Formation control and reconfiguration through synthetic imaging formation flying testbed (SIFFT) S. Mohan*a, H. Sakamotob, D. W. Millera MIT Space Systems Laboratory, 70 Vassar St, Cambridge, MA, USA 02139; b Nihon University, 7-24-1 Narashinodai, Funabashi, Chiba 274-8501 Japan;
a
ABSTRACT The objective of the Synthetic Imaging Formation Flying Testbed (SIFFT) is to develop and demonstrate algorithms for autonomous centimeter-level precision formation flying. This technology helps to enable synthetic aperture imaging systems, especially when combined with second stage precision optical control systems. Preliminary tests have been conducted on SIFFT at the Flat Floor facility at NASA’s Marshall Space Flight Center (MSFC). The goal of the testing at MSFC was to demonstrate formation control of three “apertures” in an equilateral triangle and to reconfigure the formation by rotation and expansion of the triangle. Results were very successful and demonstrate the ability to position and reconfigure separate apertures. The final configuration was with three satellites floating. The two Follower satellites expand the formation with respect to the Master satellite, which executes a 10° rotation. This test was performed successfully under various initial conditions: initial Follower rotation, initial Follower drift, and initial significant position error of each Follower. Simple PID controllers were implemented to maintain position and attitude. Results show roughly 10cm steady state error and ±5cm precision. In addition, formation capturing technique, where satellites search for each other without prior knowledge of the position of the other satellites, is developed and demonstrated both on the 2D flat table and in the 3D International Space Station environment. Future work includes using a minimum set of beacons for estimation and implementing a search algorithm so satellites can acquire each other from any initial orientation. Keywords: Formation flight, Reconfiguration, Lost-in-space, SPHERES
1. INTRODUCTION The current capability of space telescopes is strictly limited by the payload fairing of the launch vehicle. The telescope must fit within the fairing, both in mass and in size. This greatly limits the capability of the telescope since the distance observable is related to the baseline of the telescope. In traditional telescopes, the baseline is the diameter of the primary mirror. Some effort has been made to generate deployable or segmented telescopes, but they are still fundamentally limited by the fairing size. Alternate options are launch in multiple pieces and to assemble the telescope on-orbit, or the fly separated spacecraft in formation. For a formation flown telescope, the effective baseline of the telescope is the distance between the spacecraft. The light is collected from separated apertures, and then combined to form the image. Formation flown telescopes allows for very large baselines, with very little mass. However, the formation must be maintained to fractions of a wavelength in order to be accurately combined.
Figure 1: Artist Conceptions of formation flown missions. LISA 3 spacecraft (left) [1], TPF-I 5 spacecraft (center) [2], SI multi-spacecraft (right) [3]
There are several formation flown telescope proposed or under study. Examples of missions are Terrestrial Planet Finder – Interferometer (TPF-I), Laser Interferometer Space Antenna (LISA), and Stellar Imager (SI). The objective of TPF-I is to study the development and features of extra solar planets. LISA is the first space-based gravitational wave observatory. LISA detects gravitational waves by measuring the change in distance of free floating test masses. Each LISA spacecraft must control its position to within 10 nanometers. SI is UV/Optical deep-space telescope proposed to image stars and observe the universe with 0.1 milli-arcsec resolution. The current concept design for SI is a 500m diameter Fizeau Interferometer comprised of reconfigurable array of 10-30 one-meter small satellites, distributed over a parabolic virtual surface. The diameter of the virtual surface can vary from 100m up to as much as 1000m, depending on the angular size of the target object. The focal length linearly scales with the diameter of the primary array, with focal lengths of 1 km and 10 km corresponding to diameters of 100m and 1000m, respectively. The mission concept requires the expansion (or contraction) of the entire array every time a new target is selected. The array is located in a Lissajous orbit at Earth-Moon Lagrange point 2. Several formation flight technology development projects are under way and this work contributes directly to technology development research for SI. There are three testbeds in development to support SI: Fizeau Interferometer Testbed (FIT); Formation Flying Testbed (FFTB); and Synthetic Imaging Formation Flight Testbed (SIFFT). Together, these three testbeds develop and demonstrate staged control methodologies both in software and hardware. FIT develops and demonstrates nanometer level, closed-loop control of a multi-element sparse array. FFTB models the micrometer level formation flying in simulation, with specific attention to formation acquisition. SIFFT provides the development and demonstration of centimeter level precision formation flying of multi-element arrays.
2. SIFFT OVERVIEW The objective of SIFFT is to develop and demonstrate, on hardware, control algorithm for multi-element formation flown systems. There are four main research areas: formation capture, formation maintenance, formation reconfiguration, and synthetic imaging maneuvers (retargeting and reconfiguration). These four research themes are delineated into tasks over a three year development period. The tasks for Year 1 are: critical design of a precision pointing payload; fabrication and integration of two precision pointing payloads; testing of staged formation flight and precision pointing control for lost-in-space maneuvers, formation reconfiguration, and laser acquisition. This paper focuses on testing results for lost-in-space maneuvers and formation reconfiguration. Control and estimation algorithms are developed in these areas and implemented on the SPHERES testbed. SPHERES (Synchronized Position Hold, Engage, Reorient Experimental Satellites) (Figure 2) is a formation flight testbed designed to provide a fault-tolerant environment for the development and maturation of control and estimation algorithms for formation flight, docking, autonomy, and reconfiguration[4][5]. The SPHERES testbed consists of two parts: a ground testbed at MIT and a flight testbed operated by astronauts on the International Space Station (ISS). The flight testbed utilizes the unique microgravity environment of the ISS and creates a laboratory to develop and validate algorithms. The SPHERES testbed consists of six self-contained, identical, free-flyer satellites: three on the ground at MIT and three satellites on the ISS. Ground satellites are used to test algorithms prior to uplink to the ISS. Each SPHERES satellite is a complete spacecraft bus, equipped with sensing, propulsion, computing, and communication capability. Additionally, each satellite has an expansion port, which is used on the ground satellites to augment the functionality of the satellites by attaching external payloads. The components of the SPHERES testbed are the satellites, a laptop computer that serves as a ground station, and five small beacons that form the Position and Attitude Determination System (PADS). The five beacons create the working area and provide the global reference frame. Each SPHERES satellite is a generic satellite bus. SPHERES satellites communicate through RF communication on a satellite-to-satellite frequency (916 MHz) and a satellite-to-laptop frequency (868 MHz). Each satellite has 24 ultrasound receivers, 3 gyroscopes, 3 accelerometers, and 12 thrusters that enable full six degree of freedom control. The SPHERES testbed has three main operational environments: simulation, MIT-SSL facility, and the ISS. Ground testing also occurs at NASA’s Marshall Space Flight Center (MSFC) Flight Robotics Laboratory, especially for testing that requires more space than is provided by the MIT-SSL facility. The SIFFT testing primarily used the SPHERES ground testbed, with some later tests implemented on the ISS. Three ground satellites were used for testing at the NASA MSFC flat floor. The satellites also used a Universal Docking Port
(UDP) was connected to the expansion port and used to provide relative estimation. Each UDP has 3 ultrasound beacons and 3 ultrasound receivers, which enables full relative state determination between satellites. This was used in place of the PADS global reference frame for two reasons. First, the global reference frame was not available at MSFC. Second, one objective of the formation capture research area was to acquire the satellite targets using directional beacons. This more closely represents the on-orbit scenario.
3. METHODS 3.1
Formation Initialization (Lost-in-Space)
This “lost-in-space” program develops and tests formation capturing techniques using the SPHERES facility. Some researchers discussed the formation capturing problem analytically and some numerical simulations were carried out [6]. However, no hardware demonstration has been reported yet. The SPHERES facility enables the experimental demonstration in both 2D and 3D environment. The formation capturing technique discussed herein consists of the following four stages: 1.
The multiple satellites are deployed without a prior knowledge of the other satellites’ positions, emulating a release from a launch vehicle or a case of contingency.
2.
The satellites capture the other satellites within their relative sensor range, which typically has a limited fieldof-view (FOV).
3.
The satellites null their range rates.
4.
The satellites position themselves within an array.
The problem assumes that no high-accuracy terrestrial navigation aids like GPS are available, thus the formation relies only on relative measurements between the satellites. In Year 1, the first two stages in the above list are explored. Each SPHERE satellite has an onboard ultrasound (U/S) transmitter, which emulates a sensor of an actual system with a limited FOV. Also 24 U/S receives (4 on each of 6 faces) are equipped on the satellites, enabling the satellite to receive the signal from any direction. The satellites have an omni-directional satellite-to-satellite communication channel. Using these setups, the formation capturing techniques have been developed in the 2D flat table in the MIT Space Systems Lab oratory and in the 3D environment of the International Space Station (ISS). Figure 3 shows a picture from the ISS test, which was conducted in March 2007.
Figure 3: “Lost-in-space” experiment at ISS (March 2007) 3.2
Formation Selection
The use of the SPHERES ground testbed constrains the number of satellites to three satellites. Thus, it is important to determine the optimal three satellite formation to set as the target. The optimization for compactness of the array’s auto-
correlation function leads to a Golay-3 formation (equilateral triangle). The advantage of having compact arrays is that one can obtain full u-v coverage with apertures of the smallest size, when the array is used in Fizeau interferometric mode. The cone of coverage of the UDPs is 30˚ full angle. Therefore, forming a perfectly equilateral triangle is not quite possible because perturbations in the position could cause a satellite to move outside the cone of visibility. To recover from this case would require significant acquisition maneuvers that have not yet been implemented. Thus, the chosen formation target was a slightly isosceles triangle. 3.3
Formation Reconfiguration
Formation reconfiguration is defined as the method by which an array of satellites, in an initial configuration, is safely and optimally moved to achieve a desired configuration. The movement is a series of maneuvers commanded to the spacecraft that transition it from the initial state to the final state. Hence, important aspects that should be considered are: target state update, path planning from initial to final state, and obstacle avoidance. This work focuses on the target update, with future work intended to address the remaining two aspects. Work culminated in a hardware demonstration at MSFC Flight Robotics Laboratory’s flat floor facility in September, 2006. The week long testing incrementally added complexity, with a final demonstration of a three-satellite triangle configuration that rotates and expands. This paper presents the implementation methodology, test description, and results from the MSFC testing. In order to demonstrate reconfiguration, the desired transformation of the array must be selected, ideally in accordance with the mission concept. The SI mission concept and requirements calls for a reconfigurable array of spacecraft that form a Fizeau interferometer with a variable diameter of 100m up to 1000m. Other potential maneuvers would include repositioning spacecraft within the array, array rotation, and array plane change; this would enable full u-v plane coverage. The combination of these maneuvers would allow for all possible arrangements of the array and the spacecraft within the array. Among these potential maneuvers, array expansion and rotation were selected as the maneuvers to demonstrate reconfiguration for this study. Test Sequence: The final test sequence consisted of four phases: estimator initialization, attitude control enabled, formation acquisition, and formation reconfiguration. Estimator Initialization: Out of the three satellites, one satellite was designated the Master. In a true space environment, such as the Lagrange points, there is no global reference frame or sensing system, as there is in LEO or on Earth (i.e., GPS). Therefore, a reference within the system must be identified in order to define the state of the system. This is also the scenario at the MSFC flat floor. In the lab facility at MIT, the five beacons provide a test volume, inside which the full state can be determined relative to the center of the test volume. In MSFC, this frame is not available, so all state determination must be done relatively. Thus, one satellite is designated as the Master, and the two remaining satellites are labeled the Followers. The Followers estimate their states by sensing the beacons on the Master satellite. Relative estimation is done using the beacons and receivers on the UDP. A ten second initialization period is allotted for the filter to converge. The initialization is an important step because the beacons on the UDP have a finite cone of reception of 30 half-angle. All sensors on the Follower satellites are used, thus the satellites can be in any initial attitude, as long as their position is within the cone of reception of the reference beacons. For this work, it is assumed that the non-Master satellites start positioned within the cone of reception of the beacons. Attitude Control Enabled: After Kalman Filter initialization, the Follower satellites enable attitude control. This initial maneuver allows for a small maneuver to assess the controllability of the satellites. For example, beginning with attitude control only allows for a check of the convergence of the state, prior to large slewing maneuvers to change positions, which can help to prevent collisions. The attitude control is enabled using a simple PD attitude controller. The Follower satellites are oriented to match their desired attitude with respect to the Master. A PD
Master Rotation Master Initial Radius Final Radius
Initial Formation
Reconfigured Formation
Figure 4: Schematic of formation reconfiguration with a three satellite array
controller was selected because of its simplicity, since this maneuver only needs to get the satellites in the ballpark of the desired attitude. This maneuver terminates after six seconds. Formation acquisition: After achieving the proper attitude, the satellites move into the first formation. A PID position controller is used in conjunction with a PD attitude controller. The satellites are given a fixed time in this maneuver to traverse to the desired position and hold there. In a real scenario, there would likely be two maneuvers, one to get there and one to stay there. Since our test was primarily to determine if we could get there and leave, the length of time spent at each formation was not critical. Formation reconfiguration: This maneuver updates the position targets to correspond to the second formation. The same controllers are run to control to this new position target. In the demonstration test, the Master SPHERE employs an estimator using solely gyroscope information to execute a 10˚ rotation. This rotation in the Master causes a rotation of the entire array.
4. RESULTS 4.1
Formation Initialization
Lab 2D Flat Table: Figure 5 shows the initial setting of the test carried out on the 2D flat table in MIT Space Systems Laboratory. As illustrated in the figure, the beacon transmitter on each SPHERES satellite faces away from the other satellite position, that is, the other satellite is outside the U/S signal’s FOV. This emulates the “lost-in-space” condition. Starting with this condition, each satellite captures the other satellite inside their U/S signal cone by the following maneuver phases. (Phase 1) The Master satellite searches for the other satellites through a prescribed rotation. (Phase 2) The Follower satellite notifies the Master of reception of U/S signal using a direct satellite-to-satellite communication. (Phase 3) The Follower satellite starts the state estimation using the U/S signal emitted from the Master, and then points its beacon-transmitter face toward the Master according to the estimated states. (Phase 4) Once the Master receives the U/S signal emitted from the Follower, the Master also estimates its states relative to the Follower, and then points the beacon-transmitter face toward the Follower.
Onboard Beacon Onboard Beacon
Master Follower y x
Figure 5: Initial condition for 2D flat table test The initial relative velocity was set at zero in the test. During this test, both satellites actively hold their positions using global metrology. This is to prevent the translation of the satellites caused by slopes in the table. Note that the global
position and attitude data are recorded and used for position hold, but not used for the formation capturing algorithm. In this study, the formation capturing control is implemented using only relative sensing data. Figure 6 shows the attitude quaternion and angular velocity with respect to the global coordinate frame fixed to the laboratory, of the Master satellite and of the Follower satellite respectively. These data are obtained using the global metrology system, using the five beacon transmitters fixed to the laboratory. Again these global metrology data are shown here just for presentation purpose and neithertelemetry satellite theseSphere global Background statesuse (standard), 1 attitude data for formation capturing 1 control. Px
Vel, m/s
Pos, m
Py Figure 6 also shows the maneuver phases itemized above. In this test, 15 seconds were allocated for the convergence of 0 Pz the global estimator. Then the Master satellites rotate counterclockwise with a prescribed rotation rate (Phase 1) until the Follower satellite receives the U/S signal at t = 28.6 s. The direct satellite-to-satellite communication is used for this notification -1(Phase 2). 10 Then the 20Master is30commanded to rotate more70 (for margin) and stop. The100 Follower 0 40 50 30 degrees 60 80 90 starts state 0.1 estimation using the U/S signal emitted from the Master, and at t = 30 s, the Follower starts pointing its Vx beacon-transmitter face toward the Master (Phase 3). At t = 36.5 s, the Master receives the U/S signal emitted from the Vy Follower. Finally, the Master also points its beacon-transmitter face toward the Follower (Phase 4). This completes the 0 Vz formation capturing maneuver. -0.1
0 satellite10 Master
20
30
40
Attitude
1
50
60
80
90
100
q4 = cos(θ / 2 )
q1 q2
0
-1
70
qq33 == sin sin(θθ // 22)
q3 q4
0
10
20
30
40
50
60
70
ωz
Rot rates, rad/s Pos, m
Background telemetry states (standard), Sphere 2 1 0.2
0 0 -0.2 10
20
20
Global estimator convergence
0.1 Vel, m/s
cos(θ / 2 )
ωz
-0.4 -1 0 0
0
-0.1 Follower satellite 0
30
40
40
60
Attitude
60
80
70
100
120 Vx Vy Vz
(3) Follower points Master 80
1
100
120
q3 = sin (θ / 2)
q1 q2 q3 q4
0
q4 = cos(θ / 2 ) -1
Rot rates, rad/s
50
(4) Master points follower
(1) Master search maneuver
20
40 60 s Test time,
0
10
20
30
40
50
60
70
ωz
0.2
wx wy wz
0 -0.2 -0.4
0
10
20
30
40 Test time, s
50
wx wy wz
Px Py Pz
60
Figure 6: Attitude quaternion and angular rates of the leaser and Follower satellites
70
ISS 3D Environment: The formation capturing experiment with 2 SPHERES was conducted at ISS in March 2007. Two satellites searched each other by following the same four phases described in the preceding subsection. Figure 7 depicts the initial condition of the test. Figure 8(a) shows the time history of the angle between the relative position vector that points from the Master to the Follower and the body –x axis (beacon face) of the Master. Figure 9 shows the two vectors and the angle between them. This angle represents a pointing error. Similarly, Figure 8(b) shows the pointing error for the Follower satellite. These angles are calculated using global state estimates sent from the satellites during the test. Since the crew was asked to position the two satellites facing away from each other, the pointing errors were initially 167˚ for the Master and 158˚ for the Follower. By the present relative search strategy, the two satellites successfully found each other and pointed to each other, making the pointing errors within 5˚ for both satellites after t = 60 s.
Follower
Master
Figure 7: Initial condition for ISS experiment
Figure 8: Pointing-error angles in each satellite during the ISS experiment (after the convergence of the global estimator)
Figure 9: Pointing-error angle α between the pointing vector and the U/S transmitter direction (-x)
Note that each satellite has only one U/S transmitter onboard, thus the relative estimation using the single U/S signal does not provide information in the roll direction in the 3D environment. As a result, in the final configuration, their rotational orientations about the x-axis are not necessarily aligned while the two satellites can point the U/S transmitter (x) face toward the other satellite. The close examination of the test result indicates that there was a “multi-path” problem of the U/S signal during this test. Figure 10 displays the Master satellite’s data obtained in the test. Figure 10(a) shows the Master satellite’s attitude and rotation rate with respect to the inertial coordinate frame, estimated by the global estimator. Figure 10(b) shows the Master satellite’s position and velocity in the body coordinate frame, estimated by the single beacon relative estimator. Figure 11 shows the data for the Follower satellite. As seen in Figure 11(b), the Follower carried out the relative estimation between t = 0 and 4.5 s and also after t = 10.1 s by receiving the U/S signal from the Master satellite. However, the U/S transmitter (-x) face of the Master satellite is initially pointed away from the Follower as shown in Figure 8. This suggests that the U/S signal was reflected on the ISS wall. As a result, the Follower’s relative estimator converged to a proper state only after t ~ 55 s, enabling the pointing maneuver. The Follower sent a flag to the Master via the direct satellite-to-satellite communication channel at t = 16 s (Phase 2), and carried out the pointing maneuver (Phase 3). At t = 19.3, the Master received the U/S signal emitted from the Follower and started the relative estimation (Phase 4) as seen in Figure 10(b). Although the estimator did not converge quickly due to the Follower’s unstable attitude, it converged to a proper state after t ~ 58 s. Consequently, as seen in Figure 10(a) and Figure 11(a), the attitudes of both Master and Follower were properly controlled and the rotation rates were nullified. As Figure 8 shows, the pointing errors were less than 5˚ for both satellites.
Figure 10: ISS experimental results for the Master satellite
Figure 11: ISS experimental results for the Follower satellite Figure 12 displays the time history of the angle between the Master’s body z-axis and the Follower’s body z-axis during the test. This angle is calculated using the data given by the global estimators. As mentioned above, the present relative search strategy does not control the rotational orientation of the satellite along the body x-axis. Hence, the body z-axes of the two satellites do not necessarily align. As Figure 12 shows, the angle was converged to approximately 20˚ in this run. This angle should vary depending upon initial conditions and disturbances during the tests. For example, the same relative search strategy was used in another test during the same ISS session and the angle was approximately 85˚ in the end of the test. More of the ISS test results are presented and discussed in a separate paper [7]. Although the test suffered from the multi-path problem of the U/S signal, it successfully demonstrated the adequacy of the present search strategy on hardware in 3D space.
Figure 12: Angle between body z-axes of the two satellites during the ISS experiment
4.2
Formation Reconfiguration
Testing was performed incrementally. The incremental tests were as follows: •
2 satellite tests, expansion only
•
3 satellite tests expansion only
•
2 satellite tests, expansion and rotation
•
3 satellite tests, expansion and rotation
Legend: Kalman Filter Initialization Attitude Control Only Formation Acquisition
The legend for Figure 14 and 15 is given by Figure 13. One satellite is Formation Reconfiguration designated as the Master (or reference) satellite, and is the (0,0) point on the following figures. Figure 14 shows a 3 satellite expansion only test. The blue Target Position diamonds represent the target locations, while the different colors represent the Figure 13: Legend for Figure different maneuver phases. As seen from the Figure 14, the steady state error in 14 and 15 the formation reconfiguration can be as large as 10 cm, while the position error is about ± 5cm. The straight black line indicates good movement between the two formations. Figure 15 shows the results for 2 satellite tests with expansion and rotation. The curved trajectory represents the rotation of the satellite. As the Master satellite rotates clockwise, the relative distance between the Master and Follower increases. The increase occurs because the Master rotates faster than the Follower can track. Thus, the curve slopes upward until the Master stops rotating, and then slopes down towards the target position. For the 3 satellite expansion and rotation, the second Follower is a mirror image of the first Follower, as seen in Figure 14.
Figure 15: 2reconfiguration Satellite reconfiguration and Figure 14: 3 Satellite expansionexpansion only rotation The tests were deployed with a variety of initial conditions. Though the Follower satellites always started located within the cone of the visibility of the Master’s beacons, the attitude and angular rate of the Follower was varied. The following initial conditions were successfully attempted. •
Follower pointing to Master
•
Follower pointing randomly
•
Follower with slight initial spin
•
Followers started in opposite position (Follower 1 in Follower 2’s position)
Also, since all estimation is relative, the absolute position of the formation cannot be controlled. Testing demonstrated good formation maintenance even under a moderate formation translation (~2-5 cm/s).
5. CONCLUSIONS The first two stages of the formation capturing maneuver (search maneuver and pointing maneuver) were successfully developed and tested using SPHERES facility on the 2D flat table and in the 3D International Space Station environment. Future work includes implementation of two satellites in 3D, including nulling the range rates and array
positioning. In addition, the number of satellites will be increased from two to three. Formation geometry maintenance and reconfiguration were also demonstrated for three satellites, using relative estimation and simple PID controllers. The reconfiguration maneuvers of expansion and rotation were demonstrated, to a steady state error of about 10cm and a precision of ±5cm. Future work includes integrating the formation capture phase with the geometry maintenance and reconfiguration phase. The relative estimation will also be updated to use a minimal set of beacons to acquire sufficient state knowledge, instead of full state estimation.
ACKNOWLEDGEMENTS This work was funded by Goddard Space Flight Center, with COTR Kenneth Carpenter. Additionally, H. Sakamoto was supported by the Japan Society for the Promotion of Science, Postdoctoral Fellowships for Research Abroad 2005.
REFERENCES 1. Laser Interferometer Space Antenna. http://lisa.nasa.gov/TECHNOLOGY/LISA_interfer2.html 2. Terrestrial Planet Finder Interferometer artist concept. http://planetquest.jpl.nasa.gov/gallery/tpfBrowseImages.html 3. Stellar Imager mission concept. http://hires.gsfc.nasa.gov/si/ 4. A. Saenz-Otero and D. Miller. “SPHERES: a platform for formation-flight research,” UV/Optical/IR Space Telescopes: Innovative Technologies and Concepts II conference, San Diego, CA, August 2005. 5. E. Kong et. al. “SPHERES as a Formation Flight Algorithm Development and Validation Testbed: Current Progress and Beyond,” 2nd International Symposium on Formation Flying Missions & Technologies, September 14-16, 2004, Washington, DC. 6. D. Scharf, S. Ploen, F. Hadaegh and G. Sohl, “Guaranteed Spatial Initialization of Distributed Spacecraft Formations,” AIAA-2004-4893, presented at AIAA Guidance, Navigation, and Control Conference and Exhibit, Providence, Rhode Island, Aug. 16-19, 2004. 7. C. P. Mandy et al., “Implementation of Satellite Formation Flight Algorithms Using SPHERES Aboard the International Space Station,” 20th International Symposium on Space Flight Dynamics, Annapolis, Maryland, Sept. 2428, 2007.