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(12) United States Patent SS43 USOO7744036B2 (12) United States Patent (10) Patent No.: US 7,744,036 B2 Kawaguchi et al. (45) Date of Patent: Jun. 29, 2010 (54) METHOD FOR DESIGNING AN ORBIT OFA ESA/NASA, Schematic Diagram of SOHO's Orbit, Jan. 9, 2004, SPACECRAFT ESA, p. 1.* NASA, SOHO Orbit Schematic, p. 1.* (75) Inventors: Junichiro Kawaguchi, Sagamihara (JP); NASA, RLP Coordinate System, p. 1.* Kohta Tarao, Sagamihara (JP) Min Xin, et al., “Stationkeeping of an La Libration Point Satellite with 0-D Technique'. Proceeding of the 2004 American Control (73) Assignee: Japan Aerospace Exploration Agency, Conference, Jun. 30, 2004-Jul. 2, 2004, pp. 1037-1042, Tokyo (JP) XPO02407945, Boston, Massachusettes, USA. Notice: Kulkarni J. Campbell M. “Asymptomatic Stabilization of Motion (*) Subject to any disclaimer, the term of this about an Unstable Orbit. Application to Spacecraft Flight in Halo patent is extended or adjusted under 35 Orbit”. Proceeding of the 2004 American Control Conference, Jun. U.S.C. 154(b) by 61 days. 30, 2004-Jul. 2, 2004, pp. 1025-1030, XP002407946, Boston, Mas (21) Appl. No.: 11/490,344 sachusettes, USA. Gomez G. et al., “Libration Point Orbits: A Survey from the Dynami (22) Filed: Jul.19, 2006 cal Point of View'. Proceedings of the International Conference on Libration Point Orbits and Applications, Jun. 10, 2002-Jun. 14, 2002, (65) Prior Publication Data pp. 1-9 and pp. 30-37, XP002407947. Begur, Girona, Spain. US 2010/O108819 A1 May 6, 2010 (Continued) (30) Foreign Application Priority Data Primary Examiner Timothy D Collins Assistant Examiner Valentina Xavier Jul. 20, 2005 (JP) ............................. 2005-210266 (74) Attorney, Agent, or Firm Blakely, Sokoloff, Taylor & Zafman LLP (51) Int. C. B64G L/10 (2006.01) (57) ABSTRACT (52) U.S. Cl. ................................................... 244/1584 (58) Field of Classification Search .............. 244/158.1, 244/1584 1588 A method is disclosed for designing an orbit of a spacecraft See application file for complete search history. which allows the spacecraft to take a small-radius halo orbit (56) References Cited near a Lagrange point while avoiding the prohibited Zone where the spacecraft may be shadowed or might be prevented U.S. PATENT DOCUMENTS from making communication. The method makes it possible 4,795, 113 A 1, 1989 Minovitch to have a closed orbit although being similar to a Lissajous 5, 183,225 A * 2/1993 Forward ..................... 244,168 orbit, under a restricted condition where a propulsion force 6,385,512 B1 5, 2002 Belbruno magnitude applied to a spacecraft is fixed, and where it rotates 6,809,818 B2 10, 2004 Hillis et al. at a constantangular Velocity, based on the equation of motion close to a Lagrange point. The method also provide a theory OTHER PUBLICATIONS for guiding/controlling the orbit of a spacecraft, that is, the Roberts, Craig, Sun-Earth L1 Region Halo-to-Halo and Halo-to embodiment of the above orbit design method. Lissajous Orbit Transfers, 2004, AI Solutions, pp. 1-6.* AI Solutions, Technical Paper Library, p. 4.* 9 Claims, 13 Drawing Sheets TOWARDS THE SUN (PERIOD BEING SIX MONTHS) N 'N N-1- - -1 MOON ’s 2*, n . ) VEARTHN N-1 N. N: O Z-AXIS IS PERPENDICULAR SS43, TO THE ECLIPTIC SURFACE, n US 7,744,036 B2 Page 2 OTHER PUBLICATIONS Farquhar, R. W., The Flight of ISEE-3/ICE: Origins, Mission History, Dunham D. W. Farquhar R. W. Libration Point Missions, 1978 and a Legacy, , AIAA/AAS Astrodynamics Specialists Conference, 2002, Proceedings of the International Conference on Libration Point Boston, MA, Aug. 10-12, 1998, pp. 1-59. Orbits and Applications, Jun. 10, 2002-Jun. 14, 2002, XP002407948, Begur, Girona, Spain. European Search Report, EP Application No. 06117482.7. Dec. 13, Farquhar, R. W. et al., “Mission Design for a Halo Orbiter of the 2006, 3 pages. Earth', AIAA/AAS Astrodynamics Conference, San Diego, CA, Aug. 18-20, 1976, pp. 170-177. * cited by examiner U.S. Patent Jun. 29, 2010 Sheet 1 of 13 US 7,744,036 B2 FIG. SUN gy) U.S. Patent Jun. 29, 2010 Sheet 2 of 13 US 7,744,036 B2 case S 326SNS 'N n A. SSnextor s2.O U.S. Patent Jun. 29, 2010 Sheet 3 of 13 US 7,744,036 B2 FIG.3 TOWARDS THE SUN f t K (PERIOD BEING SIX MONTHS) Ya voos-2.( N. J s VEARTH\5 a X 3. Z-AXIS IS PERPENDICULAR s NgŠ3, TO THE ECLIPTIC SURFACE, N U.S. Patent Jun. 29, 2010 Sheet 4 of 13 US 7,744,036 B2 FIG.4 LINEARIZE THE EGUATION OF MOTION OF A SPACECRAFT AROUND POINT L2 (EGUATION 2). SET AN ASSUMED ORBIT APPROXIMATING ALISSAJOU ORBIT IN THE FORM OF A CLOSED CIRCUIT OBTAINED BASED ON GIVEN INITIAL VALUES (EQUATION 5). SETA PROPULSION FORCE CORRESPONDING TO THE AFOREMENTIONED ORBIT (EQUATION 6). DEDUCE A SECOND-ORDEREQUATION IN TERMS OF k" WITHATTENTION BEING PAID TO THE CONDITION THE AFOREMENTIONED EGUATION OF MOTION SATISFIES AT ANY GIVEN TIME AND TO THE CONDITION THAT THE PROPULSIVE FORCE IS FIXED (EQUATION 11). DETERMINE BY CALCULATION THE CURVED TRACE OF k" AS A FUNCTION OF OU FROM EQUATION (11) WHEN THE ANGULARVELOCITY OU IS (UXO. DETERMINEBY CALCULATION, FROM THE ABOVE CURVED TRACE,THE PROPULSION FORCE VECTORA AS A FUNCTION OF THE ANGULARVELOCITY OU U.S. Patent Jun. 29, 2010 Sheet 5 of 13 US 7,744,036 B2 0.29 0.28 o o 1.65 1.7 1.75 18 1.85 19 195 2 2.05 2.1 U.S. Patent Jun. 29, 2010 Sheet 6 of 13 US 7,744,036 B2 FIG.6 x 10 AMPLITUDE OF CONTROL INPUT U.S. Patent Jun. 29, 2010 Sheet 7 of 13 US 7,744,036 B2 FIG.7 U.S. Patent Jun. 29, 2010 Sheet 8 of 13 US 7,744,036 B2 FIG.8 O x(km) XZ(NONLINEAR) (U=20082, Z:MINUS U.S. Patent Jun. 29, 2010 Sheet 9 of 13 US 7,744,036 B2 FIG.9 (UE20082, Z:PLUS U.S. Patent Jun. 29, 2010 Sheet 10 of 13 US 7,744,036 B2 FIG.10 2 - x(km) X to x 10' x-z(NONLINEAR) ORBIT(NONLINEAR) ---. (U-17136, Z:MINUS U.S. Patent Jun. 29, 2010 Sheet 11 of 13 US 7,744,036 B2 FIG.11 NUMERICAL VERIFICATION OF AN ORBIT DESIGNED BY THE INVENTIVE METHOD IN THE REAL CELESTIAL SYSTEM COMPRISING THE SUN, EARTH AND MOON (C)=1.7136, Z:MINUS) U.S. Patent Jun. 29, 2010 Sheet 12 of 13 US 7,744,036 B2 FIG.12 x(km) x 10 NUMERICAL VERIFICATION OF AN ORBIT DESIGNED BY THE INVENTIVE METHOD IN THE REAL CELESTIAL SYSTEM COMPRISING THE SUN, EARTH AND MOON ((U=2.0082, Z:MINUS) U.S. Patent Jun. 29, 2010 Sheet 13 of 13 US 7,744,036 B2 US 7,744,036 B2 1. 2 METHOD FOR DESIGNING AN ORBIT OFA shadow by the Earth, and if the spacecraft is positioned close SPACECRAFT to point L1, can avoid passage in front of the Sun. When viewed from the Earth, it looks like ahalo around the Sun, and CROSS-REFERENCE TO RELATED thus it is called a halo orbit. APPLICATIONS Following patent documents deal with the orbits around the Lagrange points and the halo orbits, i.e., U.S. Pat. Nos. 6,809, The present patent application claims priority from Japa 818; 6,385,512; 4,795,113; and 5,183,225. nese Patent Application No. 2005-210266, filed on Jul. 20, 2005. SUMMARY OF THE INVENTION 10 FIELD OF THE INVENTION As stated above, the natural halo orbit is advantageous on the fact that it is stable and draws a closed locus without any The present invention relates to a method for designing an special artificial correction operation maintaining the motion. orbit of a spacecraft to move in a cosmic space, particularly to However, the natural halo orbit disadvantageously leads to an a method for designing an orbit of a spacecraft which enables 15 extraordinarily large orbit which must depart from a co-linear the spacecraft to be on a small-radius halo orbit near a line connecting two celestial bodies. The distance may be as Lagrange point while avoiding the prohibited Zone where the large as, for example, about one million km from the spacecraft may be shadowed or might be prevented from Lagrange point referred to here. The resulted natural Halo making communication. orbits around point L1 or L2 are too far and very much apart from the aimed points originally. Because of this, the merit BACKGROUND OF THE INVENTION associated with the halo orbits, that is, no maintenance of the position of a spacecraft is needed, is inevitably lost, if the Take, as an example, a system consisting of two celestial trajectory size is controlled and maintained artificially. Fur bodies where both bodies turn around the bury-center, a cen thermore, if it is required for a spacecraft to make a docking ter of masses of two bodies, taking a circular orbit like the 25 with another spacecraft moving along the same natural halo Earth and the Sun, and a vary light third body flying in the orbit, the period required until docking will be prolonged. system. It has been known that there are five equilibrium Thus, the merits of the natural halo orbit around a Lagrange points in the system where gravity from those two bodies and point are not utilized for various applications. Currently a a centrifugal force acting on a third body are balanced. They number of spacecrafts take natural halo orbits, for example, are known as Lagrange points.
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