## 1 Introduction  
The research and development of the ocean is of great significance due to the abundant resources in the ocean environment [1; 2; 3; 4]. With the development of advanced robotics and automation, the capacity to explore the ocean is enhanced by various underwater robots, which can replace human beings to achieve underwater detection and operations [5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17]. However up to now, most existing underwater manipulators are rigid and with fixed 5 or 7 degrees of freedom (DOF) [18; 19], which may not be very suitable for some complicated underwater scenarios (e.g., pipe networks, narrow deep cavities, etc.). In view of the above limitation, the biomimetic concept of muscles and tendons is considered as continuum manipulators [20; 21; 22; 23; 24], which can achieve the flexible motion by elastic deformation. However, this kind of manipulator has slight load capacity, low operation accuracy, and is difficult to be precisely controlled due to its infinite DOFs and limited measurements. In recent years, the cable-driven technique has been also developed for the manipulators. With numerous joints and hyper-redundant DOF to execute tasks with dexterity and adaptability, the cable-driven hyper-redundant manipulator (CDHRM) has good obstacle avoidance capability, but also better load capacity and control precision. Thus, it has wide potential application fields, such as underwater rescue, marine exploration, minimally invasive surgery, aircraft assembly, nuclear inspection, etc. [25; 26; 27]. Different from the traditional rigid manipulator with serially connected links and actuated joints [28], the CDHRM no longer has the driving actuator in each joint. Instead, the cables, which pass through the tubular structure from the base to the end-effector, are used to drive joints, and all actuators are located in the base to drive these cables. Therefore, the manipulator joints do not need any seal, and can be used directly in the underwater environment. Moreover, the joint numbers of the CDHRM can be extended with regard to the specific underwater task requirements. Thus, various compact and ingenious motions can be achieved, especially when entering into confined spaces to implement tasks [29]. However, several issues need be solved for CDHRM's practical application, including the mechanical design, kinematics, control, etc. [30]. For the mechanical design, due to the CDHRM's unique operation mode in some typical confined underwater scenarios, the connections of each joint, the distribution of actuators and cables, and the lightweight design are difficult. Therefore, the mechanical design of CDHRM with flexible obstacle avoidance and strong load carrying capacity is challenging [31]. For the kinematics and control, since the motion of each joint with two DOFs is controlled by three cables, there exists complex mapping relationships and hyper-redundancy among the cables, joints and end-effector, which results in the complexity of kinematics and control [32; 33]. Therefore, in view of the above issues, some relevant research has been conducted previously. Namely, [PERSON] et al. [34; 35] designed a multi-section continuous-backbone robot, where the relationship between the cable length and joint angle was derived in detail, and the mapping relationship between the rope speed and joint angular velocity for kinematic modeling was established. The inverse kinematics were solved via the Jacobian matrix and the trajectory optimization of the grabbing process was carried out via the zero space vector of the Jacobian matrix [36; 37; 38; 39]. [PERSON] et al. [40; 41; 42] put forward a 3D hyper-redundant chain robot including the linked, identical modules and base module, where all joints are driven by cables and state controllable. But the drawback of such a manipulator is its long action cycle, where all passive joints share the common driving input, which have to be operated one by one. [PERSON] et al.