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Acta Astronautica

Acta Astronautica 68 (2011) 1964–1972

0094-57

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/actaastro

Study on the stability of Tethered Satellite System

Yong He n, Bin Liang, Wenfu Xu

The Institute of Space Intelligent System, Harbin Institute of Technology, Harbin, PR China

a r t i c l e i n f o

Article history:

Received 13 May 2010

Received in revised form

21 November 2010

Accepted 22 November 2010Available online 5 January 2011

Keywords:

Tethered Satellite System

Stability

Equilibrium state

Control

65/$ - see front matter & 2011 Elsevier Ltd. A

016/j.actaastro.2010.11.015

esponding author. Tel.: +86 10 58858773; fax

ail address: Heyong2004@163.com (Y. He).

a b s t r a c t

Tethered Satellite System has turned into a reality from theory, and has become a new

research focus in the field of space science. However, the system’s stability will directly

affect the success of the mission. In this paper, the system’s stability is deeply studied.

First, the system’s equilibrium conditions are derived according to the system’s dynamics

model, and the maximum extent that these conditions can be reached is determined.

Second, the stability of the system equilibrium state is analyzed by utilizing Taylor

expansion formula and stability criterion, and the required conditions that the tether

parameters should be satisfied during a stable equilibrium state are also deduced. Finally,

based on the stability of studied above, a range-rate control algorithm is developed to

achieve the stable control of tether’s deploying, keeping, and retrieving. Simulation

results verify the validity of these conclusions, which provide a good reference for future

experiments of Tethered Satellite System.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Tethered Satellite System (TSS) is composed of two or moresatellites (one is the main satellite and the rest are sub-satellites), which are connected with each other by thin andlong members known as tethers. The system’s structure isnot complicated, but it has broad application prospects, suchas: space power generation, tether rendezvous, artificialgravity, deep space observation, upper atmospheric research,etc. [1–5].

Several missions have already been carried to verify theTSS concept, Such as: NASA’s Gemini 11 and 12 in 1967[6,7], which had already proved the feasibility of TetheredSatellite System’s technology, but the twice flight testsjointly developed by NASA and the Italian Space Agency(ASI) had failed, which indicated the difficulty of this area.However, the twice flight tests had shown the appliedpotential of Tethered Satellite System, thus some countriesstill spend a lot of fund on TSS research and spaceexperiments [1,7–9].

ll rights reserved.

: +86 10 58858770.

In order to make TSS operate normally on-orbit, it isnecessary to study the system’s stability, because unstablesystem will eventually make the tether break or wind withthe main satellite, leading to the failure of the mission. Tothis end, it is necessary to study the stability of TSSmovement. There are many ways to make the systemstable, such as: the use of electrodynamics tether, spacemanipulator, thrust gravity gradient force, etc. [10–13].

In the investigation of TSS stability, some literatureassumed the center of mass of TSS moved in a circle orbit,and obtained many valuable conclusions neglected tethermass. Such as: Liaw and Abed [14] had presented designs oftension feedback control laws by using Hopf bifurcation forthe stabilization of TSS during station-keeping; Pradeep[15] proposed linear tension controller for the deploymentof TSS, and the tension control law can be determinedwithout resorting to trial and error; Burov and Troger [16]investigated the existence of relative equilibriums and thesufficient conditions for TSS stability, similar studies wereperformed by Sanyal et al. [17]. Issues associated with themotion of TSS in the neighborhood of equilibrium pointwere addressed by Padgett and Mazzoleni [18–20], whopresented a method for analyzing the spin-up of TSS, anddescribed the motion of TSS in the neighborhood of

Nomenclature

l tether lengthx infinitesimal length of tether lengtha tether in-plane swing angleb tether out-of-plane swing angleli tether length to main or sub-satellite from

tether system center of mass (if i=1, then itis correspond to the parameters of the mainsatellite; if i=2, then it is correspond to theparameters of the sub-satellite. Below is thesame, no longer explained)

ri distance from the center of earth and main orsub-satellite

rx distance from the center of earth and tetherinfinitesimal length

m gravitational constant of central planetR distance between the center of earth and the

center of Tethered Satellite Systeme orbit eccentricity of tether system center

of mass

y orbit true anomaly of tether system centerof mass

m01 initial mass of main satellite

mi mass of main or sub-satellitemt tether massr tether mass per unit lengthTt tether tension_p the first derivative of parameter p with respect

to time€p the second derivative of parameter p with

respect to timep0a the first derivative of parameter p with respect

to parameter a

p00a the second derivative of parameter p withrespect to parameter a

Uð Þe Uð Þ at equilibriumpmax maximum of function p

pmin minimum of function p

Uð Þmax maximum of Uð Þ

Uð Þmin minimum of Uð Þ

C C=cosS S=sin

OIXIYIZI (Geocentric equatorial inertial coordinate system, showed as Fig. 1): it represents an inertial frame with originat the Earth’s center OI. The XIYI plane is the equatorial plane and the ZI—axis is towards the North Pole. The XI—axis isalong the direction of mean vernal equinox of epoch, while the YI—axis completes the right-hand triad.

OoXoYoZo (centroid orbit coordinate system, showed as Fig. 1): its origin is the center of mass of Tethered SatelliteSystem, Oo. The YoZo plane coincides with the orbital plane and the Yo—axis points towards Oo, from OI, the Zo—axis isalong the direction of velocity. The Xo—axis is normal to the orbital plane so as to complete the right-hand triad.

Y. He et al. / Acta Astronautica 68 (2011) 1964–1972 1965

equilibrium point by using a combination of analytical andnumerical techniques.

In Ref. [21], the author discussed the stability ofdeployment/retrieval by means of a three-dimensionalrigid body model of a dumb-bell tether in both circularand elliptical orbits, and showed that the deployment ofthe TSS was generally stable whereas the retrieval is not. Yu[22,23] analyzed the equilibrium and stationary state ofTSS operating in an elliptic orbit, and computed theperiodic motion along with its stability and domain ofattraction. In Ref. [24], Yu gave a further research to hisprevious researches [22,23], and adopted the concept of

Fig. 1. (a and b) Geometry

linearized variation equations to analyze the stability of thestationary configuration, considering the tether mass.

In this paper, we will give a further investigation on thefollowing two issues: (1) What are the system equilibriumconditions when considering the tether mass in the casethat the system center of mass moved in both circular andelliptical orbits? (2) Whether this equilibrium state isstable? If not, how to stabilize the TSS?

The paper is organized as follows. Section 2 gives theassumptions and derives the equations of the motion forthe system. In Section 3, we address the system equilibriumconditions and their maximum extent. Then, we analyze

l

S1

S2

α βYo

Zo

Xo

Oo

of the system model.

Y. He et al. / Acta Astronautica 68 (2011) 1964–19721966

the stability of the system according to the equilibriumconditions in Section 4, and determine the required con-ditions that the tether parameters should satisfy during astable equilibrium state. In Section 5, we propose a range-rate control algorithm based on the stability of abovestudies to stabilize the TSS in the process of tetherdeploying, station keeping and retrieving. The validity ofthe proposed method will be verified in Section 6.

2. Dynamic model

Although the elastic and flexible tether is more to thereal condition; it is difficult and inefficient to be solved.However, these factors must be considered when thesystem’s dynamics need to be investigated in details, butin this study, we neglect these factors. Such models areoften used in preliminary tether researches for theirsimplified mathematical expressions and computationalefficiency [25,26].

For the above reason, we make the following assumptions:

(1)

The main satellite and sub-satellite are point masses,and the tether is uniform in mass.

(2)

The elasticity and flexibility of tether are ignored. (3) The Earth’s gravitational field is regarded as a central

gravitational field, and only gravity gradient force isconsidered.

We consider the system comprising of two satellites, themain satellite S1 and the sub-satellite S2 connected throughtether in a Keplerian orbit (Fig. 1), and the kinetic energiesof the main satellite, sub-satellite and tether are given by

T1 ¼1

2m1

(2 _yRs_l1�

_Rsl1 _u� �

Saþ2 _Rs_l1þ_yRsl1 _u

� �Ca

h iCb

�2sl1 _b�_RCaþ _yRSa

�Sbþsl21

_b2þ _u2sl21C2bþ _y

2R2þ _R

2þs_l

2

1

T2 ¼1

2m2

(2 _yRs_l2�

_Rsl2 _u� �

Saþ2 _Rs_l2þ_yRsl2 _u

� �Ca

h iCb

�2sl2 _b�_RCaþ _yRSa

�Sbþsl22

_b2þ _u2sl22C2bþ _y

2R2þ _R

2þs_l

2

2

�

T3 ¼1

2

Z sl2

sl1

r_r*

xU_r*

xdx

8>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>:

ð1Þ

where

_u ¼ _yþ _a_r*

xU_r*

x ¼ 2 _yR _x� _Rx _u� �

Saþ2 _R _xþ _yRx _u� �

Cah i

Cbn

�2x _b _RCaþ _yRSa� �

Sbþx2 _b2þ _u2x2C2bþ _y

2R2þ _R

2þ _x

2o

l1 ¼ ðm2þmt=2Þl=M

l2 ¼ ðm1þmt=2Þl=M, sli ¼ ð�1ÞiUli

(

Hence, the kinetic energy of the whole system can bewritten as

T ¼1

2m1ðm2þmtÞ=M� �_l2

þ1

2M _R

2þ _y

2R2

� �þ

1

2M�l2 _b

2þ _y

2þ _a2

� �C2b

ihð2Þ

where

M� ¼ðm1þmt=2Þðm2þmt=2Þ

M�

mt

6

The potential energy of the main satellite, sub-satelliteand tether are given by

V1 ¼�mm1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R2þ2Rsl1CaCbþ sl21

pV2 ¼�

mm2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2þ2Rsl2CaCbþ sl2

2

pV3 ¼�

R sl2sl1

mrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2þ2RxCaCbþx2p dx

8>>>>><>>>>>:

ð3Þ

Using Taylor’s expansion formula, we can get thepotential energy of the whole system as follows:

V ��mM

RþmM�l2

2R3ð1�3C2aC2bÞ ð4Þ

Utilizing Lagrange’s equations, the equations of motionabout l, a and b can be expressed as follows:

€l ¼�m1_l2

l þm2l _b2þð _yþ _aÞ2C2bþ m

R3 ð3C2aC2b�1Þh i

� TtMm1ðm2þmt Þ

€a ¼�ð _yþ _aÞ ð2þm3Þ_ll�2 _b tanb

h i�

3m2R3 Sð2aÞ� €y

€b ¼�ð2þm3Þ_ll_b� 1

2 ð_yþ _aÞ2Sð2bÞ� 3mC2aSð2bÞ

2R3

8>>>><>>>>:

ð5Þ

m refers to gravitational constant of central planet; R refersto distance between the center of earth and the center ofTethered Satellite System; _y refers to orbit rate. l refers totether length; a refers to tether in-plane angle. b refers totether out-of-plane angle; _R, _l, _a and _b is differentiationwith respect to time; M=m1+m2+mt refers to system mass.m1 refers to mass of main satellite. m2 refers to mass of sub-satellite. mt refers to tether mass(mt=rl. r: mass per unitlength of strained tether). C=cos; S=sin

m1 ¼ ð2m1�MÞmt=½2m1ðm2þmtÞ�

m2 ¼ ðm2þmt=2Þ=ðm2þmtÞ

m3 ¼ 2½m1ðm2þmt=2ÞðMM�Þ�1�

8><>:

3. The existence conditions of equilibrium

When the system’s center of mass is located at a certainpoint in a circle or elliptical orbit, the equilibrium state ischosen as Xe=(ae,be), and the equilibrium state occurswhen €X e ¼

_X e ¼ 0, then the following formula will beobtained:

Sð2aeÞ ¼2R3

3m2e _y

2Sy

1þ eCy�ð2þm3Þ_ll_y

h i� _y

2sin2be�

3m cos2 aesin 2be

R3 ¼ 0

8><>: ð6Þ

According to Eq. (6), the equilibrium point of be shouldsatisfy: S(2be)=0. Considering the range ofb as bA[�p/2,p/2],then the equilibrium of be exists only in three cases: be=0,be=p/2, and be=�p/2, that is, the tether is coincided withthe orbital plane, or the tether is perpendicular to theorbital plane.

Next, we will show the satisfied conditions of a inequilibrium state. According to the knowledge of system

Y. He et al. / Acta Astronautica 68 (2011) 1964–1972 1967

centroid orbit kinematics, we have

Sð2aeÞ ¼2

32eSy�ð2þm3Þ

_l

ol

ð1�e2Þ3=2

1þeCy

" #ð7Þ

where o refers to the average angular velocity of TSS orbit,and its value is

ffiffiffiffiffiffiffiffiffiffiffim=a3

p. Combined with the character of the

sine function (9S(2ae)9r1), the existence conditions ofequilibrium about a can be deduced as

�1r2

32eSy�ð2þm3Þ

_l

ol

ð1�e2Þ3=2

1þeCy

" #r1 ð8Þ

It can be noted from Eq. (8) that

F2ðyÞð1�e2Þ

3=2r ð2þm3Þ

_l

olr

F1ðyÞð1�e2Þ

3=2ð9Þ

and

F1ðyÞ ¼ 1:5þ1:5eCyþ2eSyþ2e2SyCyF2ðyÞ ¼ �1:5�1:5eCyþ2eSyþ2e2SyCy

(ð10Þ

During the deploying phase, i40.Therefore, Eq. (9) canbe written as

0o ð2þm3Þ_l

olr

F1ðyÞð1�e2Þ

3=2ð11Þ

While during the retrieving phase, io0. Then, Eq. (9)becomes

F2ðyÞð1�e2Þ

3=2r ð2þm3Þ

_l

olo0 ð12Þ

In order to make the equilibrium existing when thesystem center of mass moves at an arbitrary point on orbit,the eccentricity should be 0reo0.75, and in the deploy-ing and retrieving phase, we have

0o ð2þm3Þ_lol

r ½F1ðyÞ=ð1�e2Þ3=2�min

½F2ðyÞ=ð1�e2Þ3=2�maxr ð2þm3Þ

_lol

o0

8<: ð13Þ

From the analysis of Eq. (13), we can get the followingconclusion: the extent of these equilibrium conditions isthe most widest when the system center of mass moves in acircle orbit.

Fig. 2. (a and b) Relationship among eccentricity, true a

Theorem 1. the necessary and sufficient condition of the

maximum extent that system equilibrium can reach is e=0.

Proof. () sufficient): This case can be proven by means ofreduction ad absurdity. That is, we select Cy=�1,eA(0,0.75], then we have

0oeð2�2e2�e3Þ ¼ ð1þeÞ3ð1�eÞ�1

Then, we can get

0o1:5ð1�eÞ=ð1�e2Þ3=2o1:5

�1:5o�1:5ð1�eÞ=ð1�e2Þ3=2o0

(ð14Þ

Eq. (14) verified the validity of the sufficient.((neces-

sary) we can use the way of reduction ad absurdity to prove

this case. Supposing there exists the case that the extent of

these equilibrium conditions is the most wide when the

system center of mass moves in an elliptical orbit., and we

select Cy=�1, eA(0,0.75], then we get the same results as

Eq. (14), which is conflict with the above hypothesis,

therefore the original conclusion is established. &

Compared with eccentricity of 0, the extent of equili-brium conditions will be reduced when the eccentricitybelong to (0,0.75], and the bigger the eccentricity is, thenarrower the extent is.

Let

Deplðe,yÞ ¼ ½F1ðyÞ=ð1�e2Þ3=2�

Retrðe,yÞ ¼ ½F2ðyÞ=ð1�e2Þ3=2�

(

when the eccentricity is not equal to zero, the parameter yn

(it is the parameter yr when F001(yr)40 or F002(yr)o0 issatisfied) can be obtained by following two steps: (1) getthe value of yr (it is the parameter y when dF1(y)/dy=0 ordF2(y)/dy=0 is satisfied) by solving equation dF1(y)/dy=0(solving dF2(y)/dy=0 during retrieving phase); (2) take thevalue of y* by judging whether F001(yr)40 or not (judgingwhether F002(yr)o0 or not during retrieving phase). Afterthat, we can get the equilibrium conditions by substitutingy* into Eq. (13).

When the value of eccentricity changes from 0 to 0.75and the value of y changes from �p to p, the results ofDepl(e,y) and Retr(e,y) are presented as Fig. 2.

nomaly and the function Depl(e,y) and Retr(e,y).

0 0.25 0.5 0.750

0.5

1

1.5

e

min

(F1)

0 0.25 0.5 0.75-1.5

-1

-0.5

0

e

max

(F2)

Fig. 3. (a and b) The extent of equilibrium conditions during deploying and retrieving phase.

Y. He et al. / Acta Astronautica 68 (2011) 1964–19721968

The e-axis of Fig. 2(a) and (b) refers to the value ofeccentricity, changing from 0 to 0.75; the y-axis of Fig. 2(a)and (b) refers to the value of true anomaly, changing from�p to p; Depl-axis of Fig. 2(a) refers to the value ofDepl(e,y); Retr-axis of Fig. 2(b) refers to the value ofRetr(e,y). Fig. 2(a) shows that the value of Depl is greaterthan 0, then we can get the information that the equili-brium conditions exists during deploying phase. ForFig. 2(b), it shows the value of Retr is less than 0, thenwe can reach the conclusion that the equilibrium condi-tions also exist during retrieving. From Fig. 2(a) and (b), wecan find that the difference between the maximum andminimum of Depl(e,y) or Retr(e,y) will be increased withthe increase of eccentricity, but the extent of equilibriumconditions will be reduced. The change of the extent ofequilibrium conditions is shown in Fig. 3.

The e-axis of Fig. 3(a) and (b) refer to the value ofeccentricity, changing from 0 to 0.75; the max(F1)-axis inFig. 2a and min(F2)-axis in Fig. 2(b) refer to the maximumextent that these conditions can be reached during deploy-ing (see in Fig. 3(a)) and retrieving phase (see in Fig. 3(b)).The maximum extent of these conditions are also equal tothe value of 9Deplmin9 and 9Retr(e,y)max9. From these twofigures, we can clearly find that the extent of the equili-brium conditions is maximal when the system’s center ofmass moves in a circle orbit, and the maximum extent ofthese equilibrium conditions will be reduced with theincreasing of eccentricity during deploying or retrieving.When e is equal or greater than 0.75, there have noequilibrium sates.

During station-keeping, from the note of Eq. (7), thesatisfied conditions of a in equilibrium state can be given in

9Sð2aeÞ9¼4

3eSy

ð15Þ

Analyzing Eq. (15), we can obtain the equilibriumconditions during the station-keeping phase that is theeccentricity must satisfy this expression—0rer0.75.

4. Stability of the equilibrium state

From the views of mechanics, equilibrium reflects thebalance relationship of tether tension, gravity, centrifugal

force, and Coriolis force acting on the system center of massin the direction of the phase angle (the angle between thedirection of tether length and the direction of gravitygradient). However, not all of the equilibrium states arestable, thus, it is necessary to analyze stability of theprevious equilibrium states.

Since l5R, using Taylor’s expansion, from Eq. (5), wecan derive the linear equations of motion from smallderivation ofDXT ¼ ½Da Db �. The equations are written as

D €a ¼�D _að2þm3Þ_y _l

l�3mR3 Cð2aeÞDa

D €b ¼�Cð2beÞð_y

2þ3mC2ae=R3ÞDb�ð2þm3Þ

_ll D_b

8<: ð16Þ

And the characteristic equations are given as

l2þð2þm3Þ

_y _ll lþ

3mR3 Cð2aeÞ ¼ 0

l2þð2þm3Þ

_ll lþCð2beÞð

_y2þ3mC2ae=R3Þ ¼ 0

8<: ð17Þ

m refers to gravitational constant of central planet; R refersto distance between the center of earth and the center ofTethered Satellite System; _y refers to orbit rate. l refers totether length; ae and be are the equilibrium state; _l isdifferentiation with time; l is the characteristic root.C=cos; S=sin.

The characteristic roots of Eq. (17) are given by

lð1,2Þ ¼�½ð1þm3=2Þ _y_l=l�7ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½ð1þm3=2Þ _y_l=l�2�3mCð2aeÞ=R3

q

lð3,4Þ ¼� ð1þm3=2Þ _y_l=lh i

7

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þm3=2Þ _y _l

l

h i2�Cð2beÞ

_y2þ

3mC2ae

R3

� �r8>><>>:

ð18Þ

The linear system stability criteria show that theequilibrium state is stable only when the real parts ofthe characteristic equation are negative. Then, we candeduce the following three conclusions: (1) during retriev-ing phase, the equilibrium state is unstable; (2) duringdeploying and station-keeping phase, the equilibrium stateis stable when C(2ae)40 and C(2be)40 (that is, the anglebetween the direction of tether length and the direction ofgravity gradient should be within 451); and (3) duringdeploying and station-keeping phase, the equilibrium stateis also stable when D1o0 and D2o0.

Y. He et al. / Acta Astronautica 68 (2011) 1964–1972 1969

where

D1 ¼ ½ð1þm3=2Þ _y_l=l�2� 3mR3 Cð2aeÞ

D2 ¼ ½ð1þm3=2Þ _y_l=l�2�Cð2beÞð_y

2þ3mC2ae=R3Þ

8<: ð19Þ

Obviously, the stable equilibrium state of b is be=0.In the following we will carry out concentrated research

in studying the stable equilibrium state of a. The existenceconditions of equilibrium during deploying were presentedin Eq. (13), that is

0o ð2þm3Þ_l

olr

F1ðyÞð1�e2Þ

3=2ð20Þ

From Eqs. (7), (10) and (20), we have

�1rSð2aeÞo4eSy=3 ð21Þ

Eq. (21) shows that the system is stable when ae changesin ð0:5pþ0:5asinð4e=3Þ,p�0:5asinð4e=3ÞÞ or ð1:5pþ0:5a

sinð4e=3Þ,2p�0:5asinð4e=3ÞÞ. In other words, sub-satelliteshould be located in back-upper or anterior-upper of mainsatellite at the equilibrium state. On the other hand, theequilibrium state should satisfy C(2ae)40 if the system isstable. Thus, the stable equilibrium state of a is ae 2 ð3 p= 4,p�0:5asinð4e=3ÞÞ or ae 2 ð7p=4,2p�0:5asinð4e=3ÞÞ.

If D1o0 and D2o0, the system is also stable. FromEqs. (7), (10), (19) and (20), we have

Cð2aeÞ4 ð0:75þeSyÞð1þeCyÞ ð22Þ

Considering eo0.75, Eq. (22) can be simplified as

Cð2aeÞ40 ð23Þ

Eq. (23) shows that the stable equilibrium state in thecase of D1o0 and D2o0 is the same as the case ofC(2ae)40.

5. Control law of tether motion

Based on the analysis of above system stability, themotion of tether deploying is stable as long as

0o ð2þm3Þ_l=ðolÞr ½F1ðyÞ=ð1�e2Þ

3=2�min ð24Þ

While the motion of tether retrieving is unstable if onlyguarantee

½F2ðyÞ=ð1�e2Þ3=2�maxr ð2þm3Þ

_l

olo0 ð25Þ

But we can build a stable second-order system tostabilize TSS by introducing the information of a, _a, b,and _b. Similar methods have been studied in Refs. [22,24],but we will do a further study on these methods to enablethem to adapt the case that tether mass cannot be ignored,and enlarge the stability of the domain. Same as theseabove literatures, this paper also choose in-plane motion asan example.

5.1. Construction of controlling algorithm

As an example, we select the case that the system centerof mass moves on a circle orbit (the case in elliptical orbitcan be similarly studied), then we have _y

2¼ m=R3, €y ¼ 0. Let

d=a�ar, where ar is in-plane’s reference angle, then Eq. (5)

can be simplified as

€dþð2þm3Þð_yþ _dÞ

_l

lþ

3m2R3

Sð2dþ2arÞ ¼� €ar�ð2þm3Þ _ar

_l

lð26Þ

Take length rate _l as

_l ¼k1_dþk2d�1:5 _y

2sinð2dþ2arÞ

_yþ _dl

2þm3

ð27Þ

where k1 and k2 are controlling parameters. SubstitutingEq. (27) to Eq. (26), we have

€dþk1_dþk2d¼� €ar�ð2þm3Þ _ar

_l

lð28Þ

Note that if ar is a constant, then Eq. (28) has become asecond-order equation. The in-plane angle a can be stabi-lized with the neighborhood of ar by setting the controllingparameters—k1, and k2 to make imaginary parts ofcharacteristic root.

5.2. Adjustment of controlling algorithm

The tether rate depends on d, _d, ar and l. From Eq. (27),but tether initial rate depends heavily on the releasemechanism. Generally speaking, the initial rate does notmeet the relationship of Eq. (27), thus it is necessary to addthe modified term to Eq. (27) for linking with the actualsituation. The modified term will be chosen as

_lreal ¼_lþð_l0�

_lÞe�ct ð29Þ

where c refers to the decay rate of differences between _lreal

and _l; t is time of tether rate control. _lreal is equal to _l, whent=0, and the result indicates that this method can beeffective used to link the initial value with the actual value._lreal is very close to _l, when the time is longer, thus Eqs. (29)and (27) have the same stability.

From Eq. (29) we can obtain length acceleration asfollowing:

€lreal ¼lð1�e�ctÞ

ð2þm3Þ_yþ _d

k1�ð2þm3Þ_l=l

h i€dþk2

_dn

�3 _y2ð _dþ _arÞcosð2dþ2arÞ

o� ð1�e�ctÞ_lrealþcle�cth i _l

l�c_l0e�ct�

_m3_l

2þm3

ð1�e�ctÞ

ð30Þ

and

_m3 ¼ _mt�m0

1ðm2þmt=2Þ2þ 13 Mðm0

1m2þm2t =2Þ

ðMM�Þ2

From now, we can make equation of motion become theequations composed of Eqs. (30) and (26).

5.3. Controlling characteristic

System in the steady state should satisfy the followingrelationship, that is: d-0, _d-0, and €d-0, then tether ratecan be written as

_lreal ��1:5 _ylð1�e�ctÞSð2arÞ

ð2þm3Þþ _l0e�ct ð31Þ

Y. He et al. / Acta Astronautica 68 (2011) 1964–19721970

When t is longer, Eq. (31) can be simplified as

_lreal ��1:5 _ylSð2arÞ

ð2þm3Þð32Þ

The relationship between the reference angle ar andtether motion can be obtained by Eq. (32), which isconcluded as following:

_lreal

40 ar 2 ðp=2,pÞ [ ð�p=2,0Þ

o0 ar 2 ð0,p=2Þ [ ð�p,�p=2Þ

-0 ar 2 f0,7pg

8><>: ð33Þ

Eq. (33) indicates: (1) the tether is in the deployingphase when ar is located in 1 or 3 quadrant; (2) the tether isin the retrieving phase when ar is located in 2 or 4 quadrantand (3) the tether is in the station-keeping phase when ar isequal to 0, p, or �p.

6. Numerical simulations

To validate the proposed conclusions and the controllaw, numerical simulations are presented in this part. Thesystem is assumed to be in an orbit with the radius of7078 km. The mass of main satellite and sub-satellite aretaken as 10,000 and 100 kg. The initial parameters of l, _l, dand _d are selected as 1 m, 0.1 m/s, p/18 and 0, respectively.

The two stages are considered in this numerical simulation:(1) deployment and station keeping; (2) station-keeping andretrieval. In the first stage, when the sub-satellite is released to30 km from the position in back-upper of main satellite, thesystem begins station-keeping. Eight hours later, the sub-satellite is retrieved to 2 km from the position in anterior-upper of the main satellite

The controlling parameters of k1 and k2 are all chosen ask1=0.0057 and k2=9.8696�10�6. The value of ar and c inthis two stages are selected as

(1)

The stage of deployment and station-keeping. In thisstage, coefficient of Eq. (31) is c=10�3, and ar satisfy

ar ¼

� p4 lr10km

kðl�3� 104Þ lZ30km

f ðlÞ 10kmr lr30km

8><>: ð34Þ

0 5 10 15 200

5

10

15

20

25

30

35

t (h)

L (k

m)

0

5

10

15

20

T t (N

)

Fig. 4. (a and b) The change of tether length and

where k=p�10�5; f(l) is a transition function, and thebelow relationship are satisfied in l=10 km, and l=30 km

f ¼� p4 ,flu0,fl

00 ¼ 0 l¼ 10km

f ¼ 0,flu¼ k,fl00 ¼ 0 l¼ 30km

(

(2)

The stage of station-keeping and retrieval. In this stage,coefficient of Eq. (31) is c=5�10�4, and ar satisfy

ar ¼

p10 lZ5km

kðl�2� 103Þ lr2km

f ðlÞ 2kmr lr5km

8><>: ð35Þ

where k=5p�10�5; f(l) is also a transition function, andthe below relationship are satisfied in l=2 km, and l=5 km.

f ¼ p10 ,flu¼ 0,fl

00 ¼ 0 l¼ 5km

f ¼ 0,flu¼ k,fl00 ¼ 0 l¼ 2km

(

Fig. 4(a) and (b) refers to the change of tether length andtension, respectively, during the motion of TSS. Where theunits of the horizontal axis are hour, the unit of the vertical axisis km and N, respectively. From these two figures, whether thetether length or tether tension changes smoothly; at thebeginning of tether releasing, the change of tether length isrelatively slow, the reason is the difference of gravity gradientforce between the sub-satellite and the main satellite is verysmall when the tether length is very short; tether tension willappear larger shake in the switching process, this is because ofthe change of _a.

Table 1 is partial data of the two stages in station-keeping, and its truncation error is 0.01 m. The data of thefirst stage is considered when time t changes from 7 to 8 h,while the data of second stage is considered when time t

changes from 14 to 15 h. Two stages of time intervals are all0.05 h to take a number, that is, take the value of tetherlength when the next 180 s is arriving, the results areshowed in Table 1. For the convenience of analysis andcomparison, the unit of time and tether length in Table 1are selected as h and m.

The results in Table 1 indicate that there exists wellstability by using the proposed method in both the two

0 5 10 15 20t (h)

tension during the motion of TSS.

0 5 10 15 20-50

-40

-30

-20

-10

0

10

20

t (h)

α r (d

eg)

α (d

eg)

Fig. 5. (a and b) The change of reference and actu

0 10 20 30 40-50

-40

-30

-20

-10

0

10

20

L (km)

α r (d

eg)

α (d

eg)

Fig. 6. (a and b) The change of reference and actual in-plane a

Table 1The value of tether length during station keeping.

The first stage The second stage

t,h l,m t,h l,m

7 29999.965 14 2000.512

7.05 29999.959 14.05 2000.477

7.1 29999.958 14.1 2000.444

7.15 29999.961 14.15 2000.415

7.2 29999.967 14.2 2000.388

7.25 29999.975 14.25 2000.363

7.3 29999.982 14.3 2000.341

7.35 29999.990 14.35 2000.321

7.4 29999.996 14.4 2000.302

7.45 30000.002 14.45 2000.285

7.5 30000.005 14.5 2000.270

7.55 30000.008 14.55 2000.256

7.6 30000.009 14.6 2000.243

7.65 30000.009 14.65 2000.232

7.7 30000.008 14.7 2000.221

7.75 30000.007 14.75 2000.212

7.8 30000.005 14.8 2000.203

7.85 30000.003 14.85 2000.195

7.9 30000.002 14.9 2000.188

7.95 30000.000 14.95 2000.181

8 29999.999 15 2000.175

Y. He et al. / Acta Astronautica 68 (2011) 1964–1972 1971

stages. The error of tether length in the first stage is lessthan 0.1 m during steady state, while in the second stagethe error of tether length is less than 0.6 m, and maintainsthe downward trend.

The reference and actual angle are showed in Fig. 5(a)and (b).

Fig. 5(a) shows the change of in-plane reference angle.The value of reference angle ar includes the change of theabove two stages. In the first stage, ar maintains a period oftime at �451. Until lZ10 km, the value of ar changes from�451 to 01 according to Eq. (34), and then the state of TSSstarts to step into the state of station-keeping at theneighborhood of 0. Eight hours later, the tether will beretrieved, and ar will be taken as 181. In order to avoid themutation of ar, some modified works are acted on thereference angle ar, that is we make ar change from 0 to 181in 500 s along the slope of 6.2932�10�4. The mutationtrend still appears in Fig. 5(a), this is because 500 s is asmall spacing compared with the time of 10 h. Whenlr5 km, ar will change from 181 to 01 according to Eq.(35), and then the system will step into the state of station-keeping at the neighborhood of 0.

Fig. 5(b) is the variation of the in-plane angle by usingthe proposed control algorithm. From this figure, we can

0 5 10 15 20-80

-60

-40

-20

0

20

t (h)

al in-plane angle during the motion of TSS.

0 10 20 30 40-80

-60

-40

-20

0

20

L (km)

ngle relative to tether length during the motion of TSS.

Y. He et al. / Acta Astronautica 68 (2011) 1964–19721972

find: in the first stage, the value of a can approach to theneighborhood of reference angle ar=�451, when tetherlength lZ10 km occurs, awill change from �451 to 01, andthen start to station-keeping at the neighborhood of 0.Eight hours later, a can approach to the neighborhood ofreference angle ar=181, then change from 181 to 01 as thechange of ar, finally, a will be stabilized in the vicinity of 0.The change of reference and actual in-plane angle relativeto tether length are shown in Fig. 6.

7. Conclusions

In order to make the stability of the results of TSS moregeneral, this paper deeply studies the system stability withconsidering tether mass and the system center of massmoving in elliptical orbit. The following conclusions areconcluded: the extent of system equilibrium conditions isthe most widest when the system center of mass moves in acircle orbit; compared with e=0, the extent of equilibriumconditions will be reduced when the eccentricity belong to(0,0.75], and the bigger the eccentricity is, the narrower theextent is; in order to stabilize TSS during deploying,station-keeping and retrieving, the eccentricity mustsatisfy 0reo0.75, otherwise, some extra control forceshould be acted on the system to achieve a stable system;the equilibrium state in deployment phase is stable as longas the equilibrium conditions exist, while the equilibriumin retrieval is unstable even if the equilibrium conditionsexist, but unstable equilibrium state can be stabilized byintroducing the information of a, _a, b, and _b.

Based on the theoretical analysis above, a range-ratecontrol algorithm is developed to achieve the stable controlof tether’s deploying, keeping, and retrieving. Simulationresults show that the presented method can be used tostabilize TSS during the process of deploying, station-keeping and retrieving. On the other hand, from the studyon change of in-plane angle, we find that the state of systemstability is well consistent with the results of theoreticalanalysis.

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