Second-order state transition for relative motion near perturbed, elliptic orbits

This paper develops a tensor and its inverse, for the analytical propagation of the position and velocity of a satellite, with respect to another, in an eccentric orbit. The tensor is useful for relative motion analysis where the separation distance between the two satellites is large. The use of nonsingular elements in the formulation ensures uniform validity even when the reference orbit is circular. Furthermore, when coupled with state transition matrices from existing works that account for perturbations due to Earth oblateness effects, its use can very accurately propagate relative states when oblateness effects and second-order nonlinearities from the differential gravitational field are of the same order of magnitude. The effectiveness of the tensor is illustrated with various examples.

[1]  S. R. Vadali,et al.  An intelligent control concept for formation flying satellites , 2002 .

[2]  P. Gurfil,et al.  Canonical Modelling of Relative Spacecraft Motion Via Epicyclic Orbital Elements , 2005 .

[3]  R. Broucke,et al.  On the equinoctial orbit elements , 1972 .

[4]  An Analytical Solution for Relative Motion of Satellites , 2005 .

[5]  E. A. Euler,et al.  Second-order solution to the elliptical rendezvous problem. , 1967 .

[6]  J. Turner.  Automated Generation of High-Order Partial Derivative Models , 2003 .

[7]  Dirk Brouwer,et al.  SOLUTION OF THE PROBLEM OF ARTIFICIAL SATELLITE THEORY WITHOUT DRAG , 1959 .

[8]  W. H. Clohessy,et al.  Terminal Guidance System for Satellite Rendezvous , 2012 .

[9]  Prasenjit Sengupta,et al.  Satellite Orbit Transfer and Formation Reconfiguration via an Attitude Control Analogy , 2005 .

[10]  P. Gurfil.  Relative Motion between Elliptic Orbits: Generalized Boundedness Conditions and Optimal Formationkeeping , 2005 .

[11]  T. Carter.  State Transition Matrices for Terminal Rendezvous Studies: Brief Survey and New Example , 1998 .

[12]  W. M. Kaula.  Theory of satellite geodesy , 1966 .

[13]  J. Junkins,et al.  Optimal Spacecraft Rotational Maneuvers , 1986 .

[14]  R. Melton.  Time-Explicit Representation of Relative Motion Between Elliptical Orbits , 2000 .

[15]  K. Yamanaka,et al.  New State Transition Matrix for Relative Motion on an Arbitrary Elliptical Orbit , 2002 .

[16]  S. Vadali,et al.  Formation Flying: Accommodating Nonlinearity and Eccentricity Perturbations , 2003 .

[17]  D. Richardson,et al.  A THIRD-ORDER ANALYTICAL SOLUTION FOR RELATIVE MOTION WITH A CIRCULAR REFERENCE ORBIT , 2003 .

[18]  S. Vadali,et al.  Modeling and Control of Satellite Formations in High Eccentricity Orbits , 2004 .

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

[20]  G. Hill.  Researches in the Lunar Theory , 1878 .

[21]  R. Sedwick,et al.  High-Fidelity Linearized J Model for Satellite Formation Flight , 2002 .

[22]  R. Gottlieb,et al.  Generalization of Lagrange's implicit function theorem to N-dimensions , 1971 .

[23]  Prasenjit Sengupta,et al.  Periodic relative motion near a keplerian elliptic orbit with nonlinear differential gravity , 2006 .

[24]  T. Carter,et al.  Fuel-Optimal Rendezvous Near a Point in General Keplerian Orbit , 1987 .

[25]  T. Carter.  New form for the optimal rendezvous equations near a Keplerian orbit , 1990 .

[26]  R. Broucke,et al.  Solution of the Elliptic Rendezvous Problem with the Time as Independent Variable , 2003 .

[27]  Kyle T. Alfriend,et al.  Nonlinear Considerations In Satellite Formation Flying , 2002 .

[28]  Derek F Lawden,et al.  Optimal trajectories for space navigation , 1964 .

[29]  Jeffrey A. Weiss,et al.  Strategies and schemes for rendezvous on geostationary transfer orbit , 1982 .

[30]  K. Alfriend,et al.  State Transition Matrix of Relative Motion for the Perturbed Noncircular Reference Orbit , 2003 .

[31]  Pini Gurfil,et al.  Euler Parameters as Nonsingular Orbital Elements in Near-Equatorial Orbits , 2005 .

[32]  Frederick H. Lutze,et al.  Second-Order Relative Motion Equations , 2003 .

[33]  Jean Albert Kechichian,et al.  Motion in General Elliptic Orbit with Respect to a Dragging and Precessing Coordinate Frame , 1998 .

[34]  Kyle T. Alfriend,et al.  Satellite Relative Motion Using Differential Equinoctial Elements , 2005 .