## 1 Introduction
As the number of matched feature points increases, this compensation becomes more obvious and the system shows better robustness to image noise [39]. It is worth mentioning that the dimensions of the homography matrix and task function remain constant regardless of the number of matching feature points, so the real-time performance of system is not affected. However, the original homography-based task function fails to take full advantage of the eight degrees of freedom of the homography matrix. In addition to further improving the real-time performance, this property is of great significance to the application of PHUVS in hyper-redundant manipulators. The online estimation of the image Jacobian is another pivotal point of an uncalibrated visual servo. Classic image Jacobian estimation methods include the dynamic Broyden-Gauss-Newton method [37], Broyden-RLS method [29], group-based Broyden method [38], Kalman filter (KF) method [23; 26; 27; 36], and particle filter [33; 38]. [PERSON] [22] showed that the KF method performs better than other methods when tracking targets with changing directions of motion. [PERSON] [29] also compared these methods and showed that the particle filter is slightly more accurate when the noise level is high, but the Kalman filter has better real-time and dynamic performance for mechanical systems. A limitation of Kalman filtering is that its accuracy may decrease or even diverge when there is modeling uncertainty and the statistical characteristics of noise are unknown [31; 32]. To solve this problem, researchers have proposed several methods such as the fuzzy adaptive Kalman filter [31; 32], Sage-Husa adaptive Kalman filter and Kalman neural network [34], but the performances of these methods are not always stable. A recently proposed variational Bayesian-based adaptive Kalman filter can improve accuracy to a certain extent, but its process is so cumbersome that it takes more than 20 times as much calculation time as the Kalman filter [40]. Considering the complexity of on-orbit maintenance tasks and the need for flexibility, a redundant manipulator with more degrees of freedom is a reasonable choice [41; 42]. Redundancy is often used in space manipulators; for example, to minimize the reactions transferred from the manipulator to the spacecraft [43] or to minimize the kinetic energy and joint torques [44]. Therefore, a hyper-redundant manipulator with nine degrees of freedom has been designed to implement on-orbit maintenance tasks for large space facilities. However, its kinematic parameters are affected by factors such as thermal expansion and contraction, corrosion, etc., and often change. It is almost impossible to recalibrate a hyper-redundant manipulator with a complex structure when its parameters change due to working in the harsh space environment for a long time. Meanwhile, a controller that is constructed with the estimated total Jacobian and directly controlled manipulator joints is more accurate and robust to the changes of manipulator parameters. In addition, the estimated total Jacobian of the homography-based task function tends to present an unstable conditional number. This results in a sudden increase of the manipulator velocity in some directions, which causes the manipulator to shake and results in serious errors [45]. On the basis of the above description, an improved uncalibrated visual servo strategy for hyper-redundant manipulators in on-orbit automatic assembly is proposed. In this strategy, an improved task function with robustness to image noise and better real-time performance is constructed; another advantage of this is that the joint space of hyper-redundant manipulators has redundancy. To improve the system accuracy and make the system robust to the variable parameters of manipulator in space, the estimated total Jacobian, which maps the task function and manipulator joints, is used to construct a controller to control the manipulator joints directly without a kinematics model.