# Orbit Propagation and Estimation

## Sections

The University of Surrey, along with Surrey Satellite Technology Ltd (SSTL), were among the first to demonstrate automated estimation of the orbits of their satellites onboard the satellite itself without use of a ground segment. This work was a combination of the GPS receiver technology being developed at SSTL with the Astrodynamics group development of orbit propagators and estimators. This was a major world-wide coup for Surrey and has enabled us to perform a number of experiments during subsequent satellite missions.

## Symplectic Integrators

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.

To see how these symplectic methods enable orbit determination onboard a satellite, the figure below shows the computational time on satellite hardware for predicting the motion of a GTO satellite into the future. The propagations have the same routines apart from the integrator and all the propagations are computed to the same level of accuracy. This means lower order integrators needed smaller timesteps. For this reason the 4th order Runge-Kutta method is off the top of the plot. The other methods tested are the Bulirsch-Stoer method, of which there were two variants, and regularised methods KS and KBS. Each of these methods reduce the computational time for the orbit propagation. The symplectic method SYM clearly outperforms all the others.

## 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 effect 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 enables us to place satellites in precise orbits such as a repeat groundtrack orbit:

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.

## Propulsion Error Determination

The ability to determine the orbit of a satellite autonomously enables us to characterise the performance of experimental propulsion systems. On UoSat-12 a resistojet propulsion system was being tested and used for orbit changes. For small duration firings the system produces large errors in thrust due to establishing a steady flow rate of propellant through the system and incomplete warming up the propellant in the combustion chamber.

A series of firings were made with the experimental propulsion system to provide nominal changes of satellite velocity along-track between 1 and 18 mm/sec. Before and after each firing 12 hours of GPS data were assembled and processed on the satellite with the specific interest in the semi-major axis (or energy) of its orbit. If the ΔV was precisely accurate we would predict a change in semi-major axis ΔaP. The estimations before and after firing provide an estimate of the actual Δa. In the plot below we show the difference between Δa and ΔaP normalised by ΔaP. So 1 on the vertical axis corresponds to 100% error, or Δa = 2ΔaP. These percentage differences are plotted against the ΔV that the orbit controller on the satellite demanded.

The plot shows that below about 10 mm/sec there are significant errors in the thrust actually provided while above 12.5 mm/sec the thrust is reasonably accurate. Such an experiment can only be achieved if the orbit estimator is sufficiently precise to determine the change in semi-major axis.

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