Precise navigation is one of the most important challenges for developing an autonomous underwater vehicle(AUV).Aiming at the observability of the underwater navigation system for range measurement with a single beacon,we obtain the observability conditions in relation to three-dimensional trajectories of AUV through the kinematic model of AUV and the local observability theory of a linear time-varying system.Subsection 1.3 explains mathematically in some detail the observability conditions;eqs.(11) and(12)...