Use of the Lyapunov Functions for Calculating the Locally ...

ISSN 0010-9525, Cosmic Research, 2021, Vol. 59, No. 3, pp. 212–221. © The Author(s), 2021. This article is an open access publication.Russian Text © The Author(s), 2021, published in Kosmicheskie Issledovaniya, 2021, Vol. 59, No. 3, pp. 255–264.

Use of the Lyapunov Functions for Calculating the Locally Optimal Control of a Thrust Vector during Low-Thrust Interorbital Transfer

R. V. Yelnikov*Research Institute of Applied Mechanics and Electrodynamics of the Moscow Aviation Institute

(National Research University), Moscow, Russia*e-mail: [email protected]

Received June 23, 2020; revised October 19, 2020; accepted December 10, 2020

Abstract—This paper presents a method for locally optimal control of the thrust vector of the electric propul-sion system (EPS) for a spacecraft that performs a multiturn interorbital transfer from the initial ellipticalorbit into a geostationary orbit (GSO). The control represents the time dependences of the angles that char-acterize the EPS thrust vector orientation in space. Here, it is assumed that the EPS is always on. The pro-posed control algorithm belongs to the class of feedback control algorithms and is based on using the Lya-punov functions. Numerical examples are presented, which characterize the operability of the proposed con-trol technique. Considerable attention is paid to the comparison of given solutions with the optimal solutionsobtained within the framework of the maximum principle formalism.

DOI: 10.1134/S0010952521030035

INTRODUCTIONElectric propulsion systems (EPSs) are widely used

currently onboard spacecraft of various designation:from vehicles operating in low-Earth orbits to auto-matic interplanetary stations. Such propulsion systemsare used both for trajectory correction and as cruisepropulsion systems.

The high values of a thrust’s specific impulse pos-sessed by EPSs allow significant savings of propellant(fuel) onboard the spacecraft. Because of this, interestin using such propulsion systems arose almost at thevery beginning of the space age. Research in the fieldof analyzing the trajectories of EPS-equipped space-craft has been carried out in many works beginning inthe middle of the 20th century [1–12].

Most of these works were devoted to the problemsof finding optimal (or quasi-optimal) control over anopen loop. Such an approach to finding the control isquite legitimate for the tasks of designing spacecraftmotion trajectories.

At the same time, to implement the controlonboard the spacecraft, control algorithms should bedeveloped, which, in accordance with the terminologyof the automatic control theory, should be attributedto the class of feedback control algorithms. (By theway, such approaches can also be used for trajectorydesigning tasks.) One of the most effective approachesto the feedback control synthesis appears to have beenpresented in papers [13, 14].

The idea of using the Lyapunov functions to findthe laws of thrust vector control for EPS-equipped

spacecraft has also been rather fruitful, and manyauthors have used it [15–20].

The main advantage of the control algorithm usedin this work is its simplicity. The operation of the pro-posed algorithm does not require large calculationresources. In addition, if necessary, it can be imple-mented onboard a spacecraft.

MATHEMATICAL MODEL OF SPACECRAFT MOTION

The motion of a spacecraft is considered in a nonro-tating, geocentric equatorial coordinate system(GECS), the origin of which is located at the Earth’scenter of masses and the axes of which are collinear tothe axes of the International Celestial Reference System(ICRS). The International Terrestrial Reference Sys-tem (ITRS) is used to calculate the disturbances fromthe noncentral nature of the Earth’s gravitational field.

The system of equations of the spacecraft motion inthe GECS can be presented as

(1)

where is the spacecraft position vector, t is time, μ isthe gravitational parameter of the Earth, P is the valueof the EPS jet thrust. is the thrust function (

μ ∂ δ= − + + +∂

δ= −

rr

r

2

2 3 ;

;

pg

d R Pmdt r

dm Pdt w

Q a e

r

δ δ = 1

212

USE OF THE LYAPUNOV FUNCTIONS FOR CALCULATING 213

when the EPS is on, and when the EPS is off),m is the spacecraft mass, is the unit vector (ort) ofthe EPS thrust vector, w is the effective exhaust veloc-ity of the EPS, is the matrix of transition from theITRS coordinate system into the ICRS coordinatesystem, is the vector of spacecraft position inthe ITRS coordinate system, R is the disturbing func-tion caused by the noncentral nature of the Earth’sgravitational field, and a are disturbing accelerationsfrom the attraction of the Moon and Sun.

The disturbing function caused by the noncentralnature of the Earth’s gravitational field is consideredin the following form:

where rE is the equatorial radius of the Earth, cn0 arecoefficients at zonal harmonics, cnm and dnm are non-normalized coefficients at tesseral and sectorial har-monics, is the mth deriv-ative of the Legendre polynomial ,

, , ϕ is the geo-detic latitude of a subsatellite point, λ is the longitudeof a subsatellite point, and N and M are the order anddegree of the used gravitational field model (theEGM96 model is used in the work, N = M = 70).

The disturbing accelerations from the attraction ofthe Moon and Sun are

where is the gravitational parameter of the jthcelestial body (index 1 denotes the Moon, and index 2denotes the Sun), and is the radius-vector of thejth celestial body in the GECS.

To calculate celestial bodies’ positions , theephemeris software DE405 is used [21].

FEEDBACK CONTROL SYNTHESIS

The system of equations of the spacecraft disturbedmotion in the osculating classical Keplerian elementsis known [22]. In this case, the equations for the focalparameter of an orbit and for the eccentricity andinclination can be presented as follows:

δ = 0pe

Q

= Tr rg Q

( )

=

= =

μ =

+ +

0E0

2

E

2 1,

N n

n nn

M nnm

nm m nm m nn m

rR c Pr r

r c C d S Pr

( ) ( )m

nP = ϕ ϕsin sin mm

nd P d( )ϕsinnP

= ϕ λcos cosmmC m = ϕ λcos sinm

mS m

r=

−= μ − −

2

31

,j jj

j jj3

r r ra

r r

μ j

jr

jr

COSMIC RESEARCH Vol. 59 No. 3 2021

(2)

here, is the focal parameter; is eccentricity; isinclination; is the true anomaly; is the latitudeargument; , , and are, respectively, the radial,transversal, and binormal projection of the jet acceler-ation acting on a spacecraft; and S, T, and W are,respectively, the radial, transversal, and binormal pro-jection of the total disturbing acceleration from theother factors disturbing the trajectory (the noncentralnature of the Earth’s gravitational field, the Lunar–Solar disturbances, etc.).

Assuming that the EPS thrust vector is directedalong the longitudinal axis of a spacecraft, we can write

and , where is the jet accelerationmagnitude, is the pitch angle, and is the yaw angle.

We will consider the motion without turning off theEPS; i.e., we will assume that .

The interorbital transfer problem is formulated asfollows. It is required to find functions and

such that the spacecraft be transferred fromthe initial state intothe final state

In the case of transfer into the GSO that we areconsidering, = 42164 km, = 0, and = 0.

Next, we introduce into consideration the Lya-punov function in the following form:

(3)

where are some constant coefficients and are the current dimensionless values of

residuals in the focal parameter, eccentricity, andinclination (at the given point of the osculating trans-fer trajectory): and

Consider the time derivative of Lyapunov function (3):

(4)

From the necessary and sufficient conditions forthe minimum of function (4), which is twice differen-tiated with respect to and , we can obtain the lawof controlling the jet acceleration vector, which pro-

( )

( )

( )

( )

= +μ + υ= υ ⋅ +μ

υ + υ ++ + + υ

= +μ + υ

2

12 ;1 cos

sin

cos 2 cos ;1 cos

cos ;1 cos

T

S

T

W

dp pp T fdt e

pde S fdt

e e T fe

pdi u W fdt e

p e iυ u

Sf Tf Wf

= θsin ;Sf f = θ ψcos cos ;Tf f = θ ψ,cos sinWf f= δf P m = δf P m

θ ψ

δ ≡ 1

( )θ = θ t( )ψ = ψ t

( )=0 0 ;p p t ( )=0 0 ;e e t ( )=0 0i i t( )= ;f fp p t ( )= ;f fe e t ( )= .f fi i t

fp fe fi

( ) = Δ + Δ + Δ2 2 21 2 3 ;L t k p k e k i

1 2 2, ,k k kΔ Δ Δ; ;p e i

( ) ( )Δ = − ,fp t p t p ( ) ( )Δ = − ,fe t e t e( ) ( )Δ = − .fi t i t i

( )

( )θ ψ =

Δ Δ Δ= Δ + Δ + Δ1 2 3

,

2 .

dLgdt

d p d e d ik p k e k idt dt dt

θ ψ

214 YELNIKOV

vides, at each time moment, the minimum of the Lya-punov function’s derivative:

(5)

here,

Assuming that the value of the focal parameter of theinitial orbit differs from the required final value, and theinitial orbit itself is elliptical and has a nonzero inclination,the coefficients, included in (3), can be introduced in thefollowing manner: and

, where are constantcoefficients that characterize the weight of a residual withrespect to the eccentricity and inclination, respectively.

It is not difficult to find the components of thethrust ort in the GECS:

where and

are the components of orts of the

radial, transversal, and binormal in the GECS:

and

Interorbital transfer duration is determined bythe time interval during which all parameters of theosculating transfer trajectory take the required valuesat the right end. At the same time, it should be notedthat the orbital elements may come to their requiredvalues nonsimultaneously.

By varying weight coefficients , it is pos-sible to obtain different solutions to the interorbitaltransfer problem. All of them will differ in transferduration, which, therefore, can be considered as somefunction of selected weight coefficients:

(6)By performing numerical minimization of (6) in

the space of weight coefficients, it is possible to reachthe decrease of the duration of a transport operation ofthe transfer and to ensure simultaneous fulfillment ofall boundary conditions.

At the same time, it should be noted that the accu-racy of fulfillment of boundary conditions at the right

−λ −λ= =λ + λ + λ λ + λ + λ

−λ=λ + λ + λ

2 2 2 2 2 2

2 2 2

; ;

;

S TS T

T S W T S W

WW

T S W

f f f f

f f

( )( )

λ = Δ υ + υ λ = Δ + Δ υ + + υ

λ = Δ = Δ + = Δ +

�

� �

� �

22

1 2

3

sin 1 cos ;

2 cos 1 2 cos ;

cos ; ; .

S

T

W f f

k e e

k pp k e e

k i u p p p e e e

( )= − 21 01 ,fk p p = 2

2 0 ,ek k e

= 23 0ik k i ( ) ( )Δ = − fi t i t i ,e ik k

= ×

,х x x S

p y y y T

z z z W

r n b fr n b fr n b f

e

( ) = 0r ;Tx y zr r r ( ) = 0n ;T

x y zn n n

( ) = 0bT

x y zb b b

=0

rr ,

r= ×0 0 0n b r , ×=

×0

r rb

r r

�

�

.

ft

ande ik k

( )= , .f f e it t k k

end of the trajectory, unfortunately, depends on thevalues of selected weight coefficients themselves. Thisresults in the situation in which, in the process ofnumerical searching for the minimum of function (6),solutions appear that do not satisfy the required accu-racy. These solutions need to be rejected. As a result,function (6) will have regions where it is not actuallydefined. Therefore, to find its minimum, it is highlydesirable to use any nongradient searching techniques,such as those presented in papers [23, 24].

OPTIMAL CONTROL

The operability of the presented approach will beillustrated by several numerical examples. However,here, the natural question arises of how close theobtained solutions are to optimal ones, for example, interms of the transfer duration criterion.

To answer this question and to find the optimal (interms of the minimum-time action criterion) trajecto-ries of transfer into the GSO, we have used the indirectmethod of optimizing the trajectories of motion of theEPS-equipped spacecraft. This method, which is basedon applying the Pontryagin maximum principle for-malism [25], allows us to reduce the problem of findingthe optimal control to the boundary value problem forthe system of ordinary differential equations.

At the first stage, we will analyze the motion of theEPS-equipped spacecraft in the central Newtonianfield of the Earth’s gravity. In this case, it is convenientto write the equations of motion in the equinoctial ele-ments that do not have singularities in the vicinity ofzero eccentricity and inclination:

(7)

( )[ ]

( )[ ]{ }

( )[ ]{ }

( )

( )

= ξ + −γμ

γ=ξ μ

× + ξ + + ξ + ηγ=

ξ μ× + ξ + − ξ − η

γ φ= −μ ξ

γ φ= −μ ξ

μ= ξγ

γ− −ξ μ

δ= −

3

1 2

1

1 2

2

2 1

3

4

23

3 4

2 sin cos ;

1

1 cos sin ;

1

1 sin cos ;

cos ;2

sin ;2

1 sin cos ;

;

T S

T S W

T S W

W

W

W

dA A f g F g F fdt

dg Adt

F g f Ff g f

dg Adt

F g f Ff g f

dg A F fdtdg A F fdt

dFdt A

A g F g F f

dm Pdt w



here, is the semimajor axis of the osculating transfertrajectory and is the true longitude of a spacecraft.

The following notations are introduced in system (7):

We note that the equinoctial variables are relatedwith classical elements in the following way:

here, is the pericenter argument and is the longi-tude of the ascending node of an osculating trajectory.

The spacecraft with initial mass m0 has to be trans-ferred from the pericenter of some initial ellipticalorbit, the ascending node longitude and the pericenterargument of which are zero,

into the GSO for the minimum time.The final conditions will take the form

The thrust vector control should provide the min-imum of an interorbital transfer time. Thus, we con-sider the problem of minimizing the functional:

In this case, the Hamiltonian of the optimal con-trol problem can be written as follows:

(8)

here, is the variable conjugate to the semimajoraxis, is the variable conjugate to the true longitude,and (j = 1 … 4) are the variables conjugate to theequinoctial elements g1, …, g4.

The jet acceleration vector orientation is found fromthe condition of the maximum of Hamiltonian (8):

(9)

here,

(10)

The conjugate variables included in the optimalcontrol law (9)–(10) can be found from the system ofthe following form:

(11)

here, is the vector of phase

variables and is the vector ofconjugate variables.

It can be shown that, for the minimum-time actionproblem, we have on the entire trajectory; that is,the spacecraft moves with the EPS turned on perma-nently.

Systems (7) and (11), when substituting controllaw (9) into (7), jointly form the system of equationsof optimal motion of a spacecraft. To integrate thissystem, it is necessary to find the vector of conjugatevariables at initial time moment and timeof interorbital transfer .

The point of entering the final orbit is not fixed; so,from the transversality condition, we have .The interorbital transfer time can be found from thecondition . Thus, the vector of residuals atthe right end of the transfer trajectory can be presentedas follows:

here, is the semimajor axis of the final orbit (GSO)and x is the vector of unknown parameters:

Thus, the problem of searching for the optimalcontrol is reduced to the numerical solution of the sys-tem of equations of the following form:

To solve this system, we can use the Powell’s hybridmethod (the HYBRID algorithm [26]).

AF

ξ = + + η = −φ = + + γ = − −

1 2 3 42 2 2 23 4 1 2

1 cos sin ; sin cos ;

1 ; 1 .

g F g F g F g F

g g g g

( ) ( )= ω + Ω = ω + Ω

= Ω = Ω

= Ω + ω + υ

1 2

3 4

cos ; sin ;

tan cos ; tan sin ;2 2

;

g e g ei ig g

Fω Ω

( ) ( ) ( )( ) ( ) ( )

= = == = =

0 0 1 0 10 2 0 20

3 0 30 4 0 40 0

; ; ,; ; 0,

A t A g t g g t gg t g g t g F t

( ) ( )( ) ( ) ( )

= == = =

1

2 3 4

42164 km; 0;0; 0; 0.

f f

f f f

A t g tg t g t g t

= →0

min .ft

tJ dt

== − + λ + λ + λ

4

11 ;j

A F jj

dgdA dFHdt dt dt

λA

λF

λ j

= =+ + + +

=+ +

1 22 2 2 2 2 2

1 2 3 1 2 3

32 2 2

1 2 3

; ;

;

T S

W

b bf f f fb b b b b b

bf fb b b

( )

( ) ( )

λ ξ += + λ + γ ξ

++ λ + ξ λ= +γ

+ λ − λφ = η λ − λ − λ + λ

γ

11 1

22

2 1 2

1 2

3 1 2 2 1 3 4

2 coscos

sinsin .

2 sin cos

sin cos .1 cos sin .

2

A

A

A g Fb F

g FF

Ab g F g F

F F

b g g F F

∂= −∂x

;d Hdtλ

( )=x 1 2 3 4TA g g g g F

( )= λ λ λ λ λ λ1 2 3 4T

A Fλ

δ = 1

( )= 0t0λ λ

ft

( )λ = 0F ft

( ) = 0fH t

( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]= − λg x 1 2 3 4( ) * ;Tf f f f f F f fA t A g t g t g t g t t H t

*A

( ) =

x0 .

f

ttλ

=g x( ) 0.


216 YELNIKOV

Table 1. Versions of intermediate elliptical orbits

No. Pericenter height, km Apocenter height, km Inclination, degrees Spacecraft mass on the intermediate orbit, kg

1 17793 62800 7 1326.4142 13543 70800 8.5 1460.1633 10043 73800 12 1581.8364 7293 78800 15.5 1696.0445 5293 84800 20 1802.960

Table 2. Results of solution of the problems of interorbital transfer into the GSO using feedback control based on the Lya-punov functions

No. Time of transfer into the GSO, days

Implemented residuals at the right end of the trajectory

Weight coefficients of the Lyapunov function

in eccentricity in inclination, degree

1 97.73 0.00327 0.005 1.5173 0.36322 125.12 0.00034 0.003 2.6314 0.54243 155.11 0.00191 0.000 2.9277 1.84514 185.43 0.00177 0.000 2.1535 1.37345 213.25 0.00019 0.024 3.0913 1.5251

ek ik

NUMERICAL RESULTS

The operability of the proposed control technique,based on applying the Lyapunov functions, will beanalyzed for several numerical examples of a low-thrust spacecraft transfer into the GSO. Let us com-pare the obtained results with the results found withinthe framework of solution of an optimal minimum-time action problem. For both control versions, thespacecraft motion was analyzed within the frameworkof the model of the central Newtonian field of theEarth’s gravity.

As an example, we consider the transport systembased on the Soyuz-2-1B launch vehicle (LV) and theFregat upper stage (US). The scheme of payload inser-tion into the GSO is as follows. The LV provides thehead unit insertion into the parking circular orbit. Thehead unit includes the US, the payload adapter, andthe payload itself, which represents the spacecraftintended for inserting into the GSO. With the help ofthe US, the head unit is transferred into some inter-mediate elliptical orbit, the pericenter argument ofwhich is assumed to be zero. On this orbit, the US isseparated from the spacecraft together with the pay-load adapter. Further insertion of a spacecraft isaccomplished under an effect of the EPS thrust force.

The mass of the head unit, inserted into the parkingcircular orbit with altitude of 200 km and inclinationof 51.7° is 8320 kg when launching from the Vostochny(Eastern) cosmodrome. (https://www.samspace.ru/products/launch_vehicles/rn_soyuz_2/, with the dateof access being June 8, 2020). The final mass of the US

is 1050 kg, and the specific impulse of thrust of itscruise propulsion system is 333.2 s [27]. The payloadadapter mass is assumed to be 50 kg. The spacecraft’sEPS includes two SPD-100D Hall’s stationary plasmathrusters, which operate simultaneously. The thrust ofone engine is 90 mN, and the specific impulse ofthrust is 1740 s (https://fakel-russia.com/index.php/ru/produktsiya, with the date of access beingJune 8, 2020).

The versions of intermediate elliptical orbits thatprovide various times of transfer into the GSO are pre-sented in Table 1. The mass of the spacecraft deliveredinto the intermediate orbit by the US was estimatedunder the assumption that the interorbital transfer is atwo-pulse apsidal one. The value of gravitationallosses and losses for control at implementing the firstimpulse is 2.5%. The second impulse is implementedperfectly.

The EPS thrust vector control for the interorbitaltransfer from the indicated intermediate orbits into theGSO was found by applying two approaches describedabove: the feedback control based on the Lyapunovfunctions and optimal control within the framework ofPontryagin’s maximum principle.

Table 2 presents the results of solution of interor-bital transfer problems within the framework of thefirst approach. The numbers of solutions in this tablecorrespond to the numbers of intermediate orbits fromTable 1. The exit from the integration of spacecraft’smotion equations took place when the required valueof the semimajor axis was reached. The residuals in the



Fig. 1. Time dependences of the transfer trajectory elements.

Optimal control Feedback control

2001500

Incl

inat

ion,

deg

Transfer time, days

0

2

10050

18

16

14

12

10

8

6

4

20

Semimajor axis

Eccentricity

Inclination

42

Sem

imaj

or a

xis,

thou

sand

km

52

45

44

43

51

50

49

48

47

46

0

Ecc

entr

icity

1.0

0.3

0.2

0.1

0.9

0.8

0.7

0.6

0.5

0.4

eccentricity and inclination implemented at the finaltime moment are also presented in Table 2.

Now we estimate how close the obtained solutionsare to optimal ones. Table 3 presents the main resultsof calculating the section of transfer from the interme-diate orbit into the GSO for the two approaches underconsideration. The numbers of solutions in the tablecorrespond to the numbers of intermediate orbits fromTable 1.

It can be seen from the analysis of Table 3 that thesolutions obtained within the framework of the pro-posed approach were very close to optimal in terms ofthe transfer duration and, accordingly, in terms of thefinal mass of a spacecraft. Indeed, the transfer dura-tion in the optimal solutions is only 2–4% less than inthe solutions obtained with the use of the discussedcontrol technique based on the Lyapunov functions.The final mass of the spacecraft delivered into theGSO is almost the same for both control versions: thedistinction does not exceed 4.51 kg.


Table 3. Main results of calculating the section of transfer frunder consideration

No.Control with use of the Lyapunov functi

Spacecraft weight on the GSO, kg Transfer duration

1 1237.34 97.732 1346.13 125.123 1440.47 155.114 1527.04 185.435 1608.60 213.25

We will also analyze the evolution of basic orbitalelements of the transfer trajectory for the case of usingfeedback control based on the Lyapunov functionsand for the case of optimal control. As an example, weconsider the fourth solution (in Tables 1–3, it is high-lighted in italics). The duration of transfer into theGSO is about 6 months in this case.

Figure 1 shows the time dependences of a semima-jor axis, eccentricity, and inclination of the osculatingtransfer trajectory for the optimal control and for thecase of using the feedback control on the basis of theLyapunov functions.

As we can see, the character of changing the eccen-tricity and inclination is, generally, similar in both solu-tions, with the strongest difference being observed inthe character of changing the semimajor axis of an orbit.

Now, we analyze the resulting control for both ver-sions of solutions. Figures 2 and 3 present the depen-dences of the angles of declination and right ascensionof the EPS thrust vector on the flight time in the GECS.

om the intermediate orbit into the GSO for two approaches

ons Optimal control within the framework of the Pontryagin maximum principle

, days Spacecraft weight on the GSO, kg Transfer duration, days

1239.09 95.811350.64 120.171443.00 152.331530.00 182.181610.31 211.37

218 YELNIKOV

Fig. 2. Dependences of the angle of declination of the EPS thrust vector on the transfer time.

50

30

100

20015010050

Transfer time, days

Dec

linat

ion,

deg

30

10

50

Fig. 3. Dependences of the angles of right ascension of the EPS thrust vector on the transfer time.

20200150100500

Transfer time, days

Rig

ht a

scen

sion

, deg

140

160

40

60

80

100

120

The changes of declination and right ascensionangles have an oscillatory character in both types ofsolutions. It can be noted that the character of controlin the optimal solution differs from that within theframework of the proposed control approach. Despitethis, the transfer durations, both in the optimal solu-tions and in the solutions corresponding to feedbackcontrol based on applying the Lyapunov function arequite close in magnitude.

DISTURBED MOTION OF A SPACECRAFTOne advantage of the proposed approach to the

EPS thrust vector control is its simplicity of adaptationto the case of disturbed motion. In the final section ofthe paper, we will give an example of implementing thecontrol technique under consideration within theframework of the updated model of motion (1) takinginto account the gravitational effect on the trajectoryof transfer from the Moon and Sun, as well as the non-central nature of the Earth’s gravitational field.

We now compare the solutions obtained within theframework of the model of the central field of theEarth’s gravity and within the framework of theupdated model. As an example, we will consider theabove-analyzed version of the interorbital transfer intothe GSO (version 4 from Tables 1–3).

In the case of disturbed motion, it is necessary tospecify the initial time moment to calculate the dis-turbing accelerations. Such a moment, i.e., the timemoment of passage through the pericenter of inter-mediate orbit 4, is considered to be 00.00.00 UTC onJanuary 1, 2025 (see Table 1). Weight coefficients

included in the control law are the same asfor the case of undisturbed motion (they are pre-sented in Table 2).

Analysis of disturbed motion of a spacecraft hasshown that the time dependences of change of a semi-major axis, eccentricity and inclination, during thetransfer, do not virtually differ from similar depen-dences for the case of motion in the central field.

ande ik k



Fig. 4. Evolution of the longitude of the ascending node.

30

10

200150100500

Transfer time, days

Asc

endi

ng n

ode

long

itude

, deg

10

30

70

50

90With disturbances Central field

Fig. 5. Dependences of the angles of declination and the EPS thrust vector on the transfer time for the cases of disturbed andundisturbed motion.

30

0155154151150

Transfer time, days

Dec

linat

ion

and

righ

t asc

ensi

on, d

eg

150

153152

30

60

90

120

Right ascension of a thrust vector

Thrust vector declination

The most significant distinctions are observed inthe evolutions of the ascending node longitude andpericenter argument (see Fig. 4).

The character of control in the disturbed andundisturbed solution also turns out to be close; how-ever, in the case of disturbed motion, one can observethe shift in the phase of oscillations of the angles ofdeclination and right ascension of the EPS thrust vec-tor. This phase shift gradually grows towards the end ofthe f light. As an example, Fig. 5 shows the interval of150–155 days of f light.

The inclusion of disturbances into the model led to aslight increase of the transfer duration (up to 186.82 daysvs. 185.43 days of transfer in the central field). This


resulted in a certain decrease of the final mass of aspacecraft, which, however, turned out to be insignif-icant. Figure 6 shows the projections of the obtainedtransfer trajectory in the GECS.

CONCLUSIONS

So, within the framework of this work, a techniquehas been proposed for controlling the thrust vector ofa spacecraft performing interorbital transfer into theGSO with the use of an EPS. This control technique israther simple and is based on using the Lyapunovfunctions. Within the framework of the consideredclass of interorbital transfer problems, this technique

220 YELNIKOV

Fig. 6. Transfer trajectory projections in the GECS.

0 40 00020 000 60 000

20 000

40 000

60 000

40 000

20 000

60 000

–20 000–40 000–60 000–80 000–100 000

Y, km

X, km

0 40 00020 000 60 0002000

6000

10 000

6000

2000

10 000

–20 000–40 000–60 000

Z, km

Y, km

makes it possible to find the solutions that are veryclose to optimal in terms of the transfer duration.

Another advantage of the proposed approach is thefact that it can easily be used for accurate models ofspacecraft motion with a various set of disturbing factors.

Due to the fact that the proposed control law isspecified as a function of the current phase state of aspacecraft, this control belongs to the class of feedbackcontrol algorithms, while the algorithm itself does notrequire large calculational resources and can beapplied onboard the spacecraft.

FUNDING

This research was supported by a grant of the RussianScience Foundation, project no. 16-19-10429.

OPEN ACCESS

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Translated by Yu. Preobrazhensky


Use of the Lyapunov Functions for Calculating the Locally ...

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