## 1 Introduction
[14] proposed an attitude tracking controller for rigid spacecraft based on predefined time, combining sliding mode control and predefined time theory to achieve high-precision control of spacecraft attitude. [PERSON] et al. [15], addressing the tracking control problem of free-floating space robots in the task space, introduced a predefined time-based backstepping control method, enabling the end-effector position and attitude of the robot to track the desired trajectory within a predetermined time, even with initial errors. [PERSON] et al. [16] presented a predefined time-based terminal sliding mode control method, demonstrating through rigorous mathematical formulation the system's singularity-free and globally predefined time convergence. This method achieves high-precision trajectory tracking control for dual-arm space robots. However, this work did not further investigate the trajectory planning problem considering kinematics. However, predefined-time control methods can guarantee convergence of errors within a given time in any situation, but they lead to the requirement of larger input gains. This is because the predefined-time theory considers convergence time even from infinitely large initial states, leading to overly conservative estimates of convergence time. In trajectory planning, a higher input gain represents larger joint angular velocities, which can lead to increased vibration for continuum space robots. In trajectory tracking control, traditional predefined time inputs that are relatively large can cause actuator saturation and faults. In engineering applications, it is rare for system states to tend towards infinity; most of the time, they are bounded. For the case where the initial conditions are bounded, [PERSON] et al. [17] proposed a generalized explicit time stability system with bounded gains, which significantly reduces the initial control input. However, there is currently relatively little research on specific applications based on explicit time stability. The control theory can achieve system convergence within predefined time while initiating smaller initial control inputs to reduce the damage caused by excessive control inputs to the actuators. However, in designing controllers, the mathematical expression forms are complex, especially when applied to sliding mode control, making it difficult to derive. Research on how to design more concise and practical trajectory planning and control algorithms based on explicit time theory is still needed. In summary, this paper proposes a high-precision control scheme for trajectory planning and tracking control of a continuum space robot. By employing explicit time theory in trajectory planning and tracking for continuous space robots, smoother joint trajectories and smaller initial control inputs can be obtained. The main contributions of this paper are as follows:  
(1) The model of continuum space robots is established by introducing the arc length parameter. Based on this, the pose feedback kinematic model of continuous space robots is established using quaternions to describe the end effector's pose of the robotic arm. In the modeling process, the momentum conservation and nonholonomic constraints of the robotic system are taken into account. This modeling approach has more accurate model parameters. (2) Based on the kinematic model of the continuum space robot, a trajectory planning method utilizing explicit time stability theory is proposed. Compared to other methods, this approach can generate smoother joint trajectories. (3) Building upon the dynamic model of the continuum space robot, a trajectory tracking control method is devised that integrates explicit time stability theory and sliding mode control. This trajectory tracking control method achieves high-precision tracking of the desired trajectory within explicit time. Compared to other methods, this approach requires lower control inputs.