Andrew Sinclair, AFRL; Edwin G. W. Peters, University of New South Wales Canberra; Joseph peterson, Texas A&M University; Melrose Brown, UNSW Canberra Space
Keywords: orbit determination, passive radio frequency measurements, Gaussian-mixture model
Abstract:
Radio frequency (RF) observations are an attractive source for characterization of transmitting satellites. For purposes of orbit determination relevant RF measurements include time difference of arrival (TDOA) and frequency difference of arrival (FDOA). TDOA measurements are related to the difference in range from the transmitter to two receivers. FOA measurements are related to the range rate from the transmitter to each receiver. Extensive work in the literature has developed signal processing methods to extract these measurements from received RF signals. These types of measurements have been broadly studied for the purposes of geolocation. The use of these measurements for orbit determination has also been considered. These studies have fallen into two categories: initial orbit determination without full characterization of probability density function of the state, or iterative refinement of a current estimate or initial guess.
This paper addresses initial orbit determination using TDOA and FOA measurements with no a priori knowledge of the transmitter’s orbit. An initial set of a TDOA measurement and two FOA measurements define a three-dimensional subspace, of the six-dimensional state space of the transmitter’s orbit, over which the probability distribution is uniform. This distribution is approximated via Gaussian mixture (GM) using Cartesian coordinates of the transmitter’s position and velocity. The use of GM approximation is similar to previous literature, but that work focused on tracking of terrestrial emitters.
In the position space, the uniform distribution is supported on the hyperboloid defined by the TDOA measurement with the receivers at the foci. A method for GM approximation of a uniform distribution over a hyperboloid is developed. Conditions are first developed for the GM approximation of a circle. A selected number of L reference points are equally spaced on the circle. A Gaussian component is associated with each reference point, with mean location on the radial line between the center of the circle and the reference point, and principal components of covariance in radial and transverse directions. In this case, the radial variance and the radial separation between the reference point and the mean are included to account for the approximation error of approximating the uniform distribution over a circle with a finite number of Cartesian Gaussian components. To define the specific mean location and principal variances, two conditions are defined. A uniformity approximation condition sets the probability density of each Gaussian component at its reference point equal to twice its probability density at the midpoints to the neighboring reference points. A radial symmetry condition for each Gaussian component places half the probability mass inside the circle and half outside the circle. These conditions lead to a nonunique, numerical solution for the GM.
The GM approximation over a circle is used as a kernel to approximate the distribution along the two dimensions of the hyperboloid. A grid of reference points on the hyperboloid are distributed along the generating hyperbolas and the circles of revolution. A Gaussian component is associated with each reference point. To select the mean and covariance parameters, the approximation on a circle is evaluated for both the circle of revolution and the local radius of curvature of the generating hyperbola.
For a given transmitter position, the FOA measurements describe the range rate to each receiver. This leaves a uniform distribution along the direction of velocities perpendicular to the relative positions from the transmitter position to each receiver. To approximate this dimension of the distribution via GM, a third dimension is added to the grid of components defined over the hyperboloid. For each Gaussian component in position space, a selected number of components are added in velocity space, with parameters for the GM approximation of a univariate uniform distribution previously defined in the literature.
The inclusion of Gaussian measurement errors results in a target distribution that is uniform-Gaussian. The distribution is uniform across the subspace of positions and velocities that satisfy the measurement values, and is Gaussian along dimensions perpendicular to that subspace. For the GM approximation the measurement errors are captured by adding additional variance in the direction normal to the hyperboloid surface, and defining variance in the velocity directions parallel with the relative positions from the trans- mitter to each receiver.
Having initialized the GM approximation of the probability distribution, the distribution is updated to account for the transmitter’s orbital dynamics and subsequent TDOA and FOA measurements using a GM extended Kalman filter (GMEKF). In this filter, the mean values are propagated using a selected nonlinear dynamics model, and the covariances are propagated using a linearized model such as the DeVries equations evaluated at the component’s mean. Similarly, measurement updates utilize the Jacobian of the measurement model evaluated at the component’s mean. The GMEKF describes how the probability distribution evolves from the uniform distribution over a three-dimensional subspace to a precise estimate for the transmitter’s orbit.
Date of Conference: September 19-22, 2023
Track: Astrodynamics