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Real-time accurate positioning is a key technology for autonomous underwater vehicles to perform close-range seafloor observations such as photo mosaicing. This paper proposes a real-time positioning method that realizes robust and drift-free positioning in a local area based on a passive acoustic landmarks set in the environment and a profiling sonar (profiler) mounted on the vehicle. The method stochastically updates the vehicle's position based on all the sensory data available. Although the position of landmarks is generally unknown, this paper assumes the position is known without error for simplification. As an observation model of the profiler we propose a radial-angular representation with a non-Gaussian distribution based on experimental data. The performance of the proposed method is verified through tank experiments using the AUV Tri-Dog 1. The effects of the number of landmarks and observation model on the positioning accuracy is discussed. Through comparison with ground truth and off-line estimations based on actual data obtained during the experiments, the positioning accuracy of the proposed method remains within 5 centimeters at all times during the 2 hour duration of the experiment with a traveled distance of 600 meters, with two landmarks at a distance of 10 meters.