## 1 Introduction  
Autonomous underwater vehicles (AUVs) play important roles in underwater missions. However, the working time is limited by on-board energy and storage capability of AUVs. The autonomous recovery of AUVs is used for recharging, data transfer, and relaunch. Docking stations can either be stationary [1; 2; 3; 4; 5; 6] or mobile [7; 8; 9; 10; 11]. The process of AUV sailing from far-end to the docking station is called homing. The homing process aims at the precise physical contact between AUVs and docking stations. Physical contact can be passive or active. A common kind of passive contact is that an AUV is captured by a robotic manipulator. A common kind of active contact is that an AUV catches a cable or pole [12; 13; 14] with a latch or hook. The other kind of active contact is using funnel-like [15; 16; 17] docking stations. In this work, We address the recovery of an AUV by actively docking with a funnel-like docking station. One of the key technologies in homing is navigation. The positions of AUV and DS should be simultaneously sensed by AUV. Three kinds of sensors are available for the position sensing of DS: (1)electromagnetic sensors, (2) acoustic sensors, and (3) optical sensors. Acoustic sensors have the longest working range, but the lowest precision. Woods Hole Oceanographic Institution(WHOI) [18; 19] conducted a research on the AUV homing method based on acoustic guidance. The transceiver installed in the head of the AUV receives an acoustic signal from the beacon installed on the DS, and measures its position relative to the beacon. The absolute position of AUV is calculated indirectly as it knows the absolute position of the beacon. Electromagnetic sensors have a better precision, but much less range, about 30-50 m. Massachusetts Institute of Technology (MIT) [20] proposed the electromagnetic guidance method. In this way, the final docking accuracy can reach 20 cm. However, due to some physical factors, the effective docking range is 20-30 m. Optical sensors have the shortest range, but the highest precision. National Radiology Solutions (NRaD) [21] proposed an optical docking system, which was demonstrated to be accurate and robust for vehicle terminal guidance and provided targeting accuracy on the order of 1 centimeter under real-world conditions, even in turbid bay water. According to the characteristics of the sensors, they usually combined acoustic and optical sensors for AUV recovery. In this work, we focused on the development of acoustic-based AUV recovery algorithms since the vision guidance technology is maturing. More importantly, if the positional error of acoustic sensor is large, it may result in the inability to enter the terminal guidance phase based on vision, resulting in a failed docking. Acoustic methods can be used at large distance. One such method for localization, is range-only localization. Range-only measurements are extensively used in many underwater homing application [22; 23; 24; 25; 26]. However, these measurements present a high-nonlinear and standard Extended Kalman Filters (EKFs) cannot cope with it. The solution to the range-only localization for homing problem using a sum of Gaussian filter is proposed in Reference [27]. The other method for localization, is range-bearing localization. In this work, the range and bearing of beacon are provided by the acoustic sensor. In other words, we can obtain the projections of distance from the transceiver to the beacon in the x and y directions of the observation coordinate system. Some SLAM algorithms for AUV have been applied to sense the position of a robot and map its surroundings simultaneously [28; 29; 30; 31]. These algorithms are not suitable for our recovery system as the statistical characteristics of our measurement error are non-Gaussian and unknown. We adopt the improved SLAM algorithm to simultaneously sense the positions of AUV and DS.