## 1 Introduction
### Related Works  
Scholars have proposed various control strategies, such as full-state feedback control (FSFB), backstep control, compensation control based on state observer, synovial membrane control, and neural network control, to address the motion control of flexible joints with nonlinear time-varying load inertia [3; 4; 5]. The backstep method compensates for the interference of the contained severe uncertainty by building a state feedback controller [6]. The state observer is capable of compensating for model disturbances and thereby reducing their impact on joint control. [PERSON], for a class of uncertain flexible joint robots with variable stiffness actuators (VSAs), explicitly constructed a state feedback controller through backstepping and disturbance learning mechanism [7], and designed a key switching mechanism to adjust online parameters in the controller, which achieved good control effect. However, when the order of the controlled system is high, the calculation of the partial derivative of the virtual control quantity will have the disadvantage of differential explosion [8], which is not suitable for space occasions with limited computing resources [9]. The FSFB method has a simple structure and a small calculation amount; however, due to the general effect on the flexible time-varying inertial control system, the flexible vibration phenomenon occurs easily when the gain is large [10; 11; 12]. Feedback linearization control is an effective means to solve the nonlinear links in the control system; however, this method relies on high-precision mathematical models, and the control effect is greatly reduced once the model is biased [12]. The synovial membrane controller has the characteristics of fast response and strong robustness to disturbances. However, when the state trajectory reaches the sliding modal plane, it does not easily slide strictly along the sliding mode towards the equilibrium point. Rather, it crosses back and forth on both sides to approach the equilibrium point, resulting in jitter [13]. The intelligent algorithm based on a neural network also has the problem of excessive computation and is difficult to apply in space engineering [14]. The state observer can compensate the model to reduce the impact of model disturbance on the control system, and there is no differential explosion problem. [PERSON] et al. propose an accurate estimation of the uncertainty of the velocity state of the manipulator and the stiffness of the joint by the Extended State Observer (ESO) [15]. [PERSON] et al. use the inverse method to generate adaptive controllers, in which dynamic uncertainty is compensated for by a radial basis function neural network (RBFNN) and joint-stiffness uncertainty is eliminated by ESO estimation. The articulated motor side interference ob  
Figure 1: Schematic of a space manipulator with flexible joints. server is proposed and fed back to the proportion differential controller, which, in turn, introduces a nominally stable controller input [16]. [PERSON] et al. achieve linear state observation of the plastic deformation of the flexible joint reducer by measuring the joint and motor position [17]. [PERSON] et al. propose a terminal sliding mode control (TSMC) strategy based on fuzzy interference observer to target liquid shaking interference of a flexible liquid-filled spacecraft [18]. [PERSON] et al. design a state observer for single-degree-of-freedom flexible joint manipulator disturbance to compensate for interference, turning the nonlinear system into a linear integral sequence system [19]. [PERSON] et al. propose a linear expansion state observer for industrial robotic arms to estimate joint flexibility and torque [20]. [PERSON] developed two low-order extended state observers to handle the external load force and the impact of the pressure compensator in a high-order nonlinear hydraulic system. Additionally, backstepping methods were designed to ensure robust stability of the system [21].