- Email: [email protected]
Optimal Control of Payload Tossing Using Space Tethered System
20th 20th IFAC IFAC Symposium Symposium on on Automatic Automatic Control Control in in Aerospace Aerospace August 21-25, 2016. Sherbrooke, Sherbrooke, Quebec, Canada 20th IFAC Symposium on Automatic Control Aerospace August 21-25, 2016. Quebec, Canada 20th IFAC Symposium on Automatic Available Control in inonline Aerospace at www.sciencedirect.com August 21-25, 21-25, 2016. 2016. Sherbrooke, Sherbrooke, Quebec, Quebec, Canada Canada August
ScienceDirect IFAC-PapersOnLine 49-17 (2016) 272–277
Optimal Optimal Control Control of of Payload Payload Tossing Tossing Using Using Space Space Tethered Tethered System System Optimal Control of Payload Tossing Using Space Tethered System Optimal Control of Payload Tossing Using Space Tethered System Lu Hong Shi*, Wang Changqing**, Lu Hong Shi*, Wang Changqing**, Lu Hong Lu Hong Shi*, Shi*, Wang Wang Changqing**, Changqing**, Zabolotnov Zabolotnov Yuriy***, Yuriy***, Li Li Aijun**** Aijun**** Zabolotnov Yuriy***, Li Aijun**** Zabolotnov Yuriy***, Li Aijun**** 710072, China ** Northwestern Northwestern Polytechnical Polytechnical University, University, Xi’an Xi’an Shanxi Shanxi 710072, China ** Northwestern Polytechnical University, Xi’an Shanxi Northwestern Polytechnical University, Xi’an Shanxi 710072, 710072, China China PRC PRC (Tel: (Tel: 86-029-888431383; 86-029-888431383; e-mail: e-mail: [email protected]) [email protected]) PRC e-mail: PRC (Tel: (Tel: 86-029-888431383; 86-029-888431383; e-mail: [email protected]) [email protected]) ** Northwestern Northwestern Polytechnical University, University, Xi’an Shanxi Shanxi 710072, 710072, China China ** Polytechnical Xi’an ** Northwestern Northwestern Polytechnical Polytechnical University, University, Xi’an Xi’an Shanxi Shanxi 710072, 710072, China China ** PRC (Tel: (Tel: 86-029-888431383; 86-029-888431383; e-mail: e-mail: [email protected]) [email protected]) PRC PRC (Tel: (Tel: 86-029-888431383; e-mail: e-mail: [email protected]) PRC *** Samara86-029-888431383; State Aerospace Aerospace University, University,[email protected]) Samara 443086, 443086, Russia Russia *** Samara State Samara *** Samara Samara State State Aerospace Aerospace University, University, Samara Samara 443086, 443086, Russia Russia *** RUS RUS (Tel: (Tel: 7-846995-62-65; 7-846995-62-65; e-mail: e-mail: [email protected]) [email protected]) RUS RUS (Tel: (Tel: 7-846995-62-65; 7-846995-62-65; e-mail: e-mail: [email protected]) [email protected]) ***** ***** Northwestern Northwestern Polytechnical Polytechnical University, University, Xi’an Xi’an Shanxi Shanxi 710072, 710072, China China ***** Northwestern Polytechnical University, Xi’an Shanxi 710072, China ***** Northwestern Polytechnical University, Xi’an Shanxi 710072, China PRC PRC (Tel: (Tel: 86-029-888431383; 86-029-888431383; e-mail: e-mail: [email protected]) [email protected]) PRC PRC (Tel: (Tel: 86-029-888431383; 86-029-888431383; e-mail: e-mail: [email protected]) [email protected]) Abstract: To To transport transport payload payload into into lunar lunar transfer transfer orbit orbit using using spinning spinning tether tether system, system, reusable reusable payload payload Abstract: tossing was based cislunar transfer and law was Abstract: To transport transport payload intoon lunar transfer orbitmission, using spinning spinning tethercontrol system, reusable payload tossing system system was proposed proposed based onlunar cislunar transfer mission, and optimal optimal control law was developed developed Abstract: To payload into transfer orbit using tether system, reusable payload for tossing stage. case insufficient thrust, Hamilton equation was solve tossing system was based on transfer mission, and law for payload payload tossing stage. In In the the case ofcislunar insufficient thrust, Hamilton equationcontrol was adopted adopted todeveloped solve the the tossing system was proposed proposed based onof cislunar transfer mission, and optimal optimal control law was wasto developed problem. The numerical result shows that the tossing design is able to transport payload into desired for payload tossing stage. In the case of insufficient thrust, Hamilton equation was adopted to solve the problem. Thetossing numerical result shows the tossing design is able to transport into desired orbit, for payload stage. In the casethat of insufficient thrust, Hamilton equation payload was adopted to solveorbit, the and law meet mission Compared rocket transport, problem. The result that the is transport desired orbit, and the the control control law could could meetshows mission requirement. Compared toto conventional rocketinto transport, much problem. The numerical numerical result shows thatrequirement. the tossing tossing design design is able ableto toconventional transport payload payload into desiredmuch orbit, less propellant is required by the designed system with reusability. and the control law could meet mission requirement. Compared to conventional rocket transport, much less propellant required the mission designedrequirement. system with Compared reusability.to conventional rocket transport, much and the controlislaw could by meet less propellant is by the system with reusability. less propellant is required required byFederation the designed designed system withOptimal reusability. © 2016, IFAC (International of Automatic by Hamilton Elsevier Ltd. All rights reserved. Keywords: Space Tethered Tossing, Control, Energy. Keywords: Space Tethered System, System, Payload Payload Tossing,Control) OptimalHosting Control, Hamilton Energy. Keywords: Keywords: Space Space Tethered Tethered System, System, Payload Payload Tossing, Tossing, Optimal Optimal Control, Control, Hamilton Hamilton Energy. Energy.
1. 1. INTRODUCTION INTRODUCTION 1. 1. INTRODUCTION INTRODUCTION Space Tethered Tethered System System (STS) (STS) consists consists of of two two or or Space multiple satellites connected by (STS) thin, long long tether. of It has has great Space Tethered System (STS) consists of twogreat or multiple satellites connected by thin, tether. It Space Tethered System consists two or potential in transportation, etc. multiple satellites satellites connected by thin, thin,space longdebris tether.removal It has has great great potential in payload payload transportation, space debris removal etc. multiple connected by long tether. It (Van Pelt et. al.) potential transportation, (Van Peltin et.payload al.) potential in payload transportation, space space debris debris removal removal etc. etc. (Van Pelt et. al.) (VanNow, Pelt et.the al.) most likely way to execute deep space Now, the most likely way to execute deep space investigation payload is Now, likely way execute space investigation ormost payload transfer mission is through through Now, the theor most likely transfer way to to mission execute deep deep space conventional rocket. However, rocket is extremely costing investigation or payload transfer mission is through conventional However, rocket mission is extremely costing investigation rocket. or payload transfer is through and vast space spacerocket. debris would would be produced produced for large scale scale deep conventional However, rocket extremely costing and vast debris be large deep conventional rocket. However, rocket is isfor extremely costing space missions up ahead. Compared to conventional rockets, and vast space debris would be produced for large scale space missions up ahead. Compared to conventional rockets, and vast space debris would be produced for large scale deep deep much less (or even even no evident) evident) propellant is required required rockets, through space less missions up ahead. ahead. Compared to conventional conventional rockets, much (or no propellant is through space missions up Compared to with Therefore, STS et. 2000) much (Lorenzini less (or (or even even noal., evident) propellant is required required through with reusability. reusability. Therefore, STS (Lorenzini et.no al., 2000) propellant much less evident) is through much attention is paid to toss payload through Space with reusability. Therefore, STS (Lorenzini et. al., 2000) much (Lorenzini attention et. is paid to toss through Space withpayload reusability. Therefore, STS al., 2000) Tethered Systems. much attention is paid to toss payload through Tethered Systems.is paid to toss payload through Space much attention Space Tethered Systems. Tethered Systems. The The concept concept of of momentum momentum exchange exchange tethered tethered system system was firstly introduced by Schechter et al. Feasibility of The concept of momentum exchange system was The firstlyconcept introduced by Schechter et al.tethered Feasibility of of momentum exchange tethered system cislunar transfer mission was discussed by Forward et al, was firstly introduced by Schechter et al. Feasibility cislunar transfer mission was discussed by Forward et al, was firstly introduced by Schechter et al. Feasibility of of proving the goodmission potentialwas of spinning spinning tether system. et Two cislunar the transfer mission was discussed tether by Forward Forward etTwo al, proving good potential of system. cislunar transfer discussed by al, tossing schemes into Geostationary Orbittether were system. proposedTwo by proving schemes the good goodinto potential of spinning spinning tether system. Two tossing Geostationary Orbit were proposed by proving the potential of Lorenzini (2000) and Hoyt mission tossing schemes schemes into Geostationary Orbitsystematic were proposed proposed by Lorenzini (2000)into andGeostationary Hoyt et et al al with with systematic mission tossing Orbit were by design. These two schemes systems revolve Lorenzini (2000) Hoyt et al design. These two and schemes set spinning systems to to mission revolve Lorenzini (2000) and Hoyt set et spinning al with with systematic systematic mission along orbits to better tossing design. These schemes set revolve along elliptic orbits to achieve achieve bettersystems tossingto abilities, design. elliptic These two two schemes set spinning spinning systems toabilities, revolve however, Misra found that in-plane motion will be chaotic in along elliptic orbits to achieve better tossing abilities, however, Misra orbits found that in-plane motion be chaotic in along elliptic to achieve better will tossing abilities, large orbit eccentricity proposed in schemes mentioned above however, Misra found that in-plane motion will be chaotic in large orbitMisra eccentricity proposed in schemes mentioned above however, found that in-plane motion will be chaotic in large orbit eccentricity proposed in schemes mentioned above large orbit eccentricity proposed in schemes mentioned above
Williams Williams et.al et.al considered considered dynamic dynamic and and control control for for rendezvous stage using optimal theories. Zhu Z.H. al Williams et.al considered dynamic and rendezvous stage considered using optimal theories.and Zhucontrol Z.H. et et for al Williams et.al dynamic control for proposed some important issues during manoeuver, valuing rendezvous stage using optimal theories. Zhu Z.H. et proposed important issues during manoeuver, rendezvoussome stage using optimal theories. Zhu Z.H.valuing et al al influence of perturbations on system. Further proposed important issues manoeuver, influence of different different perturbations on STS STS system. valuing Further proposed some some important issues during during manoeuver, valuing properties of tapered tapered tether structure were were studied by Yang Yang Y., influence different perturbations on system. Further properties of structure studied by Y., influence of of differenttether perturbations on STS STS system. Further proving that tapered tether has good performance in reducing properties of tapered tether structure were studied by Yang proving that tapered tether has good performance in reducing properties of tapered tether structure were studied by Yang Y., Y., system mass and ensuring ensuring strength at performance the same same time. time. provingmass that tapered tapered tetherstrength has good goodat performance in reducing reducing system and the proving that tether has in system mass mass and and ensuring ensuring strength strength at at the the same same time. time. system Despite Despite promising promising potential potential of of STS STS payload payload transportation, problems are to solved Despite promising potential of transportation, some problems are necessary necessary to be bepayload solved Despite some promising potential of STS STS payload before its application: Choice of tossing orbit parameters transportation, some problems are necessary to be before its application: Choice of orbitto parameters transportation, some problems are tossing necessary be solved solved (Lorenzini 2000, Rendezvous and before Choice of orbit (Lorenzini 2000, Hoyt); Hoyt); Rendezvous dynamics and its its before its its application: application: Choice of tossing tossingdynamics orbit parameters parameters elongation payload acceleration before tossing (Lorenzini 2000, Rendezvous dynamics its elongation (Williams); payload acceleration before and tossing (Lorenzini (Williams); 2000, Hoyt); Hoyt); Rendezvous dynamics and its and its control. Extensive work has been performed on first elongation (Williams); payload acceleration before tossing and its control. Extensive work has been performed on first elongation (Williams); payload acceleration before tossing two problems. Orbital design design in been system level and and its and its its control. Extensive Extensive work has has been performed on first first two problems. Orbital in system level its and control. work performed on economic analysis has already already been maturelylevel studied, and two problems. Orbital design been in system system level and and its economic analysis has maturely studied, two problems. Orbital design in and its rendezvous manoeuvre extended by economic analysis analysis has been already been maturely maturely studied, and rendezvous manoeuvre been extended by both both controller controller and economic has already been studied, rendezvous device However, little attention has been by rendezvous manoeuvre device approaches. approaches. However, littlecontroller attention and has manoeuvre been extended extended by both both controller and been paid to the tossing stage. Although orbital dynamics rendezvous device approaches. However, little attention has been paid todevice the tossing stage.However, Althoughlittle orbital dynamics rendezvous approaches. attention has were analysed by Lorenzini(2000) and Hoyt, control of been paid to the tossing stage. Although orbital dynamics were analysed Lorenzini(2000) and Hoyt, of been paid to thebytossing stage. Although orbitalcontrol dynamics acceleration maneuvre was not considered in their schemes. were analysed by Lorenzini(2000) and Hoyt, control of acceleration maneuvre was not considered in theircontrol schemes. were analysed by Lorenzini(2000) and Hoyt, of Possible chaotic motion in such elliptic orbits also implies acceleration maneuvre was not considered in their schemes. Possible chaotic motion in such elliptic orbits also implies acceleration maneuvre was not considered in their schemes. necessity of setting setting spinning systems in circular circular orbits. Possible chaotic chaotic motion in such such elliptic orbitsorbits. also implies implies necessity of spinning systems in Possible motion in elliptic orbits also necessity of of setting setting spinning spinning systems systems in in circular circular orbits. orbits. necessity Based on on review review above, above, transport transport mission mission design design from from Based Low Earth Orbit (LEO) into Earth-Moon Transfer Based on review above, transport mission design Orbit from Low Based Earth on Orbit (LEO) intotransport Earth-Moon Transfer Orbit review above, mission design from (EMTO) proposed in with Low (LEO) into Earth-Moon Transfer Orbit (EMTO) is Orbit proposed in this this paper, with spinning spinning tether Low Earth Earthis Orbit (LEO) into paper, Earth-Moon Transfer tether Orbit systems circular Analysis of (EMTO) proposed in paper, with systems revolving in circular orbits. Analysistether of (EMTO) is isrevolving proposed in in this this paper, orbits. with spinning spinning tether systems revolving in circular orbits. Analysis systems revolving in circular orbits. Analysis of of
Copyright 2016 IFAC 272 Copyright©© ©2016, 2016IFAC IFAC (International Federation of Automatic Control) 272Hosting by Elsevier Ltd. All rights reserved. 2405-8963 Copyright © 2016 IFAC 272 Peer review under responsibility of International Federation of Automatic Copyright © 2016 IFAC 272Control. 10.1016/j.ifacol.2016.09.047
IFAC ACA 2016 August 21-25, 2016. Quebec, Canada
Lu Hong Shi et al. / IFAC-PapersOnLine 49-17 (2016) 272–277
273
Π = m* / mtotal ;
rendezvous/tossing manoeuvre are performed, so that boundary conditions for rendezvous/tossing could be calculated. Non-linear optimal theories are adopted to generate optimal trace for acceleration of system motion. For the case of insufficient thrust, Hamilton on/off control of spinning motion energy is proposed to drive system to desired motion.
m m m m* = Lk 2 (m1 + t )(m 2 + t ) / mtotal − t ; 2 2 6 mtotal = m1 + m2 + mt ;
(2)
The mass of rendezvous/toss device changes when payload is captured or tossed, i.e.:
2．MODEL OF SPINNING TETHER SYSTEM
m20 = m2 , before rendezvous and after toss m12 = m2 + m p , after rendezvous and before toss
The system configuration is shown as Fig. 1, including spacecraft, tether and rendezvous/toss device m1 , mt , m2 . Tether is considered as a massive rigid rod in this paper, and system mass center revolves around unperturbed Keplerian orbit with true anomaly υ . Effect of spacecraft, device, payload ( m p ) and tether mass on spinning motion is
(3)
The dynamic model before/after rendezvous takes the same form of (2),with different values of m 2 described in (3). To analyse the dynamic performance of space tethered system, Hamilton energy of STS spinning motion is adopted in this paper (Ellis, J.R.):
considered in this paper. Spinning motion around center mass is described by in-plane angle and angular speed θ , θ . Spinning motion is controlled by engine thrust with in-plane install angle α . Tether is fully deployed for spinning motion ( L = Lk , L = L = 0 ). Size of spacecraft, payload and device are omitted compared to the length of tether (100km).
H = θ '2 + 3sin 2 θ
(4)
where, θ ' = dθ / dτ denotes in-plane angular speed in nondimensional form. According to (4), in-plane motion is specified by 3 parts. When H < 3 , system is in pendulum motion (oscillation around vertical position θ = 0, π ). When H = 3 , system becomes spinning but highly unstable because angular velocity ω = θ is zero at θ = π / 2,3π / 2 , tether may become slack around this level. When H > 3 , system spins, and higher spinning velocity promises better stability due to its centrifugal force. System in same Hamilton energy will oscillate/spin in identical periodical motion, despite the different transient states of θ , θ (Ellis, J.R.).
3．PAYLOAD TOSSING AND OPEN-LOOP TRACE DESIGN Fig1.Configuration of STS 3.1 General idea of payload transport mission
Equation of in-plane motion is described through differential equation below (Aslanov et.al.): 2 2 = R υ 2 − m + 3mΠLk (1 − 3cos θ ) R s s 2 Rc 4 Rs 2 2υ R s 3m sin 2θ M ΠLk 2 (1 ) + *θ θ = − + 3 2 Rc 2 Rc Rc m
υ =
m a (1 − e 2 )3 3
Unlike conventional rocket, payload transport by STS is completed by its spinning motion like slinger, rather impulse thrust. One general transport mission is as follows: Payload revolves in Low Transfer Orbit (LTO) near Earth, and captured by STS in LEO orbit, and tossed into elliptical transfer orbit (MTO). Payload would be captured again by second STS in MEO orbit, and tossed into the elliptical GTO orbit. Third STS in GEO orbit will pick up payload and toss it into the final cislunar transfer orbit. The idea of transport mission is presented in Fig.2:
(1)
(1 + e cos θ ) 2
where υ , e, a is the true anomaly, eccentricity and semimajor axis radius, M θ = FL cos α is the thrust moment produced by thrust force couple F , F ′ . Mass parameters are given by: 273
IFAC ACA 2016 274 August 21-25, 2016. Quebec, Canada
Lu Hong Shi et al. / IFAC-PapersOnLine 49-17 (2016) 272–277
radius. L1 = Lk m1 / mtotal is the distance from rendezvous/toss device to system mass centre before rendezvous. Payload eccentricity is limited to be less than 0.5, so that 30s rendezvous window is ensured (Lorenzini, 2004). Therefore, the orbital parameters calculated from (5) should be checked. Generally speaking, Single stage tossing is best in rendezvous accuracy, but tether mass will be much heavier due to the large velocity increment of cislunar transfer, and rendezvous/tossing impact could be unacceptable. Multistage system is more practical for the consideration of decreasing tether mass and rendezvous/tossing perturbations, but too many stages will cause rendezvous difficulties (Lorenzini 2000, Hoyt). Fig 2.Trasport mission 3.3 Rendezvous/tossing conditions
Spinning tethers systems revolve in circular orbits and transfer orbits of payload are elliptic. The number of revolving STSs is variable referring to mission requirements, it could be either single, double or multiple stage.
To achieve rendezvous/toss through STS, there are two requirements must be satisfied: 1.match of tether tip (rendezvous/toss device) and payload position; 2.match of tip and payload velocity, such requirements could be satisfied through careful selection of payload and system orbits, and calculation of spinning motion.
3.2 Choice of orbit parameters Payload and STS rendezvous in certain points (perigee or apogee usually, where tether in vertical position ( θ = 0, π )), the difference in orbit motion could be compensated by the spinning tether. Therefore, careful selection of payload and system orbits is one of the most important issues.
Match of tether tip (rendezvous/toss device) and payload position could be solved through orbital design satisfying (5), tether tip will pick up payload at vertical position when STS and payload reach perigee/apogee of payload orbit at the same moment(Lorenzini, 2004). To match speed of tether tip and payload, it quires proper STS spinning motion to let tether tip revolve together with payload:
All orbits are considered equatorial in this paper (orbital inclination iorb = 0 ). As stated in introduction, circular orbit is better in dynamic stability than elliptic orbit. The most ideal rendezvous/toss points are perigee or apogee of payload orbit, when STS spinning across its vertical position ( θ = 0, π ). Rendezvous/toss in such condition ensures STS tether tip(rendezvous/toss device) is revolving exactly at the same speed with payload, and reduces manoeuvre impact to minimum level (Williams).To ensure repeatable rendezvous opportunities, spinning tether system and payload should also meet periodically (Lorenzini, 2000).
vtip = v pc where
vtip = θ L;
Rapog
(7)
Ω s Rs vs Vi = = m ; v p a (1- e 2 ) p p
3
R peri = 2a p − Ra ; ep =
v pc = ΦVi ;
Rs
; M2 = Rs − L1 ;
v pc is the orbital velocity relative to STS system.
As verse 3.2, STS system revolves in circular orbit, and payload revolves in elliptical transfer orbit. The motion of tether tip and payload are given by:
If the orbit of STS is decided (e.g. geostationary orbit), payload transfer orbit parameters are given by:
ap =
(6)
(5)
here, Vi is the motion of payload and system described in inertial coordinate XOY. Φ is the coordinate conversion matrix:
Rapog − R peri Rapog + R peri
where Rapog , R peri are the apogee and perigee radius of payload transfer orbit. M is the period ratio of STS and payload orbit, for example, M=1.5 means system will meet payload 3 orbits later, when payload revolves 2 orbits. m , a p
T
cos θ (sin 2 ϑs − cos 2 ϑs ), Φ = [(e p + cos ϑ p )(sin ϑs sin θ + cos ϑs cos θ ) − (8) sin ϑ (sin ϑ cos θ − cos ϑ sin θ )] p s s
denote gravity constant and payload semi-major axis Eq(6) then takes the form: 274
IFAC ACA 2016 August 21-25, 2016. Quebec, Canada
θrendezvous =
v pc L1
Lu Hong Shi et al. / IFAC-PapersOnLine 49-17 (2016) 272–277
=
Φ1Vi L1
system will spin in desired motion at tossing/rendezvous position.
(9)
Using(4), Hamilton energy on/off control takes the form:
For tossing stage, payload is tossed from device, thus position match is automatically satisfied. The way of providing velocity increment for payload (match of velocity) is also through spinning motion. Eq(6) takes the form:
θtoss =
vtoss =
F max , if H < H1 FH = F min , if H > H1 0, if H = H1
vtoss − υs (R s + L2 cos θ ) ; L2
L2 = Lk
m1 ; mtotal + m p
m R peri
+
(13)
where H1 is the desired Hamilton energy level. (10)
4．NUMERICAL RESULTS
2 Rapog
Based on cislunar transfer mission, an orbit scheme and optimal control is developed according to chapter 3.
Rapog + R peri
orbital parameters R peri , Rapog should be calculated by Eq(5).
4.1 Orbit design and system configuration
L2 is the distance from tip to mass centre before toss. The current position of payload/tether tip ( Rs + L2 ) is the perigee of new transfer orbit, and apogee which is referred to the position to the next target position (second STS or Moon).
According to analysis in verse 3.2 and 3.3, the tossing mission is divided into 3 stages, with 3 STS systems revolving in LEO, MEO and GEO circular orbits. Mass of STS system is set to be 24,000kg, which is the capacity of single rocket launch, while payload and rendezvous/toss device assumed to be mp = 1000kg , m2 = 500kg . Ion engine
3.4 Open-loop trace design Once orbit design is completed, control for angular speed transition from rendezvous to tossing is required. In payload tossing mission, the trace of angular speed should be de/accelerated into desired state, with minimum fuel and sufficient small time. Therefore, optimal control of angular speed is necessary for open-loop reference control.
is adopted to provide external torque M θ for transport mission, with very little propellant requirements and smallcontinuous thrust force (Lorenzini, 2000). The nearly operational VX-200 of NASA provides thrust as large as 5N (type of Electro-thermal Plasma) (Squire et.al.), while 15N (accelerated by electric arcs) as recorded in Australian laboratory. Tapered-tether of 100km is adopted rather than conventional cylindrical tether, and is made of Spectra-2000 material. Detail of tapered-tether could be seen from research of Lorenzini(2000) et.al and Yang Y et.al. In the numerical simulation of this paper, safety factor of tether is set to be 1.75, maximum thrust is set to be ±14N , Engine install angle α is π / 2 .
Non-linear optimal control problem could be expressed as: finding control and state pair {U (t), X(t)} , so that Bolza cost functions is min/maximized within time range: tf
minJ = G(x(t0 ),t0 ,x(t f ),x(t f ))+ ∫ L(x(t),u(t),t)dt (11) to
where J is the performance index, x(t ), u (t ) are state and control variations, which are the non-linear function of time.
Orbital parameters of each STS system are shown in Table. 1:
State, path and border constraints should also be met:
= f(x(t),u(t),t); x(t) C(x(t),u(t),t) ≤ 0;
275
Table1. Orbital parameters of 3 stages toss mission (12)
ϕ ( x(t0 ), t0 , x(t f ), t f ) = 0;
Perigee/apogee
Eccentricity/P
θ (Rendezvous/
Radius(km)
eriod ratio M
toss, rad/s)
0.148/1.25
0.00501/0.0117
0.361/1.6
0.00969/0.00819
Orbit
Non-linear optimal control problem described above could be calculated by transforming complicated non-linear control into non-linear programming (NLP). Such algorithm could be completed by discrete pseudo-spectral approximation method, numerical calculations could be completed by Gpops based on SNOPT algorithm (Rao et,al.). When system mass is too large compared to rather small thrust case, trace designed by conventional non-linear optimal control maybe not practical. For this consideration, Hamilton energy on/off control provides an alternative option. It drives system into desired Hamilton energy, which ensures 275
LTO
6607/8904
LEO
9000
MTO
9095/19385
MEO
19480
GTO
19577/42068
GEO
42164
EMTO
42566/384000
0.365/1.6 0.00651/0.0115 0.802
IFAC ACA 2016 276 August 21-25, 2016. Quebec, Canada
Lu Hong Shi et al. / IFAC-PapersOnLine 49-17 (2016) 272–277
4.2 Open-loop trace of angular speed To transport payload into desired cislunar orbit, payload should be tossed according to the spinning velocity shown in table1, which are viewed as the constraints of optimal problems shown in equation (11,12). According to mission requirements, cost function is set tf
as: J = ∫ (t 2 + u (t) 2 )dt . It requires system to reach desired to
tossing angular velocity from rendezvous state with relatively short time and minimal propellant at the same time. Optimal traces in GEO, MEO and LEO orbits are presented in Fig.3~5:
Fig5. Optimal control in GEO orbit In GEO and MEO orbits, optimal traces indicate the similar performance, and thrust outputs do not exceed the maximum thrust. For example of GEO STS system, in the first 0.7 orbit, angular speed is growing with maximum thrust 14N; in the final period of 0.7 ~ 0.75 orbit, angular speed has already reached desired state, but in-plane angle θ has not reached tossing position( θ = nπ , n = 1,3,5,..., N ) yet, therefore thrust is still required to drive system into the nearest tossing position, and control output is vibrating around 0N to keep spinning motion in constant value under periodical perturbation of gravity. When the nearest tossing position is reached ( θ = 191π ), the acceleration manoeuvre is completed, and payload is ready for tossing. Fig 3. Optimal control in LEO orbit
However, in LEO STS system, insufficient thrust problem occurs. To control angular speed in constant value, maximum thrust is140N, which is almost 10 times larger than available thrust. If thrust is limited to be 14N in Fig4, angular speed is not able to be constant value to reach the nearest tossing position. Therefore, Hamilton on/off control is adopted. Numerical result is presented in Fig.6
Fig4. Optimal control in MEO orbit Fig 6. Hamilton control in LEO orbit 276
IFAC ACA 2016 August 21-25, 2016. Quebec, Canada
Lu Hong Shi et al. / IFAC-PapersOnLine 49-17 (2016) 272–277
System reaches desired motion (at tossing position) under Hamilton on/off control law. However, it is worth noting that such control is proposed for insufficient thrust, and does not ensure to be the optimal one. Therefore, it is better to use control law calculated by pseudo-spectral method when thrust level could meet the requirement of station-keeping.
277
Lorenzini, E. C. (2004). Error-tolerant technique for catching a spacecraft with a spinning tether. Journal of Vibration and Control, 10(10), 1473-1491. Misra, A. K. (2008). Dynamics and control of tethered satellite systems.Acta Astronautica, 63(11), 1169-1177. Rao, A. V., Benson, D. A., Darby, C., Patterson, M. A., Francolin, C., Sanders, I., & Huntington, G. T. (2010). Algorithm 902: Gpops, a matlab software for solving multiple-phase optimal control problems using the gauss Transactions on pseudospectral method. ACM Mathematical Software (TOMS), 37(2), 22.
5．CONCLUSIONS Complete cislunar payload tossing mission using spinning tether system was proposed, and optimal control for tossing stage was calculated. For the case of insufficient thrust of Ion engine, control law was modified by Hamilton equation. Simulation shows that payload is able to be tossed into lunar transfer orbit under mission design and optimal control, with economy and reusability.
Schechter, H. B. (1964). Dumbbell librations in elliptic orbits. AIAA Journal,2(6), 1000-1003. Squire, J. P., Olsen, C. S., Chang Díaz, F. R., Cassady, L. D., Longmier, B. W., Ballenger, M. G., ... & Bering III, E. A. (2011, September). VASIMR® VX-200 operation at 200 kW and plume measurements: future plans and an ISS EP test platform. In 32nd International Electric Propulsion Conference, Weisbaden.
However, further research is still required. The closeloop tracking control under multiple perturbations, including elastic deformation, gravitational perturbation, rendezvous/toss error etc., is still necessary to drive functioning system in realistic space environment.
Van Pelt, M. (2009). Space tethers and space elevators. Springer Science & Business Media.
6. ACKNOWLEDGEMENTS
Williams, P. (2006). Dynamics and control of spinning tethers for rendezvous in elliptic orbits. Journal of Vibration and Control, 12(7), 737-771.
The research method and writing of this paper is benefited much from inspiring advice of Prof. Zhu Z.H., York University.
Yang, Y., Qi, N., Liu, Y., Xu, Z., & Ma, M. (2015). Influence of tapered tether on cislunar payload transmission system Science and and energy analysis. Aerospace Technology, 46, 210-220.
The research is supported by 2011 International Science and Technology Cooperation Program of China, also Research and Development Program of Science and Technology of Shanxi Province(2013KW09-02).
Zhu, Z. H., & Zhong, R. (2011). Deorbiting dynamics of electrodynamic tether. International Journal of Aerospace and Lightweight Structures (IJALS), 1(1).
REFERENCES Aslanov, V. S., & Ledkov, A. S. (2012). Dynamics of tethered satellite systems. Elsevier. Ellis, J. R. (2010). Modeling, Dynamics, and Control of Tethered Satellite Systems (Doctoral dissertation, Virginia Polytechnic Institute and State University). Forward, R. L. (1991). Tether transport from LEO to the lunar surface. AIAA paper, 91, 2322. Hoyt, R. P., & Uphoff, C. (2000). Cislunar tether transport system. Journal of Spacecraft and Rockets, 37(2), 177186. Lorenzini, E. C., Cosmo, M. L., Kaiser, M., Bangham, M. E., Vonderwell, D. J., & Johnson, L. (2000). Mission analysis of spinning systems for transfers from low orbits of Spacecraft and to geostationary. Journal Rockets, 37(2), 165-172.
277
ScienceDirect IFAC-PapersOnLine 49-17 (2016) 272–277
Optimal Optimal Control Control of of Payload Payload Tossing Tossing Using Using Space Space Tethered Tethered System System Optimal Control of Payload Tossing Using Space Tethered System Optimal Control of Payload Tossing Using Space Tethered System Lu Hong Shi*, Wang Changqing**, Lu Hong Shi*, Wang Changqing**, Lu Hong Lu Hong Shi*, Shi*, Wang Wang Changqing**, Changqing**, Zabolotnov Zabolotnov Yuriy***, Yuriy***, Li Li Aijun**** Aijun**** Zabolotnov Yuriy***, Li Aijun**** Zabolotnov Yuriy***, Li Aijun**** 710072, China ** Northwestern Northwestern Polytechnical Polytechnical University, University, Xi’an Xi’an Shanxi Shanxi 710072, China ** Northwestern Polytechnical University, Xi’an Shanxi Northwestern Polytechnical University, Xi’an Shanxi 710072, 710072, China China PRC PRC (Tel: (Tel: 86-029-888431383; 86-029-888431383; e-mail: e-mail: [email protected]) [email protected]) PRC e-mail: PRC (Tel: (Tel: 86-029-888431383; 86-029-888431383; e-mail: [email protected]) [email protected]) ** Northwestern Northwestern Polytechnical University, University, Xi’an Shanxi Shanxi 710072, 710072, China China ** Polytechnical Xi’an ** Northwestern Northwestern Polytechnical Polytechnical University, University, Xi’an Xi’an Shanxi Shanxi 710072, 710072, China China ** PRC (Tel: (Tel: 86-029-888431383; 86-029-888431383; e-mail: e-mail: [email protected]) [email protected]) PRC PRC (Tel: (Tel: 86-029-888431383; e-mail: e-mail: [email protected]) PRC *** Samara86-029-888431383; State Aerospace Aerospace University, University,[email protected]) Samara 443086, 443086, Russia Russia *** Samara State Samara *** Samara Samara State State Aerospace Aerospace University, University, Samara Samara 443086, 443086, Russia Russia *** RUS RUS (Tel: (Tel: 7-846995-62-65; 7-846995-62-65; e-mail: e-mail: [email protected]) [email protected]) RUS RUS (Tel: (Tel: 7-846995-62-65; 7-846995-62-65; e-mail: e-mail: [email protected]) [email protected]) ***** ***** Northwestern Northwestern Polytechnical Polytechnical University, University, Xi’an Xi’an Shanxi Shanxi 710072, 710072, China China ***** Northwestern Polytechnical University, Xi’an Shanxi 710072, China ***** Northwestern Polytechnical University, Xi’an Shanxi 710072, China PRC PRC (Tel: (Tel: 86-029-888431383; 86-029-888431383; e-mail: e-mail: [email protected]) [email protected]) PRC PRC (Tel: (Tel: 86-029-888431383; 86-029-888431383; e-mail: e-mail: [email protected]) [email protected]) Abstract: To To transport transport payload payload into into lunar lunar transfer transfer orbit orbit using using spinning spinning tether tether system, system, reusable reusable payload payload Abstract: tossing was based cislunar transfer and law was Abstract: To transport transport payload intoon lunar transfer orbitmission, using spinning spinning tethercontrol system, reusable payload tossing system system was proposed proposed based onlunar cislunar transfer mission, and optimal optimal control law was developed developed Abstract: To payload into transfer orbit using tether system, reusable payload for tossing stage. case insufficient thrust, Hamilton equation was solve tossing system was based on transfer mission, and law for payload payload tossing stage. In In the the case ofcislunar insufficient thrust, Hamilton equationcontrol was adopted adopted todeveloped solve the the tossing system was proposed proposed based onof cislunar transfer mission, and optimal optimal control law was wasto developed problem. The numerical result shows that the tossing design is able to transport payload into desired for payload tossing stage. In the case of insufficient thrust, Hamilton equation was adopted to solve the problem. Thetossing numerical result shows the tossing design is able to transport into desired orbit, for payload stage. In the casethat of insufficient thrust, Hamilton equation payload was adopted to solveorbit, the and law meet mission Compared rocket transport, problem. The result that the is transport desired orbit, and the the control control law could could meetshows mission requirement. Compared toto conventional rocketinto transport, much problem. The numerical numerical result shows thatrequirement. the tossing tossing design design is able ableto toconventional transport payload payload into desiredmuch orbit, less propellant is required by the designed system with reusability. and the control law could meet mission requirement. Compared to conventional rocket transport, much less propellant required the mission designedrequirement. system with Compared reusability.to conventional rocket transport, much and the controlislaw could by meet less propellant is by the system with reusability. less propellant is required required byFederation the designed designed system withOptimal reusability. © 2016, IFAC (International of Automatic by Hamilton Elsevier Ltd. All rights reserved. Keywords: Space Tethered Tossing, Control, Energy. Keywords: Space Tethered System, System, Payload Payload Tossing,Control) OptimalHosting Control, Hamilton Energy. Keywords: Keywords: Space Space Tethered Tethered System, System, Payload Payload Tossing, Tossing, Optimal Optimal Control, Control, Hamilton Hamilton Energy. Energy.
1. 1. INTRODUCTION INTRODUCTION 1. 1. INTRODUCTION INTRODUCTION Space Tethered Tethered System System (STS) (STS) consists consists of of two two or or Space multiple satellites connected by (STS) thin, long long tether. of It has has great Space Tethered System (STS) consists of twogreat or multiple satellites connected by thin, tether. It Space Tethered System consists two or potential in transportation, etc. multiple satellites satellites connected by thin, thin,space longdebris tether.removal It has has great great potential in payload payload transportation, space debris removal etc. multiple connected by long tether. It (Van Pelt et. al.) potential transportation, (Van Peltin et.payload al.) potential in payload transportation, space space debris debris removal removal etc. etc. (Van Pelt et. al.) (VanNow, Pelt et.the al.) most likely way to execute deep space Now, the most likely way to execute deep space investigation payload is Now, likely way execute space investigation ormost payload transfer mission is through through Now, the theor most likely transfer way to to mission execute deep deep space conventional rocket. However, rocket is extremely costing investigation or payload transfer mission is through conventional However, rocket mission is extremely costing investigation rocket. or payload transfer is through and vast space spacerocket. debris would would be produced produced for large scale scale deep conventional However, rocket extremely costing and vast debris be large deep conventional rocket. However, rocket is isfor extremely costing space missions up ahead. Compared to conventional rockets, and vast space debris would be produced for large scale space missions up ahead. Compared to conventional rockets, and vast space debris would be produced for large scale deep deep much less (or even even no evident) evident) propellant is required required rockets, through space less missions up ahead. ahead. Compared to conventional conventional rockets, much (or no propellant is through space missions up Compared to with Therefore, STS et. 2000) much (Lorenzini less (or (or even even noal., evident) propellant is required required through with reusability. reusability. Therefore, STS (Lorenzini et.no al., 2000) propellant much less evident) is through much attention is paid to toss payload through Space with reusability. Therefore, STS (Lorenzini et. al., 2000) much (Lorenzini attention et. is paid to toss through Space withpayload reusability. Therefore, STS al., 2000) Tethered Systems. much attention is paid to toss payload through Tethered Systems.is paid to toss payload through Space much attention Space Tethered Systems. Tethered Systems. The The concept concept of of momentum momentum exchange exchange tethered tethered system system was firstly introduced by Schechter et al. Feasibility of The concept of momentum exchange system was The firstlyconcept introduced by Schechter et al.tethered Feasibility of of momentum exchange tethered system cislunar transfer mission was discussed by Forward et al, was firstly introduced by Schechter et al. Feasibility cislunar transfer mission was discussed by Forward et al, was firstly introduced by Schechter et al. Feasibility of of proving the goodmission potentialwas of spinning spinning tether system. et Two cislunar the transfer mission was discussed tether by Forward Forward etTwo al, proving good potential of system. cislunar transfer discussed by al, tossing schemes into Geostationary Orbittether were system. proposedTwo by proving schemes the good goodinto potential of spinning spinning tether system. Two tossing Geostationary Orbit were proposed by proving the potential of Lorenzini (2000) and Hoyt mission tossing schemes schemes into Geostationary Orbitsystematic were proposed proposed by Lorenzini (2000)into andGeostationary Hoyt et et al al with with systematic mission tossing Orbit were by design. These two schemes systems revolve Lorenzini (2000) Hoyt et al design. These two and schemes set spinning systems to to mission revolve Lorenzini (2000) and Hoyt set et spinning al with with systematic systematic mission along orbits to better tossing design. These schemes set revolve along elliptic orbits to achieve achieve bettersystems tossingto abilities, design. elliptic These two two schemes set spinning spinning systems toabilities, revolve however, Misra found that in-plane motion will be chaotic in along elliptic orbits to achieve better tossing abilities, however, Misra orbits found that in-plane motion be chaotic in along elliptic to achieve better will tossing abilities, large orbit eccentricity proposed in schemes mentioned above however, Misra found that in-plane motion will be chaotic in large orbitMisra eccentricity proposed in schemes mentioned above however, found that in-plane motion will be chaotic in large orbit eccentricity proposed in schemes mentioned above large orbit eccentricity proposed in schemes mentioned above
Williams Williams et.al et.al considered considered dynamic dynamic and and control control for for rendezvous stage using optimal theories. Zhu Z.H. al Williams et.al considered dynamic and rendezvous stage considered using optimal theories.and Zhucontrol Z.H. et et for al Williams et.al dynamic control for proposed some important issues during manoeuver, valuing rendezvous stage using optimal theories. Zhu Z.H. et proposed important issues during manoeuver, rendezvoussome stage using optimal theories. Zhu Z.H.valuing et al al influence of perturbations on system. Further proposed important issues manoeuver, influence of different different perturbations on STS STS system. valuing Further proposed some some important issues during during manoeuver, valuing properties of tapered tapered tether structure were were studied by Yang Yang Y., influence different perturbations on system. Further properties of structure studied by Y., influence of of differenttether perturbations on STS STS system. Further proving that tapered tether has good performance in reducing properties of tapered tether structure were studied by Yang proving that tapered tether has good performance in reducing properties of tapered tether structure were studied by Yang Y., Y., system mass and ensuring ensuring strength at performance the same same time. time. provingmass that tapered tapered tetherstrength has good goodat performance in reducing reducing system and the proving that tether has in system mass mass and and ensuring ensuring strength strength at at the the same same time. time. system Despite Despite promising promising potential potential of of STS STS payload payload transportation, problems are to solved Despite promising potential of transportation, some problems are necessary necessary to be bepayload solved Despite some promising potential of STS STS payload before its application: Choice of tossing orbit parameters transportation, some problems are necessary to be before its application: Choice of orbitto parameters transportation, some problems are tossing necessary be solved solved (Lorenzini 2000, Rendezvous and before Choice of orbit (Lorenzini 2000, Hoyt); Hoyt); Rendezvous dynamics and its its before its its application: application: Choice of tossing tossingdynamics orbit parameters parameters elongation payload acceleration before tossing (Lorenzini 2000, Rendezvous dynamics its elongation (Williams); payload acceleration before and tossing (Lorenzini (Williams); 2000, Hoyt); Hoyt); Rendezvous dynamics and its and its control. Extensive work has been performed on first elongation (Williams); payload acceleration before tossing and its control. Extensive work has been performed on first elongation (Williams); payload acceleration before tossing two problems. Orbital design design in been system level and and its and its its control. Extensive Extensive work has has been performed on first first two problems. Orbital in system level its and control. work performed on economic analysis has already already been maturelylevel studied, and two problems. Orbital design been in system system level and and its economic analysis has maturely studied, two problems. Orbital design in and its rendezvous manoeuvre extended by economic analysis analysis has been already been maturely maturely studied, and rendezvous manoeuvre been extended by both both controller controller and economic has already been studied, rendezvous device However, little attention has been by rendezvous manoeuvre device approaches. approaches. However, littlecontroller attention and has manoeuvre been extended extended by both both controller and been paid to the tossing stage. Although orbital dynamics rendezvous device approaches. However, little attention has been paid todevice the tossing stage.However, Althoughlittle orbital dynamics rendezvous approaches. attention has were analysed by Lorenzini(2000) and Hoyt, control of been paid to the tossing stage. Although orbital dynamics were analysed Lorenzini(2000) and Hoyt, of been paid to thebytossing stage. Although orbitalcontrol dynamics acceleration maneuvre was not considered in their schemes. were analysed by Lorenzini(2000) and Hoyt, control of acceleration maneuvre was not considered in theircontrol schemes. were analysed by Lorenzini(2000) and Hoyt, of Possible chaotic motion in such elliptic orbits also implies acceleration maneuvre was not considered in their schemes. Possible chaotic motion in such elliptic orbits also implies acceleration maneuvre was not considered in their schemes. necessity of setting setting spinning systems in circular circular orbits. Possible chaotic chaotic motion in such such elliptic orbitsorbits. also implies implies necessity of spinning systems in Possible motion in elliptic orbits also necessity of of setting setting spinning spinning systems systems in in circular circular orbits. orbits. necessity Based on on review review above, above, transport transport mission mission design design from from Based Low Earth Orbit (LEO) into Earth-Moon Transfer Based on review above, transport mission design Orbit from Low Based Earth on Orbit (LEO) intotransport Earth-Moon Transfer Orbit review above, mission design from (EMTO) proposed in with Low (LEO) into Earth-Moon Transfer Orbit (EMTO) is Orbit proposed in this this paper, with spinning spinning tether Low Earth Earthis Orbit (LEO) into paper, Earth-Moon Transfer tether Orbit systems circular Analysis of (EMTO) proposed in paper, with systems revolving in circular orbits. Analysistether of (EMTO) is isrevolving proposed in in this this paper, orbits. with spinning spinning tether systems revolving in circular orbits. Analysis systems revolving in circular orbits. Analysis of of
Copyright 2016 IFAC 272 Copyright©© ©2016, 2016IFAC IFAC (International Federation of Automatic Control) 272Hosting by Elsevier Ltd. All rights reserved. 2405-8963 Copyright © 2016 IFAC 272 Peer review under responsibility of International Federation of Automatic Copyright © 2016 IFAC 272Control. 10.1016/j.ifacol.2016.09.047
IFAC ACA 2016 August 21-25, 2016. Quebec, Canada
Lu Hong Shi et al. / IFAC-PapersOnLine 49-17 (2016) 272–277
273
Π = m* / mtotal ;
rendezvous/tossing manoeuvre are performed, so that boundary conditions for rendezvous/tossing could be calculated. Non-linear optimal theories are adopted to generate optimal trace for acceleration of system motion. For the case of insufficient thrust, Hamilton on/off control of spinning motion energy is proposed to drive system to desired motion.
m m m m* = Lk 2 (m1 + t )(m 2 + t ) / mtotal − t ; 2 2 6 mtotal = m1 + m2 + mt ;
(2)
The mass of rendezvous/toss device changes when payload is captured or tossed, i.e.:
2．MODEL OF SPINNING TETHER SYSTEM
m20 = m2 , before rendezvous and after toss m12 = m2 + m p , after rendezvous and before toss
The system configuration is shown as Fig. 1, including spacecraft, tether and rendezvous/toss device m1 , mt , m2 . Tether is considered as a massive rigid rod in this paper, and system mass center revolves around unperturbed Keplerian orbit with true anomaly υ . Effect of spacecraft, device, payload ( m p ) and tether mass on spinning motion is
(3)
The dynamic model before/after rendezvous takes the same form of (2),with different values of m 2 described in (3). To analyse the dynamic performance of space tethered system, Hamilton energy of STS spinning motion is adopted in this paper (Ellis, J.R.):
considered in this paper. Spinning motion around center mass is described by in-plane angle and angular speed θ , θ . Spinning motion is controlled by engine thrust with in-plane install angle α . Tether is fully deployed for spinning motion ( L = Lk , L = L = 0 ). Size of spacecraft, payload and device are omitted compared to the length of tether (100km).
H = θ '2 + 3sin 2 θ
(4)
where, θ ' = dθ / dτ denotes in-plane angular speed in nondimensional form. According to (4), in-plane motion is specified by 3 parts. When H < 3 , system is in pendulum motion (oscillation around vertical position θ = 0, π ). When H = 3 , system becomes spinning but highly unstable because angular velocity ω = θ is zero at θ = π / 2,3π / 2 , tether may become slack around this level. When H > 3 , system spins, and higher spinning velocity promises better stability due to its centrifugal force. System in same Hamilton energy will oscillate/spin in identical periodical motion, despite the different transient states of θ , θ (Ellis, J.R.).
3．PAYLOAD TOSSING AND OPEN-LOOP TRACE DESIGN Fig1.Configuration of STS 3.1 General idea of payload transport mission
Equation of in-plane motion is described through differential equation below (Aslanov et.al.): 2 2 = R υ 2 − m + 3mΠLk (1 − 3cos θ ) R s s 2 Rc 4 Rs 2 2υ R s 3m sin 2θ M ΠLk 2 (1 ) + *θ θ = − + 3 2 Rc 2 Rc Rc m
υ =
m a (1 − e 2 )3 3
Unlike conventional rocket, payload transport by STS is completed by its spinning motion like slinger, rather impulse thrust. One general transport mission is as follows: Payload revolves in Low Transfer Orbit (LTO) near Earth, and captured by STS in LEO orbit, and tossed into elliptical transfer orbit (MTO). Payload would be captured again by second STS in MEO orbit, and tossed into the elliptical GTO orbit. Third STS in GEO orbit will pick up payload and toss it into the final cislunar transfer orbit. The idea of transport mission is presented in Fig.2:
(1)
(1 + e cos θ ) 2
where υ , e, a is the true anomaly, eccentricity and semimajor axis radius, M θ = FL cos α is the thrust moment produced by thrust force couple F , F ′ . Mass parameters are given by: 273
IFAC ACA 2016 274 August 21-25, 2016. Quebec, Canada
Lu Hong Shi et al. / IFAC-PapersOnLine 49-17 (2016) 272–277
radius. L1 = Lk m1 / mtotal is the distance from rendezvous/toss device to system mass centre before rendezvous. Payload eccentricity is limited to be less than 0.5, so that 30s rendezvous window is ensured (Lorenzini, 2004). Therefore, the orbital parameters calculated from (5) should be checked. Generally speaking, Single stage tossing is best in rendezvous accuracy, but tether mass will be much heavier due to the large velocity increment of cislunar transfer, and rendezvous/tossing impact could be unacceptable. Multistage system is more practical for the consideration of decreasing tether mass and rendezvous/tossing perturbations, but too many stages will cause rendezvous difficulties (Lorenzini 2000, Hoyt). Fig 2.Trasport mission 3.3 Rendezvous/tossing conditions
Spinning tethers systems revolve in circular orbits and transfer orbits of payload are elliptic. The number of revolving STSs is variable referring to mission requirements, it could be either single, double or multiple stage.
To achieve rendezvous/toss through STS, there are two requirements must be satisfied: 1.match of tether tip (rendezvous/toss device) and payload position; 2.match of tip and payload velocity, such requirements could be satisfied through careful selection of payload and system orbits, and calculation of spinning motion.
3.2 Choice of orbit parameters Payload and STS rendezvous in certain points (perigee or apogee usually, where tether in vertical position ( θ = 0, π )), the difference in orbit motion could be compensated by the spinning tether. Therefore, careful selection of payload and system orbits is one of the most important issues.
Match of tether tip (rendezvous/toss device) and payload position could be solved through orbital design satisfying (5), tether tip will pick up payload at vertical position when STS and payload reach perigee/apogee of payload orbit at the same moment(Lorenzini, 2004). To match speed of tether tip and payload, it quires proper STS spinning motion to let tether tip revolve together with payload:
All orbits are considered equatorial in this paper (orbital inclination iorb = 0 ). As stated in introduction, circular orbit is better in dynamic stability than elliptic orbit. The most ideal rendezvous/toss points are perigee or apogee of payload orbit, when STS spinning across its vertical position ( θ = 0, π ). Rendezvous/toss in such condition ensures STS tether tip(rendezvous/toss device) is revolving exactly at the same speed with payload, and reduces manoeuvre impact to minimum level (Williams).To ensure repeatable rendezvous opportunities, spinning tether system and payload should also meet periodically (Lorenzini, 2000).
vtip = v pc where
vtip = θ L;
Rapog
(7)
Ω s Rs vs Vi = = m ; v p a (1- e 2 ) p p
3
R peri = 2a p − Ra ; ep =
v pc = ΦVi ;
Rs
; M2 = Rs − L1 ;
v pc is the orbital velocity relative to STS system.
As verse 3.2, STS system revolves in circular orbit, and payload revolves in elliptical transfer orbit. The motion of tether tip and payload are given by:
If the orbit of STS is decided (e.g. geostationary orbit), payload transfer orbit parameters are given by:
ap =
(6)
(5)
here, Vi is the motion of payload and system described in inertial coordinate XOY. Φ is the coordinate conversion matrix:
Rapog − R peri Rapog + R peri
where Rapog , R peri are the apogee and perigee radius of payload transfer orbit. M is the period ratio of STS and payload orbit, for example, M=1.5 means system will meet payload 3 orbits later, when payload revolves 2 orbits. m , a p
T
cos θ (sin 2 ϑs − cos 2 ϑs ), Φ = [(e p + cos ϑ p )(sin ϑs sin θ + cos ϑs cos θ ) − (8) sin ϑ (sin ϑ cos θ − cos ϑ sin θ )] p s s
denote gravity constant and payload semi-major axis Eq(6) then takes the form: 274
IFAC ACA 2016 August 21-25, 2016. Quebec, Canada
θrendezvous =
v pc L1
Lu Hong Shi et al. / IFAC-PapersOnLine 49-17 (2016) 272–277
=
Φ1Vi L1
system will spin in desired motion at tossing/rendezvous position.
(9)
Using(4), Hamilton energy on/off control takes the form:
For tossing stage, payload is tossed from device, thus position match is automatically satisfied. The way of providing velocity increment for payload (match of velocity) is also through spinning motion. Eq(6) takes the form:
θtoss =
vtoss =
F max , if H < H1 FH = F min , if H > H1 0, if H = H1
vtoss − υs (R s + L2 cos θ ) ; L2
L2 = Lk
m1 ; mtotal + m p
m R peri
+
(13)
where H1 is the desired Hamilton energy level. (10)
4．NUMERICAL RESULTS
2 Rapog
Based on cislunar transfer mission, an orbit scheme and optimal control is developed according to chapter 3.
Rapog + R peri
orbital parameters R peri , Rapog should be calculated by Eq(5).
4.1 Orbit design and system configuration
L2 is the distance from tip to mass centre before toss. The current position of payload/tether tip ( Rs + L2 ) is the perigee of new transfer orbit, and apogee which is referred to the position to the next target position (second STS or Moon).
According to analysis in verse 3.2 and 3.3, the tossing mission is divided into 3 stages, with 3 STS systems revolving in LEO, MEO and GEO circular orbits. Mass of STS system is set to be 24,000kg, which is the capacity of single rocket launch, while payload and rendezvous/toss device assumed to be mp = 1000kg , m2 = 500kg . Ion engine
3.4 Open-loop trace design Once orbit design is completed, control for angular speed transition from rendezvous to tossing is required. In payload tossing mission, the trace of angular speed should be de/accelerated into desired state, with minimum fuel and sufficient small time. Therefore, optimal control of angular speed is necessary for open-loop reference control.
is adopted to provide external torque M θ for transport mission, with very little propellant requirements and smallcontinuous thrust force (Lorenzini, 2000). The nearly operational VX-200 of NASA provides thrust as large as 5N (type of Electro-thermal Plasma) (Squire et.al.), while 15N (accelerated by electric arcs) as recorded in Australian laboratory. Tapered-tether of 100km is adopted rather than conventional cylindrical tether, and is made of Spectra-2000 material. Detail of tapered-tether could be seen from research of Lorenzini(2000) et.al and Yang Y et.al. In the numerical simulation of this paper, safety factor of tether is set to be 1.75, maximum thrust is set to be ±14N , Engine install angle α is π / 2 .
Non-linear optimal control problem could be expressed as: finding control and state pair {U (t), X(t)} , so that Bolza cost functions is min/maximized within time range: tf
minJ = G(x(t0 ),t0 ,x(t f ),x(t f ))+ ∫ L(x(t),u(t),t)dt (11) to
where J is the performance index, x(t ), u (t ) are state and control variations, which are the non-linear function of time.
Orbital parameters of each STS system are shown in Table. 1:
State, path and border constraints should also be met:
= f(x(t),u(t),t); x(t) C(x(t),u(t),t) ≤ 0;
275
Table1. Orbital parameters of 3 stages toss mission (12)
ϕ ( x(t0 ), t0 , x(t f ), t f ) = 0;
Perigee/apogee
Eccentricity/P
θ (Rendezvous/
Radius(km)
eriod ratio M
toss, rad/s)
0.148/1.25
0.00501/0.0117
0.361/1.6
0.00969/0.00819
Orbit
Non-linear optimal control problem described above could be calculated by transforming complicated non-linear control into non-linear programming (NLP). Such algorithm could be completed by discrete pseudo-spectral approximation method, numerical calculations could be completed by Gpops based on SNOPT algorithm (Rao et,al.). When system mass is too large compared to rather small thrust case, trace designed by conventional non-linear optimal control maybe not practical. For this consideration, Hamilton energy on/off control provides an alternative option. It drives system into desired Hamilton energy, which ensures 275
LTO
6607/8904
LEO
9000
MTO
9095/19385
MEO
19480
GTO
19577/42068
GEO
42164
EMTO
42566/384000
0.365/1.6 0.00651/0.0115 0.802
IFAC ACA 2016 276 August 21-25, 2016. Quebec, Canada
Lu Hong Shi et al. / IFAC-PapersOnLine 49-17 (2016) 272–277
4.2 Open-loop trace of angular speed To transport payload into desired cislunar orbit, payload should be tossed according to the spinning velocity shown in table1, which are viewed as the constraints of optimal problems shown in equation (11,12). According to mission requirements, cost function is set tf
as: J = ∫ (t 2 + u (t) 2 )dt . It requires system to reach desired to
tossing angular velocity from rendezvous state with relatively short time and minimal propellant at the same time. Optimal traces in GEO, MEO and LEO orbits are presented in Fig.3~5:
Fig5. Optimal control in GEO orbit In GEO and MEO orbits, optimal traces indicate the similar performance, and thrust outputs do not exceed the maximum thrust. For example of GEO STS system, in the first 0.7 orbit, angular speed is growing with maximum thrust 14N; in the final period of 0.7 ~ 0.75 orbit, angular speed has already reached desired state, but in-plane angle θ has not reached tossing position( θ = nπ , n = 1,3,5,..., N ) yet, therefore thrust is still required to drive system into the nearest tossing position, and control output is vibrating around 0N to keep spinning motion in constant value under periodical perturbation of gravity. When the nearest tossing position is reached ( θ = 191π ), the acceleration manoeuvre is completed, and payload is ready for tossing. Fig 3. Optimal control in LEO orbit
However, in LEO STS system, insufficient thrust problem occurs. To control angular speed in constant value, maximum thrust is140N, which is almost 10 times larger than available thrust. If thrust is limited to be 14N in Fig4, angular speed is not able to be constant value to reach the nearest tossing position. Therefore, Hamilton on/off control is adopted. Numerical result is presented in Fig.6
Fig4. Optimal control in MEO orbit Fig 6. Hamilton control in LEO orbit 276
IFAC ACA 2016 August 21-25, 2016. Quebec, Canada
Lu Hong Shi et al. / IFAC-PapersOnLine 49-17 (2016) 272–277
System reaches desired motion (at tossing position) under Hamilton on/off control law. However, it is worth noting that such control is proposed for insufficient thrust, and does not ensure to be the optimal one. Therefore, it is better to use control law calculated by pseudo-spectral method when thrust level could meet the requirement of station-keeping.
277
Lorenzini, E. C. (2004). Error-tolerant technique for catching a spacecraft with a spinning tether. Journal of Vibration and Control, 10(10), 1473-1491. Misra, A. K. (2008). Dynamics and control of tethered satellite systems.Acta Astronautica, 63(11), 1169-1177. Rao, A. V., Benson, D. A., Darby, C., Patterson, M. A., Francolin, C., Sanders, I., & Huntington, G. T. (2010). Algorithm 902: Gpops, a matlab software for solving multiple-phase optimal control problems using the gauss Transactions on pseudospectral method. ACM Mathematical Software (TOMS), 37(2), 22.
5．CONCLUSIONS Complete cislunar payload tossing mission using spinning tether system was proposed, and optimal control for tossing stage was calculated. For the case of insufficient thrust of Ion engine, control law was modified by Hamilton equation. Simulation shows that payload is able to be tossed into lunar transfer orbit under mission design and optimal control, with economy and reusability.
Schechter, H. B. (1964). Dumbbell librations in elliptic orbits. AIAA Journal,2(6), 1000-1003. Squire, J. P., Olsen, C. S., Chang Díaz, F. R., Cassady, L. D., Longmier, B. W., Ballenger, M. G., ... & Bering III, E. A. (2011, September). VASIMR® VX-200 operation at 200 kW and plume measurements: future plans and an ISS EP test platform. In 32nd International Electric Propulsion Conference, Weisbaden.
However, further research is still required. The closeloop tracking control under multiple perturbations, including elastic deformation, gravitational perturbation, rendezvous/toss error etc., is still necessary to drive functioning system in realistic space environment.
Van Pelt, M. (2009). Space tethers and space elevators. Springer Science & Business Media.
6. ACKNOWLEDGEMENTS
Williams, P. (2006). Dynamics and control of spinning tethers for rendezvous in elliptic orbits. Journal of Vibration and Control, 12(7), 737-771.
The research method and writing of this paper is benefited much from inspiring advice of Prof. Zhu Z.H., York University.
Yang, Y., Qi, N., Liu, Y., Xu, Z., & Ma, M. (2015). Influence of tapered tether on cislunar payload transmission system Science and and energy analysis. Aerospace Technology, 46, 210-220.
The research is supported by 2011 International Science and Technology Cooperation Program of China, also Research and Development Program of Science and Technology of Shanxi Province(2013KW09-02).
Zhu, Z. H., & Zhong, R. (2011). Deorbiting dynamics of electrodynamic tether. International Journal of Aerospace and Lightweight Structures (IJALS), 1(1).
REFERENCES Aslanov, V. S., & Ledkov, A. S. (2012). Dynamics of tethered satellite systems. Elsevier. Ellis, J. R. (2010). Modeling, Dynamics, and Control of Tethered Satellite Systems (Doctoral dissertation, Virginia Polytechnic Institute and State University). Forward, R. L. (1991). Tether transport from LEO to the lunar surface. AIAA paper, 91, 2322. Hoyt, R. P., & Uphoff, C. (2000). Cislunar tether transport system. Journal of Spacecraft and Rockets, 37(2), 177186. Lorenzini, E. C., Cosmo, M. L., Kaiser, M., Bangham, M. E., Vonderwell, D. J., & Johnson, L. (2000). Mission analysis of spinning systems for transfers from low orbits of Spacecraft and to geostationary. Journal Rockets, 37(2), 165-172.
277