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Uniform rotations of tethered system connected to a moon surface
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Uniform rotations of tethered system connected to a moon surface Alexander A. Burov, Anna D. Guerman, Ivan I. Kosenko
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S0094-5765(15)00200-3 http://dx.doi.org/10.1016/j.actaastro.2015.05.018 AA5446
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Acta Astronautica
Received date: 9 July 2014 Revised date: 17 April 2015 Accepted date: 12 May 2015 Cite this article as: Alexander A. Burov, Anna D. Guerman, Ivan I. Kosenko, Uniform rotations of tethered system connected to a moon surface, Acta Astronautica, http://dx.doi.org/10.1016/j.actaastro.2015.05.018 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
UNIFORM ROTATIONS OF TETHERED SYSTEM CONNECTED TO A MOON SURFACE Alexander A. Burov∗, Anna D. Guerman†, Ivan I. Kosenko‡ ABSTRACT We consider the problem of in-plane rotations of a space elevator with variable tether length attached to a surface of one of the primaries in a double system. The planet and its moon (or two asteroids) move about their center of mass in unperturbed elliptic Keplerian orbits. We discuss the possibilities to cause a prescribed motion of the system by changing the tether’s length. Periodic solutions of the equation for the tether length control are studied using the method of small parameter. The stability of these solutions is studied numerically. The analysis shows that there exists a control law that implements tether rotations which are uniform with respect to true anomaly; one can indicate conditions when the above rotations are stable in the first approximation. These results can be used for development of a planet elevator or a system for payload transportation to and from asteroid surface. INTRODUCTION Use of tethers for space transportation is currently discussed in several research groups and agencies. Tethers provide promising possibilities for orbital and attitude spacecraft control, distributed spacecraft missions, space debris removal, etc. [1, 2]. One of the possible applications for space tethers is a space elevator. The project of building a space elevator at the Earth, though much awaited by numerous fans, still faces serious difficulties. Meanwhile, similar systems for Moon, Mars, or asteroid exploration look much more feasible. Studies of spacecraft tethered to the Moon surface began long ago [3, 4]. Several authors discuss possible applications of such structure for lunar exploration [1, 2]. Recently some results on equilibria of such spacecraft and their stability have been obtained in [5–7]. Research on systems with variable mass distribution arise probably to [8–10] and to dissertation of V.A. Sarychev ∗ (see also [11]). Analysis of the necessary conditions of stability for relative equilibria of a satellite with variable mass distribution has been done in [12], some of these results have been re-discovered in [13, 14] (see also, e.g., [15]). ∗ Professor,
National Research University ”Higher School of Economics,” Myasnitskaya 20, 101000 Moscow RUSSIA; Senior Researcher, Department of Mechanics, Dorodnicyn Computing center of the RAS, Vavilova 40, 119333 Moscow RUSSIA † Associate Professor, C-MAST: Centre for Mechanical and Aerospace Science and Technologies, University of Beira Interior, Calcada Fonte do Lameiro, 6201-001 Covilha PORTUGAL, e-mail: [email protected]; tel.: +351 275 329 915; FAX: +351 275 329 972. ‡ Senior Researcher, Department of Mechanics, Dorodnicyn Computing center of the RAS, Vavilova 40, 119333 Moscow RUSSIA ∗ according to the author’s communication
1
Various aspects of dynamics of orbital tethered system have been studied in [16–20]. Parametric analysis of orbiting tethers is performed in [21]. Dynamics of tethered systems in the vicinity of libration points is analyzed in [5, 22]. Motions of a multi-tether system are considered in [23–30]. In [7, 31–33] we consider dynamics of a tether anchored to the Moon surface and the possibilities to keep its orientation with respect to the Earth – Moon direction despite the eccentricity of the Moon orbit. A proper control of the tether’s length can keep the fixed orientation of the tether for several modes of the system functioning. The results of [7,31,32] can be applied for other systems of two primaries, e.g., for planet’s satellites or binary asteroids whenever the point of the tether’s attachment maintains its orientation with respect to the other primary. Meanwhile, the above results are only applicable for systems, where the moon’s proper rotation is synchronized with its orbit motion so as the position of the tether’s anchor is fixed with respect to the planet – moon direction. Relative rotation of the moon’s surface requires that the tether follow the motion of the anchor. Here we examine the possibility to create a proper rotation of the tether via control of its length. POSING THE PROBLEM Consider a system of two primaries, e.g., a planet E and a moon M , that move about their center of mass O in elliptic Keplerian orbits −−→ |OM | = r =
pM , 1 + e cos ν
−−→ |OE| = μr =
pE , 1 + e cos ν
μ=
mM . mE
(1)
Here mE and mM are the masses of the planet and the moon respectively, pE and pM are parameters of their orbits, e is eccentricity, ν the true anomaly. We assume that the sizes of the primaries are negligible. Figure 1 shows the plane of the primaries’ orbits π. The spacecraft of mass m is connected to the moon surface by a tether which length can be
Figure 1. Main notations
2
changed according to some control law. Only in-plane motions of the tether are considered; the tether orientation is described by angle ϕ (Fig.1). For in-plane motion of the tethered spacecraft its kinetic energy can be written as T =
T =
m 2
x˙ 2S + y˙ S2 or
m 2 2 r˙ + r2 ν˙ 2 + ˙2 + 2 (ν˙ + ϕ) ˙ + m r˙ ˙ + rν˙ (ν˙ + ϕ) ˙ cos ϕ + m ν˙ r˙ − r˙ − r˙ ϕ˙ sin ϕ. 2
Its potential energy is
U = −Gm
mE mM + rE
(2)
.
(3)
Here G is the universal gravitational constant, 1/2 rE = |SE| = ρ2 + 2ρ cos ϕ + 2 is the distance between the planet and the spacecraft, and ρ = |EM | = (1 + μ)r is the distance between the planet and the moon. Assuming that ν, r, and are given as functions of time, one can write down the Lagrangian equations d ∂L ∂L = , dt ∂ ϕ˙ ∂ϕ
L = T − U.
(4)
Introducing the true anomaly as a new independent variable and denoting by strike the respective derivative d d d = ν˙ = ω(1 + e cos ν)2 , dt dν dν
(5)
where ω 2 = GmE /(1 + μ)2 p3M , the equation of motion can be rewritten as ϕ + 2 (1 + ϕ ) −
+
2e (1 + ϕ ) sin ν + 1 + e cos ν
(6)
(1 + μ)3 p3M pM sin ϕ 1 − = 0, (1 + e cos ν)2 f 3/2
f = 2 (1 + e cos ν)2 + (1 + μ)2 p2M + 2(1 + μ)pM cos ϕ(1 + e cos ν). It is possible to consider equations (6) from two different perspectives. In the framework of the direct problem, one can look for the system’s motions that correspond to a specific variation of the tether length; in this case = (ν) is given and one has to study the second order ordinary differential equation for ϕ(ν). Considering the inverse problem, one can find the control law for the tether length = (ν) that results in a specific variation of the tether orientation; in this case ϕ = ϕ(ν) is defined beforehand and one has to find = (ν) analyzing the first-order differential equation (6. Here we examine the possibility to implement rotations of the tether, which are uniform with respect to the true 3
anomaly. First we consider the inverse problem and find the control law = (ν) that causes such motions. Afterwards we analyze the direct problem and find the necessary conditions of stability for the above rotations. Similar problem regarding existence and stability of rotations of a tethered system has been studied in [34].
FORCES IN TETHER WITH VARIABLE LENGTH Consider the Lagrangian ˙ = T − U, Λ(ϕ, , ϕ, ˙ )
(7)
Here r and ν are given functions of time t. The equations of motion are d ∂Λ ∂Λ = , dt ∂ ϕ˙ ∂ϕ
∂Λ d ∂Λ = + F , dt ∂ ˙ ∂
(8)
where F is the generalized force correspondent to the coordinate . Since ∂Λ = m ˙ + r˙ cos ϕ + νr ˙ sin ϕ , ∂ ˙ ∂Λ = m ((ν˙ + ϕ) ˙ + rν˙ cos ϕ − r˙ sin ϕ) (ν˙ + ϕ)− ˙ ∂ mM mE (ρ cos ϕ + ) −Gm , + 3 2 rE the equation correspondent to coordinate can be written as ¨ + r¨ cos ϕ − r˙ ϕ˙ sin ϕ + ν¨r sin ϕ + ν˙ r˙ sin ϕ + ν˙ ϕr ˙ cos ϕ =
(9)
= ((ν˙ + ϕ) ˙ + rν˙ cos ϕ − r˙ sin ϕ) (ν˙ + ϕ)− ˙ mM mE (ρ cos ϕ + ) + + f , −G 3 2 rE
f = F /m.
Assume that during the motion the following constraint should hold true: − (ν) = 0.
(10)
Differentiation of equation (10) with respect to time gives d ν˙ = 0, ˙ − dν
d2 d ¨ − 2 ν˙ 2 − ν¨ = 0. dν dν 4
(11)
Substitution of expression (11) for ¨ to (9) yields
f =
d2 2 d ν¨ + r¨ cos ϕ − r˙ ϕ˙ sin ϕ + +¨ ν r sin ϕ + ν˙ r˙ sin ϕ + ν˙ ϕr ˙ cos ϕ− ν˙ + dν 2 dν
− ((ν˙ + ϕ) ˙ + rν˙ cos ϕ − r˙ sin ϕ) (ν˙ + ϕ) ˙ +G
(12)
mM mE (ρ cos ϕ + ) . + 3 2 rE
The above expression permits one to find the force in tether for a given motion of the system. UNIFORM ROTATIONS OF THE TETHER Partial solutions which correspond to the uniform in-plane rotations of the tether with respect to EM direction are described as [34] ϕ = ων + ϕ0 ,
ϕ = ω = const .
(13)
Here ω is the constant angular velocity of the rotation. Note that the absolute angular velocity of the tether’s rotation is Ω = ω + 1. Substitution of (13) to (6) allows one to obtain the following differential equation: (1 + ω) −
pM sin(ων + ϕ0 ) e (1 + ω) sin ν (1 + μ)3 p3M + 1 − = 0, 1 + e cos ν 2(1 + e cos ν)2 f 3/2
(14)
f = 2 (1 + e cos ν)2 + (1 + μ)2 p2M + 2(1 + μ)pM cos(ων + ϕ0 )(1 + e cos ν). If ω = −1, the tether keeps its orientation in the absolute reference frame. In this case the above equation becomes algebraic since the term with vanishes:
(1 + e cos ν) [(1 + e cos ν) + 2(1 + μ)pM cos(−ν + ϕ0 )] = 0.
(15)
This equation has two solutions:
I. :
= 0,
II. :
= −2(1 + μ)pM
cos(−ν + ϕ0 ) . 1 + e cos ν
The first solution is trivial; the second one is meaningful for intervals of ν when the respective right-hand term is positive, i.e, for ϕ0 + π/2 + 2πn ≤ ν ≤ ϕ0 + 3π/2 + 2πn, n = 0, ±1, ±2, ... The control laws for the tether lengths for a number of initial phases ϕ0 and e = 0.5 are shown on Fig. 2. Despite these solutions only exist for limited time intervals, they correspond to large variations of the tether’s length with permanent tether orientation in the absolute space: depending on the initial phase ϕ0 , the maximum achievable distances vary 5
between
2(1+μ)pM 1+e
and
2(1+μ)pM 1−e
.
The above class of the tether motions may turn out to be useful for delivering payload to and from the surface of an asteroid or a moon.
Figure 2. Tether length control for ω = −1, e = 0.5
Now we assume that ω + 1 = 0. One can further simplify equation of motion (6) using new variable η(ν) [32]:
= (ν) = η(ν)ρ(ν) = η(ν)
pM (1 + μ) , 1 + e cos ν
η(ν) > 0,
1 e sin ν = pM (1 + μ) η +η 1 + e cos ν (1 + e cos ν)2
(16)
.
Substitution to (6) results in
1 sin ϕ (1 + μ) [ηϕ + 2η (1 + ϕ )] + 1 − 3/2 = 0, (1 + e cos ν) f0 (η, ϕ)
f0 (η, ϕ) = η 2 + 1 + 2η cos(ϕ). 6
(17)
Uniform rotations (13) of the tether can be therefore described by equation
1 1 (1 + e cos ν)η (1 + μ) (1 + ω) + sin(ων + ϕ0 ) 1 − 3/2 = 0, 2 f0 (η, ων + ϕ0 ) or
) 1 sin(ων + ϕ 0 η = − 1 − 3/2 2(1 + μ)(1 + ω)(1 + e cos ν) f0 (η, ων + ϕ0 )
(18)
When the tether’s length follows the control law (18), the orientation angle satisfies equation (17) which takes the form
1 sin(ων + ϕ0 ) (1 + ϕ ) + 1 − 3/2 η(1 + μ)ϕ − (1 + e cos ν)(1 + ω) f0 (η, ων + ϕ0 )
sin ϕ 1 + = 0. 1 − 3/2 (1 + e cos ν) f (η, ϕ)
(19)
0
Contrary to the problem studied in [34], in this case analysis of system dynamics requires an explicit description of the control law for the tether length.
EXISTENCE OF PERIODIC SOLUTIONS For implementation of rotations (19), it is reasonable to focus on periodic variations of tether length defined by equation (18). Let’s begin with analysis of the case when the tether length is small: η 1. Using the series in derivatives of the Legendre polynomials ∞
1 = η k Pk+1 (z), (1 − 2ηz + η 2 )3/2 k=0
P1 (z) = 1,
P2 (z) = 3z,
z = − cos(ων + ϕ0 ),
P3 (z) =
15 2 3 z − ,.... 2 2
one can transform equation (18) to η = a1 (ν)η + a2 (ν)η 2 + . . . , where
15 3 2 cos (ων + ϕ0 ) − , a2 (ν) = a(ν) 2 2
a1 (ν) = −3a(ν) cos(ων + ϕ0 ), a(ν) =
sin(ων + ϕ0 ) . 2(1 + μ)(1 + ω)(1 + e cos ν)
Introducing a normalizing coefficient ε=η(0) and a new variable ξ as η = εξ, one obtains the differential equation that 7
depends analytically on small parameter ε: ξ = a1 (ν)ξ + εa2 (ν)ξ 2 + . . . .
(20)
We consider solutions of the Cauchy problem for (20) with initial condition ξ(0) = 1 as function of ε and ϕ0 ; here the variation of parameter ε corresponds to the change of the tether length in the initial problem.
For zero approximation with respect to the small parameter, one obtains the following linear homogeneous differential equation ξ = a1 (ν)ξ.
(21)
Its solution for ξ(0) = 1 can be calculated as
ν a1 (σ)dσ.
ξ(ν) = exp 0
This solution is periodic if ξ(2π) − 1 = 0; this condition coincides with
2π a1 (ν)dν = 0, 0
which can be written as
2π 0
or
2π cos 2ϕ0 0
sin(2ων + 2ϕ0 ) dν = 0, 1 + e cos ν
sin 2ων dν + sin 2ϕ0 1 + e cos ν
2π 0
cos 2ων dν = 0. 1 + e cos ν
Function sin 2ων/(1 + e cos ν)−1 is 2π-periodic and impair. Therefore
2π 0
sin 2ων dν = 1 + e cos ν
π −π
sin 2ων dν = 0. 1 + e cos ν
So the condition of periodicity for the linear approximation can be written as
sin 2ϕ0 I(e, ω) = 0 8
(22)
where
2π I(e, ω) = 0
cos 2ων dν. 1 + e cos ν
The above integral can be calculated via residue method resulting in 2ω −2ω ⎤ √ √ 2 2 1− 1−e π ⎦ = 0 ⎣ 1− 1−e + I(e, ω) = √ e e 1 − e2 ⎡
Thus the 2π-periodic solution exists if
sin 2ϕ0 = 0
or
ϕ00 = 0,
3π π , π, . 2 2
In these cases equation (21) has periodic solution (22). This solution generates family of periodic solutions parametrized by ε > 0. The complete proof of existence of 2π - periodic solutions for small ε > 0 can be done based on standard methods of construction of periodic solutions depending on parameter (see, e.g., [35]); however the procedure is quite cumbersome and we don’t represent it here. TETHER LENGTH CONTROL FOR SOME ROTATIONS When η(0) cannot be considered as small, the control law for the tether’s length should be found numerically. Figures 3 – 6 represent the results of numerical analysis of the control for ω ∈ {1, 2} and several values of η(0). This study shows that the required control can be periodic or not, and the length of the tether can increase or decrease depending on the initial phase of the rotation. The control turns out to be 2π-periodic for ϕ0 = 0 and ϕ0 = π and aperiodic for ϕ0 = π/2 and ϕ0 = 3π/2. NECESSARY CONDITIONS OF STABILITY As shown before, to perform the stability analysis of this problem it is necessary to substitute the respective control law to the equation in variations on the solution in study.The equation in variations has the following form:
)δϕ 1 sin(ων + ϕ 0 η(1 + μ)δϕ − + 1 − 3/2 (1 + e cos ν)(1 + ω) f0 (η, ων + ϕ0 )
δϕ cos(ων + ϕ0 ) sin(ων + ϕ0 ) 3η sin(ων + ϕ0 ) 1 + − 1 − 3/2 δϕ = 0. 5/2 1 + e cos ν 1 + e cos ν f0 (η, ων + ϕ0 ) f0 (η, ων + ϕ0 ) To study stability, we calculate the value Ind = 2 − |tr A| , 9
(23)
Figure 3. Periodic solutions for ω = 2, ϕ0 = 0
Figure 4. Aperiodic solutions for ω = 2, ϕ0 = π/2
10
Figure 5. Periodic solutions for ω = 2, ϕ0 = π
Figure 6. Aperiodic solutions for ω = 2, ϕ0 = 3π/2
11
Figure 7. Stability indicator for ω = 2
where A is the monodromy matrix at the solution in question; if Ind > 0 the respective solution is stable. We consider here only solutions with ϕ0 = 0 and ϕ = π for which the length control is periodic. The results of numerical analysis show that rotations with ω = 1 are always unstable. For ω = 2 (see Fig. 7) and ω = 3 (Fig. 8) the necessary conditions of stability are satisfied for quite large intervals of initial values of the tether lengths. CONCLUSIONS A simple model for space elevator for a planet moon or an asteroid system is considered. Dynamics of spacecraft in a system of two primaries is studied in the framework of plane restricted elliptical three-body problem. Spacecraft is tethered to the surface of one of the primaries (e.g., a moon). It is assumed that the proper rotation of the moon is not necessarily synchronized with its orbital motion; to avoid wrapping of the tether around the moon, the tethered spacecraft should rotate together with the moon’s surface. The study is focused on the possibility to implement such motion via control of the tether length. There are several types of control that realize uniform (with respect to the true anomaly) rotation of the tether; their properties depend significantly on the motion parameters. It is shown that during a limited time interval it is possible to keep the tether orientation in the absolute space. For a number of initial conditions and angular velocities of the desired rotation, there exists 2π- periodic control. Stability properties of the respective solutions have been studied using Floquet theory; the rotations are stable for various system parameters. The developed approach can be useful for analysis of applications of tether systems for asteroids and planet satellites exploration. For example, the above controls can be applied during limited time intervals, e.g., half-period of asteroid rotation, for tether deployment or retrieval when permanent tether orientation either with respect to the asteroid surface 12
Figure 8. Stability indicator for ω = 3
or in absolute space is required. ACKNOWLEDGMENTS This research is supported by the Russian Foundation for Basic Research (grants 12-01-00536-a, 12-08-00637-a), Russian Federal Programme (grant 14.740.11.0995), and the Portuguese Foundation for Science and Technologies, FCT. REFERENCES [1] V. V. Beletsky and E.M. Levin, Dynamics of Space Tether Systems, Advances in the Astronautical Sciences. 1993. Vol. 83. [2] E. Levin, “Dynamic Analysis of Space Tether Missions,” Advances in the Astronautical Sciences. 2007, Vol. 126. [3] J. Pearson, “Anchored lunar satellites for cislunar transportation and communication”, Journal of the Astronautical Sciences, 1979, Vol.27, No.1, 39 – 62. [4] V. V. Beletsky and E.M. Levin, “Mechanics of lunar tether system,” Cosmic Research, 1982, Vol. 20, No.5, 760 – 764. [5] A.A. Burov and I.I. Kosenko, “On relative equilibria of an orbital station in regions near the triangular libration points,” Doklady Physics, 2007, Vol.52, No. 9, 507 – 509. [6] A.A. Burov, O.I. Kononov, and A.D. Guerman, “Relative equilibria of a Moon - tethered spacecraft,” Advances in the Astronautical Sciences, 2011, Vol.136, 2553 – 2562. [7] A.A. Burov, I.I. Kosenko, and A.D. Guerman, “Dynamics of a moon-anchored tether with variable length,” Advances in the Astronautical Sciences, 2012, Vol. 142, 3495 – 3507. ¨ [8] W. Schiehlen, “Uber die Lagestabilisirung k¨unstlicher Satelliten auf elliptischen Bahnen,” Diss. Dokt.-Ing. technische Hochschule Stuttgart. 1966. 13
¨ [9] W. Schiehlen, “Uber den Drallsatz fur Satelliten mit im Innern bewegten Massen,” Zeitschrift f¨ur Angewandte Mathematik und Mechanik, 1966. Bd.46, Sonderheft. S. T132 – T134. [10] W. Schiehlen and O. Kolbe, “Gravitationsstabilisierung von Satelliten auf elliptischen Bahnen,” Ingenieur-Archiv, 1969. Bd.38. Nr.6. S.389 – 399. [11] V. A. Sarychev, Problems of Orientation of Artificial Satellites, Moscow, VINITI Ed., 1978, Vol. 11, 224 pp. [12] I. P. Polyanskaya, “Oscillations of satellite with compensating device in elliptic orbit,” Cosmic Research, 1982, Vol.20, No.5, 674 – 681. [13] A.A. Burov, “Oscillations of a vibrating dumbbell on an elliptic orbit,” Doklady Physics, 2011, Vol.56, No.3, 182 – 185. [14] A.A. Burov and I. I. Kosenko, “Planar vibrations of a solid with variable mass distribution in an elliptic orbit,” Doklady Physics, 2011, Vol. 56, No. 10, 548 – 552. [15] A. Narayan and M. D. Pandey, “Some non-linear resonance oscillations of dumbbell satellite in elliptical orbit,” International Journal of Pure and Applied Mathematics, 2012, Vol.78, No.6, 931 – 944. [16] A. Djebli, L. El Bakkali, and M. Pascal, “On fast retrieval laws for tethered satellite systems,” Acta Astronautica, 2002, Vol.50, No.8, 461 – 470. [17] A. Djebli, M. Pascal, and L. El Bakkali, “Laws of deployment/retrieval in tether connected satellites systems,” Acta Astronautica, 1999, Vol.45, No.2, 61 – 73. [18] A. Djebli, M. Pascal, and L. El Bakkali, “On deployment dynamics of tethered satellite systems,” Revue de M´ecanique Appliqu´ee et Th´eorique, 2000, Vol. 1, No. 1, 13 – 39. [19] M. Pascal, A. Djebli, and L. El Bakkali, “A new deployment/retrieval scheme for a tethered satellite system, intermediate between the conventional scheme and the crawler scheme,” Journal of Applied Mathematics and Mechanics, 2001, Vol.65, No.4, 689 – 696. [20] A. Djebli and M. Pascal, “A new method for the orbital modification of a tether connected satellite system,” Acta Mechanica, 2004, Vol.167, Nos. 1-2, 113 – 122. [21] P. Mantri, A. P. Mazzoleni, and D. A. Padgett, “Parametric study of tethered satellite systems,” Journal of Spacecraft and Rockets, 2007, Vol.44, No.2, 421 – 424. [22] B. Wong and A. Misra, “Planar dynamics of variable length multi-tethered spacecraft near collinear Lagrangian points,” Acta Astronautica, 2008, Vol.63, NNos. 11-12, 1178 – 1187. [23] A. Pizarro-Chong and A. K. Misra, “Dynamics of multi-tethered satellite formations containing a parent body,” Acta Astronautica, 2008, Vol.63, NNos. 11-12, 1188 – 1202. [24] A. D. Guerman, “Equilibria of multibody chain in orbit plane,” Journal of Guidance, Control, and Dynamics, 2003, Vol.26, No.6, 942 – 948. [25] A. D. Guerman, “Spatial equilibria of multibody chain in a circular orbit,” Acta Astronautica, 2006, Vol.58, No.1, 1 – 14. [26] A. D. Guerman, G. V. Smirnov, P. Paglione, and A. M. Seabra, “Stationary configurations of tetrahedral tethered satellite formation,” Journal of Guidance, Control, and Dynamics, 2008, Vol.31, No.2, 424 – 428. [27] A. A. Burov, A. D. Guerman, and R. S. Sulikashvili, “Relative equilibria of a tetrahedral structure with rigid and tethered elements,” Advances in the Astronautical Sciences, 2008, Vol.129,1665 – 1674. 14
[28] A. A. Burov, A. D. Guerman, and R. S. Sulikashvili, “The orbital motion of a tetrahedral gyrostat,” Journal of Applied Mathematics and Mechanics, 2010, Vol. 74, No.4, 425 – 435. [29] A. A. Burov, A. D. Guerman, and R. S. Sulikashvili, “Dynamics of a tetrahedral satellite-gyrostat,” AIP Conference Proceeding: Numerical Analysis and Applied Mathematics. American Institute of Physics, 2010, Vol. 1281, 465 – 468. [30] A. A. Burov, A. D. Guerman, and R. S. Sulikashvili, “On steady-state motions of gyrostats with equal moments of inertia in a central force field,” Journal of Applied Mathematics and Mechanics, 2011, Vol. 75, No. 5, 517 – 521. [31] A.A. Burov, A. D. Guerman, and I.I. Kosenko, “Equilibrium configurations and control of a Moon-anchored tethered system,” Advances in the Astronautical Sciences, 2013, Vol. 146, 251-266. [32] A. A. Burov, A. D. Guerman, and I. I. Kosenko, “On in-plane oscillations of a pendulum of variable length attached to a surface of a planet satellite,” Cosmic Research, 2014, Vol. 52, No. 4, 289-294 [33] A.A. Burov, A. D. Guerman, I. I. Kosenko, “Tether orientation control for lunar elevator,” Celestial Mechanics and Dynamical Astronomy, 2014, Vol. 120, Issue 3, 337-347. [34] A. A. Burov and I. I. Kosenko, “The existence and stability of orbitally uniform rotations of a vibrating dumbbell on an elliptic orbit,” Doklady Physics, 2013, Vol. 58, No. 7, 305 – 308. [35] M.M. Vainberg and V.A. Trenogin, Theory of Branching of Solutions of Nonlinear Equations (Series Monographs and textbooks on pure and applied mathematics). Wolters-Noordhoff B.V. 1974, 510 pp.
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Uniform rotations of tethered system connected to a moon surface Alexander A. Burov, Anna D. Guerman, Ivan I. Kosenko
www.elsevier.com/locate/actaastro
PII: DOI: Reference:
S0094-5765(15)00200-3 http://dx.doi.org/10.1016/j.actaastro.2015.05.018 AA5446
To appear in:
Acta Astronautica
Received date: 9 July 2014 Revised date: 17 April 2015 Accepted date: 12 May 2015 Cite this article as: Alexander A. Burov, Anna D. Guerman, Ivan I. Kosenko, Uniform rotations of tethered system connected to a moon surface, Acta Astronautica, http://dx.doi.org/10.1016/j.actaastro.2015.05.018 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
UNIFORM ROTATIONS OF TETHERED SYSTEM CONNECTED TO A MOON SURFACE Alexander A. Burov∗, Anna D. Guerman†, Ivan I. Kosenko‡ ABSTRACT We consider the problem of in-plane rotations of a space elevator with variable tether length attached to a surface of one of the primaries in a double system. The planet and its moon (or two asteroids) move about their center of mass in unperturbed elliptic Keplerian orbits. We discuss the possibilities to cause a prescribed motion of the system by changing the tether’s length. Periodic solutions of the equation for the tether length control are studied using the method of small parameter. The stability of these solutions is studied numerically. The analysis shows that there exists a control law that implements tether rotations which are uniform with respect to true anomaly; one can indicate conditions when the above rotations are stable in the first approximation. These results can be used for development of a planet elevator or a system for payload transportation to and from asteroid surface. INTRODUCTION Use of tethers for space transportation is currently discussed in several research groups and agencies. Tethers provide promising possibilities for orbital and attitude spacecraft control, distributed spacecraft missions, space debris removal, etc. [1, 2]. One of the possible applications for space tethers is a space elevator. The project of building a space elevator at the Earth, though much awaited by numerous fans, still faces serious difficulties. Meanwhile, similar systems for Moon, Mars, or asteroid exploration look much more feasible. Studies of spacecraft tethered to the Moon surface began long ago [3, 4]. Several authors discuss possible applications of such structure for lunar exploration [1, 2]. Recently some results on equilibria of such spacecraft and their stability have been obtained in [5–7]. Research on systems with variable mass distribution arise probably to [8–10] and to dissertation of V.A. Sarychev ∗ (see also [11]). Analysis of the necessary conditions of stability for relative equilibria of a satellite with variable mass distribution has been done in [12], some of these results have been re-discovered in [13, 14] (see also, e.g., [15]). ∗ Professor,
National Research University ”Higher School of Economics,” Myasnitskaya 20, 101000 Moscow RUSSIA; Senior Researcher, Department of Mechanics, Dorodnicyn Computing center of the RAS, Vavilova 40, 119333 Moscow RUSSIA † Associate Professor, C-MAST: Centre for Mechanical and Aerospace Science and Technologies, University of Beira Interior, Calcada Fonte do Lameiro, 6201-001 Covilha PORTUGAL, e-mail: [email protected]; tel.: +351 275 329 915; FAX: +351 275 329 972. ‡ Senior Researcher, Department of Mechanics, Dorodnicyn Computing center of the RAS, Vavilova 40, 119333 Moscow RUSSIA ∗ according to the author’s communication
1
Various aspects of dynamics of orbital tethered system have been studied in [16–20]. Parametric analysis of orbiting tethers is performed in [21]. Dynamics of tethered systems in the vicinity of libration points is analyzed in [5, 22]. Motions of a multi-tether system are considered in [23–30]. In [7, 31–33] we consider dynamics of a tether anchored to the Moon surface and the possibilities to keep its orientation with respect to the Earth – Moon direction despite the eccentricity of the Moon orbit. A proper control of the tether’s length can keep the fixed orientation of the tether for several modes of the system functioning. The results of [7,31,32] can be applied for other systems of two primaries, e.g., for planet’s satellites or binary asteroids whenever the point of the tether’s attachment maintains its orientation with respect to the other primary. Meanwhile, the above results are only applicable for systems, where the moon’s proper rotation is synchronized with its orbit motion so as the position of the tether’s anchor is fixed with respect to the planet – moon direction. Relative rotation of the moon’s surface requires that the tether follow the motion of the anchor. Here we examine the possibility to create a proper rotation of the tether via control of its length. POSING THE PROBLEM Consider a system of two primaries, e.g., a planet E and a moon M , that move about their center of mass O in elliptic Keplerian orbits −−→ |OM | = r =
pM , 1 + e cos ν
−−→ |OE| = μr =
pE , 1 + e cos ν
μ=
mM . mE
(1)
Here mE and mM are the masses of the planet and the moon respectively, pE and pM are parameters of their orbits, e is eccentricity, ν the true anomaly. We assume that the sizes of the primaries are negligible. Figure 1 shows the plane of the primaries’ orbits π. The spacecraft of mass m is connected to the moon surface by a tether which length can be
Figure 1. Main notations
2
changed according to some control law. Only in-plane motions of the tether are considered; the tether orientation is described by angle ϕ (Fig.1). For in-plane motion of the tethered spacecraft its kinetic energy can be written as T =
T =
m 2
x˙ 2S + y˙ S2 or
m 2 2 r˙ + r2 ν˙ 2 + ˙2 + 2 (ν˙ + ϕ) ˙ + m r˙ ˙ + rν˙ (ν˙ + ϕ) ˙ cos ϕ + m ν˙ r˙ − r˙ − r˙ ϕ˙ sin ϕ. 2
Its potential energy is
U = −Gm
mE mM + rE
(2)
.
(3)
Here G is the universal gravitational constant, 1/2 rE = |SE| = ρ2 + 2ρ cos ϕ + 2 is the distance between the planet and the spacecraft, and ρ = |EM | = (1 + μ)r is the distance between the planet and the moon. Assuming that ν, r, and are given as functions of time, one can write down the Lagrangian equations d ∂L ∂L = , dt ∂ ϕ˙ ∂ϕ
L = T − U.
(4)
Introducing the true anomaly as a new independent variable and denoting by strike the respective derivative d d d = ν˙ = ω(1 + e cos ν)2 , dt dν dν
(5)
where ω 2 = GmE /(1 + μ)2 p3M , the equation of motion can be rewritten as ϕ + 2 (1 + ϕ ) −
+
2e (1 + ϕ ) sin ν + 1 + e cos ν
(6)
(1 + μ)3 p3M pM sin ϕ 1 − = 0, (1 + e cos ν)2 f 3/2
f = 2 (1 + e cos ν)2 + (1 + μ)2 p2M + 2(1 + μ)pM cos ϕ(1 + e cos ν). It is possible to consider equations (6) from two different perspectives. In the framework of the direct problem, one can look for the system’s motions that correspond to a specific variation of the tether length; in this case = (ν) is given and one has to study the second order ordinary differential equation for ϕ(ν). Considering the inverse problem, one can find the control law for the tether length = (ν) that results in a specific variation of the tether orientation; in this case ϕ = ϕ(ν) is defined beforehand and one has to find = (ν) analyzing the first-order differential equation (6. Here we examine the possibility to implement rotations of the tether, which are uniform with respect to the true 3
anomaly. First we consider the inverse problem and find the control law = (ν) that causes such motions. Afterwards we analyze the direct problem and find the necessary conditions of stability for the above rotations. Similar problem regarding existence and stability of rotations of a tethered system has been studied in [34].
FORCES IN TETHER WITH VARIABLE LENGTH Consider the Lagrangian ˙ = T − U, Λ(ϕ, , ϕ, ˙ )
(7)
Here r and ν are given functions of time t. The equations of motion are d ∂Λ ∂Λ = , dt ∂ ϕ˙ ∂ϕ
∂Λ d ∂Λ = + F , dt ∂ ˙ ∂
(8)
where F is the generalized force correspondent to the coordinate . Since ∂Λ = m ˙ + r˙ cos ϕ + νr ˙ sin ϕ , ∂ ˙ ∂Λ = m ((ν˙ + ϕ) ˙ + rν˙ cos ϕ − r˙ sin ϕ) (ν˙ + ϕ)− ˙ ∂ mM mE (ρ cos ϕ + ) −Gm , + 3 2 rE the equation correspondent to coordinate can be written as ¨ + r¨ cos ϕ − r˙ ϕ˙ sin ϕ + ν¨r sin ϕ + ν˙ r˙ sin ϕ + ν˙ ϕr ˙ cos ϕ =
(9)
= ((ν˙ + ϕ) ˙ + rν˙ cos ϕ − r˙ sin ϕ) (ν˙ + ϕ)− ˙ mM mE (ρ cos ϕ + ) + + f , −G 3 2 rE
f = F /m.
Assume that during the motion the following constraint should hold true: − (ν) = 0.
(10)
Differentiation of equation (10) with respect to time gives d ν˙ = 0, ˙ − dν
d2 d ¨ − 2 ν˙ 2 − ν¨ = 0. dν dν 4
(11)
Substitution of expression (11) for ¨ to (9) yields
f =
d2 2 d ν¨ + r¨ cos ϕ − r˙ ϕ˙ sin ϕ + +¨ ν r sin ϕ + ν˙ r˙ sin ϕ + ν˙ ϕr ˙ cos ϕ− ν˙ + dν 2 dν
− ((ν˙ + ϕ) ˙ + rν˙ cos ϕ − r˙ sin ϕ) (ν˙ + ϕ) ˙ +G
(12)
mM mE (ρ cos ϕ + ) . + 3 2 rE
The above expression permits one to find the force in tether for a given motion of the system. UNIFORM ROTATIONS OF THE TETHER Partial solutions which correspond to the uniform in-plane rotations of the tether with respect to EM direction are described as [34] ϕ = ων + ϕ0 ,
ϕ = ω = const .
(13)
Here ω is the constant angular velocity of the rotation. Note that the absolute angular velocity of the tether’s rotation is Ω = ω + 1. Substitution of (13) to (6) allows one to obtain the following differential equation: (1 + ω) −
pM sin(ων + ϕ0 ) e (1 + ω) sin ν (1 + μ)3 p3M + 1 − = 0, 1 + e cos ν 2(1 + e cos ν)2 f 3/2
(14)
f = 2 (1 + e cos ν)2 + (1 + μ)2 p2M + 2(1 + μ)pM cos(ων + ϕ0 )(1 + e cos ν). If ω = −1, the tether keeps its orientation in the absolute reference frame. In this case the above equation becomes algebraic since the term with vanishes:
(1 + e cos ν) [(1 + e cos ν) + 2(1 + μ)pM cos(−ν + ϕ0 )] = 0.
(15)
This equation has two solutions:
I. :
= 0,
II. :
= −2(1 + μ)pM
cos(−ν + ϕ0 ) . 1 + e cos ν
The first solution is trivial; the second one is meaningful for intervals of ν when the respective right-hand term is positive, i.e, for ϕ0 + π/2 + 2πn ≤ ν ≤ ϕ0 + 3π/2 + 2πn, n = 0, ±1, ±2, ... The control laws for the tether lengths for a number of initial phases ϕ0 and e = 0.5 are shown on Fig. 2. Despite these solutions only exist for limited time intervals, they correspond to large variations of the tether’s length with permanent tether orientation in the absolute space: depending on the initial phase ϕ0 , the maximum achievable distances vary 5
between
2(1+μ)pM 1+e
and
2(1+μ)pM 1−e
.
The above class of the tether motions may turn out to be useful for delivering payload to and from the surface of an asteroid or a moon.
Figure 2. Tether length control for ω = −1, e = 0.5
Now we assume that ω + 1 = 0. One can further simplify equation of motion (6) using new variable η(ν) [32]:
= (ν) = η(ν)ρ(ν) = η(ν)
pM (1 + μ) , 1 + e cos ν
η(ν) > 0,
1 e sin ν = pM (1 + μ) η +η 1 + e cos ν (1 + e cos ν)2
(16)
.
Substitution to (6) results in
1 sin ϕ (1 + μ) [ηϕ + 2η (1 + ϕ )] + 1 − 3/2 = 0, (1 + e cos ν) f0 (η, ϕ)
f0 (η, ϕ) = η 2 + 1 + 2η cos(ϕ). 6
(17)
Uniform rotations (13) of the tether can be therefore described by equation
1 1 (1 + e cos ν)η (1 + μ) (1 + ω) + sin(ων + ϕ0 ) 1 − 3/2 = 0, 2 f0 (η, ων + ϕ0 ) or
) 1 sin(ων + ϕ 0 η = − 1 − 3/2 2(1 + μ)(1 + ω)(1 + e cos ν) f0 (η, ων + ϕ0 )
(18)
When the tether’s length follows the control law (18), the orientation angle satisfies equation (17) which takes the form
1 sin(ων + ϕ0 ) (1 + ϕ ) + 1 − 3/2 η(1 + μ)ϕ − (1 + e cos ν)(1 + ω) f0 (η, ων + ϕ0 )
sin ϕ 1 + = 0. 1 − 3/2 (1 + e cos ν) f (η, ϕ)
(19)
0
Contrary to the problem studied in [34], in this case analysis of system dynamics requires an explicit description of the control law for the tether length.
EXISTENCE OF PERIODIC SOLUTIONS For implementation of rotations (19), it is reasonable to focus on periodic variations of tether length defined by equation (18). Let’s begin with analysis of the case when the tether length is small: η 1. Using the series in derivatives of the Legendre polynomials ∞
1 = η k Pk+1 (z), (1 − 2ηz + η 2 )3/2 k=0
P1 (z) = 1,
P2 (z) = 3z,
z = − cos(ων + ϕ0 ),
P3 (z) =
15 2 3 z − ,.... 2 2
one can transform equation (18) to η = a1 (ν)η + a2 (ν)η 2 + . . . , where
15 3 2 cos (ων + ϕ0 ) − , a2 (ν) = a(ν) 2 2
a1 (ν) = −3a(ν) cos(ων + ϕ0 ), a(ν) =
sin(ων + ϕ0 ) . 2(1 + μ)(1 + ω)(1 + e cos ν)
Introducing a normalizing coefficient ε=η(0) and a new variable ξ as η = εξ, one obtains the differential equation that 7
depends analytically on small parameter ε: ξ = a1 (ν)ξ + εa2 (ν)ξ 2 + . . . .
(20)
We consider solutions of the Cauchy problem for (20) with initial condition ξ(0) = 1 as function of ε and ϕ0 ; here the variation of parameter ε corresponds to the change of the tether length in the initial problem.
For zero approximation with respect to the small parameter, one obtains the following linear homogeneous differential equation ξ = a1 (ν)ξ.
(21)
Its solution for ξ(0) = 1 can be calculated as
ν a1 (σ)dσ.
ξ(ν) = exp 0
This solution is periodic if ξ(2π) − 1 = 0; this condition coincides with
2π a1 (ν)dν = 0, 0
which can be written as
2π 0
or
2π cos 2ϕ0 0
sin(2ων + 2ϕ0 ) dν = 0, 1 + e cos ν
sin 2ων dν + sin 2ϕ0 1 + e cos ν
2π 0
cos 2ων dν = 0. 1 + e cos ν
Function sin 2ων/(1 + e cos ν)−1 is 2π-periodic and impair. Therefore
2π 0
sin 2ων dν = 1 + e cos ν
π −π
sin 2ων dν = 0. 1 + e cos ν
So the condition of periodicity for the linear approximation can be written as
sin 2ϕ0 I(e, ω) = 0 8
(22)
where
2π I(e, ω) = 0
cos 2ων dν. 1 + e cos ν
The above integral can be calculated via residue method resulting in 2ω −2ω ⎤ √ √ 2 2 1− 1−e π ⎦ = 0 ⎣ 1− 1−e + I(e, ω) = √ e e 1 − e2 ⎡
Thus the 2π-periodic solution exists if
sin 2ϕ0 = 0
or
ϕ00 = 0,
3π π , π, . 2 2
In these cases equation (21) has periodic solution (22). This solution generates family of periodic solutions parametrized by ε > 0. The complete proof of existence of 2π - periodic solutions for small ε > 0 can be done based on standard methods of construction of periodic solutions depending on parameter (see, e.g., [35]); however the procedure is quite cumbersome and we don’t represent it here. TETHER LENGTH CONTROL FOR SOME ROTATIONS When η(0) cannot be considered as small, the control law for the tether’s length should be found numerically. Figures 3 – 6 represent the results of numerical analysis of the control for ω ∈ {1, 2} and several values of η(0). This study shows that the required control can be periodic or not, and the length of the tether can increase or decrease depending on the initial phase of the rotation. The control turns out to be 2π-periodic for ϕ0 = 0 and ϕ0 = π and aperiodic for ϕ0 = π/2 and ϕ0 = 3π/2. NECESSARY CONDITIONS OF STABILITY As shown before, to perform the stability analysis of this problem it is necessary to substitute the respective control law to the equation in variations on the solution in study.The equation in variations has the following form:
)δϕ 1 sin(ων + ϕ 0 η(1 + μ)δϕ − + 1 − 3/2 (1 + e cos ν)(1 + ω) f0 (η, ων + ϕ0 )
δϕ cos(ων + ϕ0 ) sin(ων + ϕ0 ) 3η sin(ων + ϕ0 ) 1 + − 1 − 3/2 δϕ = 0. 5/2 1 + e cos ν 1 + e cos ν f0 (η, ων + ϕ0 ) f0 (η, ων + ϕ0 ) To study stability, we calculate the value Ind = 2 − |tr A| , 9
(23)
Figure 3. Periodic solutions for ω = 2, ϕ0 = 0
Figure 4. Aperiodic solutions for ω = 2, ϕ0 = π/2
10
Figure 5. Periodic solutions for ω = 2, ϕ0 = π
Figure 6. Aperiodic solutions for ω = 2, ϕ0 = 3π/2
11
Figure 7. Stability indicator for ω = 2
where A is the monodromy matrix at the solution in question; if Ind > 0 the respective solution is stable. We consider here only solutions with ϕ0 = 0 and ϕ = π for which the length control is periodic. The results of numerical analysis show that rotations with ω = 1 are always unstable. For ω = 2 (see Fig. 7) and ω = 3 (Fig. 8) the necessary conditions of stability are satisfied for quite large intervals of initial values of the tether lengths. CONCLUSIONS A simple model for space elevator for a planet moon or an asteroid system is considered. Dynamics of spacecraft in a system of two primaries is studied in the framework of plane restricted elliptical three-body problem. Spacecraft is tethered to the surface of one of the primaries (e.g., a moon). It is assumed that the proper rotation of the moon is not necessarily synchronized with its orbital motion; to avoid wrapping of the tether around the moon, the tethered spacecraft should rotate together with the moon’s surface. The study is focused on the possibility to implement such motion via control of the tether length. There are several types of control that realize uniform (with respect to the true anomaly) rotation of the tether; their properties depend significantly on the motion parameters. It is shown that during a limited time interval it is possible to keep the tether orientation in the absolute space. For a number of initial conditions and angular velocities of the desired rotation, there exists 2π- periodic control. Stability properties of the respective solutions have been studied using Floquet theory; the rotations are stable for various system parameters. The developed approach can be useful for analysis of applications of tether systems for asteroids and planet satellites exploration. For example, the above controls can be applied during limited time intervals, e.g., half-period of asteroid rotation, for tether deployment or retrieval when permanent tether orientation either with respect to the asteroid surface 12
Figure 8. Stability indicator for ω = 3
or in absolute space is required. ACKNOWLEDGMENTS This research is supported by the Russian Foundation for Basic Research (grants 12-01-00536-a, 12-08-00637-a), Russian Federal Programme (grant 14.740.11.0995), and the Portuguese Foundation for Science and Technologies, FCT. REFERENCES [1] V. V. Beletsky and E.M. Levin, Dynamics of Space Tether Systems, Advances in the Astronautical Sciences. 1993. Vol. 83. [2] E. Levin, “Dynamic Analysis of Space Tether Missions,” Advances in the Astronautical Sciences. 2007, Vol. 126. [3] J. Pearson, “Anchored lunar satellites for cislunar transportation and communication”, Journal of the Astronautical Sciences, 1979, Vol.27, No.1, 39 – 62. [4] V. V. Beletsky and E.M. Levin, “Mechanics of lunar tether system,” Cosmic Research, 1982, Vol. 20, No.5, 760 – 764. [5] A.A. Burov and I.I. Kosenko, “On relative equilibria of an orbital station in regions near the triangular libration points,” Doklady Physics, 2007, Vol.52, No. 9, 507 – 509. [6] A.A. Burov, O.I. Kononov, and A.D. Guerman, “Relative equilibria of a Moon - tethered spacecraft,” Advances in the Astronautical Sciences, 2011, Vol.136, 2553 – 2562. [7] A.A. Burov, I.I. Kosenko, and A.D. Guerman, “Dynamics of a moon-anchored tether with variable length,” Advances in the Astronautical Sciences, 2012, Vol. 142, 3495 – 3507. ¨ [8] W. Schiehlen, “Uber die Lagestabilisirung k¨unstlicher Satelliten auf elliptischen Bahnen,” Diss. Dokt.-Ing. technische Hochschule Stuttgart. 1966. 13
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