LS GEO IOD
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GEO IOD James R. Wright1
Abstract I have constructed and demonstrated a new fast-running algorithm to perform refined orbit determination for any spacecraft in GEO (geosynchronous Earth orbit), without user requirement for an a priori orbit estimate. We now have a refined method for GEO initial orbit determination (IOD). This is enabled by the use of equinoctial orbit elements, by a one-dimensional search in the equinoctial orbit element mean argument of orbit longitude, by the use of sensor locations to reduce the one-dimensional search, and by convergence boundaries of the nonlinear least squares (LS) algorithm. I shall refer herein to the new algorithm as LS_GEO_IOD. Algorithm LS_GEO_IOD uses any desired perturbative acceleration model for numerical trajectory integration, and its success appears to be independent of input measurement type.
Introduction Initial orbit determination (IOD), also called preliminary orbit determination [1], refers to a class of orbit determination methods that derive initial orbit estimates from sensor measurements and sensor locations, without user requirement for an a priori orbit estimate. However, the use of two-body orbit mechanics, and the existence of significant white noise2 on minimal sets of measurements in geocentric applications, has always created IOD estimates with very large estimation error magnitudes. Existing IOD algorithms are unlike each other, distinguished by distinct measurement types and by distinct methods to address nonlinearity. They are disparate. In addition, some IOD algorithms produce multiple distinct solutions (e.g., see [2]). Historically, the two-body IOD algorithms are associated with the names of Laplace (1780), Lagrange (1778, 1783), Gauss (1809), Gibbs (1889), Herrick (1940), Gooding (1993), and others (see [1, 2]). Two-body IOD estimates have been used to seed iterative batch least squares (LS) algorithms so as to calculate refined LS orbit estimates. LS algorithms use complete acceleration models and overdetermined sets of measurements to accomplish the refinement and to provide the unique orbit estimate. Existing nonlinear LS
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ODTK Architect, Analytical Graphics, Inc., 220 Valley Creek Blvd, Exton, PA, 19341. Laplace’s method for angles measurements fails for most applications to geocentric orbits because of white noise embedded in the angles measurements combined with the second-order Taylor’s series approximation that requires measurement time-tags to be close together and from the same sensor. 2
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algorithms, also known as Gauss–Newton algorithms, have always required an a priori orbit estimate for initialization.
Algorithm Summary Algorithm LS_GEO_IOD is unified for distinct measurement types with a nonlinear LS algorithm, with a one-dimensional search in mean argument of longitude (an element in the set of six equinoctial orbit elements),3 a user interface in Kepler orbit elements, LS differential corrections in ECI position and velocity components, and with LS iterative convergence defined by Cauchy. Sixdimensiona
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