A computationally efficient algorithm is developed for onboard planning of $n$ -impulse fuel-optimal maneuvers for establishment and reconfiguration of spacecraft formations. The method is valid in circular and elliptic orbits and includes first-order secular $J_{2}$ effects. The dynamics are expressed in terms of differential mean orbital elements, and relations are provided to allow the formation designer to transform these into intuitive geometric quantities for visualization and analysis. The maneuver targeting problem is formulated as an optimal control problem in both continuous and discrete time. The continuous-time formulation cannot be solved directly in an efficient manner, and the discrete-time formulation, which has an analytical solution, does not directly yield the optimal thrust times. Therefore, a new flight-suitable algorithm is designed by iteratively solving the discrete-time formulation while using the continuous-time necessary conditions to refine the thrust times until they converge to the optimal values. Simulation results illustrate the performance for a variety of reconfiguration maneuvers and reference orbits, including examples for the NASA CubeSat Proximity Operations Demonstration mission.

References

[1] Prussing J. E., “Optimal Four-Impulse Fixed-Time Rendezvous in the Vicinity of a Circular Orbit,” AIAA Journal, Vol. 7, No. 5, May 1969, pp. 928–935. doi:https://doi.org/10.2514/3.5246 AIAJAH 0001-1452 Link Google Scholar
[2] Prussing J. E., “Optimal Two- and Three-Impulse Fixed-Time Rendezvous in the Vicinity of a Circular Orbit,” AIAA Journal, Vol. 8, No. 7, July 1970, pp. 1221–1228. doi:https://doi.org/10.2514/3.5876 AIAJAH 0001-1452 Link Google Scholar
[3] Jezewski D. J. and Donaldson J. D., “Analytic Approach to Optimal Rendezvous Using Clohessy-Wiltshire Equations,” Journal of Astronautical Sciences, Vol. 27, July–Sept. 1979, pp. 293–310. Google Scholar
[4] Prussing J. E. and Chiu J.-H., “Optimal Multiple-Impulse Time-Fixed Rendezvous Between Circular Orbits,” Journal of Guidance, Control, and Dynamics, Vol. 9, No. 1, Jan.–Feb. 1986, pp. 17–22. doi:https://doi.org/10.2514/3.20060 JGCDDT 0162-3192 Link Google Scholar
[5] Carter T. E. and Humi M., “Fuel-Optimal Rendezvous near a Point in General Keplerian Orbit,” Journal of Guidance, Control, and Dynamics, Vol. 10, No. 6, Nov.–Dec. 1987, pp. 567–573. doi:https://doi.org/10.2514/3.20257 JGCDDT 0162-3192 Link Google Scholar
[6] Shankar G. S., “Autonomous Guidance Scheme for Orbital Rendezvous,” Ph.D. Thesis, Indian Inst. of Science, Bangalore, India, Jan. 1995. Google Scholar
[7] Bell J. L., “Primer Vector Theory in the Design of Optimal Transfers Involving Libration Point Orbits,” Ph.D. Thesis, Purdue Univ., West Lafayette, IN, Dec. 1995. Google Scholar
[8] Mailhe L. H., Schiff C. and Folta D., “Initialization of Formation Flying Using Primer Vector Theory,” International Symposium on Formation Flying Missions and Technology, CNES, Oct. 2002. Google Scholar
[9] Luo Y.-Z., Zhang J., Li H.-Y. and Tang G.-J., “Interactive Optimization Approach for Optimal Impulsive Rendezvous Using Primer Vector and Evolutionary Algorithms,” Acta Astronautica, Vol. 67, Nos. 3–4, Aug.–Sept. 2010, pp. 396–405. doi:https://doi.org/10.1016/j.actaastro.2010.02.014 AASTCF 0094-5765 Crossref Google Scholar
[10] Aubin B. S., “Optimization of Relative Orbit Transfers via Particle Swarm and Primer Vector Theory,” M.S. Thesis, Univ. of Illinois, Urbana, IL, May 2011. Google Scholar
[11] Huang W., “Optimal Orbit Transfers for Satellite Formation Flying Applications,” Ph.D. Thesis, Univ. of Missouri, Columbia, MO, July 2012. Google Scholar
[12] Arzelier D., Louembet C., Rondepierre A. and Kara-Zaitri M., “New Mixed Iterative Algorithm to Solve the Fuel-Optimal Linear Impulsive Rendezvous Problem,” Journal of Optimization Theory and Applications, Vol. 159, No. 1, Oct. 2013, pp. 210–230. doi:https://doi.org/10.1007/s10957-013-0282-z JOTABN 0022-3239 Crossref Google Scholar
[13] Kim Y., Park S.-Y. and Park C., “Semi-Analytical Global Search Algorithm for Fuel-Optimal Satellite Formation Reconfiguration: Impulsive-Thrust Approach,” Advances in the Astronautical Sciences, Vol. 150, 2014, pp. 1357–1375; also Proceedings of the AAS/AIAA Astrodynamics Specialist Conference, American Astronautical Soc. Paper 13-795, Aug. 2013. Google Scholar
[14] Vaddi S. S., Alfriend K. T., Vadali S. R. and Sengupta P., “Formation Establishment and Reconfiguration Using Impulsive Control,” Journal of Guidance, Control, and Dynamics, Vol. 28, No. 2, March–April 2005, pp. 262–268. doi:https://doi.org/10.2514/1.6687 JGCDDT 0162-3192 Link Google Scholar
[15] Dannemiller D. P., “Multi-Maneuver Clohessy-Wiltshire Targeting,” Advances in the Astronautical Sciences, Vol. 142, 2011, pp. 1–14; also Proceedings of the AAS/AIAA Astrodynamics Specialist Conference, American Astronautical Soc. Paper 00-012, Feb. 2000. Google Scholar
[16] Saunders M., “Adaptive Formation Flying Maneuver for Multiple Relative Orbits,” M.S. Thesis, Univ. of Colorado, Boulder, CO, Aug. 2011. Google Scholar
[17] Breger L. S. and How J. P., “Formation Flying Control for the MMS Mission Using GVE-Based MPC,” Proceedings of the 2005 IEEE Conference on Control Applications, IEEE, Piscataway, NJ, Aug. 2005, pp. 565–570. doi:https://doi.org/10.1109/CCA.2005.1507186 Google Scholar
[18] Breger L. S. and How J. P., “Gauss’s Variational Equation-Based Dynamics and Control for Formation Flying Spacecraft,” Journal of Guidance, Control, and Dynamics, Vol. 30, No. 2, March–April 2007, pp. 437–448. doi:https://doi.org/10.2514/1.22649 JGCDDT 0162-3192 Link Google Scholar
[19] Roscoe C. W. T., “Reconfiguration and Recovery of Formation Flying Spacecraft in Eccentric Orbits,” M.A.Sc. Thesis, Univ. of Toronto, Toronto, June 2009. Google Scholar
[20] Roth N. H., “Navigation and Control Design for the CanX-4/-5 Satellite Formation Flying Mission,” M.A.Sc. Thesis, Univ. of Toronto, Toronto, Nov. 2010. Google Scholar
[21] Anderson P. V. and Schaub H., “N-Impulse Formation Flying Feedback Control Using Nonsingular Element Description,” Journal of Guidance, Control, and Dynamics, Vol. 37, No. 2, March–April 2014, pp. 540–548. doi:https://doi.org/10.2514/1.60766 JGCDDT 0162-3192 Link Google Scholar
[22] Gaias G., D’Amico S. and Ardaens J.-S., “General Multi-Impulsive Maneuver for Optimum Spacecraft Rendezvous,” Fifth International Conference on Spacecraft Formation Flying Missions and Technologies, DLR, May 2013, Paper 4087. Google Scholar
[23] Lawden D. F., Optimal Trajectories for Space Navigation, Butterworths, London, 1963, pp. 30–124. Google Scholar
[24] D’Amico S., Ardaens J.-S. and Montenbruck O., “Navigation of Formation Flying Spacecraft Using GPS: The PRISMA Technology Demonstration,” Proceedings of the 22nd International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS-2009), Institute of Navigation, Manassas, VA, Sept. 2009, pp. 1427–1441. Google Scholar
[25] Roscoe C. W. T., Griesbach J. D., Westphal J. J., Hawes D. R. and Carrico J. P., “Force Modeling and State Propagation for Navigation and Maneuver Planning for CubeSat Rendezvous, Proximity Operations, and Docking,” Advances in the Astronautical Sciences, Vol. 150, 2014, pp. 573–590; also Proceedings of the AAS/AIAA Astrodynamics Specialist Conference, American Astronautical Soc. Paper 13-739, Aug. 2013. Google Scholar
[26] Alfriend K. T. and Schaub H., “Dynamics and Control of Spacecraft Formations: Challenges and Some Solutions,” Journal of Astronautical Sciences, Vol. 48, Nos. 2–3, 2000, pp. 249–267. Crossref Google Scholar
[27] Alfriend K. T. and Yan H., “Orbital Elements Approach to the Nonlinear Formation Flying Problem,” International Symposium on Formation Flying Missions and Technologies, CNES, Oct. 2002. Google Scholar
[28] Schaub H., “Relative Orbit Geometry Through Classical Orbit Element Differences,” Journal of Guidance, Control, and Dynamics, Vol. 27, No. 5, Sept.–Oct. 2004, pp. 839–848. doi:https://doi.org/10.2514/1.12595 JGCDDT 0162-3192 Link Google Scholar
[29] Sengupta P. and Vadali S. R., “Relative Motion and the Geometry of Formations in Keplerian Elliptic Orbits,” Journal of Guidance, Control, and Dynamics, Vol. 30, No. 4, July–Aug. 2007, pp. 953–964. doi:https://doi.org/10.2514/1.25941 JGCDDT 0162-3192 Link Google Scholar
[30] D’Amico S. and Montenbruck O., “Proximity Operations of Formation-Flying Spacecraft Using an Eccentricity/Inclination Vector Separation,” Journal of Guidance, Control, and Dynamics, Vol. 29, No. 3, May–June 2006, pp. 554–563. doi:https://doi.org/10.2514/1.15114 JGCDDT 0162-3192 Link Google Scholar
[31] Gill E., D’Amico S. and Montenbruck O., “Autonomous Formation Flying for the PRISMA Mission,” Journal of Spacecraft and Rockets, Vol. 44, No. 3, May–June 2007, pp. 671–681. doi:https://doi.org/10.2514/1.23015 JSCRAG 0022-4650 Link Google Scholar
[32] Montenbruck O., Kahle R., D’Amico S. and Ardaens J.-S., “Navigation and Control of the TanDEM-X Formation,” Journal of Astronautical Sciences, Vol. 56, No. 3, July–Sept. 2008, pp. 341–357. doi:https://doi.org/10.1007/BF03256557 Crossref Google Scholar
[33] D’Amico S., Ardaens J.-S., Gaias G., Benninghoff H., Schlepp B. and Jørgensen J. L., “Noncooperative Rendezvous Using Angles-Only Optical Navigation: System Design and Flight Results,” Journal of Guidance, Control, and Dynamics, Vol. 36, No. 6, Nov.–Dec. 2013, pp. 1576–1595. doi:https://doi.org/10.2514/1.59236 JGCDDT 0162-3192 Link Google Scholar
[34] Gim D.-W. and Alfriend K. T., “State Transition Matrix of Relative Motion for the Perturbed Noncircular Reference Orbit,” Journal of Guidance, Control, and Dynamics, Vol. 26, No. 6, Nov.–Dec. 2003, pp. 956–971. doi:https://doi.org/10.2514/2.6924 JGCDDT 0162-3192 Link Google Scholar
[35] Battin R. H., Introduction to the Mathematics and Methods of Astrodynamics, AIAA, Reston, VA, 1999, pp. 471–514. Link Google Scholar
[36] Schaub H., Vadali S. R., Junkins J. L. and Alfriend K. T., “Spacecraft Formation Flying Control Using Mean Orbit Elements,” Journal of Astronautical Sciences, Vol. 48, No. 1, Jan.–March 2000, pp. 69–87. Crossref Google Scholar
[37] Schaub H. and Alfriend K. T., “Impulsive Feedback Control to Establish Specific Mean Orbit Elements of Spacecraft Formations,” Journal of Guidance, Control, and Dynamics, Vol. 24, No. 4, July–Aug. 2001, pp. 739–745. doi:https://doi.org/10.2514/2.4774 JGCDDT 0162-3192 Link Google Scholar
[38] Hamel J.-F. and de Lafontaine J., “Linearized Dynamics of Formation Flying Spacecraft on a $J_{2}$ -Perturbed Elliptical Orbit,” Journal of Guidance, Control, and Dynamics, Vol. 30, No. 6, Nov.–Dec. 2007, pp. 1649–1658. doi:https://doi.org/10.2514/1.29438 JGCDDT 0162-3192 Link Google Scholar
[39] Hamel J.-F. and de Lafontaine J., “Neighboring Optimum Feedback Control Law for Earth-Orbiting Formation-Flying Spacecraft,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 1, Jan.–Feb. 2009, pp. 290–299. doi:https://doi.org/10.2514/1.32778 JGCDDT 0162-3192 Link Google Scholar
[40] Alfriend K. T., Schaub H. and Gim D.-W., “Gravitational Perturbations, Nonlinearity and Circular Orbit Assumption Effects on Formation Flying Control Strategies,” Advances in the Astronautical Sciences, Vol. 104, 2000, pp. 139–158; also Proceedings of the AAS Guidance, Navigation, and Control Conference, American Astronautical Soc. Paper 00-012, Feb. 2000. Google Scholar
[41] Brouwer D., “Solution of the Problem of Artificial Satellite Theory Without Drag,” Astronomical Journal, Vol. 64, No. 1274, Nov. 1959, pp. 378–397. doi:https://doi.org/10.1086/107958 ANJOAA 0004-6256 Crossref Google Scholar
[42] Schaub H. and Junkins J. L., Analytical Mechanics of Space Systems, AIAA Education Series, 3rd ed., AIAA, Reston, VA, 2014, pp. 811–814. Google Scholar
[43] Alfriend K. T. and Yan H., “Evaluation and Comparison of Relative Motion Theories,” Journal of Guidance, Control, and Dynamics, Vol. 28, No. 2, March–April 2005, pp. 254–261. doi:https://doi.org/10.2514/1.6691 JGCDDT 0162-3192 Link Google Scholar
[44] Tschauner J. and Hempel P., “Rendezvous zu einem in elliptischer Bahn umlaufenden Ziel,” Astronautica Acta, Vol. 11, No. 5, 1965, pp. 312–321. ASACAW 0004-6205 Google Scholar
[45] Hill G. W., “Researches in the Lunar Theory,” American Journal of Mathematics, Vol. 1, No. 1, 1878, pp. 5–26. doi:https://doi.org/10.2307/2369430 AJMAAN 0002-9327 Crossref Google Scholar
[46] Clohessy W. H. and Wiltshire R. S., “Terminal Guidance System for Satellite Rendezvous,” Journal of the Aerospace Sciences, Vol. 27, No. 9, 1960, pp. 653–658. doi:https://doi.org/10.2514/8.8704 JASSA7 0095-9820 Link Google Scholar
[47] McAdoo S. F., Jezewski D. J. and Dawkins G. S., “Development of a Method for Optimal Maneuver Analysis of Complex Space Missions,” NASA TN-D-7882, April 1975. Google Scholar
[48] Jezewski D. J., “Primer Vector Theory and Applications,” NASA TR-R-454, Nov. 1975. Google Scholar
[49] Lewis F. L. and Syrmos V. L., Optimal Control, 2nd ed., Wiley, New York, 1995, pp. 41–66, 281–311. Google Scholar
[50] Lion P. M. and Handelsman M., “Primer Vector on Fixed-Time Impulsive Trajectories,” AIAA Journal, Vol. 6, No. 1, Jan. 1968, pp. 127–132. AIAJAH 0001-1452 Link Google Scholar
[51] Jezewski D. J. and Rozendaal H. L., “Efficient Method for Calculating Optimal Free-Space N-Impulse Trajectories,” AIAA Journal, Vol. 6, No. 11, Nov. 1968, pp. 2160–2165. doi:https://doi.org/10.2514/3.4949 AIAJAH 0001-1452 Link Google Scholar
[52] Antsaklis P. J. and Michel A. N., A Linear Systems Primer, Birkhäuser, New York, 2007, pp. 487–497. Google Scholar
[53] Neustadt L. W., “Optimization, a Moment Problem, and Nonlinear Programming,” SIAM Journal on Control and Optimization, Vol. 2, No. 1, 1964, pp. 33–53. doi:https://doi.org/10.1137/0302004 SJCOA9 0036-1402 Google Scholar
[54] Potter J. E. and Stern R. G., “Optimization of Midcourse Velocity Corrections,” Proceedings of the IFAC Symposium on Automatic Control in the Peaceful Uses of Space, edited by Aseltine J. A., Plenum, New York, June 1965, pp. 70–84. Google Scholar

Journal of Guidance, Control, and Dynamics, volume 38 issue 9 cover

Volume 38, Number 9September 2015

Special Issue in Honor of Richard Battin

Metrics

Crossmark

Information

Copyright © 2015 by Christopher Roscoe, Jason Westphal, Jacob Griesbach, and Hanspeter Schaub. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 1533-3884/15 and $10.00 in correspondence with the CCC.

Topics

Keywords

Acknowledgments

The authors wish to thank John Bowen, Al Tsuda, and Marco Villa of Tyvak Nano-Satellite Systems, Inc. for their support and technical input to this work. Additionally, they thank Paul Anderson of the University of Colorado at Boulder for providing valuable simulation results for comparing this algorithm to the method of Anderson and Schaub [21].

Digital

Received22 August 2014
Accepted22 May 2015
Published online23 July 2015

Formation Establishment and Reconfiguration Using Differential Elements in J2-Perturbed Orbits

Abstract

References

What's Popular

Metrics

Information