*Article* **A Shrink-Branch-Bound Algorithm for eLoran Pseudorange Positioning Initialization**

**Kaiqi Liu 1,2,3,\*, Jiangbin Yuan 1,3, Wenhe Yan 1,3, Chaozhong Yang 1,3, Wei Guo 1,3, Shifeng Li 1,2,3 and Yu Hua 1,2,3**


**Abstract:** Currently, eLoran is the ideal backup and supplement for global navigation satellite systems. The time synchronization accuracy between stations in the eLoran system has improved, providing conditions for eLoran pseudorange positioning. The pseudorange positioning of eLoran is a nonlinear least-squares problem and the location of the eLoran transmitting stations may cause the above problem to be non-convex. This makes the conventional pseudorange positioning algorithm strongly depend on the initial value when solving the eLoran pseudorange positioning. We propose a shrink-branch-bound (SBB) algorithm to solve the eLoran pseudorange positioning initialization problem. The algorithm first uses a shrink method to reduce the search space of the position estimator. Then, optimization is performed using a branch and bound algorithm within the shrunk region, where a trust region reflective algorithm is used for the lower bound process. The algorithm can help the receiver to complete the initial positioning without any initial value information. Simulation experiments verify that the algorithm has a success rate of more than 99.5% in solving the initialization problem of eLoran pseudorange positioning, and can be used as an initialization algorithm for pseudorange positioning problems for eLoran or other long-range terrestrial-based radio navigation system.

**Keywords:** pseudorange positioning; branch and bound; nonlinear least squares; eLoran; trust region reflective algorithm; initialization
