Dynamics of Motorized Momentum Exchange Tether for Payloads Capture

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Advances in Space Research 59 (2017) 2374–2388 www.elsevier.com/locate/asr

Dynamics of motorized momentum exchange tether for payloads capture

Qilong Sun a,⇑, Yanfang Liu a, Naiming Qi a, Yong Yang a, Zhenpeng Chen b

a Harbin Institute of Technology, 150001 Harbin, Heilongjiang, People’s Republic of China b Air Force Aviation University, 130022 Changchun, Jilin, People’s Republic of China

Received 3 July 2016; received in revised form 16 January 2017; accepted 7 February 2017 Available online 16 February 2017

Abstract

The dynamics of motorized momentum exchange tether (MMET) during and after payloads capture is studied. The ideal velocities of payloads with same mass for capture are analyzed, and a mathematical model of capture is proposed. If the MMET captures the payloads at the ideal velocity, the MMET’s orbit and motion state are unchanged. However, considering that capture velocity error exists in practice, and analyzing the velocity error in relation to the payload approaching from the earth as an example, the impact on the MMET’s orbital parameters and state motion are investigated during payload capture for diﬀerent capture velocity errors, including errors in magnitude and direction. The simulation results show that if the payload velocity has only magnitude error, the perigees remain overlapping and the semi-major axes of the changed orbits also continue to coincide after capture; however, the changes in other orbital parameters are approximately proportional to the error. If the payload velocity has both magnitude and direction errors, then all of the orbital parameters change, including the perigee and the direction of the semi-major axis. However, the normal velocity increment caused by the capture’s normal velocity error changes the orbital parameters far less than the tangential velocity increment. Ó 2017 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: MMET; Capture mathematical model; Ideal velocity for capture; Capture velocity error; Dynamics

1. Introduction cost is considerable. In order to realize the continuous transmission with less cost, we need to establish an effective With the development of space technology, the estab- alternative transmission system. lishment of a lunar colony becomes ever closer to reality, Dynamics and control methods for tethered satellites and such an endeavor will require a continuous cislunar have been investigated for decades (Misra and Modi, transmission system to accomplish material exchanges, 1982; Lakshmanan et al., 1987; Beletsky and Levin, 1985, including transmitting construction materials, food, water, 1993). The idea of using a motorized sling on the surface and other necessities to the Moon and receiving material of a body with less atmosphere such as the Moon to accel- from lunar exploration (Murray and Cartmell, 2013). erate payloads was presented (Carroll, 1986). Transporting Employing rockets for uninterrupted transportation is payloads from LEO to the lunar surface using rotating one approach to this challenge, however, the associated asymmetric tethers was shown to be theoretically possible using no propellant (Forward, 1991). Kumar et al. (1992) ⇑ Corresponding author at: School of Astronautics, Harbin Institute of investigated out-of-plane librations on tethered payloads. Technology, 150006, People’s Republic of China. In this paper, a two-body dumbbell system moves on an E-mail addresses: [email protected] (Q. Sun), liuyanfang_hit@ elliptical orbit executing three-dimensional librational 163.com (Y. Liu), [email protected] (N. Qi), yangyong_hit@ motion during deployment. Oldson and Carroll (1995) 126.com (Y. Yang), [email protected] (Z. Chen). http://dx.doi.org/10.1016/j.asr.2017.02.013 0273-1177/Ó 2017 COSPAR. Published by Elsevier Ltd. All rights reserved. Q. Sun et al. / Advances in Space Research 59 (2017) 2374–2388 2375 presented the conceptual design of an Earth tether transport 2011). The theory behind the electrodynamic tether was facility capable of providing a 1.2 km/s Dv, which appears presented by Rasmussen and Banks (1988). A unique data to be technically feasible and could potentially significantly set was obtained from the TSS-1R mission, and the results reduce launch costs. The use of a tether as a sling for lunar showed that motion relative to the plasma must be missions was presented by Puig-Suari et al. (1995), who also accounted for in orbiting systems (Stone et al., 1999). proposed that the spacecraft attached to the end of the The robustness of electrodynamic tethers against orbital tether could achieve the required injection velocity using a debris impacts was researched (Kim et al., 2010), and the solar-powered electric motor. A rotating asymmetric tethers period of the Moon-tracking orbit was determined for system was designed to exchange payloads between low the payload transfer (Murray and Cartmell, 2013). It was Earth orbit (LEO) and the lunar surface (Hoyt, 1997). Fur- shown that if the MMET tracked an ascending (or thermore, an optimum architecture for a tether system descending) node of the Moon’s orbit around Earth, which designed to transfer payloads between a low Earth orbit could be accomplished by arranging the tether to orbit (LEO) and the lunar surface was presented (Hoyt and Earth with a critical inclination, the exchange payloads Uphoff, 2000; Hoyt, 2000a, 2000b). The system architecture could be received periodically with the Moon arriving at thus established considered the complexities of orbital either of these nodes. A new flexural model for the three- mechanics. It used an 80-km-long tether in an elliptical, dimensional dynamics of an MMET was investigated equatorial Earth orbit that exchanged payloads with a sec- (Ismail and Cartmell, 2016). The results showed that the ond 200-km-long tether in a low lunar orbit. A system with flexible model demonstrates a generally lower magnitude two rotating tethers transporting payloads between Earth of response compared with that of the rigid-body model. and Mars was also devised (Forward and Nordley, 1999). A control law for capturing payloads at the tip of a long A symmetrical motorized momentum exchange tether tether was investigated (Williams et al., 2005). In contrast, was first investigated by Cartmell (1998). Furthermore, a the dynamics of MMET after capturing payloads has not two-way payload transfer between the Earth and the Moon been studied to date. Therefore, the study presented in this was also proposed based on a symmetrically motorized paper investigates the influence of payloads capture on the momentum exchange tether system (Cartmell and Zeigler, orbit and state of the MMET. Because the tapered tether 1999, 2001; Zeigler and Cartmell, 2001), in which identical was presented to accomplish continuous cislunar transmis- payloads were configured at each end of the tether and the sion (Yang et al., 2015), this study used the tapered tether two payloads were captured or released simultaneously. instead of the uniform tether to reduce the tether mass and The system included two symmetrical tethers, identical to lower the associated energy consumption. payloads and a launch motor that consisted of a rotor In this paper, the changes of the orbit during the pay- and a stator. The rotor was attached to two propulsion load capture, and the initial conditions for the dynamics tethers, and the stator was attached to two outrigger teth- after capture are investigated by using the principle of ers that were configured with two identical masses at the momentum conservation and angular momentum conser- two ends. The motor’s output torque produced the rotating vation. Then the motion is analyzed through the dynamics angular velocity required to accomplish the exchange mis- basing on the solved initial conditions. Meanwhile, the sion. The double-ended tethers could solve the de-orbit MMET’s motion after the payload capture is compared problem for both loaded and unloaded payloads. Three- with that without the payload capture. The derived dimensional dynamics of this problem was derived by the approach in this work is limited to the main assumptions Lagrangian method, and the time evolution of attitude that the motion of the MMET is a planar motion, and parameters was analyzed. the tethers are rigid. The structure of this paper is as fol- A low-cost architecture for transferring the payloads to lows. Section 1 introduces the MMET that is used to the Moon using the MMET was presented, and the rele- accomplish continuous cislunar transmission. In Section 2, vant equations of motion were derived (Cartmell et al., the composition of the MMET is presented, and its coordi- 2004). In order to establish a more accurate model for nate system is described to obtain the mathematical model the MMET, the dynamics of the MMET was derived, con- of the capture; meanwhile, relevant assumptions are set. In sidering the effects of the tether’s axial and torsional oscil- Section 3, the capture mathematical model is proposed. In lation (Chen and Cartmell, 2007). These authors pointed Section 4, the ideal capture velocity is analyzed first, and on out that the effects of axial and torsional oscillation on this foundation, we establish the velocity error model and overall performance were significant and should not be analyze the change in the central facility’s orbital parame- ignored. Hybrid sliding mode control strategies were pro- ters during capture. Later, we analyze the dynamics with posed to mitigate the effects of the tether’s axial and tor- capture velocity error. Section 5 presents the conclusions. sional oscillation (Chen and Cartmell, 2010). The dynamic model had seven degrees of freedom (7-DOF) 2. MMET structure and coordinate systems and used the hybrid controller to meet the requirements of the 7-DOF MMET system. The MMET is shown in Fig. 1. It consists of a central An electrodynamic tether was proposed to accomplish facility (M) and two identical pieces of capture equipment continuous Earth-Moon payload exchange (Murray, (E) configured at the far ends of two symmetrical tapered 2376 Q. Sun et al. / Advances in Space Research 59 (2017) 2374–2388

z EP+ b x 11 North pole +Z b z y m b x m l y O Y m M p Orbital T R plane Z Perigee M p θ X

l p OE + Y EP22 + Earth Equator +X

Earth Vernal equinox

Fig. 1. Schematic of the MMET. Fig. 2. Coordinate systems relevant to the MMET. tethers (T). The payloads (P) are captured by the equipment. In order to capture and launch the payloads, the cen- In the coordinate system of the central facility’s body, tral facility orbits the Earth and rotates. OMxbybzb, the origin OM is in the MMET center of In order to efficiently accomplish cislunar transmission, mass. Axis OMxb coincides with the line that connects the MMET always needs to be in the orbital plane and cap- the two payloads, and is positive in the direction from ture payloads at perigee. Thus, for simplicity, we set rele- the payload in low orbit to the payload in high orbit. vant assumptions as follows: Axis OMyb is perpendicular to axis OMxb and lies in the orbital plane. Axis OMzb follows the right-handed rule of the coordinate system. The central facility of the MMET orbits the Earth on an elliptical orbit and rotates. The motion of the MMET is in the orbital plane, and the motion is planar. The planar motion can be guaran- 3. Mathematical model of capture teed by the thrusters, which can provide compensation torques. Thus, the dynamics is that of a planar model. Fig. 3 shows the process of cislunar transmission. Pay- The payloads have the same mass and the equipment load P1 comes from the cislunar transmission orbit and, captures the payloads simultaneously; thus, the position when it reaches the orbit’s perigee, the capture equipment of the MMET mass center does not vary after capture. captures it. Payload P2 comes from the Earth and is sent The tether aligns with the center of the earth and the directly to the rendezvous point by a space shuttle. The perigee at the moment of capture. rotation of the MMET provides a positive velocity incre- The bending deformation and axial elasticity deforma- ment for the equipment to capture P1 and a negative veloc- tion of the two propulsion tethers are ignored, and the ity increment for the equipment to capture P2. Thus, the tethers are rigid. rotation of MMET must guarantee that the velocity of Third-body gravitational force, aerodynamic drag, solar the capture equipment satisfies the payload capture and radiation pressure, and other parameters with small injection. effects relative to the payloads’ capture impact can be In Fig. 3, ra and rp are the apogee radius and perigee neglected. radius of the elliptical orbit of the central facility, respec-

The coordinate systems relevant to the MMET are Moon Lunar shown in Fig. 2 and are defined as follows: orbit R M In the coordinate system of the orbital plane, OEXpYp- Cislunar Z , the origin O is the center of Earth. Axis O X transmission p E E p orbit points to the perigee of the MMET. Axis OEYp is per- Hohmann pendicular to axis OEXp and lies in the orbital plane. trajectory Axis O Z follows the right-hand rule of the coordinate ra E p Earth Elliptical system. orbit In the coordinate system of the central facility’s move- v ment, O x y z , the origin O is in the MMET center rl+ B B r M m m m M p v p of mass (COM). Axis OMxm indicates the direction of c the MMET’s radius vector. Axis O y is perpendicular v M m A A to axis OMxm and lies in the orbital plane. Axis OMzm follows the right-hand rule of the coordinate system. Fig. 3. Process of cislunar transmission. Q. Sun et al. / Advances in Space Research 59 (2017) 2374–2388 2377 tively; v and v are the velocities of the tether endpoints; mv þ m ðv þ v Þ A B v c0 p p10 p20 ; v cH ¼ ð5Þ c is the speed of central facility; Rm is the radius of the ðm þ 2mpÞ Moon’s orbit; l is the length of the tether. Cislunar transmission depends on the Hohmann trajectory. The velocity where m refers to the total mass of the MMET; mP refers to of the central facility before payload capture at perigee is the mass of the payloads; From Eq. (5), the velocity of the given by MMET’s COM depends on the velocities of the payloads. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi In an exceptional case, the relationship between the pay- 2lra loads’ velocities and the central facility’s velocity satisfies v 0 j ; c0 ¼ c ð1Þ rp ðra þ rp Þ v v v : 0 0 0 p10 þ p20 ¼ 2 c0 ð6Þ

ra rp Substituting Eq. (6) into Eq. (5) we can obtain v v , where 0 and 0 are the apogee radius and perigee radius cH ¼ c0 of the elliptical orbit before payload capture, respectively, l meaning that in this case, the MMET’s capture of the two is the gravitational constant of the earth, and jc is the unit payloads has no influence on the velocity of the MMET’s v COM, with the result that the parking orbit will not be vector of c0 . Because the capture action occurs at perigee, the veloc- changed. ity of the payload P1 coming from the cislunar transmission Without considering the external torque exerted on the v MMET, because of the angular momentum conservation orbit p10 can be determined by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in the process of payloads capture, we can obtain the lR v 2 m j ; expression by p10 ¼ c ð2Þ ðrp þ lÞðRm þ rp þ lÞ 0 0 J x m li v v m li v v J HxH; 0 0 þ P½ b ð p10 c0 Þ P½ b ð p20 c0 Þ ¼

Because the other payload P2 coming from the Earth to ð7Þ the Moon is sent directly to the rendezvous point by a where J0 and JH are the MMET’s moments of inertia space shuttle; therefore this payload’s velocity vp can be 20 before and after payloads capture, respectively. J and JH adjusted as needed. 0 v v were obtained by Yang et al. (2015) as The expressions for A0 and B0 which are the velocities rffiffiffiffiffiffiffiffi of the tether endpoints before capture are given by 1 m m l2 2p1 J m r2 m l2 ð P þ EÞ 0 ¼ c c 2 p þ v ¼ v þ x li and ð3Þ 2 Dvn o A0 c0 0 b rffiffiffiffiffi oDv2 o v ¼ v x li ; ð4Þ n B0 c0 0 b exp erf Dvn and ð8Þ 21 21 where x0 is the rotational angular velocity before capture rffiffiffiffiffiffiffiffi i 1 m m l2 2p1 and b is the unit vector of the tether direction. J m r2 ð P þ EÞ H ¼ c c þ The process by which the MMET captures the payloads 2 Dvn o rffiffiffiffiffi is shown in Fig. 4. M is the central facility with velocity vc 2 0 oDvn o before payload capture and v after payload capture. E exp erf Dvn : ð9Þ cH 1 21 21 and E2 are the capture components. P1 is the payload com- v ing from the Moon with velocity p10 .P2 is the payload where mc and mE are the masses of the central facility and v coming from the Earth with velocity p20 . capture equipment, respectively. It is known that mc +2- Using the linear momentum principle, after capturing mE = m. o and 1 refer to the density and ultimate strength the two approaching payloads, the instantaneous velocity of the tapered tether, respectively. Furthermore, it is pre- of the central facility is obtained by sumed that the configuration of the central facility is a

Before capture After capture E v 1 E p1Θ 1 + v EP11 Rendezvous p10 v l Point c0 M l v v cΘ ω cΘ P Θ M 1 ω M 0 l E E l EP+ 2 2 v 22 v p2Θ p20 P2 Earth Earth

Fig. 4. Process by which the MMET captures payloads. 2378 Q. Sun et al. / Advances in Space Research 59 (2017) 2374–2388 cylinder with radius rc. Eqs. (8) and (9) are obtained by can be adjusted as needed to match the velocity of the tip optimizing the taper angle of the tethers based on the pay- of the lower tapered tether for the gentlest collision because loads’ and capture equipment’s masses as well as the con- P2 is sent directly to the rendezvous point by a space shut- D v v stant tether length l. vn is determined by the velocity of tle. Thus, the ideal A0 and B0 are given by P relative to the central facility, and it is given by 1 2 3 vA ¼ vp and ð13Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 10 6 7 v v : lR lr B0 ¼ p20 ð14Þ 6 2 m 2 a0 7 Dvn ¼ j6 7: 4 rp l R rp l rp ra rp 5 ð 0 þ Þð m þ 0 þ Þ 0 ð 0 þ 0 Þ Thus, |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V lR v 2 m j : ð10Þ A0 ¼ c ð15Þ rp l R rp l ð 0 þ Þð m þ 0 þ Þ It is known that the velocity of the tether tip relative to x the central facility determines the structure of the tapered Substituting Eqs. (1) and (15) into Eq. (3), 0 can be written as tether according to its ultimate strength. At the moment "#sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of payload capture, the velocity of P1 relative to the central lR lr 1 2 m 2 a0 facility is V. During rendezvous, in order to accomplish the x0 ¼ jx; l rp l R rp l rp ra rp ð 0 þ Þð m þ 0 þ Þ 0 ð 0 þ 0 Þ smooth capture, the velocity of E1 needs to equal the velocity of P1. After capture, the maximal velocity of the tether ð16Þ tip relative to the central facility is slightly larger than V where jx is the unit vector of the angular velocity. when the MMET moves along its orbit because of the grav- Introducing Eqs. (16) and (14) into Eq. (4), the ideal vB ity gradient. Thus, we define Dvn = 1.08V, and then use the 0 and v are obtained as optimal method (Yang et al., 2015) to design the tapered p20 tether in order to guarantee that the tapered tether does "#sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi not break. 2lra v 0 lx j ; B0 ¼ 0 c ð17Þ Substituting Eqs. (8) and (9) into Eq. (7), we can obtain rp ra rp 0 ð 0 þ 0 Þ the expression of the angular velocity after capture the two "#sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi payloads as follows: 2lra v 0 lx j ; p20 ¼ 0 c ð18Þ J x m li v v rp ðra þ rp Þ 0 0 þ P½ b ð p10 p20 Þ 0 0 0 xH ¼ : ð11Þ J H where x0 represents the length of x0. The relationship between the payloads’ velocities and Practically, it can be derived through Eqs. (3) and (4) the rotational angular velocity is assumed to satisfy that Eqs. (6) and (12) also hold true if Eqs. (13) and (14) v v x l: hold true. p10 p20 ¼ 2 0 ð12Þ Substituting Eqs. (8), (9) and (12) into Eq. (11), Eq. (11) 4.2. Capture with velocity error can be reduced to show that the angular velocities satisfy x x 0 = H. The ideal capture velocity is solved; however, in practice, velocity errors occur in both magnitude and direction. 4. Analysis of the dynamics with velocity error In this section, the velocities of E1,P1,andE2 are assumed to be ideal, which means Eqs. (13), (15), and (17) hold true. The ideal capture velocity is analyzed first; however, in We analyze the velocity error in relation to the payload P2 practice, velocity errors occur in both magnitude and direc- as an example to investigate the influence of the velocity tion. For simplicity, we establish a velocity error model in error on the orbit shown in Fig. 5, in which a is the angle relation to the payload P2 as an example to investigate the between the velocity direction of P2 and the velocity direc- influence of the velocity error on the orbit and state motion tion of the central facility before payload capture. The state of the MMET. motion of the MMET is considered starting from the following assumptions: 4.1. Ideal velocity for capture "#sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2lra v 0 lx Dv j ; p20x ¼ 0 þ c ð19Þ It is known that, during rendezvous, the smaller the rel- rp ra rp 0 ð 0 þ 0 Þ ative velocity between the payload and capture equipment D j v is, the better the rendezvous will be. If the velocities of the where v is the velocity error in the direction of c and p20x v j payloads satisfy a relative velocity of zero, we call the is the velocity component of p20 along the direction of c. v velocities ideal. Based on the geometric relationship shown in Fig. 5, p20y The ideal velocity of the tip of the higher tapered tether is given by v v can be made to match p10 and the payload’s velocity p20 Q. Sun et al. / Advances in Space Research 59 (2017) 2374–2388 2379

+ ðRv2=lÞ sin v cos v v EP11 tan h ¼ : ð26Þ p10 ðRv2=lÞ cos2 v 1

l After capture, the true anomaly hH is given by

v ω 2 c0 rp v =l v v 0 ð 0 cH Þ sin H cos H M hH ¼ arctan ; ð27Þ rp v2 =l 2 v ð 0 cH Þ cos H 1 l

+ where vH is the angle between the direction of the central EP22 v facility’s velocity and the local level after capture. Taking P20 x α v v the square of Eq. (24) and summing it with the square of P20 y P20 Earth Eq. (25), the eccentricity ratio is obtained as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Fig. 5. The velocity of the payload P2 during capture. eH rp v2 =l 2 v v ; ¼ ½ð 0 cH Þ1 cos H þ sin H ð28Þ "#sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and the semi-major axis is given by 2lra v 0 lx Dv tan aj 20 rp p20y ¼ 0 þ p y ð Þ a 0 : rp ðra þ rp Þ 20 H ¼ ð29Þ 0 0 0 rp v2 =l 2 0 cH where j is the unit vector of v , which is perpendicular p20y p20y The initial conditions before capture are shown in j v a to the vector c. Thus, p20 can be given by Table 1. Because = 0, the direction of the central facility’s "#sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi velocity is not changed during payload capture. Thus, 2lra v 0 lx Dv j hH = 0 and vH = 0. The orbital parameters of the central p20 ¼ 0 þ c rp ra rp D 0 ð 0 þ 0 Þ facility in relation to v are shown in Fig. 6. The angular "#sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi velocity after payload capture for different Dv is shown in 2lra 0 lx Dv aj ; Fig. 7. þ 0 þ tan p y ð21Þ rp ra rp 20 0 ð 0 þ 0 Þ It can be seen from Fig. 6 that the semi-major axis, eccentricity and apogee radius increase by increasing Dv. It is known that the parking orbit will not be changed if In contrast, the perigee radius is not changed during pay- a D = 0 and v = 0 because, in this situation, the velocity of loads capture despite changes in Dv. This is because P is ideal. 2 hH = 0, meaning that perigee is not changed during pay- a In Case 1, = 0; we call this situation velocity with only loads capture. In Fig. 7, the angular velocity decreases with magnitude error. The velocity directions of the central the increasing Dv. facility and the two payloads are the same. Eq. (21) can After capture, the semi-latus rectum pH is given by be written as 2 2 "#sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rp rp v =l 0 f1 ½ð 0 cH Þ1 g 2lra pH ¼ ; ð30Þ v 0 lx Dv j 2 ðrp v2 =lÞ p20 ¼ 0 þ c ð22Þ 0 cH rp ðra þ rp Þ 0 0 0 _ and the true anomaly velocity hH at perigee is obtained as v Substituting Eq. (22) into Eq. (5), cH is obtained as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "#sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _ 4 hH ¼ lpH=rp : ð31Þ 2lra m Dv 0 v 0 p j : cH ¼ þ c ð23Þ rp ra rp m m 0 ð 0 þ 0 Þ þ 2 p Table 1 The central facility is still considered as the COM of the Initial conditions and simulation parameters for the MMET. MMET, and the instantaneous radius vector is not chan- Parameters Value Unit ged during payload capture. Earth’s gravitation constant, l 3.9877848 1014 m3/s2 It is known that the eccentricity ratio e, the true anom- Radius of lunar orbit, R 3.844 108 m h R m aly , the length of the radius vector , and the velocity of Initial true anomaly, h0 0 rad 6 the central facility v satisfy the following form (Zhao and Initial radius vector, R0 6.728 10 m 6 Initial perigee radius, rp 6.728 10 m Liu, 2011): 0 r 7 Initial apogee radius, a0 1.036 10 m 2 e sin h ¼ðRv =lÞ sin v cos v ð24Þ Initial pitching angle, w0 0 rad Initial angular rate, x 0.2307959 rad/s 2 2 0 e cos h ¼ðRv =lÞ cos v 1 ð25Þ Length of the propulsion tether, l 10,000 m v Mass of the payload, mP 500 kg where is the angle between the direction of the central Mass of the central facility, m 5000 kg v c facility’s velocity and the local level; thus = 0 at the Mass of the main capture equipment, mE 200 kg moment of capture because central facility is at perigee. External radius of the central facility, rc 0.5 m Eq. (24) is divided by Eq. (25); thus, the true anomaly h Ultimate strength, 1 7.0 GPa 3 can be obtained as Density of tethers, o 1330 kg/m 2380 Q. Sun et al. / Advances in Space Research 59 (2017) 2374–2388

0.212560 8,544,150

8,544,100 0.212555 Θ (m) e Θ a 8,544,050 0.212550

8,544,000 0.212545 0 0.2 0.4 0.6 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Δv (m/ s ) Δv (m/ s ) (a) Semi-major axis for different Δv (b) Eccentricity ratio for different Δv

10,360,300 6,728,002

10,360,200 6,728,001 (m) (m) Θ Θ p a r r 10,360,100 6,728,000

10,360,000 6,727,999 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.2 0.4 0.6 0.8 Δv (m/ s ) Δv (m/ s )

Fig. 6. Orbital parameters of the central facility for diﬀerent Dv.

In Case 2, a – 0; we call this situation velocity with both 0.2308 v magnitude and direction errors. p20 is the complete form: "#sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0.230795 2lra v ¼ 0 lx þ Dv j 0.23079 p20 r r r 0 c p a p (rad/s) 0 ð 0 þ 0 Þ Θ

"#sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω 0.230785 2lra 0 lx Dv aj þ 0 þ tan p y ð32Þ rp ra rp 20 0.23078 0 ð 0 þ 0 Þ 0 0.2 0.4 0.6 0.8 Δv (m/ s ) Submitting Eq. (32) into (5), the velocity of the central facility is given by Fig. 7. The angular velocity after payloads capture for different Dv. "#sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "#sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2lra m Dv m 2lra These orbital parameters of the central facility change v 0 p j p 0 lx Dv aj cH ¼ þ c þ 0 þ tan p y rp ðra þrp Þ ðmþ2m Þ ðmþ2m Þ rp ðra þrp Þ 20 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}0 0 0 p |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}p 0 0 0 during capture. The parameters are shown in Fig. 9 for dif- v v TH nH ferent Dv and a. ð33Þ Fig. 9 shows that the semi-major axis, eccentricity, and Dv In Eq. (33) v represents the tangential speed of the cen- apogee radius are approximately proportional to . TH Moreover, the semi-major axis, eccentricity and apogee tral facility, and v represents the normal velocity of the nH radius are also approximately proportional to a; however, central facility. The angle between v and v is given by c0 cH their changes with respect to a are relatively small within v D b k nH k : the scope of the simulation conditions. v has very little ¼ arctan v ð34Þ k TH k impact on the perigee radius because the tangential velocity increment does not change the perigee, which is only influ- v b In Case 2, after capture, H = ; the resulting change enced by the normal velocity increment; and when a and Dv in the central facility’s velocity direction is shown in Fig. 8. are small, the normal velocity increment caused by Dv is The eccentricity ratio can be obtained as relatively small. It is noted that the perigee radius decreases qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi as a increases, and a indirectly reflects the normal velocity 2 2 2 2 eH ¼ ðrp v =l 1Þ cos vH þ sin vH; ð35Þ 0 cH increment. The normal velocity increment and a have an approxi- and the semi-major axis is given by mately proportional relationship, which reflects the fact rp 0 that the perigee radius of the central facility decreases with aH ¼ : ð36Þ rp v2 =l V 2 0 cH increasing normal velocity increments defined by nH : Q. Sun et al. / Advances in Space Research 59 (2017) 2374–2388 2381 "#sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 2lra V p 0 lx Dv a: nH ¼ 0 þ tan ð37Þ m m rp ra rp ð þ 2 pÞ 0 ð 0 þ 0 Þ Because Dv is relatively small, Eq. (37) can be approxi- mated by "#sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 2lra V p 0 lx a: nH 0 tan ð38Þ m m rp ra rp ð þ 2 pÞ 0 ð 0 þ 0 Þ The tangential velocity increments of the central facility V defined by TH : m Dv V p TH ¼ ð39Þ ðm þ 2mpÞ Fig. 10 shows the evolution of the tangential and normal D a Fig. 8. Change in the central facility’s velocity direction after payload velocity increments in relation to v and , respectively. capture in Case 2.

(a) Semi-major axis for different Δv and α (b) Eccentricity ratio for different Δv and α

Fig. 9. Orbital parameters of the central facility for diﬀerent Dv and a. 2382 Q. Sun et al. / Advances in Space Research 59 (2017) 2374–2388

r Fig. 10 shows that a tiny change in a causes a great 2 a0 dhH ¼ change in the normal velocity. It is noted that the changes e ðra þ rp Þ 0 0 0 hiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the semi-major axis, eccentricity, and apogee radius with m lra p 2 0 lx0 þ Dv tan a respect to a are much smaller than those with respect to Dv ðmþ2mpÞ rp ðra þrp Þ arctan qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0 0 0 : lra m Dv in Fig. 9. Thus, it can be concluded that changes in orbital 2 0 p rp ðra þrp Þ þ ðmþ2m Þ parameters caused by the tangential velocity increment, 0 0 0 p which include changes in the semi-major axis, eccentricity, ð42Þ and apogee radius, have far greater effects than changes caused by the normal velocity increment at the perigee. Fig. 11 shows the change in the true anomaly for differ- D a However, the perigee radius of the central facility is influ- ent v and after capture. As shown in the figure, the influ- D enced only by the normal velocity increment at the perigee. ence of v on the true anomaly is rather minimal because In order to analyze the MMET’s motion after capture, the component of the normal velocity increment caused D we need to obtain the orbital parameters that include the by v is relatively small; however, the change in the true a true anomaly during capture, which is given by anomaly increases quickly with increasing because the normal velocity increment caused by a is great. 2 After capture, the semi-latus rectum pH, the true anom- ðrp v =lÞ sin vH cos vH 0 cH _ _ tan hH ¼ ð40Þ aly velocity hH, and the radial velocity RH are given by rp v2 =l 2 v ð 0 cH Þ cos H 1 rp 0 2 pH ¼ ð1 eHÞ; ð43Þ rp v2 =l Differentiating both sides of Eq. (26), and then combin- 2 ð 0 cH Þ ing Eqs. (24)–(26), the change in the true anomaly during qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _ 4 capture is obtained by hH ¼ lpH=rp ; and ð44Þ 0 2 2 h dv h v rp v 2 _ 2 sin 0 cos 0 2 sin 0 0 c0 R_ p e h e h h ; : dhH ¼ þ þ dv : ð41Þ H ¼ Hð1 þ H cos HÞ H H sin H respectively ð45Þ e v e e2 l |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}0 c0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}0 0

dhT dhn 4.3. Analyzing dynamics with velocity error

In Eq. (41),dhT represents the change in the true anomaly caused by the tangential velocity increment; dhn repre- We next solve the coupled dynamic equations of the sents the increment of the true anomaly caused by the MMET after payload capture. The mass of the tether is normal velocity increment; and e0 and h0 are defined as given as presented by Yang et al. (Yang et al., 2015): rffiffiffiffiffiffi rffiffiffiffiffi the eccentricity and the true anomaly of the central facil- po oDv2 o ity’s orbit before capture, respectively. v is the angle m m m Dv n Dv ; 0 T ¼ð P þ EÞ n 1 exp 1 erf n 1 ð46Þ between the direction of the central facility’s velocity and 2 2 2 the local level before capture. It is noted that the tangential where erf(x) is the Gaussian error function. velocity increment cannot change the true anomaly at peri- In order to obtain the translational energy of the h h gee because the value of 0 is zero, and d T = 0 = 0. Only if MMET after capture, we can consider the MMET as a v – d 0 will the true anomaly change at perigee. mass point moving in an elliptical orbit. Thus, the transla- h v v v v Submitting Eq. (1), 0 =0, 0 = 0 and d = H 0 = tional energy can be expressed by b into Eq. (41); then the change of true anomaly can be expressed as 1 _ 2 2 _ 2 E ¼ mHðR þ R h Þ; ð47Þ TH 2 H H H

0.05 3.5

3 0.04 2.5 0.03 2 (m/s)

Θ 1.5 (m/s) T 0.02 v Θ n v 1 0.01 0.5

0 0 0 0.2 0.4 0.6 0.8 0 0.002 0.004 0.006 0.008 0.01 Δv (m/ s ) α (rad) (a) Tangential velocity increment caused by Δv (b) Normal velocity increment caused by α

Fig. 10. Tangential and normal velocity increments in relation to Dv and a, respectively. Q. Sun et al. / Advances in Space Research 59 (2017) 2374–2388 2383

V In these equations, cH is the potential energy of the central V V V facility after capture and EiH , PiH , and TiH (i =1,2)are the potential energies of the capture equipment, payloads, and tethers after capture, respectively. It is convenient to use the Lagrangian approach to establish the MMET’s dynamics:

C ¼ Etotal V total; ð54Þ

where Etotal and Vtotal are the kinetic and potential energies of the system after capture, respectively. These are deﬁned as E E E total ¼ TH þ RH and ð55Þ

X2 X2 X2 V V V V V : total ¼ cH þ EiH þ PiH þ TiH ð56Þ Fig. 11. Change in the true anomaly for different Dv and a. i¼1 i¼1 i¼1 where mH is the total mass of the MMET after capture; RH Substituting Eqs. (55) and (56) into Eq. (54), the and hH are the radius vector and true anomaly of the cen- Lagrangian equation can be given by tral facility after capture, respectively; and mH can be d @C @C ¼ Qq qi ¼ðRH; hH; wHÞ; ð57Þ expressed as t @q_ @q i rffiffiffiffiffiffi d i i po Q mH ¼ m þ 2ðm þ m Þþ2ðm þ m ÞDvn where qi is the generalized force, known as the work of c P E P E 21 rffiffiffiffiffi non-conservative forces, and qi represents the generalized 2 oDvn o coordinates. Ignoring the non-conservative forces such as exp erf Dvn : ð48Þ 21 21 are caused by perturbations, third-body gravitation, and Q external forces moments, qi equals (0, 0, 0). Using the method in the reference (Yang et al., 2015), Substituting the kinetic and potential energies of the sys- the rotational energy of the MMET after capture is tem into Eq. (57) yields the following equations of motion obtained by for the system: "#rffiffiffiffiffiffiffiffi rffiffiffiffiffi m m l2 p1 oDv2 o 2 2 1 1 2 ð P þ EÞ 2 n _ _ 2 l 3lm l ð3 cos wH 1Þ1 E m r exp erf Dvn hH wH € _ 2 T RH ¼ c c þ ð þ Þ RH RHh 0 58 2 2 Dvn o 21 21 H þ 2 þ 4 2 ¼ ð Þ RH mHRHDvno ð49Þ € 2mH _ _ J H € hH þ RHRHhH þ wH ¼ 0 ð59Þ The equations describing the potential energies of the 2 2 mHRH þ J H mHRH þ J H system’s components have the following form: lm l2 w 1 € € 3 T sinð2 HÞ mc wH hH 0 60 V l ; þ þ 3 2 ¼ ð Þ cH ¼ ð50Þ J HRHDvno RH "# 2 We then analyze the dynamics with velocity error. In m 1 l i l V l E 2 w w i ; ; EiH 1 þ ð3 cos H 1Þþð1Þ cos H ¼ 1 2 Case 1, after capture, the radius vector velocity is equal RH 2 RH RH to zero because the perigee is not changed and the rota- ð51Þ tional angular velocity of the tether system can be calcu- "# 2 mP 1 l i l lated from Eq. (11). Setting the orbital parameters after V l 1 3 cos2 w 1 1 cos w i 1; 2; and PiH R þ R ð H Þþð Þ R H ¼ H 2 H H capture as the initial conditions of the simulation, we can ð52Þ analyze the dynamics after payloads capture for different

( ffiffiffiffiffi rffiffiffiffiffi l m m Dv2o Dv2o p 2 3 ð P þ EÞ n n p1l o ð3cos wH 1Þl 1 V exp pffiffiffiffiffi erf Dvn þ TiH R l1 1 1 2 H 2 Dvn 2o 2 2ðRHDvnÞ o "#ffiffiffiffiffi rffiffiffiffiffi ) p Dv2o 2 Dv2o p1 o n i l cos wH1 n ffiffiffiffiffi Dvn i ; p erf exp ð1Þ 2 exp 1 ¼ 1 2 ð53Þ Dvn 2o 21 21 RHDvno 21 2384 Q. Sun et al. / Advances in Space Research 59 (2017) 2374–2388

Dv and compare them with dynamics without payloads In Fig. 12,dR = RH R0; RH represents the radius vec- capture. The initial conditions before capture are shown tor after payloads capture for diﬀerent Dv; and R0 repre- in Table 1. The time evolution of the MMET’s radius vec- sents the radius vector without payloads capture. tor parameters after capture are shown in Fig. 12, the true Similarly, in Figs. 12–14,dvRH,dhH,dxhH,dwH and anomaly parameters are shown in Fig. 13, and the pitching dxwH represent the gaps between the MMET angle parameters are shown in Fig. 14. motion parameters for diﬀerent Dv and without capture,

6 x 10

12 Δv=0 Δv=0.05(m/s) Δv=0.2(m/s) Δv=0.8(m/s) Δv=0 Δv=0.05(m/s) Δv=0.2(m/s) Δv=0.8(m/s) 10000 11 10 5000 (m) (m) Θ Θ 9

R 0 dR 8 7 -5000

0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 4 4 Time (s) x 10 Time (s) x 10

(a) Time evolution of the radius vector (b) Time evolution of the radius vector deviation

2000 Δv=0 Δv=0.05(m/s) Δv=0.2(m/s) Δv=0.8(m/s) Δv=0 Δv=0.05(m/s) Δv=0.2(m/s) Δv=0.8(m/s) 5 1000 0

(m/s) 0 (m/s) Θ Θ R v -1000 R -5 dv -2000 -10 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 4 4 Time (s) x 10 Time (s) x 10

Fig. 12. Time evolution of the MMET’s radius vector parameters after capture for diﬀerent Dv.

-3 x 10 30 2 Δv=0 Δv=0.05(m/s) Δv=0.2(m/s) Δv=0.8(m/s) Δv=0 Δv=0.05(m/s) Δv=0.2(m/s) Δv=0.8(m/s) 25 0 20 (rad) (rad) 15 -2 Θ Θ θ θ

10 d -4 5

0 -6 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 4 4 Time (s) x 10 Time (s) x 10

(a) Time evolution of the true anomaly (b) Time evolution of the true anomaly deviation

-3 x 10 -6 1.6 x 10 ΔΔv=0v=0 ΔΔv=0.05(m/s)v=0.05(m/s) ΔΔv=0.2(m/s)v=0.2(m/s) ΔΔv=0.8(m/s)v=0.8(m/s) Δv=0 Δv=0.05(m/s) Δv=0.2(m/s) Δv=0.8(m/s) 1.4 2

1.2 1 (rad/s) (rad/s) 1

Θ 0 θΘ θ ω ω 0.8 d -1 0.6 -2 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 4 4 Time (s) x 10 Time (s) x 10

Fig. 13. Time evolution of the MMET’s true anomaly parameters after capture for diﬀerent Dv. Q. Sun et al. / Advances in Space Research 59 (2017) 2374–2388 2385

0.2 Δv=0 Δv=0.05(m/s) Δv=0.2(m/s) Δv=0.8(m/s) 8000 Δv=0 Δv=0.05(m/s) Δv=0.2(m/s) Δv=0.8(m/s)

6000 0 (rad) (rad) 4000 Θ Θ ψ ψ d -0.2 2000

0 -0.4 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Time (s) 4 Time (s) 4 x 10 x 10 (a) Time evolution of the pitching angle (b) Time evolution of the pitching angle deviation

-6 x 10 0.232 Δv=0 Δv=0.05(m/s) Δv=0.2(m/s) Δv=0.8(m/s) 5 Δv=0 Δv=0.05(m/s) Δv=0.2(m/s) Δv=0.8(m/s)

0 0.2315 -5 (rad/s) (rad/s) ψΘ ψΘ ω ω

d -10 0.231

-15 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Time (s) 4 Time (s) 4 x 10 x 10 (c) Time evolution of the pitching angle velocity (d) Time evolution of the pitching angle velocity deviation

Fig. 14. Time evolution of the MMET’s pitching angle parameters after capture for diﬀerent Dv.

D respectively. It is noted that, even when v is not equal to Δv=0 zero macroscopically, the MMET still maintains stable Δv=0.05m/s periodic elliptical movement in its orbit. Therefore, the motion parameters of the central facility all change period- perigee ically. In Fig. 12, the amplitudes of RH and vRH increase apogee with increasing Dv; from Fig. 6, we know that the perigee radius is not changed during payload capture despite changes in Dv. Thus, the valleys’ value, which represents Δv= Δv=0.8m/s the perigee radius, is almost identical for different Dv and 0.2m/s 6 their value is almost 6.728 10 m; however, the valleys’ Fig. 15. Orbits of the central facility after capture for different Dv. values are slightly different because of the coupled influence of rotation, which can also be seen from Fig. 12 on a microscopic level. The peaks, which represent the apogee velocity decrease. The absolute value of the amplitude of radius, are approximately proportional to Dv, and the dhH is approximately proportional to Dv. movement cycle is also approximately proportional to Dv; The peaks and valleys of xhH appear at the perigee and as a result, the valleys and peaks are shifted to the right apogee, respectively. Similarly, the valleys and peaks of x on a timeline with increasing Dv. Furthermore, dRH and hH are shifted to the right on the timeline with increasing D xhH dvRH, the amplitudes of which exhibit approximately linear v; meanwhile, the amplitude of d increases with growth with increasing Dv, oscillate in both positive and increasing Dv because the shape of the orbit grows and negative directions. After capture, vRH becomes larger than the cycle lengthens. The pitching angles all increase period- Dv vR0 (without capture) at the starting time because the orbit ically over time for different because of the MMET size increases. However, vRH will become smaller than vR0 moving along the orbit; however, the pitching angles increase slowly over time evolution with increasing Dv. Spe- with the time growing and then larger than vR0 again, which can be seen in Fig. 12 on a microscopic level. Thus, cially, the amplitude of dxwH increases over time for differ- D dRH and dvRH present positive and negative directions with ent v. To summarize, if jc ¼ jp , then for different Dv, the time evolution. Ultimately, dRH and dvRH present positive 2 and negative directions because of the shape of the orbit MMET’s central facility moves in different orbits with growing larger and the cycle lengthening. hH increases almost overlapping perigees after capture. Moreover, the more slowly than with a bigger Dv because the period of semi-major axes of these orbits, which have different the central facility’s orbital movement increases as Dv lengths, also almost overlap. Fig. 15 shows the orbits of increases, causing the average true anomaly in the angular the central facility after capture for different Dv. 2386 Q. Sun et al. / Advances in Space Research 59 (2017) 2374–2388

In Case 2, as in Case 1, the rotating angular velocity of payload capture in relation to diﬀerent a and compare the tethered system after capture can be calculated by Eq. these dynamics with those without payload capture based (11). Setting the orbital parameters and state of the MMET on Eqs. (58)–(60). after capture as the initial conditions of the simulation and Similarly, dRH,dvRH,dhH,dxhH,dwH and dxwH repre- deﬁning Dv = 0.1 m/s, we can analyze the dynamics after sent the gaps between the parameters with and without

6 4 x 10 x 10 1.5 11 α=0 α=0.003(rad) α=0.006(rad) α=0.009(rad) α=0 α=0.003(rad) α=0.006(rad) α=0.009(rad) 1 10 0.5 (m) (m) 9 Θ Θ

R 0 8 dR -0.5 7 -1 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 4 4 Time (s) x 10 Time (s) x 10 (a) Radius vector for Δv =0.1 m/s (b) Radius vector deviation for Δv =0.1 m/s

10 2000 α=0 α=0.003(rad) α=0.006(rad) α=0.009(rad) α=0 α=0.003(rad) α=0.006(rad) α=0.009(rad) 5 1000 0

0 (m/s) (m/s) Θ Θ

R -5 R v

-1000 dv -10 -2000 -15 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 4 4 Time (s) x 10 Time (s) x 10 (c) Radial velocity for Δv =0.1 m/s (d) Radial velocity deviation for Δv =0.1 m/s

Fig. 16. Time evolution of the MMET’s radius vector parameters after capture for diﬀerent a.

-3 x 10 30 α=0 α=0.003(rad) α=0.006(rad) α=0.009(rad) α=0 α=0.003(rad) α=0.006(rad) α=0.009(rad) 0 20 -2 (rad) (rad) Θ

10 Θ -4 θ θ d -6 0 -8 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 4 4 Time (s) x 10 Time (s) x 10 (a) True anomaly for Δv =0.1 m/s (b) True anomaly deviation for Δv =0.1 m/s

-3 -6 x 10 x 10 1.5 3 α=0 α=0.003(rad) α=0.006(rad) α=0.009(rad) α=0 α=0.003(rad) α=0.006(rad) α=0.009(rad) 2

0 1 (rad/s) (rad/s) Θ θ θΘ

ω -1 ω d -2

0.5 -3 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 4 4 Time (s) x 10 Time (s) x 10

Fig. 17. Time evolution of the MMET’s true anomaly parameters after capture for diﬀerent a. Q. Sun et al. / Advances in Space Research 59 (2017) 2374–2388 2387

8000 α=0 α=0.003(rad) α=0.006(rad) α=0.009(rad) 0.01 α=0 α=0.003(rad) α=0.006(rad) α=0.009(rad) 6000 0 -0.01

(rad) 4000 (rad) Θ

Θ -0.02 ψ ψ d 2000 -0.03

-0.04 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 4 4 Time (s) x 10 Time (s) x 10 (a) Pitching angle for Δv =0.1 m/s (b) Pitching angle deviation for Δv =0.1 m/s

-6 0.2318 x 10 α=0 α=0.003(rad) α=0.006(rad) α=0.009(rad) 1 α=0 α=0.003(rad) α=0.006(rad) α=0.009(rad) 0.2316 0 0.2314 -1 (rad/s) 0.2312 (rad/s) Θ ψ

ψΘ -2 ω 0.231 ω d -3 0.2308 -4 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 4 4 Time (s) x 10 Time (s) x 10

Fig. 18. Time evolution of the MMET’s pitching angle parameters after capture for different a. payload capture for different a and Dv = 0.1 m/s, as shown in Figs. 16–18. The figures show that the changes in the Δv=0.1m/s,α =0.003rad MMET’s orbit and state increase with increasing a. After capture, the MMET maintains stable periodic motion at Δv=0.1m/s,α =0 the macroscopic level. As a increases, the normal velocity perigee apogee increment grows and the amplitudes of dRH and dvRH exhibit an approximately linear increasing trend, which is sim- Δv=0.1m/s,α =0.009rad ilar to payload capture with only different Dv as in Case 1. However, for payload capture with changes in a, the valley Δv=0.1m/s,α =0.006rad of RH changes because the perigee can be changed. On a Fig. 19. Orbits of the central facility after capture for different a and microscopic level, from Fig. 16 we can note that increasing Dv = 0.1 m/s. a decreases the valley and increases the peak of RH because perigee radius decreases and apogee radius increases after increases because of the larger orbit induced by an increase capture which can been seen from Fig. 9 apparently. in a. Fig. 19 shows the different orbits of the central facility Furthermore, xhH changes periodically over time evolu- after capture in relation to different a and Dv = 0.1 m/s. tion, and its amplitude increases with increasing a; how- x h ever, the average value of hH decreases, and thus H 5. Conclusions increases more slowly than with a bigger a. The pitching angle and pitching angle velocity also change periodically, The dynamics of the MMET during and after payloads and because it is known that when the capture is performed capture was analyzed using the mathematical model pre- at perigee, the angular momentum is influenced only by the sented in this paper under the main assumptions of the tangential velocity increment, therefore, the MMET’s ini- rigid tether and payloads of the same mass being captured tial pitching angle velocity is affected only by the tangential simultaneously. The influence on the MMET’s orbital velocity increment during payload capture. After payload parameters and motion state during payloads capture at capture, the time evolution of the pitching angles nearly different velocities was also analyzed. In the case of pay- a D equal each other for different at the same v. Because load velocity with only magnitude error, the orbit of the a the normal velocity increment caused by changes the central facility changes, including the semi-major axis, orbital parameters far less than the tangential velocity eccentricity, and apogee. Considering the case of payload D increment at perigee, therefore if v is identical, the orbit velocity with both magnitude and direction errors, the a x changes little with increasing . The amplitude of d wH orbit changes more, and the perigee as well as the direc- 2388 Q. Sun et al. / Advances in Space Research 59 (2017) 2374–2388 tions of the semi-major axis also change. However, the nor- AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, mal velocity increment caused by the capture velocity direc- Los Angeles, California, pp. 1999–2151. http://dx.doi.org/10.2514/ tion error changes the orbital parameters far less than the 6.1999-2151. Hoyt, R.P., 1997. Tether system for exchanging payloads between low tangential velocity increment at the perigees. The analysis earth orbit and the lunar surface. In: 33rd AlAA/ASME/SAE/ASEE of the dynamics of the MMET during and after payloads Joint Propulsion Conference and Exhibit, Seattle, WA, pp. 1997–2794. capture contributes to the knowledge on the dynamics of http://dx.doi.org/10.2514/6.1997-2794. MMETs and on the orbital correction that can be accom- Hoyt, R.P., Uphoff, C., 2000. Cislunar tether transport system. J. Spacecr. plished when payloads are captured in subsequent tasks so Rockets 37, 177–186. http://dx.doi.org/10.2514/2.3564. Hoyt, R.P., 2000a. Commercial development of a tether transport system. as to realize continuous transport. In: 36th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, Las Vegas, Nevada, USA, pp. 2000–3842. http://dx.doi.org/ Acknowledgement 10.2514/6.2000-3842. Hoyt, R.P., 2000b. The cislunar tether transport system architecture. 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