SERVICING AND DEPLOYMENT OF NATIONAL RESOURCES IN SUN-EARTH LIBRATION POINT ORBITS

D. C. Folta, M. Beckman, S. J. Leete, G. C. Marr, M. Mesarch, and S. Cooley

NASA Goddard Space Flight Center Greenbelt, MD, USA

53th International Astronautical Congress The World Space Congress - 2002 10-3 9 Oct 2002 I Houston, Texas

For permission to copy or republish, contact the International Astronautical Federation, 3-5 Rue Mario-Nikis, 75015 Paris, France SERVICING AND DEPLOYMENT OF NATIONAL RESOURCES IN SUN-EARTH LIBRATION POINT ORBITS

D. C. Folta, M. Beckman, G. C. Marr, M. Mesarch, S. Cooley Flight Dynamics Analysis Branch NASA Goddard Space Flight Center, Greenbelt, MD, USA dfolta@;gsfc.nasa.gov

S. J. Leete Space Science Missions Branch, NASA Goddard Space Flight Center, Greenbelt, MD, USA

Abstract Spacecraft travel between the Sun-Earth system, the Earth-Moon system, and beyond has received extensive attention recently. The existence of a connection between unstable regions enables mission designers to envision scenarios of multiple spacecraft traveling cheaply from system to system, rendezvousing, servicing, and refbeling along the way. This paper presents examples of transfers between the Sun-Earth and Earth-Moon systems using a true ephemeris and perturbation model. It shows the AV costs associated with these transfers, including the costs to reach the staging region from the Earth. It explores both impulsive and low thrust transfer trajectories. Additionally, analysis that looks specifically at the use of nuclear power in libration point orbits and the issues associated with them such as inadvertent Earth return is addressed. Statistical analysis of Earth returns and the design of biased orbits to prevent any possible return are discussed. Lastly, the idea of rendezvous between spacecraft in libration point orbits using impulsive maneuvers is addressed.

Introduction Some obstacles to human servicing are the Satellite servicing has received a great deal orbital mechanics, the cost of lifting mass, and of study and significant execution. Several the problems associated with travel time and satellites were designed for servicing using thermal and instrument environmental the Multi-Mission Modular Spacecraft design, conditions. As more ambitious missions are including Solar Max Mission, Landsat IV & planned, such as the placement of the Next V, Upper Atmosphere Research Satellite, and Generation Space Telescope (NGST) and Extreme Ultra-Violet Explorer. Rescue several other missions into a Sun-Earth L2 missions have been performed on (SEL2) or SELl libration orbit, servicing by geostationary satellites trapped in low earth the Shuttle and the use of low-Earth orbits orbit, such as WESTAR-IV, PALAPA-B, (LEOS) will be limited. Development of Intelsat-VI, and LEASAT/SYNCOM-IV. robotic satellite servicing capabilities, such as More routine human servicing work DARPA’s Orbital Express, NASA’s occurs(ed) at various space stations Robonaut, and the University of Maryland’s (International Space Station, Skylab, Salyut, Ranger, may provide for the possibility of and Mir). The servicing of the Hubble Space robotic satellite servicing at various orbital Telescope (HST) has become a successful locations in the near- or mid-range time landmark, allowing HST to become one of frame. NASA’s most productive missions. 1,2,334 An enabling set of circumstances for an Copyright @ 2002 by the American Institute of Aeronautics and Astronautics, Inc. No Copyright is expansion of satellite servicing would be the asserted in the United States under Title 17, U.S. Code. placement of humans and valuable robotic The US. Government has a royalty-fiee license to assets in close proximity to one another, A exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are reserved by the copyright owner I space architecture that includes these The servicing of national resources in the conditions is a servicing facility in a lissajous Earth-Moon and the Sun-Earth regions is or halo orbit about one of the Earth-Moon LI made difficult by the unstable dynamics of (EMLI), Earth-Moon L2 (EML2), or Earth- these regions. Trip times associated with Moon L3 (EMLJ co-linear libration points6. achieving SELl and SEL2 orbits can vary From such an orbit, spacecraft have access to from weeks to months while achieving EML, a wide variety of interesting orbits at a and EML2 may vary from days to weeks, with relatively lower AV cost. Use of the Earth- both dependent upon many initial conditions Moon stable L4 and L5 Lagrange regions such as energy and departure locations. The provides additional scenarios. These servicing design of a trajectory used to achieve locations are also an excellent staging point servicing mission requirements is intricate but for lunar surface and Earth-Moon orbital can be facilitated by the use of unique orbits exploration.’ The orbits also have ready to achieve the proper flight time or to access to geostationary orbits and transfer minimize the necessary fuel mass. The use of back to LEO orbits. Table 1 provides a brief dynamical systems (such as invariant overview of Earth-Moon libration orbit manifolds) and optimization plays a key role staging node characteristics. in designing transfer trajectories, as these tools afford the luxury of using natural Table 1. Earth/Moon Libration Orbit dynamics where possible. Results Overviei demonstrate the viability of some unique Characteristics Potential Uses transfers that meet the NASA Exploration EMLl -Unstable orbit -Assembly & Team (NEXT) goals. These goals focus on between earth and maintenance of opening the human frontier beyond low-Earth moon SIC orbit by building infrastructure robotically and -Halo:continuous -Access to lunar with humans in-situ at strategic outposts - view of the earth, surface libration points, planetary moons, planets, etc. moon, and sun, -Low Av access - and include a wide range of exploration -Earth I moon to SEL, transfer always >150° manifolds tools (e.g., space planes, balloons, human- separation constructed and maintained observatories -Unstable orbit far -Assembly & located at libration point, etc.).8 side of moon maintenance of -Halo: continuous SIC A primary purpose of this paper is to detail view of earth, -Access to lunar mission planning scenarios for servicing of moon, sun, surface national resources and to contrast them to -At full moon, -Corn relay to other transfer options. It explores the transfer surdearthlmoord lunar far side, from the Earth-Moon region to Sun-Earth in alignment -Low Av access libration orbits. The AV and trip time to SEL, transfer manifolds associated with such a mission is compared to -Unstable orbit -Assembly & a more traditional direct injection and the opposite moon maintenance of relevant differences highlighted. Analysis -Halo: continuous slc that are recently completed for NGST orbit trades are view of earth, sun, thermally used where possible.’ The servicing options -At new moon, sensitive vary over a wide range, from low thrust direct surdearthlmoord -Low Av access transfers to transfers from elliptical orbits in alignment to SEL,, transfer achieved using bi-propellant systems. This manifolds paper covers only several of the many topics -Stable orbit, low -Minimum fuel analyzed, but it represents a general stationkeeping Av, loads investigation of possibilities. The results of -Moderate -Varying the analysis presented herein will be available sunlearthlmoon transfer to SEL, geometry to aid in the support of evaluating in-space

2 servicing options for mission studies in the scenario allows a fast transfer of human and GSFC Integrated Mission Design Center. expendable resources for deployment or servicing at the EML2 vicinity, with a round Software Applications and Assumptions trip servicing time of approximately a week. For this analysis, software applications that The transfers presented here demonstrate use high fidelity perturbation modeling and feasibility and are not fully optimized. full ephemeris data were utilized. The models utilized include as a minimum: End-To-End Transfer Scenarios The following scenarios demonstrate a cis- Earth potential, 4x4 or greater lunar transfer into an EMLl or EML2 orbit Point mass bodies based on full ephemeris where a human deployment or servicing of a (DE405) national resource can occur. A subsequent Solar radiation pressure transfer to and a return from the SELl via Jacchia-Roberts atmospheric drag EML2 is then designed that would provide for Spacecraft area and mass characteristics a second servicing opportunity. The initial cis- Spacecraft engine mass depletion and lunar trajectory is modeled as injection from a accelerations 28.5' inclined 186-km circular parking orbit Runge-Kutta 8/9 integrator based on expendable launch vehicle parameters. To obtain the trajectories in this Inertial and rotating coordinate systems scenario a combination of targeting goals Differential correctors and optimization were used including: lunar B-plane methods components, rotating coordinate system position and velocities, orbital energy, and The software used consists of GSFC's epochs and propagation duration. To achieve Swingby and Generator tools and MATLAB. targets, deterministic AVs were varied at the Swingby has been used operationally by parlung orbit injection, orbit insertion, and GSFC to support four libration and two lunar trajectory mid-poin ts . missions. In addition to using Swingby to model and target the solution set, software Figures 1 and 2 present a transfer from LEO that allows robust modeling and optimization to EMLI, transfer to EML2, transfer to SELI, of the Sun-Earth and Earth-Moon dynamics is and back to EML2. The cis-lunar trajectory as used. This utility allows the generation of the shown in Figure 1 is in an earth-moon rotating full phase space while including full coordinate system. This trajectory transfers perturbation models. The Generator code the spacecraft from LEO orbit injection to developed at Purdue University was used to EMLl orbit insertion. From there, after EMLl compute invariant manifolds for EML2 to insertion and any stationkeeping, a small SEL, transfers. MATLAB using GSFC maneuver is required to achieve the EML2 developed m-files and a multi-level iteration orbit. As EML2 is a co-linear unstable scheme were used to optimize these transfer location, it is straightforward to achieve a trajectories." transfer trajectory that places the spacecraft on a departing invariant manifold. Figure 2 Transfers Between The Earth-Moon And shows the EML2 to SELl transfer in a sun- Sun-Earth Libration Orbits earth rotating coordinate system. Following Transfers between the Earth-Moon and Sun- insertion into the SELl lissajous orbit, one Earth libration points have been previously orbit revolution about the SELl point was investigated in varying consideration and used. A small maneuver then places the detail. 1 1,12m4,I5 This work builds upon 25 spacecraft on an inbound trajectory that years of GSFC libration orbit experience and results in a re-capture into the EM system. analysis and new optimal transfer studies. A The computation of the initial transfer orbit servicing scenario using co-linear Earth-Moon was not constrained to achieve any SELl libration orbits was investigated first. This lissajous orbit conditions that are

3 advantageous to any return trajectory. In fact, transfer to EMLl using a higer inclination can the transfer and final SELl orbit achieved may be computed by timing the alignment of the be considered a worst-case scenario since they outgoing velocity asymptote and the addition were unconstrained and resulted in the of a small deterministic maneuver. The maximum offset in timing for a return maneuver AV required for insertion into the trajectory. The AVs and trip times associated EMLl or EML2 libration orbit from the with this transfer scenario are presented in incoming SELl libration orbit varies as a Table 2. function of lissajous amplitude and the energy of the incoming transfer trajectory. For EMLl As shown in Figures 3 and 4, a second return and EML3 insertions, AVs on the order of 50 transfer was also achieved using the same to 300 ds were observed. A typical initial LEO to SELl outbound trajectory. The maneuver insertion for EML2 was 70 ds. In original outbound transfer results in a timing all the above scenarios, the injection energy to differential for a return lunar encounter of 14 reach EML, orbit from LEO remained days. Without a phase jump in the SELl orbit, constant at -2.13 km2/s2 (AV -3.14 Ws), the return trajectory achieves an EML3 similar in magnitude to a lunar mission such intersection. Using the dynamics of the Earth- as Lunar Prospector. Moon region, a differential correction scheme can be used to compute a maneuver that Adiustinp the Amplitude of the SELllt allows a transfer to an orbit in the Earth-Moon Orbit From EML2 region to achieve the EML2 orbit. Figure 3 Achieving the above SELl lissajous orbit shows the transfer orbit in the Earth-Moon was not dependent upon any mission region while Figure 4 shows the continuous constraints (such as y-amplitude). The size path from SELl back to the Earth-Moon and shape of the orbit was unconstrained in region. order to show the feasibility of such transfers. The orientation and amplitude of the orbit will Figures 5 and 6 show a similar transfer from vary depending upon mission requirements or to the EMLl region without a phase jump. and natural dynamics. Analysis was These return transfers demonstrate that a wide performed to determine the AV and impacts to variety of transfers are available for return temporal or spatial conditions for control of servicing missions. Segments of this full the lissajous characteristics in order to scenario can be used to size the requirements rendezvous with resources already in SELN for a returning spcaecraft from SELl and a libration orbits. rendezvous from Earth at the EML2 orbit. The impact of injection from an ISS-like Reducing the lissaious v-amtditude inclination is not significant. A similar Trajectory optimization is a broad and

EMLl Transfer to SELl Insertion into Transfer Capture Insertion at EMLz Injection SELl from SELl into and and 1 Orbit Orbit to EML1/2/3 Transfer EMLz AV (ds) 423 0.2 None None 32 to EMLl 50-293 100 to EML2 Flight Time 5 9 86 185 (includes 78 --- Segment 1 orbit) and 5 29 (includes I15 300 3 78 3 78 Cumulative I orbit) (days) Libration X - 27000 X - 27000 X - 300000 --- X - 30000 Amplitude Y - 70000 Y - 70000 Y - 800000 Y - 70000 (EML2orbit) (km) (EMLlorbit) (EML,orbit) --- 1 (SEL I orbit)

4 htbound Outbound Trajectory Fajectory

I Modn 1 1 Transfer Trajectory to EML, SEI Return Trajectory 1- 1- Return Trajectoryfrom SEL, Figure 4. Transfer to/from SELl via \ EML3in Sun Earth Rotating System Figure 1. Transfer to/from SELl via EMLllz in Earth Moon Rotating System h-Return Trajectoryj-omSEL,

Earth

\ Transfer Trajectory to EM, Outbound Trajectory I Renolz Eqlectq Figure 5. Transfer to/from EMLl in Earth =I Moon Rotating System Figure 2. Transfer to/from SELl in Sun Earth Rotating System A Outbpund Trajectory Outbound SEL ,

~ /Trajectory EML I Return Trlj A from SEL,

Earth i, \\ I i EML, Moon I SEL, Return Trajectory \ Transfer Trajectory to EML 2,\ Figure 6. Transfer to/from SELl via EMLl I in Sun Earth Rotating System Figure 3. Transfer to/from SELl via EMLJ maneuvers - at any of the patch states - are in Earth Moon Rotating System modeled by fixing time and position while complicated subject that has been studied letting the velocity vary instantaneously. Once extensively in the aerospace literature.I6 an initial guess is provided and carefully Nevertheless, fundamentally any path can be discretized, the first level of the Howell- discretized as a set of patch states that include Pernicka two level iteration scheme can be time, position and velocity. At any given used to achieve position continuity.” That is, patch state, a before and after (-/+) maneuver the patch states are connected in a (numerical) state is included. In this context, impulsive Lambert type of scheme. The result of this

5 operation is the computation of the total AV depart at the same epoch but have slightly required for the path. This total AV becomes different initial EML2 states and energy. A the cost function. If one or more components deterministic AV performed several days after (of one or more patch states) are allowed to departure on the SELl lissajous transfer vary, the total AV required for the path manifold is required to acquire a smaller SELl changes. As a result, a trajectory optimization lissajous. Also shown in Figure 7 is a transfer scheme could vary the patch states and keep into a small SEL2 orbit. This transfer was track of the total AV that is needed to achieved using an optimization process. The "connect the points". This essentially allows departure location was chosen to be the EMLz the use of any optimization strategy. libration point itself. Selection of the departing conditions as part of the overall mission design allows one to achieve any As an example, the MATLAB optimization SEL2lissajous orbit. toolbox using the fmincon function is utilized. A trajectory from the vicinity of the EMLz Table 3. EMLz Departure Characteristics point to a lissajous orbit of the SEL2 point is and AVs Transfer examined. The ephemerides model is utilized Achieved with the Earth as the central body (with 52) Lissajous to first and with the Sun and the Moon as point mass. Solar radiation pressure is also included. An Moon crossing initial estimate, with a total AV of about 700 centered) (days) SELt x:20000 AV: 0.18 115 ds, is computed by numerical Y-900,000 Y:75000 C3: 0.0002 experimentation. This initial estimate is discretized into four patch states. The patch x:2000 AV: 16 C3: -.05 state's positions and times are selected as independent variables. When the optimizer x:2000 AV: 127 converges, the total AV decreased to 180 m/s. C3: 0.03 As a comparison to the above solution of SEL2 x:0 AV: 180 120 departure from the EMLz point, two lissajous Y-200,000 Y: 0 C3: 0.75 orbits with the same epoch where constructed I I I I I that yield a transfer to two selected SELl y- amplitude orbits. These lissajous characteristics were chosen based on the full transfer scenario. The control of the SELl y- amplitude is dependent upon the selected transfer manifold and the initial conditions at the EML2 libration orbit. Unstable manifolds from the SELl orbit back to the vicinity of the Moon at a given epoch can be constructed." Also, unstable manifolds of the dynamics of the Earth-Moon region can be constructed. Transfer to Large or Medium SEL, The intersection of these two manifolds I Lissajws from EM., orbit represents a locus of transfer points. Figure 7. Changing the SELllZLissajous Y- As shown in Figure 7, a smaller SELl y- Amplitude amplitude lissajous orbit can be achieved by Low Thrust Options using a slightly different initial EML2 Another option for servicing and deployment departure condition that inserts onto a is to utilize low thrust propulsion. Low thrust different outbound unstable manifold. Table 3 propulsion has the benefit of being more provides sample AVs and trip times for these efficient than an impulsive type system. A transfers. The transfer trajectories to SELl transfer from LEO into either an EMLl orbit

6 or into a SEL2 orbit was analyzed. As compared to the AV and time associated with a high-energy direct transfer, low thrust obviously takes much longer. The amount of available mass to final EMLl orbit is increased but is limited by the launch capability. Figure 8 shows a low thrust trajectory to the EMLl in an Earth-Moon Rotating Coordinate system. An example of a low thrust trajectory from Earth parking orbit to the SEL2orbit in a Sun- Earth rotating coordinate system is shown in th Figure 9. The thrust direction is along the velocity vector. While usually called constant Figure 8. Low Thrust Trajectory to EML, thrust, the trajectory includes coasting arcs that are used to 'target' to an unstable I /7 manifold, which places the spacecraft on a trajectory that will achieve a lissajous orbit. Little additional thrust is required after lunar orbit distance; the majority of the post-lunar trajectory in Figure 9 is a coast phase. This trajectory is similar to a direct transfer to SEL2. Table 4 shows a number of different direct low thrust options considered for NGST, as well as for an EMLl mission. Obviously a high thrust to mass ratio is Figure 9 Low Thrust Trajectory from LEO required to minimize flight time. The to SELz information in the table assumed a launch No-return traiectories using the STS into a 28.5' LEO. The issue of an unplanned return of a nuclear power plant in a Sun-Earth libration point Nuclear Energy OD tions orbit is a valid one. Periodic orbits about the An option for some missions is the use of SEL, and SEL2 are unstable. Loss of control nuclear energy as a power source for low of a spacecraft will cause the spacecraft to thrust missions. Once the mission is achieved depart the periodic orbit asymptotically. The a disposal plan must be enacted and analysis direction of this divergence is locally performed to determine the possible reentry determined by the unstable eigenvector of the conditions.'* For missions in the SELl or monodromy matrix, or State Transition SEL2 region, this becomes a question of the Matrix (STM) over one period. This probability of Earth impact if a perturbation eigenvector is then globalized (non-linear places the spacecraft on an unstable manifold integration) to determine the trajectory path of that returns it to the Earth environment. the departing spacecraft. The energy of the Table 4. Sample Low Thrust Option Parameters Propulsion Information SIC Total Thrust to Transfer Time to Mass (kg) Launch Mass SEL2or EML, Thrust (N), Isp(sec), Thruster # / Type Mass (kg) Ratio (Days) 2.02, 3100, 22 DS-1 (NGST) 5633 16421 1.230e-4 672 to SEL2 2.78, 1732, 2Halls (NGST) 1408 16420 1.693e-4 365 to SELz 2.08, 2488, 2Halls (NGST) 4198 16420 1.267e-4 642 to SEL2 2.31, 3800, 14XIPS (NGST) 7200 16419 1.407e-4 6 10 to SELz 0.50, 2500, 1 Hall (non-NGST) 744 1000 1.14e-3 150 to EMLI 2.08, 2488, 2 Halls (non-NGST) 3178 4198 4.955-3 138 to EMLl

7 periodic orbit limits the space which the spacecraft can reach. This limit is called the zero velocity curve and is roughly a sphere - about the Earth at low energy levels. At E I l65-e higher energy levels, specifically the energy > Libration Point distance levels of the SELI and SEL2 points themselves, the sphere opens up via corridors near SELl and SEL2 to reveal a region either inside the Earth's orbit about the Sun or outside it. The energy levels of the periodic orbits about SELl and SEL2 are higher than the points themselves and therefore spacecraft can depart the neighborhood of the Earth and / SELI SEL2 points. Once a spacecraft departs 2b 40 $0 Sb lbo 1;o 140 160 though the SELl or SEL2 corridors, its return Unstable Manif& is considered to be statistically irrelevant, although recent events indicate that an Apollo 3rdstage accomplished this." However, some of the unstable manifolds remain in the region for some time and pass between the $ELl and E 565- 1 SEL2 points with multiple Earth flybys. It is >. LibrationPoint distance these trajectories that sometimes provide N gC Earth impacts.

Generally, half of the entire manifold leaves the system directly though a comdor with no Earth encounters. The other half exhibits some sort of Earth encounter before the majority of them leave the EM system. Monte A Carlo analysis has shown that approximately 20 40 60Unstable 80 Manlfold 100 120 140 160 0.7% of the time, a randomly perturbed spacecraft in a large periodic orbit about the Figure 11. Minimum Earth Distance, Large SELl or SEL2 points will impact the Earth Quasi Orbit within two years. Departures from smaller quasi-periodic orbits have not shown any impacts in a similar analysis. This analysis is Long Term Unstable Manlfold Evolution --- 10' -- ~- limited to Circular Restricted Three Body Problem modeling. Additional force modeling including lunar perturbations and solar radiation pressure would significantly affect the propagation of any single trajectory and could significantly increase the possibility of impact, such as with ISEE-l&2.20 However, I it's not clear if those forces would change the :o:ly overall statistical behavior of the trajectories. Figure 10 shows the minimum Earth radial 45t -1 1 I distance for 150 equally spaced displaced 012 X4m) .rn' points on a small periodic orbit over a two- Figure 12. Return Trajectory Evolution year propagation. Figure 11 shows the same data fi-om a large periodic orbit. Figure 12

8 Biased Libration Orbits small number of spacecraft with large masses Another method to ensure a non-return and extensive power systems. The libration trajectory is one that models a stationkeeping orbit rendezvous problem using high strategy that incorporates a constant (impulsive) thrust has not been addressed acceleration.21It necessitates the control of an significantly. This paper addresses some of unstable orbit in a constrained direction so the initial investigation into this problem. that an onboard propulsion failure would guarantee a drift away trajectory. As with all The primary technique used here is a libration orbits about the unstable SEL, and Lambert solution using a numerically SEL2 points, stationkeeping is required. One calculated STM and a simple differential can design a biased orbit through inclusions of correction. The STM is obtained through the deterministic AVs and accelerations that result integration of the variational equations along in a trajectory that diverges on an outbound with the equations of motion. A state vector is eigenvector direction. created consisting of the six state elements of the target spacecraft (B) and the relative state A strategy for a non-return libration orbit vector between the two spacecraft. The STM accommodation can be found by including matrix provides the sensitivity of the relative deterministic AVs in the nominal SEL, position between the two spacecraft and the lissajous orbit. This can be determined using velocity of the maneuvering spacecraft (A): the two-level differential scheme that is part ai;, of an overall dynamical systems approach. -. Given a time to rendezvous of T, the Expected gravitational and non-gravitational =A acceleration are modeled in the differential calculated AV applied to A at time 0 is equations of motion. The deterministic obtained from: d6(0) = @I’ (-FR (T)) maneuvers are pre-specified to be any ai;, appropriate value that accommodates where Os=- . This technique will accelerations and maintains the corrective =A maneuvers (i.e., stationkeeping) in a positive provide a one-impulse solution to this x direction. Without these maneuvers, the rendezvous problem. There is no expectation orbit departs into a drift away orbit. that this technique will work for most rendezvous problems, however the Libration Orbit Rendezvous And applicability of this technique will show Phase Jumps where enhanced techniques such as multiple In the above EM and SE transfer cases, the impulses, weak stability boundary, target timing of the arrival into the final mission points, or Floquet Modes are required.20~21~22 orbit was not considered. In some servicing An example case is shown below. options it may be required to rendezvous with the national resource to provide the The target spacecraft is placed in a small propulsion to return it to the EMLl or EML2 Lissajous orbit about the SEL, point. The regions, as was in the full transfer scenario. maneuvering spacecraft is placed 7 days behind the target along the same trajectory. The rendezvous problem has been The time to rendezvous is set for 60 days. In extensively studied and performed in LEO for this case, figure 13 shows the relative position many years. Additionally, the rendezvous closes to zero at a cost of 24 m/s in AV of problem for libration point orbits has been spacecraft A at time 0. However, the relative solved and optimized using continuous thrust velocity at time T was 44 m/sec. As a inputs by Marinescu, among others.” comparison STS requires less than 0.3 m/s to However, future mission concepts now rendezvous with ISS. include multiple spacecraft that may require rendezvous and docking. Low thrust propulsion systems are only feasible for a

9 of the same orbit. Phase jumping is critical in order to close the distance between spacecraft Spacecraft Trajectories to within the free-space approximation X defined above. The following figures show L1 the effect on the cost of phase jumping due to -0.5- initial phase angle and time to rendezvous. Figure 14 shows the 1’‘ AV, 2nd AV, and the - total AV for a 6 day phase jump over 60 days -1 (time to rendezvous) versus the initial location of the target spacecraft in the periodic orbit -1.5 1 (or phase angle). Figure 14 shows that the total AV is fairly insensitive to the location. 5 -2 Total AVs of 54 to 84 dsare seen about one revolution of the orbit. Figure 15 fixes the initial location (approximately 0 deg initial >I-2.5 phase angle) and shows total AV versus time Transfer ‘>‘\ of the phase jump for various T (times to -3 - 1, \Trajectory,, i i’ i rendezvous). 1, ‘. U’ -3.5- \\ / ‘\ , / ‘\---A’ -4 I

N-Total ’,

... .. -.

101

O -150 -100 -50 0 50 100 150 In-Plane Phase Angle (deg)

Figure 14. Total AV vs. Starting Phase Angle

T=lW 4507 The process of jumping ahead or behind on the same periodic orbit is called “Phase

Jumping”. This idea has been used previously T=ZCd for z-axis control to avoid the solar exclusion zone. The goal is to phase jump the T=3W 1501 T=4W maneuvering spacecraft to rendezvous with 1 T=5W the target spacecraft at a later time. This initial phase error would be due to a number of different sources including launch errors, OO 2 4 6 8 10 12 Positive Phase Jump (days) maneuver execution errors, or phasing of other systems/orbits. Obviously, two Figure 15. Total AV For Phase Jump spacecraft coming from different orbits or launches would not end up in the same phase

10 Conclusions Restricted Three Body Problem, Celestial Mechanics, We have demonstrated several concepts and 20,389-404. 12. R. Farquhar, “The Utilization of Halo Orbits in trajectory designs in support of servicing Sun- advanced Lunar Operations”, NASA TN D-6365, Earth libration missions in Earth-moon NASNGSFC, July 1971 libration orbits. Several critical areas 13. R. Farquhar,” Stationkeeping In The Vicinity Of regarding AVs, travel time, rendezvous, and Collinear Libration Points With Applications To A Lunar Communications Problem”, AAS Preprint 66-1 32, orbit dynamics were addressed. It was found AAS, July 1966. that the AVs are in a manageable range of less 14. R. Farquhar, D. Muhonen, and D. Richardson, than 500m/s, similar to direct transfer mission “Mission Design for a Halo Orbiter of the Earth”, requirements with small SEL, y-amplitudes. AIAMAAS Astrodynamics Conference, San Diego, CA, Timing associated with servicing is of August 1976. 15. E. Belbruno, “Low Energy Trajectories for Space importance and can drive the orbit mechanics Travel and Stability Transition regions, Proceedings of and rendezvous opportunities. The timing IFAC Workshop on Lagranian and Hamiltonian Methods requirement results in additional maneuvers, for Nonlinear Control, IFAC Publ, Elsevier Science phase jumping, and targeting schemes. It is LTD., Oxford, March 2000. 16. J. Breakwell, J. Brown, ‘‘ The Optimization of clear that algorithms and software tools for Trajectories”, Journal of SZAM, Vol 7, June 1959, pp. trajectory design in this regime must be 215-247 further developed to incorporate better 17. K. Howell and H. Pernicka, “Numerical understanding of the solution space, and to Determination of Lissajous Trajectories in the Restricted improve the efficiency, and to expand the Three Bode Problem”,Celestial Mechanics, Vol. 41, 1988, pp. 107-124. capabilities of current approaches. 18. M. White and D. Stenger, “The Safe Use Of Nuclear Power Systems In Outerspace: Current Safety Standards References And A Suggested Framework For The Future”, Space 1. D. Portree and R. TreviAo, Walking to Olympus: An Technology and Applications international forum, 2000, EVA Chronology, Monographs in Aerospace History American Institute of Physics CP504 Series #7, October, 1997. 19. NASA Press Release, Media relations office Jjet 2. D. Waltz, On-Orbit Servicing of Space Svstems, Propulsion Laboratory, California Institute of Krieger Publishing Company, Malabar, Florida, 1993 Technology, September 20,2002 3. D. Waltz, Supplement to On-Orbit Servicing of Space 20. J. Amenabar, “ISEE Orbital Decay Due to Lunar Svstems, Krieger Publishing Company, Malabar, Third-Body Perturbations”, Advances in Astronautical Florida, 1998 Science1 Astrodynamics, Vol. 11, AIANAAS 87-535, 4. S. Leete, “Design for On-Orbit Spacecraft Servicing”, August 1987 Proceedings of the 2001 Core Technologies for Space 21. D. Folta, S. Cooley, K. Howell, “Trajectory Design Systems Conference, Colorado Springs, CO, 1 1/28-30, Stratgies for the NGST L2 Libration Point Mission”, 2001. AAS 0 1-205, Astrodynamcis Conference, Feb. 2001 5. H. Willenberg, M. Fruhwirth, S. Potter, S. Leete and 22. A. Marinescu, A. Nicolae, and M. Dumitrache, R. Moe, “Site Selection and Deployment Scenarios for Optimal Low-Thrust Libration Points Rendez-vous in Servicing of Deep-Space Observatories”, Proceedings of Earth-moon System, AIAA 99-4050 IEEE Aerospace Conference, 2002. 23. E. Belbruno, “Analytic Estimation of Weak Stability 6. B. Kent Joosten, “Exploration in the Earth‘s Boundaries and Low Energy Transfers”, Contemporary Nnqhborhood” .4rchitecture Analysis, NASA JSC, July Mathematics, AMS Vol. 292 February 2002 1 1,2000. 24. K. C. Howell & T. M. Keeter, Station-Keeping 7. M. Lo & S. Ross, “The Lunar LI Gateway: Portal to Strategies for Libration Point Orbits: Target Point and the Stars and Beyond”, Proceedings of the AIAA Space Floquet Mode Approaches, AAS 95-1 95, AASIAIAA 2001 Conference and Exposition, pub. AIAA. Spaceflight Mechanics Meeting, Albuquerque, MN, Feb 8. Report of the NASA Exploration Team: Setting a 13-16, 1995. Course for Space Exploration in the 21R Century, June 25. G. Gomez, et al, Dynamics and Mission Design Near 1999 through December 2000. October 2001. Libration Points, Vol. I Fundamentals: The Case of 9. M. Beckman and D. Folta, NGST Servicing Trade Collinear Libration Points, World Scientific, 2001. Study, Aug. 2002 IO. J.Guzman, D. Cooley, K. Howell, D. Folta, ”Trajectory Design Strategies that Incorporate Invariant Manifolds and Swingby”, AAS 98-349, AIAAIAAS Astrodynamics Conference, Boston, August 1998. I I. J. Breakwell, J. Brown, “ The Halo Family of Three Dimensional Periodic Orbits in the Earth-Moon

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