The usage of state transition tensors (STTs) was proved as an effective method for orbital uncertainty propagation. However, orbital maneuvers and their uncertainties are not considered in current STT-based methods. Uncertainty propagation of spacecraft trajectory with maneuvers plays an important role in spaceflight missions, e.g., the rendezvous phasing mission. Under the effects of impulsive maneuvers, the nominal trajectory of a spacecraft will be divided into several segments. If the uncertainty is piecewise propagated using the STTs one after another, large approximation errors will be introduced. To overcome this challenge, a set of modified STTs is derived, which connects the segmented trajectories together and allows for directly propagating uncertainty from the initial time to the final time. These modified STTs are then applied to analytically propagate the statistical moments of navigation and impulsive maneuver uncertainties. The probability density function is obtained by combining STTs with the Gaussian mixture model. The proposed uncertainty propagator is shown to be efficient and affords good agreement with Monte Carlo simulations. It also has no dimensionality problem for high-dimensional uncertainty propagation.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Rovetto, R. J., Kelso, T. S. Preliminaries of a space situational awareness ontology. In: Proceedings of the 26th AIAA/AAS Space Flight Mechanics Meeting, 2016.
Battin, R. H. An Introduction to the Mathematics and Methods of Astrodynamics. American Institute of Aeronautics and Astronautics, 1999.
Geller, D. K. Linear covariance techniques for orbital rendezvous analysis and autonomous onboard mission planning. Journal of Guidance, Control, and Dynamics, 2006, 29(6): 1404–1414.
Sabol, C., Hill, K., Alfriend, K., Sukut, T. Nonlinear effects in the correlation of tracks and covariance propagation. Acta Astronautica, 2013, 84: 69–80.
Vittaldev, V., Russell, R. P. Space object collision probability via Monte Carlo on the graphics processing unit. The Journal of the Astronautical Sciences, 2017, 64(3): 285–309.
Junkins, J. L., Akella, M. R., Alfriend, K. T. Non-Gaussian error propagation in orbital mechanics. Guidance and Control, 1996, 1996: 283–298.
Junkins, J. L., Singla, P. How nonlinear is it? A tutorial on nonlinearity of orbit and attitude dynamics. Advances in the Astronautical Sciences, 2003, 115(SUPPL.): 1–45.
Julier, S. J., Uhlmann, J. K. Unscented filtering and nonlinear estimation. Proceedings of the IEEE, 2004, 92(3): 401–422.
Park, R. S., Scheeres, D. J. Nonlinear mapping of Gaussian statistics: Theory and applications to spacecraft trajectory design. Journal of Guidance, Control, and Dynamics, 2006, 29(6): 1367–1375.
Fujimoto, K., Scheeres, D. J., Alfriend, K. T. Analytical nonlinear propagation of uncertainty in the two-body problem. Journal of Guidance, Control, and Dynamics, 2012, 35(2): 497–509.
Park, I., Scheeres, D. J. Hybrid method for uncertainty propagation of orbital motion. Journal of Guidance, Control, and Dynamics, 2018, 41(1): 240–254.
Yang, Z., Luo, Y.-Z., Zhang, J., Tang, G.- J. Uncertainty quantification for short rendezvous missions using a nonlinear covariance propagation method. Journal of Guidance, Control, and Dynamics, 2016, 39(9): 2170–2178.
Hernando-Ayuso, J., Bombardelli, C. Orbit covariance propagation via quadratic-order state transition matrix in curvilinear coordinates. Celestial Mechanics and Dynamical Astronomy, 2017, 129(1–2): 215–234.
Yang, Z., Luo, Y.-Z., Lappas, V., Tsourdos, A. Nonlinear analytical uncertainty propagation for relative motion near J2-perturbed elliptic orbits. Journal of Guidance, Control, and Dynamics, 2018, 41(4): 888–903.
Berz, M. Modern Map Methods in Particle Beam Physics. Academic Press, 1999.
Valli, M., Armellin, R., Di Lizia, P., Lavagna, M. R. Nonlinear mapping of uncertainties in celestial mechanics. Journal of Guidance, Control, and Dynamics, 2013, 36(1): 48–63.
Morselli, A., Armellin, R., Di Lizia, P., Bernelli Zazzera, F. A high order method for orbital conjunctions analysis: Monte Carlo collision probability computation. Advances in Space Research, 2015, 55(1): 311–333.
Wittig, A., Di Lizia, P., Armellin, R., Makino, K., Bernelli-Zazzera, F., Berz, M. Propagation of large uncertainty sets in orbital dynamics by automatic domain splitting. Celestial Mechanics and Dynamical Astronomy, 2015, 122(3): 239–261.
Wittig, A., Colombo, C., Armellin, R. Longterm density evolution through semi-analytical and differential algebra techniques. Celestial Mechanics and Dynamical Astronomy, 2017, 128(4): 435–452.
Riccardi, A., Tardioli, C., Vasile, M. An intrusive approach to uncertainty propagation in orbital mechanics based on Tchebycheff polynomial algebra. In: Astrodynamics 2015. Advances in Astrnautical Sciences. American Astronautical Society, 2015: 707–722.
Jones, B. A., Doostan, A., Born, G. H. Nonlinear propagation of orbit uncertainty using non-intrusive polynomial chaos. Journal of Guidance, Control, and Dynamics, 2013, 36(2): 430–444.
Jones, B. A., Doostan, A. Satellite collision probability estimation using polynomial chaos expansions. Advances in Space Research, 2013, 52(11): 1860–1875.
Jones, B. A., Parrish, N., Doostan, A. Postmaneuver collision probability estimation using sparse polynomial chaos expansions. Journal of Guidance, Control, and Dynamics, 2015, 38(8): 1425–1437.
Vetrisano, M., Vasile, M. Analysis of spacecraft disposal solutions from LPO to the Moon with high order polynomial expansions. Advances in Space Research, 2017, 60(1): 38–56.
Bierbaum, M. M., Joseph, R. I., Fry, R. L., Nelson, J. B. A Fokker–Planck model for a two-body problem. AIP Conference Proceedings, 2002, 617(1): 340–371.
Horwood, J. T., Aragon, N. D., Poore, A. B. Gaussian sum filters for space surveillance: Theory and simulations. Journal of Guidance, Control, and Dynamics, 2011, 34(6): 1839–1851.
DeMars, K. J., Bishop, R. H., Jah, M. K. Entropybased approach for uncertainty propagation of nonlinear dynamical systems. Journal of Guidance, Control, and Dynamics, 2013, 36(4): 1047–1057.
Vishwajeet, K., Singla, P., Jah, M. Nonlinear uncertainty propagation for perturbed two-body orbits. Journal of Guidance, Control, and Dynamics, 2014, 37(5): 1415–1425.
Vittaldev, V., Russell, R. P. Space object collision probability using multidirectional Gaussian mixture models. Journal of Guidance, Control, and Dynamics, 2016, 39(9): 2163–2169.
Fujimoto, K., Scheeres, D. J. Tractable expressions for nonlinearly propagated uncertainties. Journal of Guidance, Control, and Dynamics, 2015, 38(6): 1146–1151.
Vittaldev, V., Russell, R. P., Linares, R. Spacecraft uncertainty propagation using Gaussian mixture models and polynomial chaos expansions. Journal of Guidance, Control, and Dynamics, 2016, 39(12): 2615–2626.
Luo, Y.-Z., Yang, Z. A review of uncertainty propagation in orbital mechanics. Progress in Aerospace Sciences, 2017, 89: 23–39.
Vallado, D. A. Fundamentals of Astrodynamics and Applications, 3rd edn. Microscosm Press, 2007.
Fehse, W. Automated Rendezvous and Docking of Spacecraft. Cambridge University Press, 2003.
Phillips, K. R function to symbolically compute the central moments of the multivariate normal distribution. Journal of Statistical Software, 2010, 33(1): 1–14.
Terejanu, G., Singla, P., Singh, T., Scott, P. D. Uncertainty propagation for nonlinear dynamic systems using Gaussian mixture models. Journal of Guidance, Control, and Dynamics, 2008, 31(6): 1623–1633.
Yang, Z., Luo, Y.-Z., Zhang, J., Tang, G.-J. Homotopic perturbed Lambert algorithm for long-duration rendezvous optimization. Journal of Guidance, Control, and Dynamics, 2015, 38(11): 2215–2223.
The authors acknowledge the financial support from the National Natural Science Foundation of China (Nos. 11222215 and 11572345), the National Basic Research Program of China (973 Program, No. 2013CB733100), and the Program for New Century Excellent Talents in University (No. NCET-13-0159).
Zhen Yang received his B.S., M.S., and Ph.D. degrees in aerospace engineering from National University of Defense Technology, China, in 2011, 2013, and 2018, respectively. He was a visiting scholar of Cranfield University, UK, in 2016. After graduation from doctoral, he joined National University of Defense Technology as a lecturer in 2018. His current research interests include astrodynamics, uncertainty quantification, and space trajectory optimization.
Ya-Zhong Luo received his B.S., M.S., and Ph.D. degrees in aerospace engineering from National University of Defense Technology, China, in 2001, 2003, and 2007, respectively. Since December 2013, he has been a professor in National University of Defense Technology. His current research interests include manned spaceflight mission planning, spacecraft dynamics and control, and evolutionary computation.
About this article
Cite this article
Yang, Z., Luo, YZ. & Zhang, J. Nonlinear semi-analytical uncertainty propagation of trajectory under impulsive maneuvers. Astrodyn 3, 61–77 (2019). https://doi.org/10.1007/s42064-018-0036-7
- uncertainty propagation
- state transition tensors (STTs)
- Gaussian mixture model
- rendezvous phasing