Robust Orbit Determination via Convex Optimization

Given the initial position and velocity of an object, such as a satellite in orbit around the Earth, it is possible to determine its position and velocity at any other time by integrating the equations of motion. This prediction method is called propagating. Specifying the position and velocity of the satellite at a specific time, called the epoch, is a common way of representing an orbit. In this context, the position and velocity are known as the Cartesian orbital elements. Different methods of representing an orbit are in common use and it is rare to use one representation in isolation. Classical elements are another common representation that are easily interpreted and propagated. For this reason, classical elements were used in this project. While providing a thorough background in orbital mechanics is beyond the scope of this paper, an excellent reference is [PC12], which discusses orbital elements and propagation in detail. Given the orbit of a satellite, it is straightforward to compute its distance from a point on the Earth, such as a ground station, at any time. Ground stations can measure this distance, called the range, by sending a signal to the satellite. The satellite sends back a signal after a known delay, which is finally received by the ground station. By measuring the total transmission time and factoring in atmospheric effects, the range can be estimated with high accuracy. A ground station may also measure the elevation and azimuth of a satellite. The issues involved in accurate estimation of these parameters, collectively called tracking data, are discussed in [TSB04, §3]. For simplicity, in this project we focused on range measurements only. Accommodating other measurement types is straightforward. Let fr(x; t) be the range measured by a particular ground station at a particular time t, provided the orbital elements are given by x and the measurements have no noise. Informally, fr answers the question: at time t, how far is the satellite from the ground station? Evaluating fr is nontrivial. We must propagate the orbital elements to the desired time and account for signal delays to determine the range a ground station would measure at time t. High fidelity propagators will result in an fr for which analytical formulas are not possible, and in practice, even simple forms are too complex to differentiate explicitly. Instead, numerical methods are used to compute gradients approximately.