Contents I. Introduction
Accurate navigation systems are essential for the successful operation of autonomous vehicles. Although there exist alternatives such as terrain-based navigation, most navigation systems contain an Inertial Navigation System (INS) that provides the state of the vehicle by integrating, in open-loop, the information provided by inertial sensors, e.g., accelerometers and rate gyros. Although INS provides very good short term results, its performance necessarily degrades over time, which has lead the scientific community to consider aiding devices in the design of Navigation Systems. Among the myriad of aiding devices, the Global Positioning System (GPS) is a very popular choice, see, e.g., [1], [2], and [3]. While long baseline (LBL) solutions offer more information, recent efforts have been made in the field of navigation using range measurements to a single source. This last solution, although harder from the theoretical point of view, has significant cost advantages. Moreover, the complexity of the hardware is lower and, for intensive missions, the deployment time is also much smaller. A so-called Synthetic Long Baseline (SLBL) navigation algorithm for underwater vehicles is proposed in [4]. The vehicle is assumed to have access to range measurements to a single transponder, from time to time, and between sampling instants, a high performance dead-reckoning system is used to extrapolate the motion of the vehicle. A discrete-time Kalman filter is applied to a linearized model of the system to obtain the required estimates. In [5] the authors deal with the problem of underwater navigation in the presence of unknown currents based on range measurements to a single beacon. An observability analysis is presented based on the linearization of the nonlinear system which yields, off course, local results. Based on the linearized system dynamics, a Luenberger observer is introduced but in practice an extended Kalman filter (EKF) is implemented, with no warranties of global asymptotic stability. More recently, the same problem has been studied in [6] and [7], where EKFs have been extensively used to solve the navigation problem based on single beacon range measurements.
