Vehicle Navigation with Time Difference of Arrival Method in Hexagonal Hierarchical Spatial Index

Modern transportation system gives specific requirements for performance of on-board navigation system to guarantee the safety of transportation. Integration of autonomous vehicles in transportation systems requires development of new positioning algorithms that will be computationally cheap for on-board equipment. Time Difference of Arrival (TDoA) positioning method grounds on measuring the time difference between two received radio signals from the network of synchronized ground transmitters. Time difference is proportional to the range difference which identifies a hyperbolic line of vehicle location. Navigation equation of TDoA is not linear and requires the usage of numerical methods to obtain vehicle position in case of multiple input data. In the paper, a TDoA position method has been implemented in the space of spatial indexes. Spatial index system partitions the shape of WGS-84 ellipsoidal model into a set of cells with hierarchical addresses. We use a regular hexagonal cell shape which has multiple benefits for navigation. Application of TDoA in hexagonal hierarchical spatial index provides a solution to navigation equation with the help of logical functions only. Logical operations and functions of set theory provide computational cheap identification of vehicle position that is important for modern autonomous vehicle development. In the numerical demonstration, a part of ground network of perspective eLoran system has been used for positioning with spatial indexes.