Ph.D. Student of Geodesy, Department of Surveying and Geomatic Engineering, University of Tehran, Iran
Assistant Professor, Department of Surveying and Geomatic Engineering, University of Tehran, Iran
Graduate Student of Geodesy, Department of Surveying and Geomatics Engineering, University of Tehran, Iran
Satellite orbit determination is a method of determining the state vector, i.e., position and velocity, of a Low Earth Orbiting (LEO) satellite or interplanetary spacecraft. During the past two decades, many satellites with various applications including geodetic applications have been launched at low altitudes. For instance, TOPEX / POSEIDON, CHAMP, GRACE and GOCE are the examples of the geodetic LEO satellites. They are dominantly affected by the disturbances forces, i.e., the Earth’s gravity field anomalies and the atmospheric drag. Therefore, the LEO satellites orbit determination has special complexity and challenges which needs particular consideration.
In many researches, three techniques namely dynamic, kinematic and reduced dynamics approaches are implemented for LEO satellite orbit determination. In the dynamic approach, the satellite’s motion is modeled by the equation of motion which is expressed in the Earth Centered Inertial (ECI) frame. In this method, all forces acting on a satellite are computed using the dynamical model and numerically integrated to propagate the state vector to the subsequent epochs from an initial state vector. The dynamic model describing the satellite motion with time is constructed using the forces acting on the satellite. The gravitational forces such as the Earth and the Sun and Moon and any other third body gravitational attraction and their indirect effect and non-gravitational forces such as atmospheric drag and solar radiation pressure have been modeled for orbit determination procedure.
Kinematic orbit determination is a purely geometrical approach based on the observations that requires neither dynamic force models nor the physical information. The kinematic orbit is derived from observations. In the classical implementation of this approach, the orbital elements are derived from angular observations, e.g., azimuth and elevation of satellite.
Nowadays, Global Navigation Satellite Systems (GNSS) especially Global Positioning System (GPS) could represent independent continuous kinematic orbit. The kinematic orbit derived from GPS observations is a dense and accurate orbit. It provides necessary information for many applications in satellite geodesy. However, the accuracy of kinematic orbit is limited to noise, systematic and gross errors of observations.
The mismodeling of dynamic orbit and GPS measurement errors of kinematic orbit are both reduced when dynamic and geometrical information is combined in the reduced-dynamic orbit. The reduced dynamic orbit is generated by incorporating dynamic models as the dynamic model of the dynamic system of the orbital motion with the kinematic orbit in dynamic filtering process. Using the dynamic model, the effects of observation errors, noise, systematic and gross errors, will be reduced.
Kalman filtering is the most widely used method in satellite reduced-dynamic orbit determination process. It is useable for linear dynamic system with linear observation equations. However, the Extended Kalman Filter (EKF) or the linearized form the system equations should be used. In the case of linear form application, the initial value of unknowns is required.
The problem of orbit determination is one the highly nonlinear problem in engineering applications. For the implementation of the standard form of the Kalman filter for orbit determination, the initial orbit has to be computed. Different orbit determination methods are introduced for this purpose. In this article the idea of reference orbit determination based on the numerical integration is introduced.
The reference orbit is an initial approximation of the observed satellite orbit that can be used for linearizing purposes. The reference orbit is determined using numerical integration methods. It deviates from the real orbit because of using erroneous initial values and difference in the Earth’s real and reference gravitational field. Consequently, the reference positions of satellites, derived from the reference orbit, are different from the actual positions. The differences in positions are called the location errors. In order to minimize the location errors, the reference orbit should be computed as close as possible to the real orbit.
In this paper, the least squares approach is proposed for selecting the initial conditions in a way that the total misfit of the reference orbit and to the observed orbit is minimized. When integrating a reference orbit in a time interval, the location error is zero at the initial time and it increases linearly to a maximum at the end of time interval or the so-called the v-shaped pattern of the error. It may be better to uniformly distribute the differences over the interval. In other words, the v-shaped pattern of the differences is changed in such a way that the deviation of two orbits remains constant. This orbit is called the best-fitting reference orbit.
The more accurate reference orbit the less linearization error occurs. By using the best-fitting reference orbit instead of initial one in Kalman filter algorithm, 3D RMS of reduced-dynamic orbit is reduced to 1 meter over a full day.