbit. The formula for the impulse calculation of the characteristic velocity,
that is necessary to rotate the plane of the orbit by the noncoplanar angle , has
the form of
2
sincos22
наr
V , (12)
where - is the angle between the velocity vectors and the plane of the local hori-
zon at the time of the pulse
A two-pulse Hohmann control program is used for the implementation of a
coplanar orbital flight between a circular orbit with a radius that is equal to the
radius of the apogee of the initial orbit, and a circular orbit with a radius equal to
the radius of the apogee of the final orbit. The transition trajectory represents a
semi-ellipse of Hohmann, the perigee of which is in the initial orbit, and the apo-
gee on the final. The cost of the Delta-v for the Hohmann maneuver is calculated
by the formulas:
кана
на
ка rr
r
r
V
2
13 , (13)
1
2
4
кана
ка
на rr
r
r
V , (14)
where наr , каr — are the apogee radii, respectively, of the initial and final orbits.
The flight time for the Hohmann semi-ellipse is determined by the formula
3aT , (15)
where 2кана rra - is the length of the major semiaxis of the Hohmann
ellipse.
The fifth impulse of the characteristic speed is calculated by the formula (16)
and is intended for the flight between a circular orbit with a radius equal to the
radius of the apogee of the final orbit and the final elliptical orbit with the perigee
кпr и and apogee каr :
36
1
2
5
какп
кп
ка rr
r
r
V . (16)
The total characteristic flight speed is determined by the expression:
5
1i
iVV . (17)
In the case where the radius of the apogee of the initial orbit is less than the
radius of the apogee of the final orbit, the interorbital flight between them can be
performed in a similar way.
Tables 7 and 8 show the matrices formed by the initial data of table 6 using
formulas (11) – (16) of the matrix VC and TC and the characteristic speeds and
transit times between any pair of orbits of servicing spacecraft. The units of the
matrix elements are the characteristic speeds km/s, and the matrix elements of the
minute flight.
Table 7 – Matrix of Delta-v between pairs of orbits of serviced by spacecraft
Orbit
number 1 2 3 4 5
1 0,00 0,68 1,24 0,99 0,29
2 0,68 0,00 1,18 0,68 0,82
3 1,27 1,18 0,00 1,14 1,40
4 0,99 0,68 1,14 0,00 1,12
5 0,29 0,82 1,40 1,12 0,00
Table 8 – The matrix of flight times between pairs of orbits of serviced by spacecraft
Orbit
number 1 2 3 4 5
1 0,00 56,63 64,15 58,94 52,97
2 56,63 0,00 68,48 63,15 57,04
3 64,15 68,48 0,00 70,95 64,59
4 58,94 63,15 70,95 0,00 59,36
5 52,98 57,04 64,59 59,36 0,00
As a result of VC and TC matrices standardization accordingly to formulas
(9), the additive convolution of two criteria into one in accordance with (10) and
solving the single-criterion traveling salesman problem obtained by using the
branch and bound method, the results are presented in table 9 for different values
of weights. 1 and 2 objective function.
Table 9 – Calculation results
1 2 Route
The cost route of the
characteristic speed,
km/s
Duration of
flight, min
1,0 0,0 1 2 4 3 5 1 4,18 308,29
0,0 1,0 1 3 5 2 4 1 5,15 307,87
0,5 0,5 1 2 4 3 5 1 4,18 308,29
The calculation option with 1 =1 и 2 =0 corresponds to the minimization
of the Delta-v of the flight, the calculation option with 1 =0 and 2 =1 corre-
37
sponds to the minimization of the flight time, and the calculation option with
1 =0,5 and 2 =0,5 corresponds to the two-criteria minimization with the same
weight coefficients for the Delta-v and flight time. The cost of the Delta-v for the
flight depends significantly on the chosen route, and the duration of the flight was
insensitive to the choice of route, for the considered group of close orbits of the
servicing spacecraft and the type of maneuver of interorbital flights.
Conclusion. From the point of view of the choice of ballistic parameters,
variants of OSS systems and advanced main engines SSC are considered and ana-
lyzed. An assessment of the feasibility of their use depends on the planned to per-
form service tasks. The schemes of most promising serviced objects are identified.
Method for planning a rational route of a sequential bypass of the orbits of space-
craft has been proposed. The calculation by the proposed method is presented as
an example. The obtained results can be used for the justification, planning and
implementation of space service operations.
This work was funded by the Ukrainian Budget Program “Support of the De-
velopment of Priority Lines of Research” (KPKVK 6541230).
1 Ivanov V. M. Conceptual foundations of orbital servicing for advanced automatic space vehicles (in Russian).
Spacecraft and Rockets. Vestnik Moskovskogo Aviatsionnogo Instituta. 2008. V. 15. No. 3. Pp. 5 7.
2 Barbee B. Design of Spacecraft Missions to Remove Multiple Orbital Debris Objects. Specialist Conference,
Paper AAS 15-598, August 2015.
3 Stephen J. Design for on-orbit spacecraft servicing. Specialist Conference, Paper AAS 14-374, October 2014.
4 Caveny L. Space Engines: State of the Art and Prospects (in Russian). Moscow : Mir, 1988. 454 pp.
5 Serdyuk V. K., Tolyarenko N. V., Khlebnikov N. N. Space program support vehicles (in Russian). Rocket Pro-
duction and Space Engineering: VINITI. Moscow. 1990. V. 11.
6 Gusev Yu. G., Pilnikov A. V. Pole and place of electrojet engines in the Russian Space Program (in Russian) /
Trudy MAI (electronic journal). Issue 60. https:// www.mai.ru/science/trudy/.
7 Vasin A. I., Koroteev A. S., Lovtsov A. S. Overview of the work on electrojet engines at the Keldysh Research
Center (in Russian). Trudy MAI (electronic journal). Issue 60. https:// www.mai.ru/science/trudy/
8 Kim V. Stationary plasma engines in Russia: problems and prospects (in Russian). Trudy MAI (electronic jour-
nal). Issue 60. https:// www.mai.ru/science/trudy/
9 Gorshkov O. A., Muravlev V. A., Shagaida A. A. Hall and Ion Plasma Engines for Spacecraft (in Russian).
Moscow: Mashinostroyeniye, 2008. 280 pp.
10 Narimanov G. S. Basics of Spacecraft Flight and Design Theory (in Russian). Moscow.: Mashinostroyeniye,
1972. 607 pp.
11 Petukhov V. G. Robust suboptimal feedback control for low-trust transfer between noncoplanar elliptical and
circular orbits (in Russian). Spacecraft and Rockets. Vestnik Moskovskogo Aviatsionnogo Instituta. 2010.
V. 17, No. 3. Pp. 50 58.
12 Lebedev V. N. Low-Thrust Spacecraft Flight Calculation (in Russian). Moscow: Coputing Center of the USSR
Academy of Sciences, 1968. 105 pp.
13 Grishin S. D., Zakharov Yu. A., Odolevsky V. K. Low-Thrust Spacecraft Design (in Russian). Moscow:
Mashinostroyeniye, 1990. 223 pp.
14 Ilyin V. A., Kuzmak G. E.. Optimal Spacecraft Orbit Transfers (in Russian). Moscow: Nauka, 1976. 740 pp.
15 Grozdovsky G. L., Ivanov Yu. N., Tokarev V. V. Space Flight Mechanics (Optimization Problems) (in Rus-
sian). Moscow: Nauka, 1975. 704 pp.
16 Salmin V. V. Methods for the Solution of Variational Problems in Low-Thrust Space Flight Mechanics (in
Russian). Samara : Samara Research Center of the Russian Academy of Sciences, 2006. 127pp.
17 Petukhov V. G. Optimization of interplanetary trajectories for spacecraft with ideally regulated engines
using the continuation method. Cosmic Research. 2008. V. 4. No. 3. Pp. 219–232.
18 Grigoriev I. S., Zapletin M. P. Choosing promising sequences of asteroids. Automation and Remote Control.
2013. V. 74. Issue 8. Pp. 1284 – 1296.
19 Papadimitriou C., Steiglitz K. Combinatory Optimization. Algorithms and Complexity (in Russian) / Moscow:
Mir, 1982. – 510 pp.
20 Sigal I. Kh. Algorithms for solving the two-criterion large-scale travelling salesman problems. Computational
Mathematics and Mathematical Physics. 1994. V. 34. No. 1. Pp. 33–43.
Received on 10.10.2018,
in final form on 26.11.2018
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