## 1 Introduction  
With the development of space exploration in recent decades, on-orbit space robots are increasingly gaining attention from researchers. Space robots serve as crucial tools for ensuring the safe operation and prolonged lifespan of spacecraft, contributing to tasks such as space station construction, satellite maintenance, and space debris clearance [1, 2]. However, traditional rigid robots, characterized by limited degrees of freedom and insufficient flexibility, face challenges in efficiently and safely while completing on-orbit tasks within complex and confined environments. Continuum robots, as a new type of biomimetic robotic arm, offer higher flexibility and enhanced interactive safety [3]. They can perform a variety of tasks in narrow and restricted environments, holding promising applications in the field of space robots. Continuum robots can achieve continuous bending and motion in complex and confined unstructured environments, leveraging their inherent flexibility. The driving mechanism of continuum manipulators is mostly installed in the base at the bottom. With a relatively small mass and lower stiffness compared to traditional rigid robots, continuum robots exhibit higher safety when in contact with the environment. Currently, continuum robots find widespread applications in areas such as aerospace [4], medical fields [5], power systems [6], exploration [7], and rescue operations [8]. However, the use of soft materials and structures can lead to vibrations, creating a strongly coupled nonlinear system. Modeling and controlling such systems both still lack a well-established unified approach. Additionally, the base of space robots is free-floating, introducing complex dynamic coupling between the motion of the manipulator and the satellite base, posing significant challenges in trajectory planning [9]. Currently, research either focuses solely on the trajectory planning problem of continuum space robots or solely on the dynamic tracking control problem, without effectively integrating the two. Therefore, research on the kinematics and dynamics of continuum space robots is meaningful but also confronts various challenges [10]. Inaccurate calculation of the robot system's motion deformation, kinetic energy, and momentum can affect the dynamic coupling of the base, thereby increasing trajectory planning and control errors. Current commonly used high-precision control methods for robots include adaptive control, fuzzy control, sliding mode control, backstepping control, etc., but most of these systems are asymptotically stable. [PERSON] et al. [6], addressing the uncertainty compensation issue in a cable-driven continuum robot system, proposed a novel disturbance-resistant control method based on a sliding mode controller and a linear extended state observer. While this method can compensate for the effects of system uncertainty, its sliding mode controller employs exponential reaching law, which cannot guarantee the convergence time of errors. [PERSON] et al. [11], addressing robust tracking control issues in pneumatic-driven continuum soft robots, proposed a finite-time sliding mode controller to achieve convergence of tracking errors within a finite time. For the task space tracking control problem of free-floating space robots, [PERSON] et al. [12] utilized a fixed-time extended state observer to ensure that, even in the presence of external disturbances, the convergence time of tracking errors is bounded and independent of the system's initial state. Since the relationship between fixed-time controller time and parameters is complex, [PERSON] et al. [13] introduced the predefined time stability theory, with the upper bound of stability time related to the system, allowing for the pre-definition and simplification of controller design. The predefined time stability theory ensures system response speed and allows for specification of the convergence time during controller design. [PERSON] et al.