Goddard Space Flight Center
Stationkeeping in and Design of Transfer Between Earth-Moon L1,2 Orbits: ARTEMIS Mission Results
Future In-Space Operations (FISO)
Dave Folta Goddard Space Flight Center
February, 23, 2011 Agenda
Goddard Space Flight Center
• General Libration Orbit Background • Stationkeeping Methods • The ARTEMIS Mission Results and Operations • Stationkeeping Observations • Comparison to Other Lunar Orbits • Summary Libration Orbit Dynamics
Goddard Space Flight Center Where Are They?
What Are They?? Collinear Points: L1, L2, L3 (unstable)
Equilibrium or libration points represent Triangular Points: L4, L5 (stable) singularities in the equations of motion where velocity and acceleration components are zero and the forces are balanced
Viewed in the rotating frame: centrifugal L4 (Coriolis-Type) force balances with gravitational forces of the two primaries 1 a.u. 1 a.u. Libration points are in plane with no Z component. Orbits are mapped to a rotating L3 Sun L1 L2 frame where there are no time dependent forces 0.01 a.u. A system of interest involves the Sun (m1), Earth/Moon System the Earth-Moon system (m2) and the spacecraft m3
L1 and L2 At a distance of 1.5 million km L5 L4 and L5 At a distance of 150. million km Solar Rotating Coordinates Ecliptic Plane Projection Rotating Coordinate Frames
Goddard Space Flight Center Sun-Earth & Earth-Moon
• X is towards smaller body (sun to Earth) • Y is along smaller body velocity • Z is out of Ecliptic plane
• Co-linear Unstable locations at L1, L2, and ‘L3’ • Stable locations at L4 and L5
L4
L2 L Sun - Earth 1 Sun - Earth Moon’s Orbit L1 L2
150 million km L Sun 5 L3
1.5 million km 1.5 million km General Background and ARTEMIS
Goddard Space Flight Center • Vicinity of the Earth-Moon collinear libration points as promising locations for scientific data collection and/or communications options • Orbits near the collinear locations, including quasi-periodic Lissajous trajectories, are inherently unstable and must be controlled • A variety of stationkeeping strategies have previously been investigated, most notably for applications in the Sun-Earth system; fewer studies have considered trajectories near the Earth-Moon libration points • A true four-body problem - Earth-Moon orbit maintenance is more challenging than Sun-Earth system Shorter time scales (~14 days vs. ~6 months for 1 rev) Larger orbital eccentricity of the secondary (lunar eccentricity ~0.055) Sun acts as a significant perturbing body both in terms of the gravitational force as well as solar radiation pressure • ARTEMIS Implications Baseline trajectory is defined only to design a feasible mission No reference motion that is required Minimize cost in terms of fuel is the highest priority 5 Spacecraft constraints (spinning with limited DV direction) General Background and ARTEMIS
Goddard Space Flight Center
• Only recently has anyone exploited the regions near the Earth-Moon libration points • ARTEMIS (Acceleration Reconnection and Turbulence and Electrodynamics of the Moon's Interaction with the Sun) is the first Earth-Moon libration mission
• Two spacecraft entered into Earth-Moon L1 and L2 orbits in August and October of 2010 • Stationkeeping strategies implemented for ARTEMIS support A traditional baseline orbit-targeting approach A global optimum search scheme A balancing approach as part of an orbit continuation scheme • To support ARTEMIS, orbit maintenance costs were compared for periodic and non-periodic orbits in different dynamical models; Circular Restricted Three-Body (CRTB) as well as Moon-Earth-Sun ephemeris models using high-fidelity modeling that incorporates all perturbations.
6 ARTEMIS
Goddard Space Flight Center • The ARTEMIS mission is an extension to the Time History of Events and Macroscale Interactions during Substorms (THEMIS) mission. • The ARTEMIS mission moved two (of five) spacecraft in the outer-most elliptical Earth
orbits and, with lunar gravity assists, re-directing to both the L1 and L2 via transfer trajectories that exploit the multi-body dynamical environment. • Once the Earth-Moon libration point orbits are achieved and maintained for several months, both spacecraft will be inserted into elliptical lunar orbits. • The P1 spacecraft entered Earth-Moon Lissajous orbits August 25th 2010 and P2 followed in October 2010. Artemis provides comprehensive Earth-lunar environment analysis using particles and fields instruments. P2 Transfer Trajectory P1 Transfer Trajectory
7 ARTEMIS P1 In Earth-Moon L2 Lissajous orbit
Goddard Space Flight Center
Earth-moon Libration Orbit • P1 captures into L2 Orbit • Transfer to L1 Orbit • Transfer to Lunar Orbit
ARTEMIS-P1 Spacecraft’s Orbit – Side View
Towards Earth ARTEMIS- Moon P1 Here on August L1 25th L2
ARTEMIS L1 Orbit
ARTEMIS L2 Orbit ARTEMIS P2 to Enter Earth-Moon L1 Orbit October 23
Goddard Space Flight Center ARTEMIS L1 Orbit
Earth-Moon Libration L2 to L1 Transfer Orbit Orbit Direction • P2 captures into L1 to Earth Orbit via L2 • Transfer to Lunar Orbit
Moon Sample of Unstable Mode Directions using STM of and EML Orbit Goddard Space Flight Center 2
• System dynamics show six modes, 4 oscillating, one converging, one diverging • Stable or Unstable directions can be used to maintain orbit balance
Sample of stable and unstable modes for the EML2 early orbit Earth-Moon L1 and L2 Stability
Goddard Space Flight Center • Application of a 0.1cm/s DV in the +/- velocity direction at 1 day interval locations about a sample orbit
L2 Departures -DV applied +DV applied Along Unstable Modes
-Dv Returns to Lunar Region
+Dv Departs Toward L2 Moon L2 Sun/Earth Libration Points
L 1 -DV applied +DV applied Departures Along Unstable Modes
+Dv Returns to Lunar Region
-Dv Departs L1 Moon L1 Toward Stationkeeping Strategies (1 of 2)
Goddard Space Flight Center Stationkeeping strategies must satisfy the following self-imposed conditions: o Full ephemeris with high-fidelity models o Globally optimized solutions were possible
o Methods that can be applied for any Earth-Moon orbital requirements at L1 or L2 Standard approaches cannot be employed for various reasons, o Reference orbit is required which is not necessarily available nor desired o Strategy is based on the CRTB model only o Proposed approach cannot accommodate the ARTEMIS spacecraft constraints
Strategy Goal(s) Advantage Disadvantage Selection Criteria 1 Target x-axis Velocity at -Validated in Sun-Earth - Overly constraining -Operationally constraining and larger crossing, parallel to operations - Can lead to increased DVs DV budget crossing with x- x-axis, is zero - Can use a DC or optimization - Dynamics may not result in - Sensitivity issues in computing axis velocity process to target a single reaching subsequent x-axis maneuvers to achieve targets can require parameter crossings high recovery DVs constraint - May not meet operational - Not selected for this ARTEMIS constraints 2 Unstable mode Cancel unstable - Simple design based on - Requires the use of STMs based - Reference orbit not available component of the dynamical properties of the on reference orbit or the EM - Sensitivity to mode calculation due to cancellation error libration point orbit libration point lunar eccentricity + solar gravity - Multiple algorithms available - Intensive and iterative - Not selected for this application to apply DV calculations 3 Continuous Converge onto a - Possibly less sensitive to - Requires a reference orbit - Requires a reference orbit reference orbit navigation and execution errors - Uses near continuous thrusting - Does not apply to ARTEMIS spacecraft Controllers - Maintains orbit within user- (may be discretized) operationally (near continuous control) defined small torus -Requires computation of gain - Not selected for this application from an STM that is based on the libration point or actual orbit -Linear approximations for control feedback 12 Stationkeeping Strategies (2 of 2)
Goddard Space Flight Center • Baseline orbit control-point targeting approach: employs a DC to maintain the vehicle near a nominal trajectory, determined a priori. • Global optimum search method: a strategy that can be applied to any trajectory designed within a higher-fidelity environment • Continuation: Balance the orbit by meeting goals several revolutions downstream, thereby ensuring a continuous orbit without near-term requirements or the reliance on specific orbit specifications. … Use Velocity or Energy to define goals … Employs a Boxed environment for divergence control Strategy Goal(s) Advantage Disadvantage Selection Criteria
4 Baseline Orbit Target multiple points - Rigorous method with -Selection of control points results - Reference points computed as a along orbit guaranteed results in the computation of a reference first guess from an available baseline Control-Point - Based in ephemeris model orbit - Used only as first guess utility for Targeting - Possibly larger DVs ARTEMIS 5 Boxed Define constraints in - Always converges if box - Current implementation algorithms - Used in this application only to terms of distance from size and targeting scheme are limited identify and re-direct solutions that Environment x-axis and combined properly - Logic required in s/w to check for are not converging y-axis trajectories that depart the system 6 Orbit Velocity (or energy) is - Guarantees a minimal DV - Needs accurate integration and full - Analyzed with intent to apply to determined to deliver to achieve orbit continuation ephemeris modeling ARTEMIS Continuation s/c several revs - Several control constraints - Logic required in s/w to check for downstream (see 1 above) can be applied departure trajectories (e.g., x-axis velocities - 3-D application - Optimization requires monitoring all slightly negative) of process 7 Global Search over orbital - Guaranteed minimal DV - Optimization requires monitoring - Analyzed with intent to apply to parameters at specified magnitude and direction for of process ARTEMIS Optimum orbital locations to seek several orbits - Requires accurate integration and Search minimum DV full ephemeris modeling 13 Baseline Control–Point Targeting Strategy Lissajous Trajectory Goddard Space Flight Center (Statistical Errors in All Directions; 300 Trials) • Two maneuvers per revolution for stationkeeping. • Once reaching an x-axis crossing, control points at the next x-axis crossing are targeted. • Maximum y-axis location process is actually ‘less reliable’ in general • Modified with an initial guess incorporated into the process the numerical computations are stabilized. • The third option targets x-axis crossings and includes four maneuvers per revolution. No. of Avg Total Avg DV per Std Dev Avg DV per Avg Time Maneuvers DV (m/s) Maneuver (m/s) Year (m/s) Between Man Per 129 (m/s) (days) days X-axis, 16 9.45 0.59 2.45 26.59 7.63 every crossing Max Y-Amp 17 14.54 0.86 7.26 40.92 7.21 every crossing 4 Pts/Rev 33 2.50 0.08 0.32 7.04 3.82 (~3.8 days) 14 Baseline Control–Point Targeting Strategy Sample Stationkeeping Costs for Control-Point Method Goddard Space Flight Center (Error in +x Direction Only)
• Periodic halo orbits and non-periodic Lissajous trajectories are first integrated in a barycentric Earth-Moon rotating coordinate frame consistent with the circular restricted three-body problem (CR3B) for a desired number of revolutions. • Higher-fidelity baseline orbits are then computed using operational numerical methods by discretizing the CR3BP solutions into a series of patch points and re-converging the solution in a Moon-centered Moon-Earth-Sun ephemeris model using multiple shooting.
Dynamic Orbit No. of Total DV Avg. DV per DV per Avg. Time Model Man (m/s) DV(m/s) year (m month Between /s) (m/s) Man (days) CR3B Halo 17 15.16 0.8919 41.62 3.468 7.42 Lissajous 16 12.26 0.766 34.46 2.872 7.64 Ephem Halo 17 20.19 1.188 55.15 4.596 7.42 Lissajous 16 13.25 0.8282 37.26 3.105 7.64
15 Sample Stationkeeping Costs for Global Optimum Search (Error in +x direction Only; One Trial) Goddard Space Flight Center
• At a desired maneuver location, a maneuver plane is defined parallel to the x-y plane and through the current spacecraft location. • A maneuver angle, α, is measured from the +x-axis and is varied from 0° to 180° in plane. • At each maneuver angle, a DV magnitude is computed that results in a trajectory with zero x-velocity (one option) at an x-z plane crossing several crossings in the future. • Ultimate goal is to ensure that the spacecraft remains in orbit near the libration point for the immediate future. • Simulated errors are then added and propagated forward. • Increase in cost when ephemeris model incorporated. • Employing a Lissajous trajectory noticeably increases the cost, likely due to the shape of Lissajous orbit and changing inclination.
Dynamic Orbit No. of Total DV Avg. Δv Δv per year Δv per month Avg. Time Model Man (m/s) (m/s) (m/s) (m/s) Between Man (days) CR3B Halo 17 8.983 0.5284 24.65 2.054 7.39 Lissajous 16 10.09 0.6305 28.18 2.348 7.69
Ephemeris Halo 17 12.54 0.7374 34.37 2.86 7.4 Lissajous 15 26.33 1.7552 78.13 6.51 7.69 16 Continuous Method
Goddard Space Flight Center • Each profile varied the maneuver location and then the number of revolutions • Each simulation uses statistical navigation errors (1km and 1 cm/s 1s, and constant maneuver errors of 2%. ( can also add attitude ‘pointing’ errors) • DV constrained within the ARTEMIS spin plane • Results demonstrate that maneuvers at a frequency of least once every seven days are desired to both minimize the DV budget • A more frequent maneuver plan (3.8-day updates) is only slightly better • DVs are larger for one revolution and maximum y-axis amplitude cases DV for Continuous Optimal Method using 1.5-rev (10 Trials). Maneuver Location No. of Avg Total DV Avg DV per Std Dev Avg DV per Avg Time Maneuvers (m/s) Maneuver (m/s) Year (m/s) Between Per 126 days (m/s) Maneuver (days) X-axis, every crossing 15 4.23 0.28 0.78 12.27 7.3
X-axis, once per orbit 7 34.14 4.88 7.07 106.51 15.2
Max Y-Amp Every 15 6.26 0.42 .95 18.13 7.3 crossing Max Y-Amp Once per 7 38.29 5.46 6.98 110.91 14.9 orbit 4 Pts/Rev 33 4.74 0.15 0.33 13.72 3.8 ( ~3.8 days) 17 Executed ARTEMIS EML1 / EML2 Stationkeeping Cost
Goddard Space Flight Center Stationkeeping cost since insertion into libration orbits (w/o axial corrections) • Total P1 ~ 3.2 m/s, 21 maneuvers, mean = 0.16 cm/s std = 0.13 cm/s • Total P2 ~ 2.6 m/s, 16 maneuvers, mean = 0.17 cm/s std = 0.10 cm/s Projected yearly stationkeeping cost • ~9 m/s based on routine weekly maneuvers and ARTEMIS constraints
ARTEMIS SKM Cost 3.000
2.500
2.000
1.500 Axial/Radial Burns P1 SKM Cost P2 SKM Cost
1.000 SKM Magnitude SKM [m/s]
0.500
0.000 0 5 10 15 20 25 SKM Number EML1 / EML2 Stationkeeping Observations (1 of 2)
Goddard Space Flight Center • Navigation: Navigation solution errors are less than 100 m and 0.1 cm/s Stable navigation solutions are achieved with ~ 4 days of tracking that require both north and south DSN locations (ARTEMIS uses batch least- squares operationally, a filter was also investigated)
• Maneuvers: Execution errors (performance and pointing) are ~ 1-2 % Sensitivity to total errors are magnified at max y-amplitude locations Maneuver frequency of at least once every seven days is desired to minimize the DV and to align with the navigation solution deliveries
• General: Dynamics dictate that for the lunar orbital perturbation to be satisfied integration should include at least 1.5 revolutions. This strategy will include major perturbations and the effects of the lunar orbit eccentricity The inclination of the orbit (at least for the ARTEMIS scenarios analyzed) does not affect the DV magnitude and the magnitude remains constant The DV direction with respect to the x-axis may be repeatable for a given maneuver scenario EML1 / EML2 Stationkeeping Observations (1 of 2)
Goddard Space Flight Center • General: The Solar Radiation Pressure perturbation must be accounted. ARTEMIS navigation solutions varied when s/c in lunar shadow The number of revolutions used to continue the orbit is key in minimizing the DV requirements As the ARTEMIS orbit’s inclination decreases, the y-amplitude increases and affects the selection of the orbital conditions to successfully continue the orbit Strategies investigated and implemented yield DV budgets less than 10 m/s per year Libration point orbit stationkeeping DV requirements are very reasonable and much less in comparison to low lunar orbit DV maintenance requirements, approximately 11 m/s per month
20 Comparison to Lunar Orbits
Goddard Space Flight Center Relay Orbit Pro Con Circular Orbit – Minimize DV and Range to moon Reduce % orbit visible Elliptical Polar Orbit – Increase % orbit visible High Skp DV and initial rotation DV Elliptical Frozen Orbit – Minimize Skp DV Larger range Libration Orbit – Maximize Durations Range >40k km Butterfly Orbit – Maximize Duration, Minimize DV Range > 40k km
oTotal 1-yr DV ranges from 644 m/s to 1023 m/s o Initial rotation of ~45 deg results in additional 200-300 m/s DV cost o Elliptical skp DV assumes a 10 deg LOA maintenance, apoapsis over south pole o Elliptical orbit provides coverage upwards of 70%, circular orbit maximize at 40% coverage o Elliptical change in periapsis altitude to 1000 km has minimal impact to percent coverage o Ranges are roughly the same for equal elliptical apoapsis or circular orbit radius Orbit Altitude Inc DV Insert DV LOA DV Skping Total DV Orbit Period Visibility/Orbit Visibility/Orbit Max Range (km) (deg) (m/s) Rotation (m/s) 1-yr (m/s) 1-yr (m/s) (Hrs) (Hrs) (%) (km) 100 Circ 90 840 0 150 990 2.0 0.1 5.5 334 500 Circ 90 793 0 12 805 2.6 0.4 14.6 1041 1000 Circ 90 752 0 12 764 3.6 0.7 20.7 1718 5000 Circ 90 646 0 12 658 13.8 4.6 33.7 6077 10000 Circ 90 632 0 12 644 31.7 11.8 37.3 11173 100 x 500 90 763 100 160 1023 2.3 0.4 17.5 1029 100 x 1000 90 687 124 169 980 2.7 0.8 28.0 1625 100 x 5000 90 427 260 182 869 7.0 4.7 67.6 5096 100 x 10000 90 326 305 195 826 13.9 11.5 82.7 10003 100 x 20000 90 256 326 209 791 31.9 28.9 90.7 19975 Frozen 30 x 216 90 840 40 0 880 2.0 0.1 5.0 400 Frozen 718 x 8088 58 413 240 0 653 12 8.7 67 8614 Butterfly* 90 300 0 50 350 14 days 168 95 45000 Halo* n/a 500 0 50 550 14 days 150 50 60000 Lissajous n/a >400 0 10 410 14 days 14 days 50* <60000 Lunar Orbit Options and Stationkeeping
Goddard Space Flight Center • Orbit Options Polar (90-deg inc) orbits will precess and periapsis will rise and fall with perturbations Frozen orbit with inclination at ~60-deg and sma of 7800km, (periapsis at ~ 100km, apoapsis at ~ 14000km) Yields a ‘no stationkeeping’ orbit that is frozen wrt line of apsides ( apsides do not rotate) Still allocate a small DV, ~ 20m/s (tbd), for any error corrections
• Frozen conditions for elliptical orbits vary slightly if w is not 90 or 270. So some drift will occur, but can be targeted to drift toward smaller eccentricity Other EML1 / EML2 Orbits
Goddard Space Flight Center
‘Vertical’ EM Orbits ‘Butterfly’ EM Orbits
‘Vertical’ EM Orbits
23 Summary
Goddard Space• FlightMethods Center have been demonstrated that result in low stationkeeping DV requirements and that meet the ARTEMIS mission requirements • Full ephemeris model and the associated errors from navigation and maneuvers are required to accurately model the accelerations that affect the DV. • The dynamics of the Earth-Moon environment also must be modeled over a sufficient duration (=> 21 days) • An increase in the frequency of the maneuvers tends to reduce the overall DV requirements as does the placement of the maneuvers near the x-axis crossing. • Stationkeeping cost with realistically modeled navigation errors does have a floor – a rule of thumb from ARTEMIS, ~20:1 ratio of SKM DV to nav+execution errors for ½ half rev. • Mission applications and mission constraints must also be considered • The methods developed allow a general application whether there is a reference orbit, spacecraft constraints on DV direction, or orbital parameters requirements • Investigation of robust strategies and options to improve the DV required for stationkeeping are continuing (e.g. comparing DV directions to unstable modes) • ARTEMIS (both spacecraft) on-track as the first Earth-Moon libration orbiters, Aug/Oct 2010 thru July 2011 24