Open Platform Orbit Determination Systems Using a Mixture of Orbit Estimator and Orbit Propagator

Abstract

This paper describes a flexible orbit determination method that uses a mixture of orbit estimator and orbit propagator with the ability to perform corrections if the ephemeris from GPS, ground track or other method is obtainable. The satellite uses various sensors including Global Positioning Systems (GPS) or trusted orbit propagator in order to obtain the orbital information with the minimum delay and error at the lowest cost. Orbit propagator uses an orbital dynamic model where the analytic form needs to be constantly updated in order to maintain its accuracy and the integrator needs heavy computation but both constitute the error propagation in which the accuracy depends on the complexity and selection of orbital elements. Orbit estimator uses set of sensors data to produce an estimate where the accuracy depends on measurement noise characteristic and the model used. To avoid divergence, a sensible process and measurement noise model are selected. The orbit estimate is derived from an Extended Kalman Filter (EKF) while the Variation of Parameters (VOP) is used to propagate from one state to the other. Any obtainable ephemeris will be used as an initial state. The EKF uses the Position and Velocity elements as they possess dynamics that are beneficial to the estimator. The propagator uses the Keplerian elements as it consists of slow varying elements [a, e, i] and fast varying elements [ω, Ω, υ]. The EKF will be more difficult to diverge towards any abrupt disturbance if the slow varying elements from the orbit propagator are blended with the orbital elements produced by the orbit estimator and prevent the estimator from diverging.

Appendix

$${\mathbf{f}}_{d} = \frac{1}{2}\frac{{C_{D} A}}{m}\rho v_{rel}^{2} \frac{{\vec{v}_{rel} }}{{v_{rel} }}$$

where

$C_{D}$ :

is the coefficient of drag. A dimensionless coefficient that can be adjusted to suit the result of the orbit, which depends on the shape of the satellite and materials used. A crude approximation, $C_{D} = 2$ for initial orbit determination. Usually $2 \le C_{D} \le 2.3$

A :

is the effective area of the satellite normal to velocity vector

m :

is the mass of the satellite

ρ :

is atmospheric density

$\vec{v}_{rel}$ :

is the relative velocity vector of the satellite to the atmosphere

$v_{rel}$ :

is the magnitude of the velocity vector

The perturbation due to geopotential is given by:

$${\text{f}}_{g} = \Delta \frac{GM}{r}\sum\limits_{n = 0}^{\infty } {\sum\limits_{m = 0}^{n} {\frac{{R_{{}}^{n} }}{{r^{n} }}\overline{P}_{nm} \sin \,(\phi )\left( {\overline{C}_{nm} \cos m\lambda + \overline{S}_{nm} \sin m\lambda } \right)} }$$

where

P _nm :

is the Legendre polynomial

C _nm and S _nm :

are the geopotential coefficient with degree n and order m, and the detail can be represented as

$${\text{f}}_{tb} = \mu_{M} \cdot \left( {\frac{{\vec{s} - \vec{r}}}{{\left| {\vec{s} - \vec{r}} \right|^{3} }}} \right) - \mu_{earth} \frac{{\vec{r}}}{{\left| {\vec{r}} \right|^{3} }}$$

where the mass of the satellite is negligible compared to earth and third body and

μ _M :

is the gravitational constant due to body M

s :

is the geocentric coordinate of body M

r :

is the geocentric coordinate of the satellite

$${\text{f}}_{SRP} = - \frac{{P_{SR} c_{R} A_{ \oplus } }}{m}\frac{{\vec{r}}}{{\left| {\vec{r}} \right|}}$$

where

r :

is the radius of the satellite from the Earth

m :

is the mass of the satellite

c _R 1 + ε :

is the reflective coefficient (dependent on materials). $1.2 \le c_{R} \le 1.8$

$A_{ \oplus }$ :

is the area perpendicular to the sun or the acceleration can be expressed as

$${\text{f}}_{ERP} = \sum {C_{R} \left( {v_{j} a_{j} \cos \theta_{j}^{E} + \frac{1}{4}\varepsilon_{j} } \right)P_{SR} \frac{A}{m}\cos \theta_{j}^{S} \frac{{dA_{j} }}{{\pi r_{j}^{2} }}e_{j} }$$