Abstract
In this paper, an analytical second-order state transition matrix (STM) for relative motion in curvilinear coordinates is presented and applied to the problem of orbit uncertainty propagation in nearly circular orbits (eccentricity smaller than 0.1). The matrix is obtained by linearization around a second-order analytical approximation of the relative motion recently proposed by one of the authors and can be seen as a second-order extension of the curvilinear Clohessy–Wiltshire (C–W) solution. The accuracy of the uncertainty propagation is assessed by comparison with numerical results based on Monte Carlo propagation of a high-fidelity model including geopotential and third-body perturbations. Results show that the proposed STM can greatly improve the accuracy of the predicted relative state: the average error is found to be at least one order of magnitude smaller compared to the curvilinear C–W solution. In addition, the effect of environmental perturbations on the uncertainty propagation is shown to be negligible up to several revolutions in the geostationary region and for a few revolutions in low Earth orbit in the worst case.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alfriend, K., Vadali, S.R., Gurfil, P., How, J., Breger, L.: Spacecraft Formation Flying: Dynamics, Control and Navigation, 2nd edn. Butterworth-Heinemann, Oxford (2009)
Battin, R.: An Introduction to the Mathematics and Methods of Astrodynamics. Aiaa, Reston, Virginia (1999)
Bombardelli, C., Hernando-Ayuso, J.: Optimal impulsive collision avoidance in low Earth orbit. J. Guid. Control Dyn. 38(2), 217–225 (2015). doi:10.2514/1.G000742
Bombardelli, C., Gonzalo, J.L., Roa, J.: Approximate solutions of non-linear circular orbit relative motion in curvilinear coordinates. Celest. Mech. Dyn. Astron. 127(1), 49–66 (2017). doi:10.1007/s10569-016-9716-x
Chan, F.K.: Spacecraft Collision Probability. The Aerospace Press, El Segundo (2008)
Coppola, V.T., Tanygin, S.: Using bent ellipsoids to represent large position covariance in orbit propagation. J. Guid. Control Dyn. 38(9), 1775–1784 (2015). doi:10.2514/1.G001011
Folcik, Z., Lue, A., Vatsky, J.: Reconciling covariances with reliable orbital uncertainty. In: 12th Advanced Maui Optical and Space Surveillance Technologies Conference, Maui, Hawaii (2011)
García-Pelayo, R., Hernando-Ayuso, J.: A series for the collision probability in the short-encounter model. J. Guid. Control Dyn. 39(8), 1908–1916 (2016). doi:10.2514/1.G001754
Geller, D.K., Lovell, T.A.: Angles-only initial relative orbit determination performance analysis using cylindrical coordinates. J. Astronaut. Sci. 64(1), 72–96 (2017). doi:10.1007/s40295-016-0095-z
Hill, K., Alfriend, K., Sabol, C.: Covariance-based uncorrelated track association. In: AIAA/AAS Astrodynamics Specialist Conference, Honolulu, Hawaii, AIAA 2008-7211 (2008)
Junkins, J.L., Akella, M.R., Alfriend, K.T.: Non-Gaussian error propagation in orbital mechanics. J. Astronaut. Sci. 44(4), 541–563 (1996)
Kuchynka, P., Serrano Martin MA., Catania, M., Marc, X., Kuijper, D., Braun, V., et al.: Sentinel-1A: Flight dynamics analysis of the August 2016 collision event. In: 26th International Symposium on Space Flight Mechanics, Matsuyama, Japan (2017) ISTS-2017-d-028/ISSFD-2017-028
Lane, C.M., Axelrad, P.: Formation design in eccentric orbits using linearized equations of relative motion. J. Guid. Control Dyn. 29(1), 146–160 (2006). doi:10.2514/1.13173
Lee, B.S., Hwang, Y., Kim, H.Y., Kim, B.Y.: GEO satellite collision avoidance maneuver strategy against inclined GSO satellite. In: SpaceOps 2012 Conference, Stockholm, Sweden (2012). doi:10.2514/6.2012-1294441
Lee, S., Lyu, H., Hwang, I.: Analytical uncertainty propagation for satellite relative motion along elliptic orbits. J. Guid. Control Dyn. 39(7), 1593–1601 (2014). doi:10.2514/1.G000258
Melton, R.G.: Time-explicit representation of relative motion between elliptical orbits. J. Guid. Control Dyn. 23(4), 604–610 (2000). doi:10.2514/2.4605
Newman, K., Frigm, R., McKinley, D.: It’s not a big sky after all: justification for a close approach prediction and risk assessment process. Advances in the Astronautical Sciences 135(2):1113–1132, AAS 09-369. AAS/AIAA Astrodynamics Specialist Conference, Pittsburgh (2009)
Sabol, C., Sukut, T., Hill, K., Alfriend, K.T., Wright, B., Li, Y., et al.: Linearized orbit covariance generation and propagation analysis via simple Monte Carlo simulations. In: AAS 10-134, AAS/AIAA Space Flight Mechanics Conference, San Diego, California (2010)
Yamanaka, K., Ankersen, F.: New state transition matrix for relative motion on an arbitrary elliptical orbit. J. Guid. Control Dyn. 25(1), 60–66 (2002). doi:10.2514/2.4875
Author information
Authors and Affiliations
Corresponding author
Additional information
The Ministry of Education, Culture, Sports, Science and Technology (MEXT) of the Japanese government supported Javier Hernando-Ayuso with one of its scholarships for graduate school students. The work of Claudio Bombardelli was supported by the Spanish Ministry of Economy and Competitiveness within the framework of the research project “Dynamical Analysis, Advanced Orbital Propagation, and Simulation of Complex Space Systems” (ESP2013-41634-P).
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Hernando-Ayuso, J., Bombardelli, C. Orbit covariance propagation via quadratic-order state transition matrix in curvilinear coordinates. Celest Mech Dyn Astr 129, 215–234 (2017). https://doi.org/10.1007/s10569-017-9773-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10569-017-9773-9