Experimental study on dynamics and control of tethered satellite systems with climber

Acta Astronautica 69 (2011) 96–108

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Experimental study on dynamics and control of tethered satellite systems with climber Hirohisa Kojima a,, Yohei Sugimoto b, Yoshiyasu Furukawa a a b

Tokyo Metropolitan University, Department of Aerospace Engineering, 6-6 Asahigaoka, Hino, Tokyo 191-0065, Japan School of Engineering, Aerospace Sciences, University of Glasgow, Glasgow G12 8QQ, Scotland, United Kingdom

a r t i c l e in f o

abstract

Article history: Received 26 October 2010 Received in revised form 9 February 2011 Accepted 18 February 2011 Available online 25 March 2011

Tethered satellite systems (TSSs) open up the possibility of alternative forms of transportation such as space elevators, which could conceivably replace conventional space transportation systems in the foreseeable future. A simplified dynamic model of a TSS with a single climber is used to evaluate the fundamental librational motion. An optimal acceleration climber transit control scheme is developed by solving a two-point boundary value problem (TPBVP) that minimizes the time integral of the square of the climber transit acceleration along the tether, along with placing constraints on the final tether angle, climber’s position and velocity. The numerical simulations show that if penalties are introduced with regard to the tether angles at the terminal time, residual librations resulting from optimal acceleration climber transits are smaller than for the case of constant speed climber transits. Experiments are carried out to validate the derived optimal climber acceleration profile, using the ground-based experimental setup that emulates the tether librational motion. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Tethered satellite system Optimal control Space elevator

1. Introduction The first conceptual design of a tethered satellite system (TSS) as an artificial gravity generator was proposed by Tsiolkovsky [1]. After a century had passed since his futuristic proposal, the first Shuttle-based TSS mission (TSS-1) was carried out in orbit in 1992 by NASA and ASI [2]. TSS-1 was designed to investigate the dynamical forces acting upon the TSS and to develop the capability for future tether applications on the Space Shuttle, the International Space Station, and the Space Elevator. NASA currently has a single short-term goal of achieving 100-km tether deployment by creation of new materials and developing control schemes and all the supporting subsystems.  Corresponding author. Tel.: þ81 42 585 8653; fax: þ81 42 583 5119. E-mail addresses: [email protected] (H. Kojima), [email protected] (Y. Sugimoto), [email protected] (Y. Furukawa).

0094-5765/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2011.02.009

TSS experiments basically involve three phases: deployment, station-keeping, and retrieval. Many previous studies have focused on the analysis of the dynamic behavior of the TSS and control of the deployment/ retrieval dynamics [3–7], and elastic oscillations were studied in [8,9]. Although there are several issues involved in the retrieval phase, such as slackness or very low tension in the tether, and collision between the mother ship and subsatellite, these are not taken into consideration since the retrieval phase may not be necessary in most orbital missions. During the operational phase, the risk of meteoroid and orbital debris (M/OD) impacts, and the effect of solar radiation on the tether increase in proportion to the tether length. Because astronauts may not be able to inspect more than a kilometer of tether by themselves, the multi-application survivable tether (MAST) experiment [10] proposed the use of a climber as a tether inspector. The MAST system consists of two pico-satellites, a 1-km tether and a climber. The climber inspects

H. Kojima et al. / Acta Astronautica 69 (2011) 96–108

the condition of the tether as it travels along the tether and is capable of detecting damage to the surface due to M/OD impacts and solar radiation. In regard to future TSS orbital missions, the successful use of climbers as tether inspectors will represent a large forward step, allowing safety operations to be carried out on demand. Other application of climbers on TSS is that they can provide microgravity/variable gravity environment [11,12]. One of the critical disadvantages of climber transit is that undesired oscillations are induced in the TSS by the Coriolis effect of the traveling climber. To understand this effect, the dynamical behavior of a TSS with a climber has been investigated by Misra et al. [13], Lorenzini et al. [14], Modi and Bechmann [15], Fujii et al. [16], and so on. TSS with a climber was modeled as a two bar model in [13,15], modeled as a flexible tether with a climber [16], but a control problem of a TSS with a climber has not been considered in [13,16]. On the other hand, LQR based thruster control was presented to regulate librational motion in [15]. Three simple climbing procedures were presented by Cohen and Misra [17]. Also, optimal climber trajectories for space elevators have been analyzed by Williams and Ockels [18], who found that the residual librational motion of tethers and subsatellites could be suppressed by applying an optimal control method that minimizes the variation in climber speed and acceleration, and who showed that the in-plane libration can be regulated to zero. However, the tether used in Ref. [18] was a rigid ribbon, that is, bending of the tether at the climber position was not considered, although this occurs for a flexible tether due to the Coriolis force. Even if the tether with a climber is modeled as two bars with a bend at the climber position [13,16], it is impossible to completely regulate tether librations using only a single crawling climber. In addition, in spite of the approximately one year operation period of the tether physics and survivability experiment (TiPS) [19], it is reported that the librations caused by the initial deployment of the tethered subsatellite have not yet become zero due to the perturbation of the orbit. Taking these results into account, we do not focus on complete suppression of the librational motion of a TSS in this paper, but instead try to reduce the residual librational motion as much as possible. For this purpose, we clarify the dynamic behavior of a TSS with a single climber, and develop an adaptable and more effective climber control method, minimizing the residual librations of the TSS by using a simplified mathematical TSS model. The mathematical model of the TSS derived in this paper is basically the same as that in [13,15], but it is slightly more simplified in order to compare the numerical results with the experimental results. Climber acceleration profiles are studied using an experimental setup that was developed to emulate the TSS behavior on the ground. To the authors’ best knowledge, the optimal control of a climber on a TSS has not yet been studied experimentally. In other words, the contribution of this paper is that the dynamics behavior of a TSS with a single climber and the effectiveness of the proposed optimal climber acceleration profile were validated on the ground experimentally.

97

The results show that the experimental setup is capable of emulating the fundamental librational motion of a tether with a single climber, and that the climber velocity determined by the optimal acceleration scheme is effective in suppressing the final librational motion, even if the tether is simply modeled as two bars with a bend at the climber position. 2. Mathematical model 2.1. Simplified TSS model The simplified TSS model used in this paper is shown in Fig. 1. This model is similar to the three-mass tethered satellite model employed in [20,21], with one very-significant difference; in the setup used in this study, the climber can ascend and descend the tether, whereas subsatellite 1 in the model in [20,21] could not. The equations of motion derived in this paper are basically the same as those in [13,15], but are slightly more simplified by assuming that the mother ship is much heavier than the subsatellite and climber, and considering an active force for the climber to ascend and descend along the tether, instead of the upper and lower tether tension. To study the dynamics of the climber in the TSS, a detailed mathematical model of the system should be derived. However, to determine the optimal control schemes for the climber to minimize the librational motion of the tether and the subsatellite, a simplified model is sufficient as a first step toward control design. In this study, only the in-plane motion of the TSS is considered because the experimental setup cannot accommodate out-of-plane motion. In addition, although the center of the mass of the system changes depending on the tether angles and the climber’s position, it is assumed in this study that the tether is a rigid body, but is utterly massless and has no moment of inertia, the climber and the subsatellite are point masses, and the mother ship maintains a circular orbit. The climber with mass m1 is at a distance L1 along the tether from the mother ship, and the subsatellite with mass m2 is at a distance of L2 along the tether from the climber. Note that the total tether

Fig. 1. Simplified representation of a TSS with a climber.

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length L¼L1 þ L2 is constant in this study. Hereafter, the section of tether between the mother ship and the climber is referred to as tether-1 and the section between the climber and the subsatellite is referred to as tether-2. Libration angles of the climber and the subsatellite from the local vertical are denoted by y1 and y2 , respectively. Additionally, m is the Earth’s gravitational parameter 3 ðm ¼ 3:98613  105 km =s2 Þ, R is the orbital radius of the mother ship, L_ 1 is the ascent/descent velocity of the climber along tether-1, and O is the orbital rate of the mother ship. The orbital rotating coordinate frame ða1 ,a2 ,a3 Þ has its origin at the center of the Earth. This model is comparable with the experimental setup that will be described later. 2.2. Equations of motion The equations of motion of the mathematical TSS model are obtained using Kane’s method. The basis vectors are T

ð1Þ

a2 ¼ ½0,1,0T

ð2Þ

a3 ¼ ½0,0,1T

ð3Þ

a1 ¼ ½1,0,0

The positions of the climber and subsatellite in the orbital frame are, respectively, given by R1 ¼ ðRL1 cosy1 Þa1 L1 siny1 a2

ð4Þ

R2 ¼ fRL1 cosy1 ðLL1 Þcosy2 ga1 fL1 siny1 þ ðLL1 Þsiny2 ga2

2 þ 2ðLL1 ÞOy_ 2 cosy2 2L_ 1 y_ 2 siny2 þ ðLL1 Þy_ 2 cosy2 2 fRL cosy ðLL Þcosy gO2 þL y_ cosy ga 1

u1 :¼ y_ 1

ð6Þ

u2 :¼ y_ 2

ð7Þ

u3 :¼ L_ 1

ð8Þ

Taking the orbital rate vector Xð ¼ Oa3 Þ into consideration, the velocity vectors of the climber and subsatellite in the orbital frame, V 1 and V 2 are expressed, respectively, as follows: V 1 ¼ ðL1 Osiny1 L_ 1 cosy1 þ L1 siny1 y_ 1 Þa1 þ fðRL1 cosy1 ÞOL_ 1 siny1 L1 cosy1 y_ 1 ga2

ð9Þ

V 2 ¼ fðL1 siny1 þ ðLL1 Þsiny2 ÞOL_ 1 cosy1 þ L_ 1 cosy2 þ L siny y_ þ ðLL Þsiny y_ ga 1

1 1

1

2 2

1

þ fðRL1 cosy1 ðLL1 Þcosy2 ÞOL_ 1 siny1 þ L_ 1 siny2 L1 cosy1 y_ 1 ðLL1 Þcosy2 y_ 2 ga2 ð10Þ The acceleration vectors of the climber and the subsatellite, a1 and a2 , are given by

a1 ¼ fL€ 1 cosy1 þ L1 y€ 1 siny1 þ 2L_ 1 ðO þ y_ 1 Þsiny1

1

2

þ L1 ðO þ y_ 1 Þ2 siny1 ga2

a2 ¼ fL€ 1 cosy1 þ L€ 1 cosy2 þ L1 y€ 1 siny1 þ ðLL1 Þy€ 2 siny2 þ 2L_ 1 ðO þ y_ 1 Þsiny1 2L_ 1 Osiny2 þ 2L1 y_ 1 Ocosy1

2

1 1

1

1

1

1

2

2

1

1

1

ð12Þ The partial velocities of the climber and the subsatellite with respect to the generalized velocities are @V 1 ¼ L1 siny1 a1 L1 cosy1 a2 @u1

ð13Þ

@V 1 ¼0 @u2

ð14Þ

@V 1 ¼ cosy1 a1 siny1 a2 @u3

ð15Þ

@V 2 ¼ L1 siny1 a1 L1 cosy1 a2 @u1

ð16Þ

@V 2 ¼ ðLL1 Þsiny2 a1 ðLL1 Þcosy2 a2 @u2

ð17Þ

@V 2 ¼ ðcosy1 þcosy2 Þa1 þ ðsiny1 þ siny2 Þa2 @u3

ð18Þ

The gravitational forces affecting the climber and the subsatellite are given by F g1 ¼ mm1

R1 jR1 j3

ð19Þ

F g2 ¼ mm2

R2 jR2 j3

ð20Þ

The active force vector, for the climber to ascend or descend along tether-1, F c , is expressed as F c ¼ Tðcosy1 a1 siny1 a2 Þ

ð21Þ

where T is the magnitude of the active force on the climber, which allows the climber to hang on the tether against the force of gravity, and is thus almost always positive. If the climber is required to descend at acceleration higher than that induced by gravity, this active force must become negative. However, such a situation is not considered in this paper. Employing Kane’s equation, Ki þ Ki ¼ 0,

ði ¼ 1,2,3Þ

Ki ¼ ðF g1 þ F c Þ 

Ki ¼ m1 a1 

þ 2L1 y_ 1 Ocosy1 ðRL1 cosy1 ÞO2 þL1 y_ 1 cosy1 ga1 þ fL€ 1 siny1 L1 y€ 1 cosy1 2L_ 1 Ocosy1 2L_ 1 y_ 1 cosy1

1

2L_ 1 y_ 1 cosy1 þ 2ðLL1 ÞOy_ 2 siny2 þ fL1 siny1 2 2 þ ðLL1 Þsiny2 gO2 þ ðLL1 Þy_ 2 siny2 þ L1 y_ 1 siny1 ga2

ð5Þ The generalized velocities are selected as

1

fL€ 1 siny1 þ L€ 1 siny2 L1 y€ 1 cosy1 ðLL1 Þy€ 2 cosy2 2L_ Ocosy þ 2L_ ðO þ y_ Þcosy þ 2L Oy_ siny

@V 1 @V 2 þ F g2  @ui @ui

@V 1 @V 2 þ m2 a2  @ui @ui

ð22Þ ð23Þ

ð24Þ

the equations of motion can be obtained as ð11Þ

M q€ þ f ¼ ½0,0,TT

ð25Þ

where M is the mass matrix, f is the nonlinear term vector combining the Coriolis and centrifugal forces, and q ¼ ½y1 , y2 ,L1 T is the state vector. The elements of M and f

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99

are as follows: M11 ¼ ðm1 þm2 ÞL21

ð26Þ

M12 ¼ M21 ¼ m2 L1 ðLL1 Þcosy12

ð27Þ

M13 ¼ M31 ¼ m2 L1 siny12

ð28Þ

M22 ¼ m2 ðLL1 Þ2

ð29Þ

M23 ¼ M32 ¼ m2 ðLL1 Þsiny12

ð30Þ

M33 ¼ m1 þ 2m2 ð1cosy12 Þ

ð31Þ

f1 ¼ mm1 RL1 siny1 =R31 þ mm2 RL1 siny1 =R32 ðm1 þ m2 ÞRL1 O2 siny1

Fig. 2. Overview of the experimental apparatus.

þm2 ðLL1 ÞL1 ðO þ y_ 2 Þ2 siny12 þ2ðm1 þ m2 ÞL1 L_ 1 ðO þ y_ 1 Þ 2m2 L1 L_ 1 ðO þ y_ 2 Þcosy12 mm2 ðLL1 ÞL1 siny12 =R32 f2 ¼ mm2 RðLL1 Þsiny2 =R32 þ mm2 ðLL1 ÞL1 siny12 =R32 m ðLL ÞRO2 siny m ðLL ÞL ðO þ y_ Þ2 siny 2

1

2

2

1

1

1

ð32Þ

12

2m2 ðLL1 ÞL_ 1 O þ2m2 ðLL1 ÞL_ 1 ðO þ y_ 1 Þcosy12 2m2 ðLL1 ÞL_ 1 y_ 2

ð33Þ

f3 ¼ þ mm1 ðL1 Rcosy1 Þ=R31 mm2 L=R32 mm2 Rcosy1 =R32 þ mm2 Rcosy2 =R32 þ mm2 Lcosy12 =R32 þ 2m2 L_ 1 y_ 12 siny12 þ 2mm2 L1 ð1cosy12 Þ=R32 þ m2 L1 ðO þ y_ 1 Þ2 cosy12 þ m2 ðLL1 ÞðO þ y_ 2 Þ2 ð1cosy12 Þ ðm1 þ m2 ÞL1 ðO þ y_ 1 Þ2 þ ðm1 þ m2 ÞRO2 cosy1 m2 RO2 cosy2

ð34Þ

where

y12 : ¼ y1 y2

Fig. 3. Schematic representation of the experimental setup.

ð35Þ

Note that although the active force T can be positive or negative, because the climber is capable of ascending and descending by grasping the tether, we here consider only a positive active force for consistency with the experimental conditions. 2.3. Experimental setup To validate the proposed optimal climber transit scheme which will be explained in the next section, experimental studies are carried out using the apparatus shown in Fig. 2. Fig. 3 shows a schematic representation of the experimental setup used to study the tether behavior. Because it is impossible to emulate both in-plane and out-of-plane motions on the ground, the experimental setup is intended to emulate in-plane motion only. The experimental setup consists of (1) a rotating plate driven by a motor connected to the main shaft to emulate the orbital motion, (2) an inclined turntable to emulate the Earth’s gravity, (3) a reel mechanism to control the tether length, (4) a parallel slider equipped with a motor to emulate the change of the orbital radius of the mother satellite, (5) a CCD camera to monitor the position of the climber and subsatellite, and (6) a laptop computer that calculates the tether angles from the position of the climber and subsatellite. In order to avoid twisting of the electric wires

Fig. 4. Inclined turntable of the experimental setup.

connected to the motors, a slip ring located above the main shaft is used. Fig. 4 shows the inclined turntable component of the apparatus. This setup can emulate the fundamental dynamical behavior of a TSS with a climber in a ground-based environment. Contrary to previous studies [22,23] in which the emulated orbital radius was constant, the inclination angle of the turntable can be changed using a jackscrew in accordance with the orbital radius in order to emulate the gravity gradient affecting the subsatellite, as shown in Fig. 4. The rotational axis of the inclined table is coincident with the center of the axis to emulate orbital rotational motion.

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The gravity along the inclined turntable is constant, whereas for an actual orbital system it depends on the orbital radius. In addition, the rotational axis is not perpendicular to the turntable. Therefore, there exist differences between the equations of motion for the experimental setup and those of the orbital system, which are related to the gravity and the angular velocity vector. The active forces affecting the climber and subsatellite in the experimental setup are F 0g1 ¼ m1 gsinfa1

ð36Þ

F 0g2 ¼ m2 gsinfa2

ð37Þ

where g is the gravitational acceleration constant on the ground (g ¼ 9.8 m/s2) and f is the inclination angle of the table. The angular velocity vector in the coordinate system of the experimental setup is expressed as

x ¼ osinfa1 þ ocosfa3

ð38Þ

ði ¼ 1,2,3Þ

Ki0 ¼ ðF 0g1 þ F c Þ  Ki ¼ m1 a1 

ð46Þ

ð40Þ

O is emulated by the orbital rate in the experimental

ð41Þ

f10 ¼ ðm1 þ m2 ÞgL1 sinfsiny1 ðm1 þ m2 ÞRL1 o2 cos2 fsiny1 þ m ðLL ÞL ðocosf þ y_ Þ2 siny 1

2

12

þ 2ðm1 þm2 ÞL1 L_ 1 ðocosf þ y_ 1 Þ 2m L L_ ðocosf þ y_ Þcosy 2 1 1

2

12

ðm1 þ m2 ÞL21 o2 sin2 fcosy1 siny1 m2 ðLL1 ÞL1 o2 sin2 fcosy1 siny2

ð42Þ

f20 ¼ m2 ðLL1 ÞRo2 cos2 fsiny2 þ m2 ðLL1 Þgsinfsiny2 m ðLL ÞL ðocosf þ y_ Þ2 siny 2

1

1

1

12

2m2 ðLL1 ÞL_ 1 ðocosf þ y_ 1 Þ þ 2m2 ðLL1 ÞL_ 1 ocosfcosy12 2m2 ðLL1 ÞL_ 1 y_ 2 m2 ðLL1 ÞL1 o2 sin2 fsiny1 cosy2 m2 ðLL1 Þ2 o2 sin2 fsiny2 cosy2

ð43Þ

f30 ¼ ðm1 þ m2 Þgsinfcosy1 þ m2 gsinfcosy2 m1 L1 ðocosf þ y_ 1 Þ2 m2 L1 ðocosf þ y_ 1 Þ2 ð1cosy12 Þ þ ðm1 þm2 ÞRo2 cos2 fcosy1 m2 Ro2 cos2 fcosy2 þ m ðLL Þðocosf þ y_ Þ2 ð1cosy Þ þ 2m L_ y_ siny 2

1

2

2

2

The gravity-gradient torque near the mother ship in the experimental setup, Tg2, is given by

Assuming that the tether angle is small, the orbital rate

@V 1 @V 2 þ F 0g2  @ui @ui

@V 1 @V 2 þ m2 a2  @ui @ui

1

ð45Þ

ð39Þ

the mass matrix in the equations of motion for the experimental setup is obtained, which is the same as that for the orbital system, while the nonlinear terms in the equations of motion for the experimental setup, f10 , f20 , and f30 , which are different from those of the orbital system, are obtained as follows:

2

Tg1 ¼ 3mO2 L2 sinycosy

Tg2 ¼ mgLsinfsiny þ mRLo2 cos2 fsiny þmL2 o2 sin2 fcosysiny

Employing Kane’s equation, Ki0 þKi ¼ 0,

orbital equations of motion, additional centrifugal terms related to the angular velocity term osinf appear in the equations of motion for the experimental setup. The side effects of these additional terms are considered and compensated for in the emulated gravity gradient in the experimental setup. To reduce the difference between the nonlinear terms for the experimental setup and the orbital system, and to emulate the gravity gradient affecting the climber and subsatellite as precisely as possible, the inclination angle of the turntable should be set appropriately. For this purpose, we now consider the gravity-gradient torque on the orbit and the gravity on the turntable. The gravity-gradient torque on the orbit, Tg1, is approximately expressed as

12

2

2 1 12

2

2

12

2

ðm1 þ m2 ÞL1 o sin fsin y1 þm2 ðLL1 Þo sin fsin y2 m2 ðLL1 Þo2 sin2 fsiny1 siny2 þm2 L1 o2 sin2 fsiny1 siny2

ð44Þ where ocosf is the emulated orbital rate perpendicular to the turntable, and is different from the rotation velocity o because of the inclination angle f. Compared to the

setup o by changing the time scale, and that the above two torques are almost equal, the inclination angle of the turntable f is then determined to emulate the gravity gradient as follows [24]: ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) g þ g 2 þ 4o4 ðRLÞðR þ3LÞ ð47Þ f ¼ sin1 2ðRLÞo2 It should be noted that although the gravity gradient affecting the climber changes in space in accordance with its orbital radius, for simplicity, the inclination angle of the turntable is set to be constant in this study. Despite the above mentioned, differences, the basic behavior of tether libration induced by a climber transit can be successfully emulated by appropriately setting the inclination angle of the turntable. This is because the third term in Eq. (46), which is the centrifugal torque due to the angular velocity osinf, is taken into consideration as a part of the gravity gradient emulated in the experimental setup, and then the inclination angle of the turntable is chosen appropriately. It should also be noted that the emulation of librational motion is limited to cases where the Coriolis force is the dominant force not only inducing but also suppressing tether librations. This is because as the climber speed decreases, other forces such as air drag and friction increase. Taking these undesired forces into consideration, the emulated orbital rate or period should be chosen appropriately to prevent the undesired forces from becoming larger than the Coriolis forces. Fig. 5 shows a schematic representation of the climber actuation. The reel mechanism controls the distance between the climber and the mother ship L1 using an extendable string connected to the climber. In order to keep the total tether length equal to L, the subsatellite is suspended from the reel that is representative of the mother ship by another tether of constant length which passes through the climber.

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almost always desirable in order to achieve smoother climber travel. We consider the problem of finding the optimal climber acceleration along tether-1, L€ 1 , to minimize the cost function Z tf 2 2 2 2 2 J ¼ w1 y1 ðtf Þ þw2 y2 ðtf Þ þw3 y_ 1 ðtf Þ þw4 y_ 2 ðtf Þ þ L€ 1 ðtÞ dt 0

ð48Þ subject to the equations of motion described in the previous section, and the initial and final conditions for the case of descending fy1 ð0Þ, y_ 1 ð0Þ, y2 ð0Þ, y_ 2 ð0Þ,L1 ð0Þ, L_ 1 ð0Þg ¼ f0,0,0,0, e,0g

ð49Þ

fy1 ðtf Þ, y_ 1 ðtf Þ, y2 ðtf Þ, y_ 2 ðtf Þ,L1 ðtf Þ, L_ 1 ðtf Þg ¼ f0,0,,,Le,0g ð50Þ and those for the case of ascending fy1 ð0Þ, y_ 1 ð0Þ, y2 ð0Þ, y_ 2 ð0Þ,L1 ð0Þ, L_ 1 ð0Þg ¼ f0,0,0,0,Le,0g

ð51Þ

fy1 ðtf Þ, y_ 1 ðtf Þ, y2 ðtf Þ, y_ 2 ðtf Þ,L1 ðtf Þ, L_ 1 ðtf Þg ¼ f,,0,0, e,0g

ð52Þ

Fig. 5. Schematic image of the climber actuation method.

Fig. 6. Photographs of the climber and subsatellite. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

The orbital velocity and climber transit velocity are controlled by stepper motors. A digital CCD camera is installed above the table to record data for the tether angles, by detecting colored marks on the subsatellite and climber. Fig. 6 shows two photographs of the subsatellite and climber taken by the camera. As shown in Fig. 6, the tether length between the mother ship (represented by the reel mechanics) and the climber is changed, whereas the total tether length, that is, L1 þ L2, is kept constant. To easily obtain photographs of the subsatellite and climber, the walls of the apparatus are covered by dark curtains, and the subsatellite and climber are equipped with red markers. Pieces of dry ice are set under the climber and the subsatellite to reduce friction between the satellites and the surface of the inclined turntable. 3. Optimal climber transit control The hierarchy gradient method [25] was used in this study to obtain optimal solutions. According to the findings of Williams [18], minimizing the time integral of the square of the climber’s acceleration along the tether is

where wi, (i¼1,2,3,4) is a weighting coefficient, e is a constant length to avoid retrieval instability of the tether [26], and n indicates no constraint. The value of e will be addressed in the next section. It should be noted that contrary to Ref. [18] where it was assumed that the tether was a rigid body, and the final in-plane tether angles were set to zero, in this study no constraints are placed on the tether angle y1 for the ascent case, or on the final value of y2 for the descent case. This is because it is impossible to simultaneously regulate both final tether angles y1 and y2 using one actuator such as the climber in this system. Instead of setting the final constraints on both the tether angles simultaneously, the final tether librations are included in the performance index as a penalty function. The terminal times are chosen as tf ¼ 0:2n½orbits,

ðn ¼ 3,4,5Þ

ð53Þ

The reason why values of less than 0.6n orbits are not chosen for the terminal times is that the active control force T is assumed to be positive, and a negative force may be required if the climber acceleration during descent is much higher than the acceleration due to gravity and the centrifugal force. The experimental setup cannot implement such a rapid descent. Although terminal time of about four orbits was considered in [13], one orbit was selected as the longest terminal time in this study. This is because as mentioned later, as the terminal time was longer, Coriolis force would become smaller than other disturbance forces such as air-drag, and friction between the subsatellite model and the surface of the table, as the result, it would become more difficult to observe the librational motion of the tether in the setup. In this study, the climber acceleration along tether-1, L€ 1 , is used as the virtual control input instead of T, and the actual value of T is calculated from the obtained optimal state profiles using the equations of motion (Eq. (25)). As expressed in the initial conditions, the tethered subsatellite is suspended from the mother ship along the local vertical direction with no initial librations. The climber then begins to descend along tether-1 from the

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mother ship to the subsatellite, or begins to ascend from the subsatellite to the mother ship. The terminal constraints expressed in Eqs. (50) and (52) are set by taking into account retrieval instability, that is so called ‘‘spaghetti problem,’’ involving unpredictable high-frequency lateral oscillations that arise when the climber approaches the end of the tether. In this study, the relationship between the overall transit time during climber travel and the tether librations is investigated by increasing the transit time at intervals of 0.2 orbits, as shown in Eq. (53), but the overall climber transit time is not optimized. 4. Results of numerical simulations and experiments 4.1. Numerical and experimental conditions Constant and optimal acceleration climber transits with weighting subject to the orbital equations of motion (Eq. (25)) are simulated for different transit times. It is assumed that the tether length L is 100 km, the orbital radius of the mother ship R is 6600 km, and the weights of the climber and subsatellite, m1 and m2, are both 1000 kg. In this case, the orbital period is 5330 s ( ¼ 88.8 min). The parameter indicating the initial and final climber positions is set as e ¼ 10 km, that is, 10% of the tether length. In the following figures, the distance between the climber and the mother ship is normalized by the total tether length L. The experimental parameters are set as follows. The weights of the climber and the subsatellite are 90 g, and the tether length L is 40 cm, i.e., the actual transit distance is set to 32 cm. The distance R between the end of the reel system and the main shaft along the inclined turntable is 70 cm. The angular rotation rate of the main shaft is o ¼ p=3 rad/s ( ¼601/s), i.e., the orbital period is 6.0 s. To emulate the gravity gradient affecting the climber and subsatellite, by substituting the above orbital radius and orbital rate of the mother ship into Eq. (47), the inclination angle of the table is determined to be 121. It should be noted that even if other orbits such as a geostationary orbit are selected, the inclination angle of the table is always the same for the case of circular orbits, because the gravity gradient affecting the tethered satellite system is proportional to the orbital rate, and the orbital rate is emulated by changing the time scale factor for the setup in this study. For consistency with the numerical simulations, the overall climber transit time in the experiments is considered for three cases from 3.6 to 6.0 s, at intervals of 1.2 s. By solving a two-point boundary value problem (TPBVP), Eq. (48), subject to the orbital equations of motion Eqs. (25)–(35), and the initial and terminal constrains Eqs. (49) and (50), or Eqs. (51) and (52), the optimal solutions are obtained, and are then employed for the experiments. To compensate for the retrieval instability, the optimal acceleration control scheme, which includes weighting coefficients on the terminal librations, as shown in Eq. (48), is implemented. The weighting coefficients are set to be w1 ¼ w2 ¼ 2:5  101 ,w3 ¼ w4 ¼ 2:5  106 . These values

are chosen after several iterations in which the existence of optimal solutions is confirmed. To validate the effectiveness of the weighting coefficients for suppressing the retrieval instability, constant speed control methods are also implemented, and the performance is compared. In the experiment, it took about 10 emulated orbits until the initial libration was damped out when the turntable was not rotated. This implies that the damping ratio due to the frictional force between the dry ice and the surface of the inclined turntable is 1.97  10  2. Therefore, it was confirmed in the experiments that the frictional force between the dry ice and the surface of the inclined turntable was sufficiently low to allow emulation of the librational motion of the TSS in space for a few orbits. A zero-phase filter was applied to remove noise from the CCD camera images used to determine the positions of the climber and subsatellite. The experimental setup contains a radio-controlled servo mechanism to clamp the subsatellite and prevent it from librating until the start of climber motion. Decent and ascent of the climber began after the rotational speed of the turntable had reached the specified speed to emulate the orbital motion, and the subsatellite had been released from the servo mechanism. Because of uncertainties such as noise in the color sensor and misalignment between the subsatellite and the servo mechanism, the initial tether angles were not always precisely zero. However, the errors involved are small and are not expected to have any major impact on the experimental results.

4.2. Ascent case Fig. 7 shows the numerical and experimental results for constant speed ascent. The librational angle y1 is seen to have a peak at a time of about 0.3 orbits both in the numerical and experimental results. This can be explained as follows. The period of the librational motion due to the pffiffiffi gravity gradient is about 2p= 3O  0:58 orbits, and the Coriolis force induced by the constant speed retrieval (ascent) is almost constant. As the result, the angle y1 librates with an offset in which the gravity-gradient torque induced by the offset angle balances with the torque induced by the Coriolis force, and the time taken for y1 to change from zero to its maximum value is half the period of the libration, that is, 0:58=2  0:29 orbits. The tether angle y2 has a flat response in the time range between 0.2 and 0.4 orbits. This is because the Coriolis force under constant speed retrieval (ascent) causes an increase in y2 , which is, however, balanced by the influence of the gravity gradient. The experimental results show that at the terminal time, although the tether angle y2 was successfully suppressed, y1 was not. The behavior of the tether angle y1 can be regarded as the retrieval instability. Because the tethered system treated in this study is not controllable, it is impossible to regulate both tether angles simultaneously by controlling only the climber motion. Therefore, additional actuators such as thrusters may need to be installed on the subsatellite to suppress this retrieval instability.

1

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Climber’s distance from the mother (L1/L)

H. Kojima et al. / Acta Astronautica 69 (2011) 96–108

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Fig. 7. Numerical and experimental results for constant speed ascent: (a) climber’s distance from the mother ship, (b) tether angle y1 , and (c) tether angle y2 .

Fig. 8 shows the numerical and experimental results for optimal acceleration ascent. Compared to the case for constant speed ascent, the peak in the tether angle y1 appears later, and residual librations of y1 are successfully suppressed. This is because the climber speed during the first

half of transit is lower than that during constant speed ascent and higher during the second half; in this situation the libration induced by the climber transit is canceled by the Coriolis force as much as possible at the terminal time. In addition, in contrast to the results for constant speed ascent,

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Experiment tf =0.6 orbit tf =0.8 orbit tf =1.0 orbit

0.8 0.6 0.4 0.2 0 0

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Climber’s distance from the mother (L1/L)

Climber’s distance from the mother (L1/L)

Simulation 1

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Fig. 8. Numerical and experimental results for optimal acceleration ascent with weighting: (a) climber’s distance from the mother ship, (b) tether angle y1 , and (c) tether angle y2 .

the tether angle y2 does not have a flat response in the range between 0.2 and 0.4 orbits, but has a peak at around half of the terminal time. This response of the tether angle y2 can be also explained from the speed profile of the climber transit.

From the above discussion, it can be concluded that the numerical and experimental results for tether angles y1 and y2 are in reasonable agreement for the case of ascending.

H. Kojima et al. / Acta Astronautica 69 (2011) 96–108

Climber’s distance from the mother (L1/L)

Fig. 9 shows the numerical and experimental results for constant speed descent. As shown in Fig. 9, the numerically

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Climber’s distance from the mother (L1/L)

calculated tether angle y1 is always negative because of the Coriolis force induced by the constant descent. It is numerically and experimentally observed that tether angle y1 has a peak around 0.1 orbits for all cases, and librates with a period

4.3. Descent case

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Fig. 9. Numerical and experimental results for constant speed descent: (a) climber’s distance from the mother ship, (b) tether angle y1 , and (c) tether angle y2 .

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constant speed descent is about half pffiffiffithat for constant speed ascent, that is, one fourth of 2p= 3o. This is because the climber’s librational motion is affected not only by the gravity gradient, but also by the tension on the tether connecting the

Climber’s distance from the mother (L1/L)

of about 0.28 orbits. If the climber was not connected to the subsatellite with a tether, the libration period would be the same as in the constant speed ascent case, that is, about 0.3 orbits. However, the libration period of 0.28 orbits for

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Fig. 10. Numerical and experimental results for optimal acceleration descent with weighting: (a) climber’s distance from the mother ship, (b) tether angle y1 , and (c) tether angle y2 .

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107

Table 1 Experimental data for tether librational angles at the terminal time. Terminal time

Constant speed control

Optimal acceleration control

Ascent tf ¼ 0:6 orbits tf ¼ 0:8 orbits tf ¼ 1:0 orbits

y1 ¼ 0:17 rad, y2 ¼ 0:03 rad y1 ¼ 0:04 rad, y2 ¼ 0:03 rad y1 ¼ 0:25 rad, y2 ¼ 0:02 rad

y1 ¼ 0:10 rad, y2 ¼ 0:01 rad y1 ¼ 0:09 rad, y2 ¼ 0:01 rad y1 ¼ 0:06 rad, y2 ¼ 0:02 rad

Descent tf ¼ 0:6 orbits tf ¼ 0:8 orbits tf ¼ 1:0 orbits

y1 ¼ 0:07 rad, y2 ¼ 0:06 rad y1 ¼ 0:014 rad, y2 ¼ 0:09 rad y1 ¼ 0:03 rad, y2 ¼ 0:05 rad

y1 ¼ 0:02 rad, y2 ¼ 0:08 rad y1 ¼ 0:009 rad, y2 ¼ 0:02 rad y1 ¼ 0:04 rad, y2 ¼ 0:03 rad

climber and the subsatellite. This combination may lead to a doubling of the libration frequency. The numerical results indicate that the final value of the tether angle y1 is successfully regulated, and this is also confirmed by the experimental results. On the other hand, no suppression of the tether angle y2 was observed in either the numerical or experimental results for the case of constant speed. This is because regulation of the terminal tether angle y2 was not considered in this case. It can also be seen from the figures that y1 becomes smaller as the climber approaches the subsatellite, while y2 becomes larger. As mentioned earlier, this behavior can be interpreted as the retrieval instability if deployment of the climber from the mother ship is regarded as retrieval to the subsatellite. Fig. 10 shows the numerical and experimental results for optimal acceleration descent. The peak in the tether angle y1 appears later than the case for constant speed control. In addition, the size of the y1 peak is larger than that for the case of constant speed descent with tf ¼0.6 orbits, whereas it is similar to, or less than that for constant speed descent with tf ¼ 0.8 and 1.0 orbits. This is because, compared to the case for constant speed control, the climber speed was lower at the beginning of the transit, and higher around the middle of the transit. Comparing the experimental results for constant speed control with those for the optimal acceleration control, it can be seen that y1 is successfully suppressed at all cases of the terminal time, and y2 is successfully suppressed at the terminal times of tf ¼0.8 and 1.0 orbits in the latter case. This is because the terminal tether angles are included in the performance index as penalty functions. The results also show that the librational motion of a tether with a single climber can be successfully emulated with the experimental setup used in this study. From the results of descending cases mentioned above, we can conclude that the proposed optimal acceleration control method is effective in suppressing residual librations of the tether even if control of both tether angles is impossible. 4.4. Discussion The experimental results showed that optimal acceleration control is effective at suppressing retrieval instability of the tether induced by the climber transfer. It is also shown that the climber speed for the case of optimal descent control slightly speeds up in the first half

of the transit. This is because a lower climber speed was selected in order to suppress the tether residual librations in the latter half of the climber transfer, so that the speed in the first half must, in turn, be increased to maintain the distance traveled in the latter half. The final tether angles for the constant speed and optimal acceleration control schemes are summarized in Table 1. The tether angle y1 for the case of ascent was reduced by the use of the optimal acceleration control, due to the penalty functions used. On the other hand, there were no significant differences with regard to y2 between the experimental results for constant speed ascent and optimal acceleration ascent control. This may be because the work energy used for jerking up the climber was converted to potential energy, and as a result did not contribute so much to inducing librations in tether-2. During the experiments, rigid rotational motion of the satellites occurred and was observed, which was not considered in the simplified mathematical model. To reduce this unexpected motion in the experiments, the subsatellite and climber should be made much smaller. In addition, the high-frequency vibrations observed in the experiments corresponded to longitudinal and traverse vibrations induced on the tether. The polyester sewing machine thread used for the tether is possibly too stretchy for use as a tether material. The flexibility and elasticity of the tether, which were not taken into account in the simplified mathematical model, should be considered to predict the behavior of the tether more realistically in the near future. Nevertheless, Figs. 7–10 show that the climber transit distance agrees well with the results of the numerical simulations, and that the tether librational motion is quite similar to that predicted by the simulation for the case of a short terminal time. On the other hand, as the terminal time becomes longer, the differences between the numerical and experimental results become larger. These differences most likely resulted from external disturbances such as air-drag and friction between the dry ice and the surface of the inclination table, in addition to modeling errors. Therefore, the experimental setup described in this paper is capable of emulating the dynamic behavior of a TSS with a climber over the range of terminal times treated in this paper. Furthermore, the numerical results indicated that the climber’s descent speed increased when the difference

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between the tether angles y1 and y2 was smaller in the optimal acceleration schemes during the second half of the climber transient. This may be because if the tether is rigid, then the librational motion can be controlled by the climber motion, and this would not give rise to a tether retrieval instability. In addition, the initial tether angles are not always zero in practice. Therefore, as future research, we plan to implement feedback control methods for the climber transfer in order to suppress both tether angles y1 and y2 at the terminal time, starting from non-zero tether angles, and compensate for external disturbances and modeling errors. 5. Conclusion A simple dynamic model was used to describe the fundamental dynamical behavior of a TSS with a single climber, and to develop an effective climber transit control method. The ascent or descent of the climber in the local vertical along the tether induces a librational motion of the tether, which is undesirable for practical operation of a TSS. The equations of motion for this model were derived from Kane’s equation and the optimal climber transit acceleration profile was obtained by minimizing the square of the transit acceleration of the climber using penalty functions of the terminal tether angles and angular velocities of the tether. Simulations were then carried out to investigate the relation between the transit time and the librational motion of the tether. The experimental results indicated the competitive performance of the optimal acceleration control method over the constant speed control scheme. For advanced applications, climbers will be used to control the librations and chaotic motion of tethered satellite systems in elliptic orbits, or the librations of unanchored space elevators at the initial ribbon deployment phase for construction of a space elevator. To develop more effective means of climber transit, more precise dynamic models taking into account the mass and flexibility of the tethers, and the moments of inertia of the climbers and subsatellites are required. Nevertheless, the mathematical model considered in this paper is still helpful for the preliminary design of climber transfer control. References [1] E.K. Tsiolkovsky, Speculations about earth and sky, and on vesta, Akademiia Nauk SSSR, Moscow (1895) 35. [2] M. Dobrowolny, H.N. Stone, A technical overview of TSS-1: the first tethered-satellite system mission, II Nuovo Cimento C 17 (1994) 1–12, doi:10.1007/BF02506678. [3] H.A. Fujii, S. Anazawa, Deployment/retrieval control of tethered subsatellite through an optimal path, Journal of Guidance, Control, and Dynamics 17 (1994) 1292–1298, doi:10.2514/3.21347.

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