Unified State Model theory and application in Astrodynamics

The Unified State Model is a method for expressing orbits using a set of seven elements. The elements consist of a quaternion and three parameters based on the velocity hodograph. A complete derivation of the original model is given in addition to two proposed modifications. Both modifications reduce the number of state elements from seven to six by replacing the quaternion with either modified Rodrigues parameters or the Exponential Map. Numerical simulations comparing the original Unified State Model, the Unified State Model with modified Rodrigues parameters, and the Unified State Model with Exponential Map, with the traditional Cartesian coordinates have been carried out. The Unified State Model and its derivatives outperform the Cartesian coordinates for all orbit cases in terms of accuracy and computational speed, except for highly eccentric perturbed orbits. The performance of the Unified State Model is exceptionally better for the case of orbits with continuous low-thrust propulsion with CPU simulation time being an order of magnitude lower than for the simulation using Cartesian coordinates. This makes the Unified State Model an excellent state propagator for mission optimizations.

[1]  John L. Crassidis,et al.  Sliding Mode Control Using Modified Rodrigues Parameters , 1996 .

[2]  Wenpeng Zhang,et al.  Spacecraft aerodynamics and trajectory simulation during aerobraking , 2010 .

[3]  A comparative study of Newtonian, Kustaanheimo/Stiefel, and Sperling/Burdet optimal trajectories , 1975 .

[4]  T. C. van Flandern,et al.  Low-precision formulae for planetary positions , 1979 .

[5]  S. Altman.  A unified state model of orbital trajectory and attitude dynamics , 1972 .

[6]  Gerald R. Hintz,et al.  Survey of Orbit Element Sets , 2008 .

[7]  J. Petit.  Symplectic Integrators: Rotations and Roundoff Errors , 1998 .

[8]  T. Fukushima.  New Two-Body Regularization , 2006 .

[9]  P. Hughes.  Spacecraft Attitude Dynamics , 1986 .

[10]  黄晓燕,et al.  IgG , 2010, Definitions.

[11]  Oliver Montenbruck,et al.  Satellite Orbits: Models, Methods and Applications , 2000 .

[12]  V. Bond.  The uniform, regular differential equations of the KS transformed perturbed two-body problem , 1974 .

[13]  N. Sinha,et al.  On the Orbit Determination Problem , 1985, IEEE Transactions on Aerospace and Electronic Systems.

[14]  Nico Sneeuw,et al.  Energy integral method for gravity field determination from satellite orbit coordinates , 2003 .

[15]  Stefanie R. Beaver,et al.  Orbital Targeting Based on Hodograph Theory for Improved Rendezvous Safety , 2010 .

[16]  C. Tsitouras.  A Tenth Order Symplectic Runge–Kutta–Nyström Method , 1999 .

[17]  The KS-transformation in hypercomplex form and the quantization of the negative-energy orbit manifold of the Kepler problem , 1985 .

[18]  F. Sebastian Grassia,et al.  Practical Parameterization of Rotations Using the Exponential Map , 1998, J. Graphics, GPU, & Game Tools.

[19]  E. Mooij,et al.  Performance Aspects of Orbit Propagation using the Unified State Model , 2010 .

[20]  Velocity-space maps and transforms of tracking observations for orbital trajectory state analysis , 1975 .

[21]  E. Stiefel.  Linear And Regular Celestial Mechanics , 1971 .

[22]  A short derivation of the sperling-Burdet equations , 1975 .

[23]  Victor R. Bond,et al.  Modern Astrodynamics: Fundamentals and Perturbation Methods , 1996 .

[24]  Jörg Waldvogel,et al.  Quaternions and the perturbed Kepler problem , 2006 .

[25]  James R. Wertz,et al.  Space Mission Analysis and Design , 1992 .

[26]  Theocharis A. Apostolatos.  Hodograph: A useful geometrical tool for solving some difficult problems in dynamics , 2003 .

[27]  T. Fukushima.  Numerical Comparison of Two-Body Regularizations , 2007 .

[28]  S. Altman,et al.  Orbital hodograph analysis , 1965 .

[29]  Anastassios E. Petropoulos,et al.  Shape-Based Algorithm for Automated Design of Low-Thrust, Gravity-Assist Trajectories , 2004 .

[30]  S. Breiter.  Explicit Symplectic Integrator for Highly Eccentric Orbits , 1998 .

[31]  Alan H. Karp,et al.  A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides , 1990, TOMS.

[32]  J. M. Hedo,et al.  A special perturbation method in orbital dynamics , 2007 .

[33]  J. B. Eades.  Orbit information derived from its hodograph , 1968 .

[34]  Fang Qun,et al.  Flight Vehicle Attitude Determination Using the Modified Rodrigues Parameters , 2008 .

[35]  Fang Toh Sun.  Hodograph analysis of the free-flight trajectories between two arbitrary terminal points : technical report , 1964 .

[36]  R. Battin.  An introduction to the mathematics and methods of astrodynamics , 1987 .

[37]  M. D. Vivarelli.  The KS-transformation in hypercomplex form , 1983 .

[38]  I. V. Kurcheeva.  Kustaanheimo-Stiefel Regularization and nonclassical canonical transformations , 1977 .