## 1 Introduction  
In the mid-19 th century, [PERSON], a British engineer, developed the first self-propelled torpedo, called the Whitehead torpedo [1]. Autonomous underwater vehicles (AUVs) are often developed for commercial and military purposes. The pioneer of AUVs for scientific research is the well-known EPAULARD, which was developed by the French Institute of Marine Development (Institut francais de recherche pour l'exploitation de la mer, IFREMER) [2]. Over the last three decades, AUV development has been characterized by high security, high controllability, and low costs. In particular, AUVs are suitable for exploration, inspection, or monitoring in long-term, routine, reproducible, or high-risk missions, which include hydrological surveys, marine geomorphology mapping, seabed resource surveys and environmental assessments, pollutant monitoring, and deep-sea infrastructure and pipeline inspection [3; 4; 5; 6; 7]. [PERSON] et al. [8] developed a real-time vision-based method for AUV guidance and control. [PERSON] and [PERSON] [4] demonstrated that a vision-based system is suitable for short-distance measurements. In their study, the three-dimensional (3D) reconstructed images were obtained from a camera to extract cluster depth information, and the computed time and distance from the Kalman filter were used to control the AUV. [PERSON] et al. [9] used a single camera and an acoustic sensor to calculate the relative position of an AUV and underwater cable in a 3D environment for inspecting and maintaining artificial underwater structures. The camera provided two-dimensional planar information, and the acoustic sensor provided a third dimension. The 3D spatial information of the cable was used as a control parameter for AUV operation. To perform underwater positioning and pipeline inspection, [PERSON] [10] reconstructed 3D environmental information and then defined the geometric and edge features of an image. Subsequently, an extended Kalman filter was adopted to estimate the positional relationship between an AUV and pipeline edges for developing an automatic docking station. [PERSON] et al. [11] proposed a control method combining image information acquired by the AUV's front camera with optical flow. In their experiment, a light-source array algorithm marked features of the docking station. On the basis of calculations from the image servo controller, AUV docking was achieved by automatically adjusting its posture. Since AUVs are often thought of as nonlinear coupling systems, it is difficult to present the input-output relationship simply by traditional linear control systems. Therefore, a fuzzy controller can simplify the process of determining system-model appropriateness to meet control requirements. To implement docking control, [PERSON] et al. [12] used three independent fuzzy controllers to manipulate the heading angle, pitch angle, and depth of the Ocean Voyager AUV. [PERSON] and [PERSON] [13] used a sliding-mode fuzzy controller on the pitch or heading angle. In their double-input single-output control system, the input parameters were deviation and error rate, and the output parameter was the rudder angle. [PERSON] et al. [14] used Tagaki-Sugeno-Kang fuzzy inference to control vehicle heading and speed parameters with fuzzification to enable autonomous docking under the guidance of an ultrashort baseline system. [PERSON] et al. [15] combined 3D image reconstruction with a fuzzy controller to dynamically estimate optimal routing on the basis of time or energy consumption. In general, AUVs can acquire positioning information by using the global positioning system (GPS), an inertial navigation system (INS), a Doppler velocity log, or an acoustic modem [16]. However, using these instruments may be difficult because of the rapid attenuation of radio waves in water, bandwidth limitations, blind spots, or equipment costs. Therefore, using a vision-based navigation system for AUV motion control may be beneficial.