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A new sensor-based homing integrated guidance and control law is presented to drive an underactuated autonomous underwater vehicle (AUV) toward a fixed target, in three dimensions, using the information provided by an ultrashort baseline (USBL) positioning system. The guidance and control law is first derived using quaternions to express the vehicle's attitude kinematics, which are directly obtained from the time differences of arrival (TDOA) measured by the USBL sensor. The dynamics are then included resorting to backstepping techniques. The proposed Lyapunov-based control law yields global asymptotic stability in the absence of external disturbances and is further extended, keeping the same properties, to the case where constant known ocean currents affect the dynamics of the vehicle. Finally, a globally exponentially stable nonlinear TDOA and range-based observer is introduced to estimate the ocean current and uniform asymptotic stability is obtained for the overall closed-loop system. Simulations are presented illustrating the performance of the proposed solutions.