## 1 Introduction  
Traditional robots with a fixed degree of freedom (DOF) have been primarily devised for repetitive industrial production tasks in known and structured environments with little human participation [1]. With the increasing demand for complicated exploration or operation tasks in complex scenarios, hyper-redundant continuum manipulators, which exhibit more dexterity and flexibility during motion [2; 3], have attracted much attention of researchers. Among them, inspired by the concept of biomimetic muscles and tendons, such as the snake or octopus, cable-driven hyper-redundantcontinuum manipulators (CHCM) or similar designs generally consist of multiple serially connected joint modules driven by cables, rigid linkages or elastic materials [4; 5]. Due to the redundant DOFs, CHCM can reach desired positions with different postures; thus, flexible motion, good obstacle avoidance capability, and security can be achieved for surgery, ocean exploration, underground or intra-cavity inspection etc. [6]. Various mechanical designs of continuum manipulators with different actuator and connection structures have been developed [7], where the complexity of kinematics and the control method caused by complicated and per-redundant relationships between cables, joints and end-effector positions are the key problem to be solved [8; 9]. The first snake-like continuum robot was designed about 50 years ag. Since then, research about continuum robots has been emerging in large numbers [10]. [PERSON] et al. [11; 12] proposed a serious of different modular continuum robots and corresponding control methods to achieve reliable movement in complicated terrains. In particular, the Unified Snake robot was designed with up to 16 joints and presented a variety of motion patterns and excellent obstacle avoidance ability [13; 14]. A continuum robot consisting of several same joints was developed by Mamba and the attached force/torque sensors provide the information of how the robot touches surrounding environments [15]. [PERSON], et al. of OC Robotics [5] developed a commercialized CHCM suitable for assembly tasks within wing boxes and replacing manual procedures, where a complex kinematics model was established to deal with the strong nonlinearity. [PERSON] [16] proposed a cable-driven snake-like robot consisting of 2-DOF joint modules with a flexible backbone, then an instantaneous kinematic model and numerical inverse kinematics (IK) algorithm were derived, which is solved via Newton-Raphson method. Another commonly used method for IK questions is solving the pseudoinverse of the Jacobian matrix, based on which a multi-section continuum robot was designed and its kinematics model with optimized motion trail among cable displacements and joint rotations was derived [17; 18; 19]. To cope with the challenge of solving IK for hyper-redundant robots, a modified computing method was proposed to solve IK of \((2n+1)\) DOF continuum manipulator, where the analytical approach was adopted to improve the efficiency of solvers on each joint and the IK question was handled as an optimization problem [20]. For achieving the best result of multi-solution IK problems for hyper-redundant robots, a computing algorithm based on the cyclic coordinate descent was presented, which could handle many redundant DOFs in three spatial dimensions [21]. [PERSON] et al. [22] compared an exhaustive method and several optimization algorithms considering computing time-cost and tolerable position error, where the limitations of each method were explained. Based on mechanical designs and kinematics or dynamic modeling, upper-level control strategies through optimization or model-free approaches have become another important research mission, such as the attitude control, obstacle avoidance and adaptive tracking [21; 23].