Estimation of Ballistic Coefficient from Raw Receiver Measurements
Madeline McDougal and Scott Martin, Auburn University
Recently, low Earth orbit (LEO) satellites have been deployed with more frequency than in previous years and this is only expected to increase in the future. These satellites are commonly used for communication, Earth observation, and signal monitoring. LEO satellites can be used to augment GNSS to improve positioning performance for end users. Precise LEO satellite locations are needed to calculate the position of the end user receiver. Two approaches to localizing the LEO satellite are ground station tracking and on-board GNSS receivers. Using on-board GNSS for LEO satellite positioning can potentially reduce the need for expensive ground tracking stations. However, on-board GNSS receiver failure would cause the LEO satellite to lose its positioning capabilities during orbit. For the duration of the outage, the LEO satellite would need to propagate its position forward in time to transmit accurate satellite location to the end user. One of the major forces effecting LEO satellites in orbit is atmospheric drag. Not accounting for the atmospheric drag causes the propagated positioning solution to become incorrect over time. Using the GNSS receiver measurements, the LEO satellite could calculate its drag force to correct its propagation solution on-board. Using a filtering technique on GNSS measurements to estimate the LEO satellite's perturbations over time, the ballistic coefficient will be estimated to a 10% accuracy of the true value of the coefficient.
Accurate orbit determination requires precise knowledge of perturbations affecting satellite motion. For Earth, there are two different forces that cause perturbations, conservative and non-conservative. Atmospheric drag is the largest non-conservative force affecting satellites in low Earth orbit. One major variable to determine the atmospheric drag is the satellite's ballistic coefficient that is comprised of the satellite's mass, area normal to the airflow, and drag coefficient. These variables are rarely known once the satellite is in orbit causing the ballistic coefficient to be approximated leading to errors. To improve this estimation, the LEO satellite can estimate its ballistic coefficient from GNSS receiver measurements and can transmit that variable along with its ephemeris to a ground receiver. The overall aim of this paper will be to estimate the ballistic coefficient during orbit propagation from raw GNSS receiver measurements.
The need for accurate ballistic coefficient estimation will improve the orbit propagation solution. Since satellites are multiplying in low Earth orbit, improved propagation techniques are needed to determine the satellite's orbit before being transmitted to the ground. While most satellites have the same basic structure of a central body with solar panels attached to the sides, they will vary in shape of the central body and overall size of the satellite. Johnathan Walsh, in his publication, mentions the formation and variation of the satellite geometry. This variance causes the ballistic coefficient to be difficult to calculate especially if the satellite geometry is not known. Because little is known about the ballistic coefficients of satellites in space, the ability to estimate the coefficient is crucial to determining the drag solution. Estimating the ballistic coefficient as a function of time also helps as the satellite changes orientation or as the satellite starts to decay during the deorbit process.
Prior research completed by Arrun Saunders has shown that it is possible to estimate the ballistic coefficient from two-line element set data. An ability to predict the ballistic coefficient to a relative maximum error of 14% was developed. Documented research shows that two-line element data when propagated can reduce accuracy of a solution since the sets were originally designed to be compact and use a simple propagation technique. Using raw GNSS receiver measurements to propagate an orbit and also estimate the ballistic coefficient should improve the accuracy of navigation solutions. Another publication by Il-Chul Moon estimates the ballistic coefficient of a in-atmosphere high speed target using a Gaussian Process Particle Filter which showed a possible solution to the estimation of the ballistic coefficient. Using the Gaussian particle filter, did improve the root mean square error of the estimation.
To estimate the ballistic coefficient, the orbit needs to be propagated from the last known position and velocity. To achieve the orbit propagation, the explicit Runge-Kutta formula will be used to calculate the acceleration model based on the combination of conservative and non-conservative forces. Next, the estimation of the atmospheric drag is accomplished. This is done through a filtering process that will filter out the changes between the gravitational perturbations leaving the drag force as the next largest perturbation effecting the satellite. Once the drag force is localized, the ballistic coefficient will be determined by the drag force equation. This equation includes the density of the atmosphere, the ballistic coefficient, and the relative velocity to the Earth's rotation. The value normally viewed as the ballistic coefficient in other research areas is the reciprocal when in satellite terms. Once the drag force is estimated, the ballistic coefficient will be solved to some degree of accuracy. The first steps will be to estimate the ballistic coefficient average and then progress to estimating the ballistic coefficient as a function of time.
The satellite data, simulated by Spirent GSS9000 simulator, used for propagation includes drag force. The data is comprised of position and velocity vectors outputted from a Novatel GNSS receiver. Adjustable parameters within the Spirent simulation include the classical orbital elements, the mass of the satellite, the drag coefficient, and the area of the satellite exposed to the airflow. For now, the drag coefficient will be set to 2.2 as a commonly used value over satellite research, with future work including adding a variable coefficient. From these parameters, the satellite is simulated in orbit for a set amount of time and then the satellite states are fed into the GNSS receiver whose output will be used for propagation. Using the mass, area, and drag coefficient, the known ballistic coefficient can be found and used for comparison once estimation is complete. The goal will be for the estimation to be within 10% of the known ballistic coefficient which will show improvement over previous research. First the position accuracy as a function of the dead-reckoning time will be shown to ensure that the satellite is staying within an allowable deviation. Then, the average of the ballistic coefficient will be displayed for a variety of satellite masses and areas. Finally, the ballistic coefficient over time will be calculated and displayed for a least one satellite of average size. A Monte Carlo simulation will also be performed on the true value of the ballistic coefficient with added noise over time. The results of the estimation would be plotted showing the likelihood of the estimation remaining between two sigma of the true coefficient. Future work includes showing that the ballistic coefficient could be calculated as an average of altitude to simulate a deorbit situation.