## 1 Introduction  
In the complex sea, autonomous underwater vehicles (AUVs) play an irreplaceable role for various missions, such as exploration of marine resource, environmental monitoring, seabed topographic survey, and target search [1; 2]. In recent years, with the deepening of marine development, simple detection and target search cannot meet the operational requirements. However, the underwater vehicle-manipulator system (UVMS), which consists of underwater vehicle and manipulators, and its operating technology have been rapidly developed and widely used in the fields of scientific research and ocean systems engineering for performing interactive tasks, such as opening and closing of valves, cutting in coordination, and so on [34]. An AUV when equipped with a manipulator becomes a kinematically redundant system, it has more degrees of freedom (DOF) than is required to perform a task in its operational space. In TRIDENT EU FP7 project, an underwater electric manipulator was developed and equipped on multipurpose intervention-AUV [4]. An underwater electric manipulator (MARIS 7080, 7 DOF manipulator) was equipped on semi-autonomous underwater vehicle for the intervention mission (SAUVIM) [5]. In UVMS, the underwater manipulator is a key component, and its joint control accuracy directly determines the operating accuracy of the UVMS. At present, underwater manipulators are mostly control objects with joint redundancy, and the accuracy mainly depends on the control performance of joints [6]. However, due to the nonlinear, time-variation of dynamic properties and other factors (i.e., external disturbances such as ocean current disturbance) [7; 8], the system dynamic model has uncertainties, which will affect the joint control of the manipulator based on the dynamic model. Besides, for most hydraulic manipulators, the joint control performance is also affected by joint lag [6; 9; 10], which means that the joint lag will affect the control accuracy of manipulator joint. In conclusion, under the influence of the above factors, it has great research significance and practical value to develop the joint control technology of underwater manipulators for improving the accuracy and efficiency of underwater operation. Currently, many methods have achieved high control accuracy, which uses the online approximation to deal with the uncertainty factors, and determine the controller and adaptive law of parameter through stability analysis [11; 12]. Among them, in view of the nonlinear and uncertain identification ability [11; 12; 13; 14; 15; 16], neural network and fuzzy control has been used in the joint control of manipulator. In [13], a feedforward neural network is employed to learn the parametric uncertainties, existing in the dynamical model of the robot manipulator, and the influence of weight and bias weight of the two-layer neural network are considered, but this increases the difficulty of designing the weight adjustment law, and an excellent neural network has relatively more control parameters, and selection of initial values of neural network parameters will increase the difficulty of the controller design and the complexity of the stability analysis, and also the adaptive adjustment of control parameters will increase the load on the hardware system. In [14], a novel robust decentralized control of electrically driven robot manipulators by adaptive fuzzy estimation and compensation of uncertainty, and in the fuzzy controllers, it uses expert's knowledge, the trial and error method, or an optimization algorithm such as particle swarm optimization (PSO) to design the fuzzy rules, and the design process may be more complicated and not online. Recently, some researchers have proposed the regressor-free control of manipulators based on function approximation technique (FAT) [17; 18; 19; 20], in which the uncertainty factors have been estimated or approximated using the Taylor function expansion [18], Fourier series [19], or Legendre polynomial [17; 20].