Reduced Computational Complexity with Updated Formulation of Joint Doppler and Ranging

Joint Doppler and Ranging (JDR) is a relative Doppler and range-based positioning method that can localize a surface user with a minimal navigation infrastructure. The traditional implementation of JDR can be unintuitive and require high numerical precision; this can lead to truncation errors during computation. These issues not only reduce the comprehension and readability of JDR, but can also reduce navigation performance. This report introduces an updated computational formulation of JDR that improves performance and reduces computational complexity of the navigation method. The updated formulation of JDR reduces the effects of truncation errors and increases compatibility with lower bit processors. This is done by converting the original formulation into JDR Doppler equivalent equations: predictions of Doppler measurements at the user and reference station. This measurement model is not only more intuitive but also decreases the range of computed values by orders of magnitude. The updated formulations improve compatibility with navigation filters and simplifies the creation of measurement covariance matrices. Navigation performance of the updated and original formulations of JDR are compared with 64 bit double and 32 bit single precision floats. This is done through a navigation simulation of a lunar surface user and two Lunar Relay Satellites. The new formulation reduces the required computation time for a position fix and preserves navigation accuracy even with a significant reduction in precision from a double to a single precision float. Overall, this updated formulation reduces the computational complexity of JDR and results in increased hardware compatibility, simplified navigation filters, and improved navigation performance.