A Norm-Minimization Algorithm for Solving the Lost-in-Space Problem with XNAV

An algorithm is presented for solving the lost-in-space problem using simultaneous observations of X-ray pulsars. Using a norm-minimization-based approach, the algorithm extends a previous banded-error intersection model to 3-dimensional space. Higher-fidelity X-ray pulsar signal models are considered, including the parallax effect, Shapiro delay, time dilation, and higher-order pulsar frequency models. The feasibility of solving the lost-in-space problem using X-ray pulsar navigation is revisited with the improved models and prior knowledge requirements are discussed. Monte Carlo simulation techniques are used to establish upper bounds on uncertainty and determine the accuracy of the algorithm. Results indicate that it is necessary to account for the parallax effect, time dilation, and higher-order pulsar frequency models in order to successfully determine the position of the spacecraft in a lost-in-space scenario. In a heliocentric case study, the algorithm uniquely identified a candidate spacecraft position within a spheroid search domain up to 10 \(\times\) $\times$ 10 \(\times\) $\times$ 0.01 AU \(^3\) ${}^3$ in size by simultaneously observing eight to nine pulsars. The maximum search domain size can be increased by observing additional pulsars, which may allow the lost-in-space problem to be solved anywhere within the Solar System. The median position error of the algorithm is on the order of 15 km. Results further indicate that choosing lower-frequency pulsars increases the maximum search domain size at the expense of increased position error.