Closed-form Least Squares Localization in Multiple Non-synchronized Systems Leveraging TDOA

Vehicle localization is an important part of autonomous vehicles, traditional localization methods often rely on Taylor series expansions and iterative techniques to estimate positions accurately. However, it relies on a good initial estimation and has a computational complexity that increases with the number of iterations. Aiming at this problem, we introduce a Closed-Form Least Squares (CFLS) algorithm that estimates positions within a single epoch, eliminating the need for iterative processing. In multi-system settings, a direct algebraic solution is not feasible. We reformulate the problem to obtain approximate closed-form solutions by intermediate variables. It scales to multiple systems, in contrast to existing intermediate-variable methods restricted to single or dual systems. Although squaring the measurements can increase sensitivity to noise, our CFLS algorithm achieves positioning accuracy close to the Cramér-Rao Lower Bound (CRLB). Experimental results using three satellite systems at 50 reference stations show that the CFLS algorithm provides accuracy comparable to that of conventional iterative least squares methods while reducing average processing time by 35.11%, making it highly suitable for real-time applications.