Climber Motion Optimization for the Tethered Space Elevator

Transcription

1 Climbr Motion Optimization for th Tthrd Spac lvator Paul Williams * and Wubbo Ockls Dlft Univrsity of Tchnology, Th Nthrlands Th tthrd spac lvator could provid a rvolutionary mans for nabling chap transportation to gostationary altitud and byond. Assuming that such a systm can b built, on of th dynamic dsign problms is dtrmining a mans of moving th lvator along th tthr so as to minimiz th rsidual librational motion of th lvator ribbon and countrwight. In particular, this papr studis this problm from th point of viw of dynamic optimization. A simplifid dynamic modl of th lvator systm is drivd that accounts for th fundamntal libration mod and th motion of th lvator. Th modl is usd to solv an optimal control problm that rsults in zro final librational motion of th ribbon. I. Introduction H tthrd spac lvator is a radical concpt that has xistd in th litratur vn prior to th modrn T spacflight ra []. Th basic concpt rquirs a tthr or ribbon xtnding from gostationary altitud to th arth s surfac. Howvr, in ordr to mak th systm fasibl, th ribbon is rquird to xtnd past gostationary altitud. Th systm cntr of orbit is dsignd to rmain at gostationary altitud so that th systm spins in orbit at th sam rat as th arth rotats. This mans that th tip of th ribbon is always locatd at th sam location on th arth s surfac. Although Tsiolkovsky [] is oftn crditd with its invntion, it was not until Parson [] that th concpt was cmntd in nginring trms. At th tim of publication, no nginring matrial xistd that could com clos to th rquirmnts ndd for an arth orbiting lvator. Rcnt dvlopmnts in carbon nanotubs [3] offr on potntial mans for nabling th dvlopmnt of a full scal lvator systm, albit with many tchnical challngs to ovrcom. Svral prvious works hav dalt with various aspcts of th dynamics of th lvator systm. Cohn and Misra [4] drivd a continuum modl rprsntation of th lvator ribbon to calculat th frquncis and mod shaps. Lss rigorous stimation of frquncis and mod shaps has bn prsntd lswhr [5]. Williams [6] usd a lumpd mass modl of th ribbon to calculat frquncis and mod shaps, and also simulatd th ffcts of th passag of th lvator up to gostationary altitud. Howvr, th lvator motion was prscribd according to an acclration-constant spd-dclration schdul. Cohn and Misra [7] usd a simplifid modl of th lvator systm to produc an analytic approximation for th libration rspons of th ribbon du to an lvator s motion undr th assumption of constant vlocity motion. Thy also showd that th us of multipl ascnding lvators can b usd to produc a zro nt libration angl for th lvator ribbon. Studis of th dynamics of othr lvator systms hav also bn undrtakn, using modls of varying fidlity [8-]. If th lvator systm is producd, it will b ncssary to considr activ control of th lvator during ascnt, so that svr latral oscillations ar not inducd. Furthrmor, it is likly that motion planning will b rquird so as to rduc th rsulting libration amplitud of th ribbon following passag of th lvator to th dsird altitud. Prsnt tchnology allows vry rapid computation of optimal trajctoris, which would nabl planning of trajctoris just prior to bginning ascnt, or vn ral-tim fdback control. This papr prsnts a numrical analysis of th optimal trajctoris for th ascnt of a climbr to gostationary altitud. A simplifid modl of th systm is usd to produc th optimal trajctoris, and a mor dtaild modl is usd to compar th rsults. * Applid Rsarchr, Faculty of Arospac nginring, P.O. Box Postbus 558, 6 GB Dlft, AIAA Mmbr. -mail: tthrd.systms@gmail.com. ASST Chairholdr, Faculty of Arospac nginring, P.O. Box Postbus 558, 6 GB Dlft, -mail: w.j.ockls@tudlft.nl.

2 II. Dynamic lvator Modl A. Simplifid lvator Modl To study th dynamics of th climbr on th lvator systm, dtaild modls of th systm can b drivd. Howvr, to study th optimal mans for controlling th climbr to minimiz th librations of th ribbon and countrwight, a simplifid modl of th systm is usd. Th modl trats only th in-plan motion of th ribbon. It considrs th ribbon to b compltly rigid, and th climbr and countrwight to b point masss. Th modl drivation is a simplifid vrsion of th on prsntd in [7]. Th rfrnc systm for th modl is shown in Fig.. Th origin of th rotating coordinat systm is locatd at th arth s surfac at th anchor point. Th countrwight, m, is locatd at a distanc of l from th anchor point, qual to th lngth of th ribbon. Th lvator, m, is at a c distanc d along th ribbon. Th dviation of th ribbon from th vrtical is masurd by th libration angl θ. Th radius of th arth is dnotd by R. y d x m θ m c Fig.. Simplifid rprsntation of th tthrd lvator systm anchord to th arth. Lt th position of th lvator, countrwight, and an arbitrary point along th ribbon at a distanc s from th anchor b writtn in th rotating coordinat fram as Th inrtial vlocity of th lvator in th rotating fram is givn by r = ( R + dcos θ) i + ( dsin θ) j () r = ( R + lcos θ) i + ( lsin θ) j () c r() s = ( R + scos θ) i + ( ssin θ) j (3) r = [ d cos θ d( θ + ω)sin θ] i + [ d sinθ+ dθ cos θ+ ω( R + dcos θ)] j (4) with similar xprssions for th countrwight and arbitrary point along th ribbon. Not, howvr, that th radial vlocity trms l and s do not appar in ths quations bcaus th ribbon lngth is assumd to b constant. It may b possibl to control th ribbon oscillations by rling th main lvator ribbon, but this possibility is not considrd hr. Th total kintic nrgy of th systm is calculatd by T = m r r + m r r + ρ()() s r s r () s ds c c c Th potntial nrgy du to th Nwtonian gravitational fild of th arth is calculatd as l (5)

3 V μm μm c = ( R + lcos θ) + l sin θ ( R + dcos θ) + d sin θ l μρ() s ds ( R + scos θ) + s sin θ (6) whr μ is th arth s gravitational paramtr. Aftr application of Lagrang s quations, th quation of motion for th libration of th ribbon is givn by ( θ + ω) Μ mdd θ = ω θ+ μ θ R sin R sin [ 3/ Μ Μ Μ ( R + R d cos θ + d ) ml ρ() ss + + ( R R lcos l ) ( R R scos s ) l c 3/ 3/ + θ+ + θ+ d] s md (7) whr Μ = md + ml + () ss d s (8) l ρ c Μ = md + ml + ρ() ss d s (9) c l Th quations of motion for th lvator rprsnt a dampd nonlinar oscillator. Th damping (positiv and ngativ) is providd by movmnt of th lvator along th ribbon, d. For all practical purposs, th libration angl of th ribbon will b small. quation (7) can b linarizd around th vrtical position to obtain th natural frquncy of th libration mod. Th quation shows that thr is a non-zro forcing contribution of th lvator as it movs along th ribbon. Hnc, any movmnt of th lvator along th ribbon initiats libration of th ribbon. Such libration is undsirabl as it incrass th tnsion in th ribbon and can ultimatly affct th accuracy of payload insrtion into orbit. Th nxt sction xamins mthods of liminating th rsidual libration aftr climbing th ribbon. B. Multibody lvator Modl Th us of a simpl modl for th lvator systm nglcts th latral oscillations of th ribbon. In smallr scal tthrd satllit systms, such oscillations can usually b safly nglctd during prliminary control dsign sinc thir ffct in th ral systm is oftn hard to prdict. Furthrmor, th tnsions involvd ar usually vry small. Howvr, bcaus of th xtrm scal of th tthrd lvator systm, latral oscillations of th ribbon ar strongly coupld to th motion of th countrwight. Hnc, any latral oscillations induc libration of th ntir ribbon and countrwight. For this rason, it is important to assss th dgr of coupling btwn th lvator motion and th nsuing systm librations. This is important in assssing th applicability of simplifid modls for th dsign of ascnt/dscnt trajctoris for th lvator. A lumpd mass tthr modl is usd in this papr. Th modl was originally prsntd in [6], whr it was usd to simulat prscribd acclration-constant spd-dclration motion of th of th lvator to gostationary altitud. In this work, th lvator motion is spcifid via th output of th trajctory optimizr. Th quations of motion rfrnc to a rotating rfrnc fram for N tthr lmnts ar givn by max

4 x = ( F + F + F ) R ω z ω z + ω y + ω y ω ω y + ω R s d g j x x x y j y j z j z j y x j y j j j mj + ω x + ω R+ ω x ω ω z y j z z j z x j y = ( F + F + F ) ω R ω x ω R ω x + ω z + ω z ω ω z s d g j y y y z z j z z j x j x j z y j j j j mj + ω y + ω y ω ω R ω ω x z j x j x y x y j ( s d g z = F + F + F ) ω y ω y + ω R+ ω x + ω R + ω x ω ω R j z z z x j x j y y j y y j x z j j j m j x z z y x z j x j y j y z j ωω + ω + ω ωω z y Z R θ Y δ r X φ α m m j m m n n x + m +,..., m n+ Nmax j =,..., N () max Fig.. Lumpd mass tthr modl. whr th coordinats usd ar dfind in Fig. and ω = ω, ω, ω = α sin δ, δ, α cosδ () { x y z} { } { ωx, ωy, ωz} { α sin δ αδ cos δ, δ, α cosδ αδ sin δ} ω = = + Fig. 3 illustrats th discrtization of th tthrd lvator systm to nabl prscribd motion of th lvator. Th complt systm is split into two sparat subsystms: an uppr tthr and a lowr tthr. For convninc, ths hav bn labld as tthr [] and [], rspctivly. Th motion of th lvator is achivd by changing th lngths of th tthr sgmnts closst to th lvator. Th lvator mass is numbrd as m N, which rprsnts th max maximum numbr of tthr lmnts in ithr th uppr or lowr tthr. Th lmnts in th uppr and lowr tthr subsystms ar numbrd in such a way that th xtrm lmnts start at and incras to n and n, rspctivly. Th ky forcs on th systm ar tnsion, damping, and gravity. Ths forcs ar drivd xplicitly in [6]. () [] m lvator Lowr tthr [] [] m [] n m Nmax m m [] n Uppr tthr [] [] m Fig. 3. Discrtization of tthrd lvator for prscribd lvator motion.

5 C. Th Nd for lvator Spd Optimization Fig. 4 shows th rsults of a simulation of a kg lvator climbing a 7, km long ribbon to gostationary altitud. Th lvator movs at a constant spd of 5 m/s, with acclration and dclrations of 4 m/s at th bginning and nd of th journy. Th major ffct of th motion on th dynamics is th initialization of th first libration mod. Not that th first two mods of th systm ar th rigid body oscillations, about th anchord tip, shown in Fig. 5. Th first mod is in th plan of th orbit, whras th scond mod is out of th orbital plan. Thr is almost no damping in th systm and hnc th libration continus wll aftr th lvator cass climbing. Rathr than climbing at a constant vlocity, a bttr stratgy is to control th lvator s ascnt in such a way that th librational oscillations ar minimizd. Howvr, it is not an intuitiv dsign task w utiliz optimal control to obtain a rapid solution to this problm. x (km) 6 x y (km) y (km) Altitud (km) x 4 Fig. 4. Rspons of systm to lvator climbing to gostationary altitud at 5 m/s. 8 x 4 Priod: 6.7 (hrs) 8 x 4 Priod: 3.6 (hrs) x (km) x (km) y (m) z (m) Fig. 5. Mod and for, km anchord tthr. III. Optimization of Climbr Motion In [7], Cohn and Misra analyzd th quations of motion for th ribbon libration undr lvator forcing ffcts and proposd synchronizing th ascnt of thr climbrs to null th ffcts of th rsidual ribbon libration at th nd of th climb to gostationary altitud. This assums that th climbrs maintain constant spd during th climb. Any faults in th lvator s motors or disruption in powr supply could rsult in failur of th mthod to liminat th rsidual oscillations. It is of intrst to b abl to plan th ascnt of a singl climbr to gostationary altitud such that th final ribbon libration is nullifid. This is not possibl by maintaining a singl climbr at constant spd. Th advantag of optimizing th motion of a singl climbr is that it is possibl to rplan trajctoris in cas of motor failurs or othr unschduld vnts. Furthrmor, coordinating multipl climbrs could b a logistical issu and would not b optimal from an oprational prspctiv.

6 A. Numrical Optimization of lvator Ascnt. Basic Ascnt Problm Considr th problm of finding th optimal climbr acclration u() = d () to minimiz th cost function ( ) t f d (3) t 4 J = u + c x t subjct to th quations of motion x x x x = x = θ = x = u (4) th initial conditions {,,, } {,,,} x x x x = (5) 3 4 t and th trminal constraints { x, x, x, x } {,, h, } 3 4 tf = (6) G If a solution to th problm dfind by qs. (3) through (6) xists, thn it is possibl to nullify th libration of th ribbon through propr schduling of th lvator s spd on th ribbon. Not that th cost function minimizs a wightd functional of th climbr acclration and climbr spd, whr c is a wighting paramtr.. Tim Optimal Ascnt An important class of manuvrs that can b usd to stablish lowr bounds on th lvator s ascnt is th minimum tim manuvr. In such a manuvr, both th lvator acclration and vlocity ar boundd to maximum prmissibl lvls, and th cost function is slctd as th final tim, J = t f (7) Th cost function is subjct to th constraints d d d (8) min min max d d d (9) max as wll as th quations of motion dfind by qs. (4). In addition, th manuvr is subjct to th sam initial and trminal constraints givn in qs. (5) and (6). B. Numrical Solution Mthod An fficint solution to th nonlinar optimal control problm dscribd in Sction III.A can b obtaind by application of dirct transcription tchniqus combind with nonlinar programming. In this papr, th continuous

7 optimal control problm is transcribd into a discrt paramtr optimization problm by way of th Lgndr psudospctral mthod [-3]. This procss is automatd in th softwar DIRCT [4]. IV. Numrical Rsults A. Tim Optimal Ascnt As notd in th prvious sction, it is important to stablish a lowr bound to th fasibl manuvr tim for ralistic constraints on th lvator s maximum spd and acclration. Th tim optimal ascnt problm is solvd for th cas whr th lvator is kg, th ribbon is, km long with a maximum cross-sctional ara of mm, and th matrial strngth is 35 GPa with dnsity 3 kg/m 3. Ribbon Libration Angl (dg) Fig. 6. Ribbon libration angl for tim optimal lvator ascnt to gostationary. Th optimal control problm is solvd using th Lgndr psudospctral mthod with th tim domain discrtizd into nods. Th lvator s maximum spd is limitd to 5 m/s, with a maximum acclration of 4 m/s. Th optimal solution is prsntd in Figs. 6 to 8. Fig. 6 shows that th libration angl of th ribbon paks at approximatly. dg aftr approximatly 8 days. Th optimal tim of ascnt to nullify th ribbon libration is days. Figs. 7 and 8 show that th lvator is abl to maintain a constant spd throughout th vast majority of th ascnt. Although hard to discrn on th tim scal, th initial acclration and dclration phass occur at th maximum limit. Fig. 8 shows that in ordr to bring th final libration angl and libration rat of th ribbon to zro at th final tim, th lvator s spd must b rvrsd for just ovr half a day at about 8 days. At th highr altitud, th forcing ffct is largr and is thrfor abl to forc th ribbon swing back arlir than would hav othrwis occurrd. Onc th swing back has commncd, th lvator is abl to rsum its cours to gostationary, whr it coms to a stop. Hnc, th swing back is much fastr than th initial swing away from th local vrtical.

8 lvator Distanc along Ribbon (L/H G ) Fig. 7. Tim optimal lvator ascnt to gostationary. 6 lvator Spd (m/s) Fig. 8. lvator spd schdul for tim optimal ascnt to gostationary altitud. B. ffct of Tim of Ascnt In this sction, th acclration and spd constraints on th lvator ar liftd, and smooth unconstraind optimal solutions ar sought as a function of diffrnt systm paramtrs. In this particular subsction, th tim of ascnt is varid and th lvator s motion is optimizd to liminat th final ribbon libration. Th tim of ascnt is varid from 4 days to 3 days in incrmnts of day. Th lvator mass is kg, th Young s modulus is 35 GPa, th ribbon dnsity is 3 kg/m 3, th maximum cross-sctional ara is mm, and th ribbon lngth is, km. Th rsults of numrical optimizations ar shown in Figs. 9 to. Fig. 9 shows th librational rspons of th ribbon to th lvator s motion for th optimizd trajctoris as a function of th tim of travl. For clarity, th tim scal is normalizd by tim of travl. For th fastst travl tim, th libration angl of th ribbon rachs a pak just aftr th midway point. Th optimal solution for liminating th rsidual libration dos not involv simply climbing at a constant spd. Instad, th spd paks at approximatly 5 % of th trip tim, thn slowly dcrass to zro. Th lvator rvrss dirction aftr approximatly 45 % of th trip tim, which causs th libration to rach its pak quickly. Th lvator thn acclrats back towards gostationary, raching a scond pak of m/s, aftr which

9 th lvator is slowd back to a stop. Incrasing th transit tim dcrass th vlocity paks at th bginning and nd of th transfr, and rducs th nd for th lvator to travl back down th ribbon. Incrasing th tim of travl lads to a dcras in th pak libration amplitud. In addition, th libration pak shifts towards th nd of th ascnt. For vry long travl tims, th ribbon undrgos a sris of low amplitud oscillations about a non-zro man, culminating in a slightly largr pak just prior to rturning to th local vrtical. Ribbon Libration Angl (dg) lvator Distanc along Ribbon (L/H G ) Incrasing travl tim Normalizd Tim Fig. 9. Ribbon libration angl as a function of tim of travl Normalizd Tim Fig.. Normalizd distanc of lvator along ribbon for optimal ascnt.

10 3 5 lvator Spd (m/s) Normalizd Tim Fig.. Optimal lvator spd as a function of tim of ascnt. C. ffct of lvator Mass Th Coriolis forcs xrtd on th ribbon as a rsult of th lvator s motion along it ar dirctly proportional to th lvator mass. Hnc, th torqu acting on th ribbon to disturb it away from th vrtical position dpnds dirctly on th lvator mass. In this sction, th sam basic systm is usd to simulat th ffcts of lvator mass on th optimal climbr trajctoris. Th ribbon is, km long with Young s modulus 35 GPa, th maximum cross-sctional ara is mm, and mass dnsity of 3 kg/m 3. Th travl tim is fixd at days, and th lvator mass is varid btwn and, kg. Optimal lvator trajctoris ar shown in Figs. through 4. Fig. shows that th pak ribbon angl incrass in dirct proportion to th lvator mass. This shap of th variation in ribbon angl is ssntially th sam, irrspctiv on th lvator mass. Initially th ribbon rmains in vrtical quilibrium and only starts to display noticabl dviations aftr approximatly day of travl. Th libration paks aftr approximatly 7 days, aftr which th ribbon rapidly convrgs back to th stabl quilibrium position. Figs. 3 and 4 show th variation of th lvator position along th ribbon and th lvator s spd, rspctivly. Ths figurs indicat that th optimal control inputs ar rlativly insnsitiv to lvator mass. This is an important rsult bcaus it mans that th lvator load can b changd substantially without impacting th opn-loop control. Not, howvr, that this conclusion is only valid within th assumption that th ribbon rmains straight. Incrasing th lvator mass will crat largr Coriolis forcs and hnc induc largr latral ribbon vibrations. This was stablishd through simulations in [6] and th main rsult is summarizd in Fig. 5, which shows that th maximum latral dviation of th tthr from th lin conncting th anchor and th countrwight is linarly proportional to th lvator mass.

11 Ribbon Libration Angl (dg) Incrasing lvator mass lvator Distanc along Ribbon (L/H G ) Fig.. Ribbon libration angl as a function of lvator mass Fig. 3. Normalizd distanc of lvator along ribbon for optimal ascnt as a function of lvator mass.

12 7 6 lvator Spd (m/s) Fig. 4. Optimal lvator spd as a function of lvator mass. 6 Max. Latral Oscillation (km) Mass (kg) Fig. 5. Maximum latral dflction of a flxibl lvator ribbon during transit to gostationary altitud at a constant spd of 45 m/s. D. ffct of Tthr Lngth Th final paramtr that is xamind in this work is th ribbon lngth. Using th sam paramtrs as in th prvious xampl, but with an lvator mass of kg, th ribbon lngth is varid btwn 7, and 6, km. Rsults of th optimal trajctoris ar shown in Figs. 6 through 8. It should b notd that changing th ribbon lngth changs th total mass and mass distribution in th systm. Th rsults show, howvr, that changing th ribbon lngth has only a minor ffct on th optimal trajctoris. Th ffct is mor substantial than changs in lvator mass. Th trajctoris ar only affctd bcaus th systm dynamics chang du to th chang in gravitational forcs on th longr ribbon. In rality, th ribbon lngth will b virtually constant during oprations. Hnc, vn changs on th ordr of a fw hundrd kilomtrs would hav ngligibl impact on th control of th lvator.

13 Ribbon Libration Angl (dg) lvator Distanc along Ribbon (L/H G ) Fig. 6. Ribbon libration angl as a function of tthr lngth Fig. 7. Normalizd distanc of lvator along ribbon for optimal ascnt as a function of tthr lngth.

14 7 6 lvator Spd (m/s) Fig. 8. Optimal lvator spd as a function of tthr lngth.. Numrical Simulation of an Optimal Trajctory in Flxibl Ribbon Modl Th optimal solution was implmntd in a flxibl, lastic lvator modl dscribd in Sction II. It has bn found that th Young s modulus of th tthr matrial plays a vry important rol in th dynamics. In particular, in ordr for th simpl modl to match th fundamntal frquncy of th multibody modl, th Young s modulus must b vry larg (significantly largr than would b obtainabl with Carbon nanotubs). An optimal solution was producd for a rlativly short transit to Gostationary with a, km long ribbon. Th maximum crosssctional ara is mm with a dnsity of 3 kg/m 3 and ultimat strngth of 35 GPa. Th rfrnc optimal solution for transit to gostationary altitud for a kg climbr is shown in Figs. 9 and. To implmnt th solution in th flxibl modl, th climbr s vlocity was intrpolatd. Th libration angl of th flxibl ribbon is shown in Fig.. This illustrats that th libration angl prdictd using th simpl modl is quit diffrnt from that producd using th multibody modl. Th diffrnc is almost two ordrs of magnitud, i.., tims. This has a significant impact on th dynamics bcaus th inducd vlocitis ar drastically diffrnt. Howvr, th basic natur of th libration history is prsrvd, with th tthr rturning to an approximat vrtical position. In addition, th dflction of th lvator from th ribbon cntrlin is approximatly constant at km throughout th climb to gostationary, as shown in Fig.. Th latral oscillations ar also rducd by th optimal climbr profil compard with a constant spd lvator ascnt [6].

15 Climbr distanc along ribbon (km) x Fig. 9. Climbr distanc along ribbon for rfrnc optimal solution. 6 4 Climbr vlocity (m/s) Fig.. Climbr vlocity for rfrnc optimal solution.

16 .5 Libration angl (dg) Fig.. Libration angl of ribbon using opn-loop optimal solution implmntd in flxibl ribbon modl. Climbr dflction (km) Fig.. Dflction of climbr rlativ to lin conncting anchor and countrwight for opn-loop optimal solution. V. Conclusions Th control of th ascnt of a spac lvator to gostationary altitud has bn considrd. Th fundamntal mod of vibration of an anchord ribbon is a rigid body oscillation about th anchor point on th arth s surfac. This mod lads to larg oscillations of th countrwight, which can b locatd up to, km abov th anchor point. As th lvator climbs th ribbon, Coriolis forcs crat a torqu that initiats th librational mod. Th lack of damping in spac causs this mod to prsist. Using a simplifid modl that only accounts for this mod, it is possibl to solv th ascnt problm using optimal control. Th rsults show that it is possibl to liminat th librational oscillations by rvrsing th dirction of th lvator for a short tim. For a maximum spd of 5 m/s, with acclrations limitd to 4 m/s, th optimal trip tim is on th ordr of 9.38 days.

17 Rfrncs. Tsiolkovsky, K.., Spculations Btwn arth and Sky, Isd-vo AN-SSSR, Moscow, 895, p. 35 (In Russian, rprintd in 959).. Parson, J., Th Orbital Towr: A Spaccraft Launchr using th arth s Rotational nrgy, Acta Astronautica, Vol., 975, pp Iijima, S., Hlical Microtubuls of Graphitic Carbon, Natur, Vol. 354, 99, pp Cohn, S.S., and Misra, A.K., lastic Oscillations of th Spac lvator Ribbon, Journal of Guidanc, Control, and Dynamics, Vol. 3, No. 6, 7, pp dwards, B.C., and Wstling,.A., Th Spac lvator: A Rvolutionary arth-to-spac Transportation Systm, Williams, P., Dynamic Multibody Modling for Tthrd Spac lvators, 58th Intrnational Astronautical Congrss, Hydrabad, 4-8 Spt. 7, IAC-7-D Cohn, S.S., and Misra, A.K., Th ffct of Climbr Transit on th Spac lvator Dynamics, 58th Intrnational Astronautical Congrss, Hydrabad, 4-8 Spt. 7, IAC-7-D4..4.\ 8. Williams, P., and Blanksby, C., Study on Prolongd Rndzvous using Tthrs, AAS/AIAA Astrodynamics Spcialist Confrnc, Aug. 7-5, mbassy Suits Hotl, Lak Taho Rsort, Papr AAS Williams, P., and Blanksby, C., Prolongd Payload Rndzvous using a Tthr Actuator Mass, Journal of Spaccraft and Rockts, Vol. 4, No. 5, 4, pp McInns, C.R., Dynamics of a Particl Moving Along an Orbital Towr, Journal of Guidanc, Control and Dynamics, Vol. 8, No., 5, pp Fujii, H.A., Watanab, T., Kusagaya, T., Sato, D., and Ohta, M., Dynamics of a Flxibl Spac Tthr quippd with a Crawlr Mass, Journal of Guidanc, Control, and Dynamics, Vol. 3, No., 8, pp lnagar, J., Kazmi, M.A., and Razzaghi, M., Th Psudospctral Lgndr Mthod for Discrtizing Optimal Control Problms, I Transactions on Automatic Control, Vol. 4, No., 995, pp Ross, I.M., and Fahroo, F., Lgndr Psudospctral Approximations of Optimal Control Problms, Lctur Nots in Control and Information Scincs, Vol. 95, Springr-Vrlag, Nw York, pp Williams, P., Usr s Guid to DIRCT Vrsion.7, Tchnical Rport, Mlbourn, Australia, March 5.