Celest Mech Dyn Astr (2017) 129:215–234
Orbit covariance propagation via quadratic-order state
transition matrix in curvilinear coordinates
· Claudio Bombardelli
Received: 30 January 2017 / Revised: 8 June 2017 / Accepted: 12 June 2017 /
Published online: 28 June 2017
© Springer Science+Business Media B.V. 2017
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.
Keywords State transition matrix · Curvilinear coordinates · Quadratic solution ·
Clohessy–Wiltshire solution · Orbit uncertainty
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 The online version of this article (doi:10.1007/s10569-017-9773-9)
contains supplementary material, which is available to authorized users.
The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Space Dynamics Group, Technical University of Madrid, Plaza Cardenal Cisneros 3, 28040 Madrid,