2008 aiaa/aas astrodynamics specialist...

2008 AIAA/AAS Astrodynamics Specialist Conference, 18-21 August 2008, Honolulu, Hawaii, USA

On the dynamics of a tethered systemnear the collinear libration points

M. Sanjurjo-Rivo,∗ Fernando R. Lucas,† J. Peláez,‡

Technical University of Madrid (UPM), 28040 Madrid, Spain

Claudio Bombardelli,§

ESTEC-ESA, Noordwijk, 2201 AZ, The Netherlands

Enrico C. Lorenzini,¶ Davide Curreli,‖

University of Padova, Padova, 35131, Italy

Daniel J. Scheeres,∗∗

The University of Colorado, Boulder, CO 80309-0429, USA

and M. Lara,††

Real Observatorio de la Armada, 11110 San Fernando, Spain

In this paper we investigate the dynamics of a tethered system near the collinear libration points exploitingthe benefits of the Hill approach. Rotating and non-rotatingtethers, with constant or variable length, areinvestigated. We include the mass of the tether in the formulation to obtain more accurate simulations whenvery long tethers are involved. We try simple strategies that permit, using a feedback control law, to stabilizethe system around equilibrium positions which are basically unstable.

I. Introduction

THe Lagrange equilibrium solutions of the Circular Restricted Three Body Problem (CRTBP) have turned out tobe much more important than they seemed at a first sight. Therehave been many space applications in the past,

and at present, which have used these libration points as essential elements of space missions (see Ref.1). They willlikely play a much more important role for future space exploration missions. Particularly attractive are the collinearpoints (L1, L2 andL3) given their location and accessibility. Unfortunately all of them are unstable, which means aspacecraft to be kept at or orbiting around them will requirecorrection manoeuvres typically to be performed at theexpense of propellant mass. Colombo, one of the pioneer about the use of space tethers showed in2 the feasibility ofcontrolling the unstable nature of the collinear Lagrangian points exploiting a varying-length tether system. Such aconcept was later studied more deeply by Farquhar in Refs.3,4 where he set out the following problem: «Considertwo satellites of equal mass that are connected by a light cable of adjustable length. Is it possible to stabilize theposition of the mass center of this configuration in the vicinity of a collinear libration point by simply changing thelength of the cable with an internal mechanism?». This control scheme, which exploits the capability of dumbbellsystems of shifting the centre of gravity position with respect to the centre of mass in a controlled manner, allowskeeping the position of an artificial satellite close to the Lagrangian points without using propellant.

∗PhD. Student, ETS Aeronáuticos, Pz. Cardenal Cisneros 3, 28040 Madrid, Spain, [email protected]†Student, ETS Aeronáuticos, Pz. Cardenal Cisneros 3, 28040 Madrid, Spain, [email protected]‡Professor, ETS Aeronáuticos, Pz. Cardenal Cisneros 3, 28040 Madrid, Spain, Lifetime AIAA, Member [email protected]§Research fellow, Advanced Concept Team, ESTEC, Keplerlaan1, 2201 AZ, Noordwijk, The Netherlands, [email protected]¶Professor, Faculty of Mechanical Engineering, Via Venezia1, 35131, Padova, Italy, [email protected]‖PhD. Student, Faculty of Mechanical Engineering, Via Venezia 1, 35131, Padova, Italy, [email protected]

∗∗Professor, Department of Aerospace Engineering Sciences,429 UCB Boulder, CO 80309-0429, [email protected]††Scientist, Ephemerides Section , 11110 San Fernando, Cádiz, Spain [email protected]

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American Institute of Aeronautics and Astronautics Paper 2008-7380

Misra et al.5 later studied the problem with a different approach and included the possibility of using rotatingtethered system of constant length to ease the stabilization process. In the latter study the dynamics of a passive(constant-length) rotating dumbbell system were investigated without addressing possible control strategies. The needfor further, more in-depth dynamic and control analysis of the system was highlighted.

In general, tether dynamics is complex (see10) and close to the collinear points it is influenced by many differentfactors. One of them is the reduced massν of the primary around which the tether is moving. Usually this parameteris small and the Hill approximation allows a much more simpledescription of the dynamics which permits to gainan insight into the complex evolution of the tethered system. For these cases the Hill formulation directly providesan excellent approximation to the problem. Moreover, it gives significant clues for the analysis of the general case inwhich ν is of order unity; this way the approximation makes easier the indispensable numerical analysis associatedwith a detailed description of the dynamics.

Most of the analysis performed previously neglected the tether mass. However, some cases considered involvevery long tethers; for example, Misra et al. in5 described the dynamics of a≈ 1500 km long tether. Even with verylight materials the mass of the tether could be important; the SEDS tether had a mass of 0.33 kg/km and using thesame tether (Spectra 1000, 0.7 mm of diameter) the mass wouldreach, approximately, 500 kg. Farquhar in3 considera tether 3844.05 km long made of Aluminum and with a mass about504.15 kg; assuming a wire the diameter ofthe tether would be≈ 0.25 mm. Even with such a very fine tether the mass start to be significant. Therefore, itseems appropriate to include the mass of the tether in the formulation, specially when some control strategies withvariable-length tethers required to increase the tether length in a significant way.

In this paper we investigate the dynamics of a tethered system near the collinear libration points exploiting thebenefits of the Hill approach. We try a simple strategy that permits, using a feedback control law, to stabilize thesystem around equilibrium positions which are basically unstable. Rotating tethers, with constant or variable length,are investigated.

More recently Peláez and Scheeres8,9 proposed to placeelectrodynamic tethersfor permanent power generationat points in the neighborhood of the Lagrangian points of theinner Jupiter moonlets (Metis, Adrastea, Amalthea andThebe). The electrodynamic tether will bedeorbitingthe moonlet by using its gravitational attraction; in doingso itconverts the mechanical energy of the moonlet into electrical energy that can be used onboard. As a consequence, acontinuous power can be extracted from the orbital energy ofthe moonlet. In that case, current control was proposedas the sole mean to stabilize both the position and the attitude of a constant-length non-rotating electrodynamic tethersystem placed in the vicinity of the unstable Lagrangian points of the moonlets. The dynamical analysis of9 showsthat there exist equilibrium positions where the tether could be operated appropriately. Some of these equilibriumpositions arestableand otherunstable. In this paper electrodynamic tethers are not considered; but the near future,the effects of the electrodynamics forces on the stability of the equilibrium positions close to the Lagrangian pointswould be taking into account for tethers made of conductive materials.

II. Previous Analysis

Firstly, we consider the article4 by Farquhar. He obtained the governing equations of a tethered system in theneighborhood of a collinear point: 1) taking the cartesian coordinates of the end masses as generalized coordinatesand introducing the tether length as a constraint, 2) assuming a massless tether with two equal end masses, 3) expandingthe equations around the small primary and neglecting termshigher than second order, and 4) assuming no additionalperturbation on the system.

Farquhar found unstable equilibrium positions close to thecollinear points. He studied first the one dimensionalmotion of the tether, neglecting the coupling terms. Then, he tackle the tridimensional analysis and introduces acontrol law involving the orientation of the cable; the out-of-plane motion turns out to be not controlable with this law,and a detailed analysis is carried out to obtain the gains which ensure stability for the in-plane motion.

The approach gathered in the article5 by Misra el al. is devoted to the same purpose, having one’s eye on theEarth-Moon system. However, there are differences regarding Ref.4; some of them are listed in what follows: 1) thepaper only consider the in-plane motion and it takes as generalized coordinates the cartesian coordinates of the systemcenter of mass and the tether in-plane libration angle, 2) they use a massless tether, but the end masses can be different.

In a classical analysis, similar to Farquhar’s one, they linearize the equations of motion in the neighborhood of

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a collinear Lagrangian point and present some numerical experiments to avoid the instability; they are based on thecontrol of the tether length and on the effect of rotation on tether motion. Accordingly, a control law is providedshowing its stabilizing efficacy for specific values of the control gains. Moreover, the possibility of stabilize thedynamics of a tethered system by means of its rotation is presented in a particular case.

More complex analysis have been carried out by Wong and Misrain papers,6,7 where the three-dimensional con-trolled dynamics of a spinning tether connected three–spacecraft system near the second libration point of the Sun-Earth system is examined. The center of mass follows a predefined trajectory (Lyapunov orbit, Lissajous trajectoryor halo orbit); in order for this spinning formation to be useful as an interferometer, the tether librations must be con-trolled in a way such that the observation axis of the system can be pointed in a specified direction, while the center ofmass follows one of these trajectories.

III. New Approach. Extended Dumbbell Model

In this paper a new approach facing the general treatment of atwo point tethered system moving in the neighbor-hood of a collinear libration point is presented. We extend the analysis previously cited by accommodating: 1) a threedimensional analysis allowing different mass configurations and considering the effect of the tether mass, 2) a moresolid linearization process which avoid, in some extension, the problems associated with several small magnitudes, 3)a reduced number of parameters describing the motion of the system, and 4) a more tractable set of equations whichmake easy the understanding of the main feature of the dynamics.

Our approach is based on the Hill approximation; this formulation uses a spatial scale in the neighborhood of thesmall primary where thegravity gradientof the main primary, theinertia Coriolis forceand thegravitational attractionof the small primaryare of the same order of magnitude. Such an approach allows toanalyze the dynamics of thesystem in the Lagragian points with the suitable accuracy for the majority of the binary systems of the Solar Systemand permits to present the equations of motion in an understandable way even when a general case is considered.Moreover, there is no need of carrying out a linearization toachieve the set of equations since the simplification of theoriginal equations is due to an asymptotic expansion in the parameterν, i.e., the reduced mass of the small primarywhich is quite small in a great number of cases. At last but notleast, this formulation provides aclosed modelinvolving only one non-dimensional parameter which sheds light into the importance and the organization of each andevery element implicated in the dynamics.

m

r1

r2GP

ℓ

mP1

mP2

x

y

zω

i

j

k

O

ℓ2 = (1 − ν)ℓ

νℓ

Figure 1. Reference frames

We assume a pair of primaries moving in circular orbitsaround its common center of mass. We summarize here themain results of the CRTBP involved in the analysis. Themasses of the main primary (mP1

) and the small primary(mP2

) are known. Any non rotating frame with origin atGP ,the center of mass of the primaries, is an inertial frame. Therelative motion of primaries takes place in a constant plane.Let GP x1y1 be an inertial frame embedded in this plane.

Since we are interested in the motion of the S/C in theneighborhood of the small primary (mP2

) we take the synodicframeOxyz with origin at the center of mass ofmP2

and theplaneOxy in the orbital plane of both primaries (see fig.1).This frame coincides with the orbital frame of the small pri-mary in its trajectory around the main primary and it is rotatingaround the directionOz with the angular velocity:

ω =

√

G(mP1+ mP2

)

ℓ3(1)

Primaries are at rest in this frame at positionsmP2(0, 0, 0), mP1

(−ℓ, 0, 0).The Hill approximation is introduced by means of the following change of variables:

x = ℓ ν1/3 ξ, y = ℓ ν1/3 η, z = ℓ ν1/3 ζ

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and we use an extended Dumbbell Model which allows for the mass of the tether in the analysis. In such a case, andtaking the non-dimensional timeτ = ω t as independent variable, the equations of motion of the tethered system areexpressed —in non-dimensional form— as follows:

ξ − 2η − (3 − 1

ρ3)ξ =

λ

ρ5

3N cosϕ cos θ − ξS2(N

ρ)

(2)

η + 2ξ +η

ρ3=

λ

ρ5

3N cosϕ sin θ − ηS2(N

ρ)

(3)

ζ + ζ(1 +1

ρ3) =

λ

ρ5

3N sin ϕ − ζS2(N

ρ)

(4)

θ + (1 + θ)

[

Is

Is− 2ϕ tanϕ

]

+ 3 cos θ sin θ =3N

ρ5

(−ξ sin θ + η cos θ)

cosϕ(5)

ϕ +Is

Isϕ + sin ϕ cosϕ

[

(1 + θ)2 + 3 cos2 θ]

=3N

ρ5(− sin ϕ[ξ cos θ + η sin θ] + ζ cosϕ) (6)

whereρ =√

ξ2 + η2 + ζ2 and the quantityN and the functionS2(x) are given by

N = ξ cosϕ cos θ + η cosϕ sin θ + ζ sin ϕ, S2(x) =3

2(5x2 − 1) (7)

O

m1

m2

Gr2

x

x

yy

z

z

θ

ϕ

u

Figure 2. Dumbbell Model

Here,(θ, ϕ) are the classical tether libration angles(see fig. 2). In these equations the new parameterthat captures the influence of the tether length is

λ =

[

Ld

ℓ

]2

· a2

ν2/3, a2 =

Is

m L2d

(8)

where,m = m1 + m2 + mT is the total mass ofthe system andIs is the moment of inertia about aline normal to the tether by the center of massG ofthe system; the non-dimensional parametera2 is oforder unity and takes its maximum valuea2 = 1/4for a massless tether with equal end masses (m1 =m2). For a tether of varying length the parameterλis a function of time since the deployed tether massmd and the deployed tether lengthLd(t) are chang-ing. Moreover, some terms of these governing equa-tions involve the ratio:

Is

Is= 2

Ld

LdJg, Jg = 1 − Λd (1 + 3 cos 2φ)

(

3 sin2 2φ − 2Λd

) , Λd =md

m, cos2 φ =

(

m1

m+

Λd

2

)

(9)

This formulation includes the mass of the tether through theparameterΛd and the mass angleφ. In order to neglect thetether mass, we only have to introduce the conditionΛd = 0 in the above expressions. For a tether of constant lengththe parameterλ is also constant and the quotientIs/Is vanishes, that is,Is/Is = 0. For the sake of simplicity wewill assume a massless tether in what follows but the equations (2-9) can be used to simulate the dynamics of massivetethers in a simple way.

In the search of equilibrium position we will consider the tether tension given by

T0 ≈ m1 m2

m1 + m2ω2Ld

ϕ2 + cos2 ϕ(1 + θ)2 + 3 cos2 θ − 1 +1

ρ33(

N

ρ

)2

− 1 − Ld

Ld

(10)

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This expression has been derived under the following assumptions: 1) the tether is inert, 2) the mass of the tetherhas been neglected and 3) the Hill approach has been taken into account. Obviously, at any equilibrium position thetether tension must be positive because a cable does not support compression stress. In a given equilibrium position,the above expression takes the form

T0 ≈ Tc

cos2 ϕ1 + 3 cos2 θ − 1 +1

ρ33(

N

ρ

)2

− 1

, Tc =m1 m2

m1 + m2ω2Ld (11)

We will use expression (11) to check the tether tension in a given steady solutions of the equations (2-9); if the tensionis positive the equilibrium position will exist; if negative, the equilibrium position will not exist.

IV. Non Rotating Tethers. Constant Length

As a first step we analyze the simplest case: non rotating tethers with constant length for whichIs = 0 and thesystem of equations (2-9) has several families of equilibrium positions depending on λ. The most interesting arecontained into the coordinate planes. Among these families, the one which is related to the collinear libration pointsis characterized by the following values of the variables (which are functions of the parameterλ):

ξe = ±ρe, ηe = ζe = ϕe = 0, θe = 0, π, λ =ρ2

e

3

(

3 ρ3e − 1

)

, ρ3e > 1/3 (12)

The tether lies along theOx axis and the system center of massG is separated from the libration point an amount thatdepends onλ (the square of the tether length). This is the family we will focus on. It start from the collinear librationpointsL1 andL2 with λ = 0 (equivalent to a non tethered system) and it moves away for increasing values ofλ.

10−6

10−4

10−2

2.5

2.55

2.6

2.65

2.7

2.75

2.8

2.85

R(s5)

λ

Figure 3. Real part of the unstable eigenvalue as a function of λ

The expression (12) can be expanded when thetetherlengthλ is small and provide the asymptotic solution

ξe ≈ (1

3)

1

3 + 31

3 λ − 9 λ2 + O(λ3) (13)

The convergence of this serie is poor and many termsare need to obtain a precise value forξe; however, forreally small values ofλ the two first terms give a usefulapproximation.

Nevertheless, the linear stability analysisa shows thatthe equilibrium positions of that family are always un-stable, that is, they are unstable for all values ofλ. Ineffect, two degree of freedom are always stable —theout of plane angleϕ and the coordinateζ which undergooscillatory motions— but the other degree of freedomare unstable because one eigenvalue is always real andpositive, for any value ofλ. Figure3 shows the unstableeigenvalue as a function ofλ. As it can be observe infigure, the presence of the cable makes the systemmore unstable. In spite of that behavior, the value of the parameterλ is usually too small for practical tether lengths and the majority of binary systems in the Solar System, and theincrease on the instability is practically unnoticeable.

However and before to continue with the analysis we review briefly the stabilization technique proposed by Far-quhar in3,4 and considered by Misra et al. in.5

aThe variational equations can be obtained from (14-18) by requiring a constant length for the tether (δλ = 0).

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Dim.

N. Dim. ξL2= ( 1

3)1

3

xL2= ( 1

3)1

3 Lc

31

3 λ

31

3 λ Lc

∆xL2

(1) (2)O

G

Figure 4. Distances from the small primary and from the collinear point L2

whenλ ≪ 1 (Lc = ℓν1/3)

Such a technique is based in the following rea-soning: let us assume that the tether length isL andfor that length we have the equilibrium position la-beled with(1) in figure 4. On the right of such aposition, the system center of massG is acted bya force that impulses it toward the right. By in-creasing the tether length up toL′ = L + ∆L wemove the equilibrium position up to the point la-beled with(2) in figure4; now the force acting onG impulses it toward the left. Then we decrease thetether length in order to move the equilibrium position on the left side ofG . . . Thus, by changing the tether length inan appropriate way the center of massG can bestabilizedand kept in the neighborhood of the collinear lagrangianpointL2. Following these ideas, the next paragraph is devoted to explore the variation of the tether length as a way ofcontrolling the instability.

V. Non Rotating Tethers. Variable Length

First of all, in the equations (2-9) an approximation for the gravitational potential of both primaries has been made.For example, the gravitational potential of the small primary has been expanded in powers of the ratioL/r2, where

r2 is the distancer2 = | −−→OG | (see figure2), and terms of order equal or greater than 3 have been neglected in suchan expansion. This approach works fine whenL/r2 ≪ 1. Most of the times, parameterλ is small for feasible cablelengths in the binary systems of interest in the Solar System. At the equilibrium positionsconsidered in this section,that relation can be expressed in terms of non dimensional variables as follows:

10−4

10−3

10−2

10−1

100

10−1

100

λ

Lr2

Figure 5. Ratio between tether length and distance from thesecond primary to the center of massG of the system as a func-tion of λ, for a2 = 1/4.

L

r2=

√

3 ρ3e − 1

3 a2

and it is plotted as a function ofλ in figure5 for a2 = 1/4.Note that the approach used in the deduction of the equationsof motion is fine whenλ is small but it fails whenλ is of orderunity. When tether length variations are allowed, the varia-tional equations of the system (2-9), around the equilibriumposition (12), takes the form:

δξ =

(

15 ∓ 2

ξ3e

)

δξ + 2 δη − 3 δλ

ξ4e

(14)

δη =

(

6 ± 1

ξ3e

)

δη +(3 ξ3

e ∓ 1)

ξ2e

δθ − 2 δξ (15)

δζ = −(

7 ∓ 1

ξ3e

)

δζ +

(

3 ξ3e ∓ 1

)

ξ2e

δϕ (16)

δθ = ± 3

ξ4e

δη − 3

(

1 ± 1

ξ3e

)

δθ − 3 Jg δλ

λe(17)

δϕ = ± 3

ξ4e

δζ −(

4 ± 3

ξ3e

)

δϕ (18)

whereδλ stands for the variations ofλ from its equilibrium valueλe andJg < 1 is a function of the tether massconfiguration (for a massless tetherJg = 1; see eq. (9)). These equations show that coordinates(ζ, ϕ) associated tothe out-of-plane motion are decoupled from the other coordinates. That is, the out-of-plane motion cannot be correctedby the variation of the tether length as it has been pointed out by Farquhar.4 Fortunately, the out-of-plane motion isstable (oscillatory). From now on, we focus the analysis on the in-plane motion which is the source of the instabilityexhibited by the system.

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A. Linear Approximation for small values of λ

Sinceλ is small, in general, and it plays a relevant role in the dynamics, our first analysis assumes that the distancebetween the system center of massG and the equilibrium position is small and of orderλ (in non-dimensional vari-ables). Thus we try to control deviations from the equilibrium position which are of the order ofλ. In order to do thatwe carry out the following change of variable:

ξ = ξL + 31/3 λ0(1 + u(τ)) , η = 31/3 λ0 v(τ) , λ = λ0(1 + s(τ)) (19)

whereξL = (1/3)1/3 is the distance from the lagrangian pointL2 to the center of mass of the small primary.The next step is to linearize the governing equations (2-9) neglecting terms of orderλ0 in front of unity. The Hill

equations linearized inλ0 lead to this system of equations:

d2 u

d τ2− 2

d v

d τ− 9 u(τ) +

9

2

(

3 cos2 θ cos2 ϕ − 1)

s(τ) +27

2

(

cos2 θ cos2 ϕ − 1)

= 0 (20)

d2 v

d τ2+ 2

d u

d τ+ 3 v(τ) − 9 cos2 ϕ cos θ sin θ (1 + s(τ)) = 0 (21)

d2 w

d τ2+ 4 w(τ) − 9 cosϕ cos θ sin ϕ (1 + s(τ)) = 0 (22)

(1 + s(τ))d2 θ

d τ2+

d s

d τ(1 +

d θ

d τ) − 2 tan ϕ

dϕ

d τ

(

1 +d θ

d τ

)(

1 +d s

d τ

)

+ 12 cos θ sin θ (1 + s(τ)) = 0 (23)

(1 + s(τ))d2 ϕ

d τ2+

d s

d τ

dϕ

d τ+ (1 + s(τ)) sin ϕ cosϕ

[

(

1 +d θ

d τ

)2

+ 12 cos2 θ

]

= 0 (24)

which should be integrated starting from the appropriate initial conditions.In what follows we analyze two different approaches trying the control of this linearized system. The first one is

an «ad hoc» control scheme which emerges from the analysis of the one dimensional case and later on it has beenextended to the bidimensional problem. The second one is a classical control approach for the tridimensional problemthat we call «proportional control».

1. «Ad hoc» control

In this approach, the solution of the above system of equations is forced to be bounded, avoiding directly the insta-bility. Firstly, the one dimensional problem is consideredand then we will tackle the full in-plane analysis.

ONE DIMENSIONAL. With the tether along theOx axis the variablesv(τ), w(τ), θ(τ), ϕ(τ) vanish. The equationgiving the evolution of the system and its analytical solution turn out to be:

d2 u

d τ2− 9 u(τ) + 9 s(τ) = 0 ⇒ u(τ) = C e−3 τ + D e3 τ + up(τ) (25)

The solution of (25) involves a particular solutionup(τ) for anycontrol function s(τ). We select acontrol functions(τ) which fulfills the following requirements: 1) it should be fitted to the general form

s(τ) = e−β τ (A cosΩτ + B sinΩτ)

where the parameters(A, B, β,Ω) satisfy the next conditions, 2) the particular solutionup(τ) must cancel the con-tribution of D e3 τ , and 3) initially the tether length is the one which corresponds to the equilibrium position aroundwhich we are linearizing. These conditions provide the following values for the parameters(A, B):

A = 0 B =Ω2 + (β + 3)2

9 Ω· (3 u0 + u0)

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Figure6 shows the response of the system for the valuesΩ = 1.5 andβ = 0.5 arbitrarily selected. The initialconditions areu0 = 0.15 andu0 = 0 and a massless tether has been selected. Figures show how thesystem tendsasymptotically to the equilibrium position keeping the corresponding equilibrium tether length. The maximum lengthis roughly a 15 % larger than the final one. The control manoeuvre involve small deployment and retrieval tethervelocities of the order of 1 m/s. Notice that the tether tension is small, of the order of 16 mN.

3.6

3.8

4

4.2

4.4

4.6

4.8

5

5.2

0 200 400 600 800 1000

t (hours)

x(km)

940

960

980

1000

1020

1040

1060

1080

1100

1120

1140

1160

0 200 400 600 800 1000

t (hours)

L(km)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 200 400 600 800 1000

t (hours)

L(m/s)

0.0145

0.015

0.0155

0.016

0.0165

0.017

0.0175

0.018

0.0185

0.019

0 200 400 600 800 1000

t (hours)

T (N)

Figure 6. One dimensional control scheme. Massless tether with two equal masses (500 kg) at both ends in the Earth-Moon system. Theselected parameters areβ = 0.5, Ω = 1.5, for the initial conditions u0 = 0.15 and u0 = 0. The nominal tether length isL = 1000 km

BIDIMENSIONAL . In this paragraph we deal with the in plane problemξ, η, θ, since the instability is linked to themotion that takes place within it. Additionally, a new hypothesis is made presuming the system attitude variationswould be small, i.e.,θ ≪ 1 andθ ≪ 1. Hence, the system of equations (20)-(24) yields to:

d2 u

d τ2− 2

d v

d τ− 9 u(τ) + 9 s(τ) = 0 (26)

d2 v

d τ2+ 2

d u

d τ+ 3 v(τ) − 9 θ = 0 (27)

d2 θ

d τ2+

d s

d τ+ 12 θ = 0 (28)

In this case, the general solution for(u(τ), θ(τ), v(τ)) of the system of differential equations can be expressed as:

u(τ ) = C1 sin γ1τ + C2 eγ2τ + C3 cos γ1τ + C4 e

−γ2τ + C5

12

47

√3 cos(2

√3τ ) − C6

12

47

√3 sin(2

√3τ ) + up(τ )

θ(τ ) = C5 sin 2√

3τ + C6 cos 2√

3τ

v(τ ) = C1

√7 + 4

γ1

cos γ1τ + C2

√3(2 −

√7)

γ1

eγ2τ − C3

√7 + 4

γ1

sin γ1τ − C4

√3(2 −

√7)

γ1

e−γ2τ − C5

63

47sin 2

√3τ

−C6

63

47cos 2

√3τ + vp(τ ) where γ1 =

q

−1 + 2√

7, γ2 =

q

1 + 2√

7

8 of 24


As in the previous paragraph, we select acontrol function s(τ) which fulfills the following requirements: 1) itshould be fitted to the general forms(τ) = e−β τ (A cosΩτ + B sin Ωτ) where the parameters(A, B, β,Ω) satisfythe next conditions, 2) the particular solutions ofu(τ), v(τ), θ(τ) must cancel the contribution of the termC2 eγ2τ , and3) initially the tether length is the one which corresponds to the equilibrium position around which we are linearizing.With this requirements, we obtain,A = 0 and an additional relationB = B(β, Ω). Figure7 shows the evolution ofthe system for the bidimensional motion in the same case considered in figure6, that is, the response of the system forthe valuesΩ = 1.5 andβ = 0.5, with the initial conditionsu0 = 0.15 andu0 = 0.

3

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

5.2

0 100 200 300 400 500 600 700 800 900 1000

t (hours)

x(km)

940

960

980

1000

1020

1040

1060

1080

1100

1120

1140

0 100 200 300 400 500 600 700 800 900 1000

t (hours)

L(km)

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600 700 800 900 1000

t (hours)

L(m/s)

0.007

0.0075

0.008

0.0085

0.009

0.0095

0.01

0 100 200 300 400 500 600 700 800 900 1000

t (hours)

T (N)

Figure 7. Bidimensional control scheme. Massless tether with two equal masses (500 kg) at both ends in the Earth-Moon system. Theselected parameters areβ = 0.5, Ω = 1.5, for the initial conditions u0 = 0.15 and u0 = 0. The nominal tether length isL = 1000 km

-4

-3

-2

-1

0

1

2

3

4

5

6

0 100 200 300 400 500 600 700 800 900 1000

t (hours)

θ(o)

Figure 8. Evolution of the attitude angleθ when the bidimensionalcontrol scheme is used.

The tether length and its variation are similar to theprevious case. The differences —which appear in the po-sition of the center of mass and in the tension— are due tothe impossibility of canceling, simultaneously, the contri-bution of the unstable and oscillatory terms. The residualoscillation in position, attitude and tension is, therefore,due to those terms. Figure8 shows the solution for the an-gle θ during this control phase; the values ofθ are smallconsistently with the hypothesis which has been made.

2. Proporcional control

Keeping the linear approximation, it is possible to setout more general control strategies for the tridimensional

9 of 24


problem which match the general forms(τ) =

∑

Ki xi

and where the constantsKi are the gains andxi the state variables of the system. A simple analysis provideindicationsabout the options provided by this kind of control. As previously, we suppose thatθ, ϕ and its derivatives are small(θ, ϕ, θ, ϕ ≪ 1). Consequently, the system can be expressed as:

d2 u

d τ2− 2

d v

d τ− 9 u(τ) + 9 s(τ) = 0 (29)

d2 v

d τ2+ 2

d u

d τ+ 3 v(τ) − 9 θ(τ) = 0 (30)

d2 w

d τ2+ 4 w(τ) − 9 ϕ(τ) = 0 (31)

d2 θ

d τ2+ 12 θ(τ) +

d s(τ)

d τ= 0 (32)

d2 ϕ

d τ2+ 13 ϕ(τ) = 0 (33)

(34)

Taking into account the above control function, the system of equations takes the following linear form:

d y

d t= Ay (35)

wherey =

u, v, w, θ, ϕ, u, v, w, θ, ϕT

and the matrixA is given by

AT =

0 0 0 0 0 9 − 9 Ku 0 0 − 9 Ku(1−Ku)1+Kθ

0

0 0 0 0 0 −9 Kv −3 0 9 KuKv+3Kv

1+Kθ

0

0 0 0 0 0 −9 Kw 0 −4 9 KuKw+4 Kw

1+Kθ0

0 0 0 0 0 −9 Kθ 0 0 9 KuKθ−121+Kθ

0

0 0 0 0 0 −9 Kϕ 0 99 KuKϕ−9 Kw+13 Kϕ

1+Kθ−13

1 0 0 0 0 −9 Ku −2 09 K2

u−Ku+2 Kv

1+Kθ0

0 1 0 0 0 2 − 9 Kv 0 0 −Kv+Ku(2−9 Kv)1+Kθ

0

0 0 1 0 0 −9 Kw 0 0 9 KuKw−Kw

1+Kθ

0

0 0 0 1 0 −9 Kθ 0 09 KuKθ−Kθ

1+Kθ

0

0 0 0 0 1 −9 Kϕ 0 09 KuKϕ−Kϕ

1+Kθ

0

Firstly, we analyzed the stabilization possibilities whenonly one gain is different from zero, i.e., whens(τ) = Ki xi.The analysis of the eigenvalues ofA shows that only the particular cases(τ) = Ku u(τ) with Ku > 1 leads to astabilized system since in this case all the eigenvalues arepure imaginary and the system would oscillate around theequilibrium position.

In a second step, we include a second variable with a non-vanishing gain; the idea is to obtain an asymptoticallystable behavior for the in-plane motion, studying control schemes which correspond to this more general expressions(τ) = Ku u(τ)+Ki xi. However, the analysis of the eigenvalues ofA shows that there is no successful combinationsof gains which provide asymptotic stability. In the most favorable case, whens(τ) = Ku u(τ) + Ku u(τ) withKu > 1 , Ku > 0, the matrixA has two complex conjugate imaginary eigenvalues associated to the in plane motion:therefore, we have no asymptotic stability with this more elaborated control function.

10 of 24


B. Full Problem. Proportional control

In this paragraph, we face the full problem stated by the system of equations (14,15, 17) in the case in which the lengthvariationλ(τ) is governed by a proportional control law:

10−6

10−4

10−2

2

2.5

3

3.5

4

4.5

5

K∗ξ

λe

Figure 9. Sufficient condition for Kξ to stabilize the system as afunction of λe. Values over the curve (Kξ > K∗

ξ ) provide stability.

λ = λe +∑

Ki (xi − xe,i)

whereKi are gains andxi stands for the variablesξ, η, θ.This way, we search for the gains needed to stabilize thesystem. When this control law is included in the systemof variational equations, it becomes a linear system of dif-ferential equations which can be written in vectorial form:

d y

d t= My (36)

whereyT =

δξ, δη, δθ, δξ, δη, δθ

.

The detailed stability analysis of (36) can be cumber-some. Nevertheless, taking into account the results of thepreceding section, we firstly consider the simpler case inwhich only one gainKξ is different from zero. In thatsituation, the characteristic polynomial takes the form:

s3 +

[

3 Kξ

ξ4e

−(

2 − 4

ξ3e

)]

s2 +

[(

6 Jg + 27

ξ4e

+6

ξ7e

)

Kξ −(

105 − 6

ξ3e

+4

ξ6e

)]

s+

+6

[(

9

ξ4e

− 3

ξ7e

)

Kξ −(

45 +9

ξ3e

− 2

ξ6e

)]

= 0

(37)

The Descartes rule of signs provides a sufficient condition to be fulfilled byKξ in order to stabilize the system; sucha condition is drawn in figure9 as a function of the equilibrium positionλe for a massless tether (Jg = 1).

This control scheme provides stability but not asymptotic stability. In order to do that, it seems convenient toinclude gains which affects thevelocity variables. Considering the analysis of the precedent section it is appropriate toinclude the two control gains:Kξ andKξ, even if they are not able to stabilize the system in the linear approximation.For the full problem considered here, the Routh-Hurwitz stability criterion (see Ref. 11) has been used in orderto establish whether that pair of gains can provide stability. The result indicate that they must fulfil the followingconditions

Kξ > 0, (11 + 2Jg)ξ4e − 2ξe − 3Kξ < 0,

12((J2g + 10Jg − 6)ξ3

e + 1)

ξ3e((11 + 2Jg)ξ4

e − 2ξe − 3Kξ)Kξ > 0 (38)

Unfortunately, they are incompatible; in effect, for a massless tether (Jg = 1), the numerator of the last conditionis positive for all values ofξe and the denominator negative! Therefore we can conclude that the system can not bestabilized using values ofKξ andKξ different from zero at the same time.

C. Control Drawbacks

In previous sections we have deduced control laws which permit stabilize the system within certain limits. However,these control strategies exhibit some difficulties which should be underlined. For example, if we are working in theneighborhood ofL2 (the external Lagrangian point) the above strategies only can be used when the center of massGis on the right of L2 (xG > xL2

, see Fig.4). If G becomes on the left side ofL2 (xG < xL2) there is not possible

control based on the tether line. And there exist more unsuitable characteristics that we will comment in what follows.

1. Length Variations

The philosophy underneath the control strategies is based on the Farquhar ideas. In general, the length of the tether ischanged in such a way that the system center of massG approach continuously the equilibrium position.

11 of 24


10−6

10−4

10−2

100

100

101

102

103

λ

d Ld xe

General. eq. (40)Linear. eq. (39)

Figure 10. Ratio between the variation of length needed and thedeviations of the center of mass of the system as a function oftheparameter λ. Comparison of linear and general expressions.

An important question arise in this matter: if the posi-tion of the equilibrium point must be moved an amount∆xe,what is the minimum tether length variation∆L required toproduce such a translation? In dimensional values, the equa-tion which link both variations is:

∆L

∆xe=

1

2 31

3

√a2 λ

(39)

In the general case, this ratio can be expressed as a functionof the non-dimensional equilibrium position:

dL

dxe=

15 ρ3e − 2

2√

3 a2

√

3 ρ3e − 1

(40)

Both relations are shown in figure10; note that the linear ap-proximation fits well till values ofλ of the order of∼ 10−2.The important point is that for small values ofλ (λ < 10−6)the variation∆L is very high; thus, to move the equilibrium

position 1 km require deploy (or retrieve) 1000 km of tether line. The presence of a minimum suggest that it wouldbe worth working near that point wheredL/d xe is approximately∼ 4.5. That implies searching for values ofλ∼ 1.3 · 10−1. Nevertheless, in order to be coherent with the approximations made in the model (remember figure5)the values ofλ should be smaller thanλ < 10−1. As a result, values ofλ of the order of10−2 keep the coherence withour approximation and provide feasible tether length variations.

10−2

100

102

104

102

104

106

Sun-EarthEarth-MoonMars-PhobosJupiter-AmalteaJupiter-IoSaturn-Enceladus

L (km)

d Ld xe

Figure 11. Ratio between the variation of length needed and thedeviations of the center of mass of the system as a function ofthetether length (km)

Figure11shows the ratiod Ld xe

as a function of the tetherlength (km) for the equilibrium position in different binarysystems of interest. The part of the curve above the mini-mum and on the right should not be considered since it cor-responds to too high values ofλ. This figure permits to es-tablish which systems of primaries would be more suitable tobe controlled with the strategies carried out in this paper.Onthe right hand side of the figure the tether length needed towork in the vicinity of the minimum ofdL/d xe is extremelyhigh. However, the values on the left side of the figure leadto more reasonable values of the tether length, with the ra-tio dL/d xe near its minimum. Hence, this analysis help usto evaluate «a priori» the suitability of the control strategiesbased in the variation of the tether length for each one of thebinary systems of the Solar System.

There exist other troubles in the proportional control re-garding the tether length. As we stated before, there exista limit due to the fact that the equilibrium positions are on

only one side of the Lagrangian point (on the right forL2, on the left forL1) and not on both sides. Therefore, it isnecessary to ensure that the tether is in the correct side. For theL2 case, for example, this condition can be expressedin dimensional and non dimensional values as follows

‖x − xe‖ = ∆x < ‖xe − xL2‖, ‖ξ − ξe‖ = ∆ξ < ‖ξe − 3−1/3‖ (41)

Secondly, the tether length is obviously positiveL > 0 and this leads to a restriction on the control strategy. Thetether lengthL is proportional to

√λ and this variable changes according to the control law and, in general, it depends

on the variations∆y with respect to the equilibrium position:

12 of 24


10−6

10−5

10−4

10−3

10−2

10−8

10−6

10−4

10−2

100

λ

∆ξ

10−6

10−5

10−4

10−3

10−2

0.258

0.259

0.26

0.261

0.262

0.263

0.264

0.265

λ

∆ξ|C1

max

∆ξ|C2max

Figure 12. Upper picture: ∆ξ|max in both cases,∆ξ|C2max (blue

line) and ∆ξ|C1max (red line). Lower picture, ratio between the

maximum allowable perturbations for both constraints

10−2

100

102

104

10−5

100

Sun-EarthEarth-MoonMars-PhobosJupiter-AmalteaJupiter-IoSaturn-Enceladus

L

∆x|max

Figure 13. Maximum perturbation admisible vs. tether length L(both in km) for several binary systems.

λ = λe + K · ∆y

Hence, the constraint can be written as:λe + K · ∆y > 0.An upper bound forλ can be obtained considering:

λe + K · ∆y ≥ λe − ‖K ‖ ‖∆y ‖

And, this is a sufficient condition on the modulus of the con-trol gains vector in order to fulfil the constraintL > 0:

‖K ‖ <λe

‖∆y ‖max(42)

We should note that the vector of gains must fulfil the con-dition related to the stability of the system.

It is not easy to determine, in a general manner, the value‖∆y ‖max since it depends on the dynamical evolution ofthe system. However, it seems reasonable to assume thatin a controlled system the maximum perturbation coincideswith the initial one since the control schemebring the sys-tem closerto the unstable equilibrium point. As a conse-quence,‖∆y ‖max = ‖∆y ‖0 in many situations.

For the particular case of the control law exposed in theprevious section:λ = λe + Kξ (ξ − ξe), the constraint (42)can be expressed as:Kξ < λe/(ξ − ξe)max. Assuming thatthe maximum takes place at the initial instant, the condition(42) in Kξ takes the following form:

Kξ <λe

‖ξ0 − ξe‖non-dimensional (43)

Kξ <

√λe a2

‖x0 − xe‖dimensional (44)

We obtained two requirements for the control strategy:1) positive tether lengthL and 2) center of massG on thecorrect side of the Lagrangian point. As we show in whatfollows the first one is much more restrictive than the sec-ond. In order to compare them, both restrictions can be mea-sured in terms of the maximum admisible perturbation; thisone is defined as the maximum initial perturbation whichcan be controlled fulfilling the required constraint (1) or 2)).

For the second constraint, once the equilibrium positionfor the nominal operation of the tether has been selected,we have to ensure that the initial position belongs to an in-terval of length2

∣

∣ξE − 3−1/3∣

∣ and centered inξE . Hence,the maximum allowable perturbation is the semi-interval∆ξ|(C2)

max =∣

∣ξE − 3−1/3∣

∣ and this value is fixed for eachequilibrium position.

For the first constraint, we can obtain the maximum allowableperturbation from the relation (43).

∆ξ|C1max =

λE

Kξ

13 of 24


Since the value of∆ξ|C2max depends not only onλE but also on the control gain, we will use the minimum value of

Kξ which guarantees the stability of the system, i.e.,K∗ξ (see figure9), in order to provide a real maximum of the

perturbation. Accordingly, the value of the maximum allowable perturbation for each constraint can be drawn as afunction of the equilibrium position as it is shown in the upper picture of figure12.

As a consequence, the requirementL > 0 is more restrictive than the necessity of placing the centerof mass onthe correct side of the Lagrangian point.

10−5

10−4

10−3

−5

0

5

10

15

20

TTc

∆ξ

Figure 14. Maximum and minimum tether tension as a function of theinitial perturbation ∆ξ for a given equilibrium position: λe = 10−2.

Figure13shows the maximum perturbation admis-ible in the position of the center of massG (in km)versus the tether lengthL (in km) for several binarysystems of interest in the Solar System. This figure hasbeen elaborated taking into account the above results,that is, the constrain 1) is the real limit.

2. Tension

Tethers require positive tension since a cable can notwithstand compression. As a consequence there ex-ists a natural limit for tether operation which will beexplored in this paragraph. The expression of the ten-sion in a massless cable within the Hill approximationis collected in equation (10). Hence, the tension willdepend on the tether control law through the last term:(Ld/Ld).

We analyze the stabilizing proportional controlstudied in the previous section (λ = λe + Kξ(ξ − ξe))in order to determine the influence of the different ele-ments involved in the control law: initial perturbation, equilibrium position and mass configuration. Figure14 showsthe maximum and the minimum tensions for a massless tether asa function of the initial perturbation, for a givenequilibrium position (in this case the corresponding toλe = 10−2).

10−4

10−3

10−6

10−5

10−4

λe

∆ξ

Figure 15. ∆ξ which provides zero tension as a function of the equi-librium position described by the parameter λe (blue line). ∆ξ whichprovide zero tether length when the minimum value ofKξ needed tostabilize the system is considered (red line).

As it can be seen, there exists a maximum admisi-ble initial perturbation over which the tension becomesnegative. Moreover, the maximum value of the initialperturbation in the figure has been chosen as the max-imum ∆ξ which can be controlled with the minimumgainKξ of this equilibrium position. That means thatthe positive tension constraint is more restrictive thanthe requirementL > 0 exposed in the previous para-graph.

This way, it is possible to associate a critical valueof ∆ξ to each equilibrium position. Figure15summa-rizes such a critical value as a function ofλe. Along theblue line, the zero tether tension condition is reached;as a consequence, the admisible values of∆ξ must bebelow this line. Here the red line represents the re-quirementL > 0. It is clear, from the figure, that thislast requirement is the most restrictive; therefore, thelast figure constitutes also the maximum perturbationallowed which can be controlled withfeasibletetherlength variations maintaining the tether tighten.

14 of 24


VI. Rotating Tethers

u1 ≡ u

u2

u3

G

HG

m1

m2

u1 ≡ u

u2

u3

G

x

y

z

φ3

φ2

φ1

φ1

s

Figure 16. Gu1u2u3 frame attached to the tether inthe Dumbbell Model (upper picture). Definition of theTait-Bryant ( or Cardan) angles (lower picture).

The Attitude Dynamics of a rotating tether can be analyzed moreintuitively using the Newton-Euler formulation. The core of theanalysis is the angular momentum equation:

dHG

dt= MG

HereMG is the resultant of theexternal torques applied to the cen-ter of massG of the tethered system andHG is the angular momen-tum of the system, atG, in the motion relative to the center of mass.In the extendedDumbbell Model which we use here the angularvelocity of the tether and its angular momentumHG are:

Ω = u × u + u (u ·Ω ), HG = I Ω = Is(u × u )

Attached to the tether we take a reference frameGu1u2u3 wherethe unit vectors are given by:

u1 = u, u2 =u

‖u ‖ , u3 = u1 × u2

In this body frame the angular momentum is:

HG = IsΩ⊥u3, where Ω⊥ = ‖u × u ‖ = ‖u ‖

and the angular momentum equation takes the form:

dΩ⊥dt

u3 + Ω⊥du3

dt=

1

IsMG

This way we obtain the following equations:

du1

dt= Ω⊥u2

du3

dt=

M2

Ω⊥Is

u2

dΩ⊥dt

=M3

Is

From a mathematical point of view, the orden of the system of differential equations is four. For rotating tethers theattitude dynamics is described in a better way using Euler angles in sequence 1–2–3 (Tait-Bryant or Cardan angles)instead of the classical libration angles(θ, ϕ). In terms of the Bryant angles the unit vectorsu1 andu3 are given by

u1 = (cosφ2 cosφ3, cosφ1 sinφ3 + sinφ1 sin φ2 cosφ3, sinφ1 sin φ3 − cosφ1 sin φ2 cosφ3)

u3 = (sin φ2, − sinφ1 cosφ2, cosφ1 cosφ2)

The equations governing the time evolution of the Bryant angles are:dφ1

dτ= − M2

ω2Is· 1

Ω⊥· cosφ3

cosφ2,

dφ2

dτ= − M2

ω2Is· 1

Ω⊥· sin φ3

dφ3

dτ= Ω⊥ +

M2

ω2Is· 1

Ω⊥· cosφ3 tanφ2,

dΩ⊥dτ

=M3

ω2Is

whereΩ⊥ = Ω⊥/ω is the non-dimensional form ofΩ⊥. These equations should be integrated from the initialconditions:

at τ = 0 : φ1 = φ10, φ2 = φ20, φ3 = φ30, Ω⊥ = Ω⊥ 0

15 of 24


For a fast rotating tether the value ofΩ⊥ is large,Ω⊥ ≫ 1. There are two time scales: 1) the period of the orbitaldynamics of both primaries and for whichτ is of order unity —this is the «slow time»— and 2) the period of theintrinsic rotation of the tether for whichτ1 = Ω⊥τ of order unity —this is the «fast time». To obtain the governingequations we perform to operations: 1) we average the original governing equations, and 2) we introduce the Hillapproach.

u1 ≡ u

u2

v1

v2

u3 ≡ v3

Gφ3

φ3

Figure 17. Stroboscopic frame

For the averaged process we introduce a stroboscopicframe (see figure17). For example, consider one of the gov-erning equations

dφ

dτ= f(φ1, φ2, φ3, Ω⊥)

Its averaged form is:

<dφ

dτ>=

1

2π

∫ 2π

0

f(φ1, φ2, φ3, Ω⊥)dτ1

To integrate the functionf(φ1, φ2, φ3, Ω⊥) the slow vari-ables(φ1, φ2, Ω⊥) take constant values and thefast variableφ3 is approximated byφ3 ≈ τ1 + φ30.

After averaged the governing equations and introducingthe Hill approach,for a fast rotating tether the evolutionof the center of mass and tether attitude is the same as inthe case of constant tether length; the differences betweenboth cases are: 1) nowλ = λ(τ) and 2) the time evolution ofΩ⊥(τ) are different in both cases. As a consequence,providing that the rotation rate of the tether keeps a high value (Ω⊥(τ) ≫ 1), the governing equations are:

ξ − 2η = (3 − 1

ρ3)ξ − 1

2

λ

ρ5

3N sinφ2 − ξS2(N

ρ)

(45)

η + 2ξ = − η

ρ3+

1

2

λ

ρ5

3N cosφ2 sin φ1 + ηS2(N

ρ)

(46)

ζ = −ζ(1 +1

ρ3) − 1

2

λ

ρ5

3N cosφ2 cosφ1 − ζS2(N

ρ)

(47)

dφ1

dτ= cosφ1 tanφ2 (48)

dφ2

dτ= − sinφ1 (49)

which only have a free parameterλ, defined in (8), and where the quantityN is given by:

N = ξ sin φ2 − (η sin φ1 − ζ cosφ1) cos φ2 (50)

The equations (45-50) should be integrated from the initial conditions:

at τ = 0 : ξ = ξ0, η = η0, ζ = ζ0, ξ = ξ0, η = η0, ζ = ζ0, φ1 = φ10, φ2 = φ20 (51)

When the initial conditions areφ10 = φ20 = 0 the solution for the anglesφ1 andφ2 is φ1(τ) = φ2(τ) ≡ 0, thatis, if initially the tether rotates in a plane parallel to theorbital plane of both primaries, the direction of the angularmomentum keeps a constant value.

16 of 24


In these equations the time evolution of angles(φ1, φ2) is decoupled from the other variables and it can be inte-grated separately. The solution for them turns out to be:

sin φ1 =cosα sin β

√

cos2 β + cos2 α sin2 β, sin φ2 = sin β sin α (52)

cosφ1 =cosβ

√

cos2 β + cos2 α sin2 β, cosφ2 =

√

cos2 β + cos2 α sin2 β (53)

Obviously the constant values(β, α0) are given by the initial conditions:

cosβ = cosφ10 cosφ20, β ∈ [0,π

2] (54)

sin α0 =sin φ20

sin β, cosα0 =

sinφ10 cosφ20

sin β(55)

Finally, equations (45-50) are exactly the same for tethers of constant length and for varying length tethers; thedifferences between both cases are: 1)λ is constant for a tether of constant length and it is a function of time,λ = λ(τ),for a varying length tether, and 2) the time evolution ofΩ⊥(τ) is different in both cases but for a fast rotating tetherthe particular evolution ofΩ⊥(τ) is irrelevant to a large extent, provided that it takes largevalues, that isΩ⊥(τ) ≫ 1.

A. Rotating tethers. Equilibrium positions

The steady solutions of equations (45-50) provide the equilibrium positions of a rotating tether. Equations (48-49)only have the steady solutionφ1 = φ2 = 0. Therefore, in order to have an equilibrium position for thetethered systemthe rotation of the tether should take place in a plane parallel to the orbital plane of both primaries. As a consequenceN takes the valueN = ζ, and the stationary equations become:

ξ

3 − 1

ρ3+

3

4

λ

ρ5

(

5

(

ζ

ρ

)2

− 1

)

= 0 (56)

η

1

ρ3− 3

4

λ

ρ5

(

5

(

ζ

ρ

)2

− 1

)

= 0 (57)

ζ

1 +1

ρ3+

1

4

λ

ρ5

(

9 − 15

(

ζ

ρ

)2)

= 0 (58)

Moreover, the length must be constant, that is,λ =constant.These equations have several solutions; two of them are on the Oz axis and they are spurious, that is, in fact they

are due to the expansions carried out in the model but they arenot real equilibrium positions. However, the equationsprovide two real equilibrium positions for which the centerof massG is on theOx axis; one of them is close to theexternal collinear point (L2) and the other one close to the internal Lagrangian point (L1). For the sake of simplicitywe focus on the first one which is given by:

ξe = ρe, ηe = ζe = 0, λe =4ρ5

e

3

(

3 − 1

ρ3e

)

, ρ3e > 1/3 (59)

For small values ofλ the above solution can be expanded in power ofλ and provide the asymptotic solution

ξe ≈ (1

3)

1

3 + 31

3

λ

4− 9

λ2

16+ O(λ3) (60)

Comparing these expressions with their equivalent in the non-rotating case, (12-13), an important conclusion can bededuced: in order to place the center of massG in an equilibrium position at a given distance from the LagrangianpointL2 the real length of the rotating tether must be twice the real length of the non-rotating tether. In other words,for two tether of the same length, the equilibrium position of the rotating tether is four times closer toL2 than thenon-rotating tether.

17 of 24


B. Linear stability analysis of the collinear equilibrium positions

In previous sections, it was found that the motion of a tethered system with constant length was slightly more unstablethan the motion of the particle. Now the effect of a fast angular velocity will be studied and compared with the non-rotating tether. In order to study the linear stability, thevariational equations are deduced by introducing the followingvariations in the dynamical system (45-50):

ξ = ξe + δξ η = δη ζ = δζ φ1 = δφ1 φ2 = δφ2

The resulting linear equations can be written in terms of thestate vectory = (ξ, η, ζ, φ1, φ2, ξ, η, ζ, φ1, φ2)T :

dδy

dτ= Meδy Me(ρe) =

0

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

@

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 1

k1,1 k1,2 k1,3 k1,4 k1,5 0 2 0 0 0

k2,1 k2,2 k2,3 k2,4 k2,5 −2 0 0 0 0

k3,1 k3,2 k3,3 k3,4 k3,5 0 0 0 0 0

k4,1 k4,2 k4,3 k4,4 k4,5 0 0 0 0 0

k5,1 k5,2 k5,3 k5,4 k5,5 0 0 0 0 0

1

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

A

(61)

Thek sub-matrix takes the form:

k(ρe) =

0

B

B

B

B

B

B

B

@

15ρ3

e−2

ρ3e

0 0 0 0

0 −3 0 0 0

0 0 −25ρ3

e−1

ρ3e

0 ±23ρ3

e−1

ρ2e

0 0 0 −1 0

0 0 0 0 −1

1

C

C

C

C

C

C

C

A

(62)

The stability properties of the equilibrium position are given by the eigenvalues of theMe; they turn out to be:

s1,2 = ±i, s3,4 = ±i, s5,6 = ±√

2ρe(5ρ3e − 1)

ρ2e

i, s9,10 = ±

√

ρe

(

1 − 4ρ3e +

√

61ρ6e − 14ρ3

e + 1)

ρ2e

i

2.5

2.55

2.6

2.65

2.7

2.75

2.8

0.7 0.71 0.72 0.73 0.74 0.75 0.76

ρe

s7

s7 (non-rotating)

Figure 18. Unstable eigenvalue

s7,8 = ±

√

ρe

(

−1 + 4ρ3e +

√

61ρ6e − 14ρ3

e + 1)

ρ2e

,

Excepts7,8, all of them are pure imaginary, so they do notdestabilize the system. However,s7,8 summarizes a pair ofreal eigenvalues; one of them is always positive and makesthe equilibrium position unstable. Figure18shows that forlarger tethers, the system becomes more unstable, as in thenon-rotating case.

In summary, the presence of the tether does not im-prove the stability properties of the equilibrium position.As in the case of non-rotating tethers, the instability in-creases with the tether length; however, the increase isslower than in the non-rotating tether case (see figure18).

18 of 24


C. Varying Length Tether. Linear control strategy

In spite of the instability, the variation of the tether length brings the chance of obtain a stable motion around theequilibrium position. In the Hill approximation, the governing equations for fast rotating tethers are given in (45-50).The time evolution of the angular velocityΩ⊥(τ), now it is governed by the new equation:

dΩ⊥dτ

= sin φ1 sin φ2 cosφ2 − Ω⊥Is

Is(63)

The admisible equilibrium positions appear whenφ1 = φ2 = 0. In such a case, by integrating equation (63) we obtainthe conservation of the modulus of the angular momentum:

Ω⊥Is = constant (64)

Therefore, for a rotating tether at an equilibrium position, the fast rotationΩ⊥ is determined by the variation of themoment of inertiaIs due to the variation of the tether length.

The three variational equations describing the attitude and the motion of the center of mass normal to the orbitalplane are the same that in the constant length tether case:

δζ =

2−5ρ3

e + 1

ρ3e

δζ +

2−3ρ3

e + 1

ρ2e

δφ2

δφ1 = −δφ1

δφ2 = −δφ2

They are uncoupled from the other degree of freedom and they have pure imaginary eigenvalues associated with, sothis motion is bounded. However, since these equations do not depend on the variationδλ(t), it will not be possible toobtain asymptotic stability for the out-of-plane motion with the control schemes carry out in this paper.

The remaining two variational equations, which govern the motion of the center of mass in the orbital plane, canbe expressed:

[

δξ

δη

]

= 2

[

0 1

−1 0

][

δξ

δη

]

+

[

15ξ3

e−2ξ3

e0

0 −3

] [

δξ

δη

]

+

[

−34ξ4

e

0

]

δλ

or, by using the new state vectory = (ξ, η, ξ, η):

dδy

dτ= Mδy + b0δλ (65)

where

M(ρe) =

0 0 1 0

0 0 0 115ξ3

e−2ξ3

e0 0 2

0 −3 −2 0

and b0 = (0, 0,− 3

4ξ4e

, 0)T (66)

From equation (60) can be deduced that a differential increment∆λe takes associated a variation of the equilibriumposition of the same order∆ξe (as it will be seen later, with dimensional variables does not happen the same). Thisjustify the use of a proportional control law, based on the variations:

y = ye + δy, λ(t) = λe + δλ(t)

wherey is the state vector described before. Therefore we select the following proportional control law:

δλ = K · δy where K = (Kξ, Kη, Kξ, Kξ)

is the gain vector.19 of 24


By replacingδλ with the above control law, the system becomes:

dδy

dτ= Nδy where the matrix N = M + [b0, K] (67)

takes the form

N =

0 0 1 0

0 0 0 115ξ3

e−2ξ3

e− 3Kξ

4ξ4e

− 3Kη

4ξ4e

− 3Kξ

4ξ4e

2 − 3Kη

4ξ4e

0 −3 −2 0

and its characteristic polynomial is:

s4 +

3Kξ

4ξ4e

s3 +

−32ξ4

e − 6Kη + 8ξe + 3Kξ

4ξ4e

s2 − 3

2Kη − 3Kξ

4ξ4e

s + 38ξe − 60ξ4

e + 3Kξ

4ξ4e

= 0 (68)

D. Searching oscillatory stability

In this initial stability analysis, control gains that remove the hyperbolic instability of the equilibrium point and replaceit with oscillatory stability will be searched. Once this level of stability is achieved, the center of massG of the systemwill undergoes an oscillatory motion about the equilibriumposition.

2.5

3

3.5

4

4.5

5

0.7 0.71 0.72 0.73 0.74 0.75 0.76

ρe

KξrotKξnon−rot

Figure 19. Plot of stability condition for Kξ gain

General assignment of the gains results in a 4th ordercharacteristic polynomial (see (68)) in s with all the coef-ficients different from zero. To simplify the analysis, co-efficients ofs3, ands are made null by taken:

Kξ = 0 Kη = 0 Kη = 0 (69)

The resulting characteristic polynomial is then:

a(ξe, Kξ)x4 + b(ξe, Kξ) · x2 + c(ξe, Kξ) = 0 (70)

a = 1 (71)

b =−32ξ4

e + 8ξe + 3Kξ

4ξ4e

(72)

c = 3−60ξ4

e + 8ξe + 3Kξ

4ξ4e

(73)

The discriminant of this equation is stated as:

∆(ξe, Kξ) = b2 − 4ac

If ∆ > 0 then the polynomial has three distinct, real roots ins2. From Descartes Rule of Signs, if the coefficientsa,b andc are all positive, then the roots all have negative parts. If both of these conditions are satisfied, then the rootsall have the forms2

j = −w2j , and therefore all the eigenvalues of the system are pure imaginary,sj = ±

√−1wj ,

and the oscillatory stability is achieved. Figure19shows the lower limit forKξ needed for fulfill all the requirementsmentioned. This limit can be stated explicitly:

Kξ ≥ 4

3ρe

(

15ρ3e − 2

)

(74)

Figure19 also shows gain requirement for the non-rotating tether (which were studied in previous sections); anotherimportant conclusion is remarked: for the rotating tether gain is larger than the gain for the non-rotating one. Notethat this was expected; the proposed control law is proportional to displacements, and, as can be seen from equations(13, 60), in order to vary the equilibrium position the same quantity, the rotating tether requires a∆λ larger than thenon-rotating one.

20 of 24


E. Asymptotic stability

Although a small increment of the nondimensional variable∆ξe requieres an increment of the tether length∆λe ofthe same order:

∆ξe =31/3

4∆λe,

when this relation is expressed in term of dimensional variables:

∆L =2

31/3

1√λe

√a2

∆xe,

it es found that, whenλe ≪ 1, an increment∆xe of a few kilometers requires a variation of the tether lengthtoohigh. Therefore, since the variations ofL are limited by the tether deployment capacity, it is necessary to reduce thevariations of∆xe by searching asymptotic stability.

Through the Routh-Hurwitz theorem, it has been found that asymptotic stability is guaranteed if the gains of thecontrol law satisfy the relations

Kξ ≥ 4

3ρe

(

15ρ3e − 2

)

Kξ > 0 Kη = 0 Kη = 0 (75)

Note that the condition forKξ is the same that the one required for oscillatory stability.Just adding the dissipationKξ > 0 the asymptotic stability is achieved.

Consider the linearized one-dimensional motion, that is, when the center of massG is limited to theOξ axis byneglecting the other coordinates and the coupling effects (Coriolis terms). Such motion is described by the equation:

δξ +3

4ξ4e

· Kξ δξ + w2nδξ = 0 (76)

where the natural frequencywn is given by

wn = wn(ξe, Kξ) =

√

3Kξ

4ξ4e

− 15ξ3e − 2

ξ3e

Equation (76) has two stable eigenvalues whenKξ > 0 given by

s1,2 = −3Kξ

8ξ4e

±√

[3Kξ

8ξ4e

]2 − w2n (77)

The gainKξ is introducing damping in the system; the critical value which produce critical damping is given by

Kξcr=

8

3ξ4e

√

3Kξ

4ξ4e

− 15ξ3e − 2

ξ3e

(78)

The analysis performed here for the one-dimensional case can be translated almost directly to the bi-dimensionalcase as the simulations collected in what follows highlight.

Figure20 shows the results obtained in two simulations of a rotating tether with different values ofKξ andKξ.The characteristics of the simulations are:λe = 0.0401 (ξe = 0.707); ξ0− ξe = 10−3, ηe = 10−3; Ω⊥ = 50. Pictureson the left side corresponds to a control law without dissipation (Kξ = 0) and for whichKξ = 4. Pictures on the rightside of the figure corresponds to the same case but now with a control law including dissipation; two different valuesof Kξ has been taken (Kξ = 0.5 subcritical andKξ = 3 supercritical).

As a consequence, the fast rotating tethers are able to be asymptotically controlled; and this is a significativedifference with the non-rotating tethers which are not ableto be controlled in that way. From this point of view, figure20 is good example.

21 of 24


0.7055 0.706 0.7065 0.707 0.7075 0.708 0.7085 0.709−3

−2

−1

0

1

2

3x 10

−3

Kξ = 4

δξ

δξ

0.7065 0.707 0.7075 0.708−15

−10

−5

0

5x 10

−4

Kξ

= 3

Kξ

= 0.5

δξ

δξ

−1.5 −1 −0.5 0 0.5 1 1.5

x 10−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

−3

Kξ = 4

δη

δη

−5 0 5 10

x 10−4

−1.5

−1

−0.5

0

0.5

1x 10

−3

Kξ

= 3

Kξ

= 0.5

δη

δη

0.7055 0.706 0.7065 0.707 0.7075 0.708 0.7085 0.709−1.5

−1

−0.5

0

0.5

1

1.5x 10

−3

Kξ = 4

η

ξ

0.7065 0.707 0.7075 0.708−5

0

5

10x 10

−4

Kξ

= 3

Kξ

= 0.5

η

ξ

Figure 20. Nondimensional simulation;λe = 0.0401 (ξe = 0.707); ξ0 − ξe = 10−3, ηe = 10−3; Ω⊥ = 50

22 of 24


Related with these simulations figure21describes the time evolution of other important parameters: 1) the angularvelocity Ω⊥ of the rotating tether (the initial value has been selected close toΩ⊥ = 50) and 2) the evolution ofthe parameterλ. Note that without dissipation both parameters oscillate around some averaged values with smallamplitudes (roughly the 10% of the initial values). However, when the control law includes dissipation both parameterstend asymptotically to constant values. Note that the time evolutions of both parameters are not independent since theyare linked by the relation (64).

0 5 10 15 20 25 3048

50

52

54

56

58

60

62

64

66

τ

Ω⊥

Kξ = 4

0 5 10 15 20 25 3050

51

52

53

54

55

56

57

58

τ

Ω⊥

Kξ

= 3

Kξ

= 0.5

0 5 10 15 20 25 300.185

0.19

0.195

0.2

0.205

0.21

0.215

τ

√λ

Kξ = 4

0 5 10 15 20 25 300.19

0.195

0.2

0.205

0.21

0.215

τ

√λ

Kξ

= 3

Kξ

= 0.5

Figure 21. Nondimensional simulation;λe = 0.0401 (ξe = 0.707); ξ0 − ξe = 10−3, ηe = 10−3; Ω⊥ = 50

VII. Conclusions and future works

From the analysis carried out in this paper some significant conclusion can be drawn. First of all, we deduced arobust formulation including in the analysis the Hill approach which gives place to simpler expressions and simulta-neously keeps the necessary accuracy on the results.

With this new formulation, we found a new non-dimensional parameter which captures the influence of the tetherlength and play a central role in the dynamics. We also found several equilibrium positions close to the Lagrangiancollinear points where a tethered satellite can be placed toexploit the benefits of these special positions. These resultsagree with the previous ones given in the literature.

Since the equilibrium positions are unstable we obtained several control laws which permit the operation of thetethered system in the neighborhood of the Lagrangian point, using rotating and non-rotating tethers. This stabilizationtechniques open the door to interesting applications of tethered systems in different fields of the Astrodynamics andthe Space Exploration.

In general, the use of tethers is more appropriate in binary systems whose secondary primary is small, because thetether length is smaller. We also underline some difficulties associated with the stabilization techniques carried outin this paper; for example, the stabilization proposed hereonly can be used on one side of the selected Lagrangianpoint, since on the other side there is no equilibrium positions. Therefore, it is forbidden to cross the collinear point.

23 of 24


However, we show that there is a very high sensitivity of the tether length with the deviation from the equilibriumposition; from this point of view the most restrictive requirement is to keep a non-vanishing tether length.

Rotating tethers exhibit a better behavior from the controlpoint of view; however, its operation require moreaccuracy because the ranges of operations are narrower for them.

In the future this analysis will be extended in order to take into account the power needed to operate the tetherwhen the tether length is varied to provide control on the system.

Acknowledgments

The work for this paper was supported by the ARIADNA researchscheme established by the Advanced ConceptsTeam of the European Space Agency. The work of M. Sanjurjo-Rivo, F. R. Lucas, M. Lara, D. J. Scherees and J. Peláezhas been also partially supported by the research project entitled Propagation of orbits, advanced orbital dynamicsand use of space tethers(ESP2007-64068) supported by the DGI of the Spanish Ministry of Education and Science.

References1B. Wie, Space Vehicle Dynamics and Control, AIAA Education Series, AIAA, Reston VA, 1998. p.2902G. Colombo, The Stabilization of an Artificial Satellite at the InferiorConjunction Point of the Earth-Moon System, Smith-

sonian Astrophysical Observatory Special Report No. 80, November 1961.3R. W. Farquhar , The Control and Use of Libration-Point Satellites, NASA TR R-346, September 1970. pp. 89-1024R. W. Farquhar , Tether Stabilization at a Collinear Libration Point, The Journal of the Astronautical Sciences, Vol. 49, No.

1, January-March 2001, pp. 91-106.5A. K. Misra, J. Bellerose, and V. J. Modi, Dynamics of a Tethered System near the Earth-Moon Lagrangian Points, Proceed-

ings of the 2001 AAS/AIAA Astrodynamics Specialist Conference, Quebec City, Canada, Vol. 109 of Advances in the AstronauticalSciences, 2002, pp. 415–435.

6B. Wong and A. K. Misra, Dynamics of a multi-tethered system near the Sun-Earth Lagrangian point, 13th AAS/AIAASpace Flight Mechanics Meeting, Ponce, Puerto Rico, February 2003, Paper No. AAS-03-218.

7B. Wong and A. K. Misra, Dynamics of a Libration Point Multi-Tethered System, Proceedings of 2004 International Astro-nautical Congress, Paper No. IAC-04-A.5.09.

8J. Peláez and D. J. Scheeres, A permanent tethered observatory at Jupiter. Dynamical analysis, Proceedings of the 17thAAS/AIAA Space Flight Mechanics Meeting Sedona, Arizona, Vol. 127 of Advances in the Astronautical Sciences, 2007, pp.1307–1330

9J. Peláez and D. J. Scheeres, On the control of a permanent tethered observatory at Jupiter, Paper AAS07-369 of the 2007AAS/AIAA Astrodynamics Specialist Conference, Mackinac Island, Michigan, August 19-23, 2007

10E. M. Levin , Dynamic Analysis of Space Tether Missions, Advances in The Astronautical Sciences, Vol. 126. AmericanAstronautical Society, 2007.

11 F.R. Gantmacher, Applications of the Theory of Matrices, Ed. Dover, 2005.

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