Astrodynamics Group

Orbit Propagation and Estimation

Predicting the orbits of satellites is an essential part of mission analysis and has impacts on the power system, attitude control and thermal design. It is the starting point in planning whether a proposed mission is feasible and how the satellite(s) need to be designed. The computation of the orbits of artificial satellites around planets such as the Earth has been studied in detail in the last century. The problem of computing the orbits of satellites, however, is not straight forward. The main factors affecting the orbit of a satellite are: the non-spherical Earth, atmospheric drag, perturbative effects from the gravitational pull of the Sun and other planets and radiation pressure. These effects are important considerations for different types of satellite orbits. For example satellites in LEO are strongly affected by the non-spherical nature of the Earth and even atmospheric drag. Satellites out in geostationary orbit, however, are sufficiently far from the Earth for these effects to be ignorable. The gravitational pull of the Sun and Moon, however, does play a significant role in the evolution of their orbits. Effects such as atmospheric drag and radiation pressure are also very dependent upon the shape, size and mass of the satellite.

The purpose of satellite orbit propagators is to provide high accuracy in predicting the position of a satellite. This is usually achieved by employing a very short timestep. The calculation of the forces acting on a satellite at each timestep, however, slows down the computation which makes it prohibitive to propagate anything but crude force models on the satellite itself.

New geometric integration schemes have been developed recently which exploit the geometric properties of the orbital dynamics of a satellite. These symplectic methods preserve very accurately conserved quantities such as the energy and angular momentum associated with the orbital motion. Explicit schemes have been developed which also enables very fast computation of the satellite orbit for a given required level of accuracy.

As well as the accuracy, symplectic schemes also provide a method for splitting timesteps for different types of force, and this can be exploited for satellites to great effect, greatly reducing the expensive force calculations. The resulting propagation schemes we have developed are now able to compute accurate orbit trajectories onboard the satellite and therefore enable real time accurate orbit estimation to be performed in an autonomous way.

Orbit Estimation

Satellites which orbit in LEO are permanently within the constellation of GPS satellites and therefore specially adapted GPS receivers, developed here at Surrey, are used to estimate the position and velocity of the satellite with respect to this constellation. These measurements provide accurate position fixes but have no predictive power on where the satellite will be in the future. For this we need to estimate the orbit parameters.

Orbit estimation exploits the fact that satellite orbits can be determined by a set of constants - called the orbital elements . We can use several GPS measurements to try and determine these six constants and this produces a best estimate subject to the fact that each measurement is subject to noise.

The affect of perturbative forces acting on satellites causes the orbital elements to evolve slowly in time. This means that estimating the orbit of a satellite can only be performed within a certain level of accuracy dictated by the accuracy of the dynamical model.

The ability to compute orbital trajectories on a satellite enables a recursive estimation filter to be run on the satellite to estimate the satellite's orbit and thus predict future positions.

This figure shows the position estimates based on GPS on the night when Selective Availability (SA) was switched off. SA degrades the position accuracy artificially and the predictions show that the model used was more accurate than the degradation of the signal, hence the predictions improved once the measurement noise was reduced.

Epicycle Models

An alternative approach to numerical propagation of orbits is the development of analytic models. These models are of necessity lower accuracy, but the nature of the Geopotential lends itself to reasonably accurate analytic models being developed. The models are based upon the fact that nearly all satellites in LEO are on almost circular orbits with the variation in altitude due to orbital eccentricity being comparable to the effects of Earth's oblateness.

These analytic models are extensions of simple epicycle models that have been developed for modelling small eccentricity Keplerian orbits, where the eccentricity is considered the small parameter. The models take account of all the effects of the Earth's non-spherical shape as well as atmospheric drag. The models are accurate enough to predict the position of a satellite to within 1 km over a period of 1 week.

The analytic models are very useful for orbit estimation on the satellites as the epicycle elements (equivalent to the standard orbital elements) are indeed constants and therefore can be estimated reasonably accurately. Not having to compute the forces at each timestep also means that the recursive estimation is much faster.

Orbit estimation using the analytic epicycle model has been successfully demonstrated on all Surrey's satellites since 1998. Having an analytic model also enables control algorithms to be developed based on this dynamics which provides reasoanbly accurate autonomous control.