###### Abstract
A smaller output force will not execute the operation of insertion and extraction while a bigger output force may shut down these processes. In addition, the accuracy of the position input will in turn affect the value of output force. Then, the impedance control [20, 21, 22, 23, 24] has become an effective means to solve this problem. Based on admittance control, [PERSON] et al. [25] proposed a control scheme for the problem of fast and sensorless peg in hole insertion of double arm robot, and completed the experimental verification on an ABB double arm robot. However, there are still considerable technical difficulties in the on-orbit operation of insertion and extraction of space robots. In the process of inserting and extracting of space robot, there must be uncertainty caused by unmodeled error, external disturbance and violent impact, which undoubtedly improves the design difficulty of the control system. In view of the above uncertain effects, not only the traditional classical nonlinear methods such as robust control [26], sliding mode control [27], and adaptive control based on neural network [28], some new control methods and theories such as LQR method [29], MPC method [30], and reinforcement learning method [31] have been fully studied and achieved good control results in the field of aerospace. However, these controllers generally have complex design and a large calculation. Limited by the computer performance of space robot, it is worth mentioning that computational efficiency also has an important impact on the success or failure of space task. Furthermore although these controllers have achieved a high control accuracy in terms of steady-state performance (control accuracy), it fails to take into account the transient performance (error range, approaching rate, overshoot, etc.) at the same time. In addition, some of them can only ensure one or two transient performances, such as error range, convergence rate, etc. Moreover, the controlled system is expected to converge in limited task time and limited operation space according to the on-orbit task requirements. Thus, a high-quality attitude control system can largely ensure that the space manipulator and its end replaceable instruments enter the hole vertically, and reduce the adverse impact effect as much as possible. This would directly determine the success or failure of the space robot on-orbit insertion and extraction operation. The prescribed performance control method (PPC), which was jointly proposed by Greek scholars [PERSON] and [PERSON] [32] in 2008, has attracted researchers' extensive attention. Its core idea is to artificially set the performance envelope for the state of the controlled system, and describe the transient and steady-state performance of the controlled system through the convergence characteristics of the performance envelope function. Based on these advantages, the PPC method is widely used in the spacecraft attitude control [33; 34; 35]. Therefore, based on the above discussion, the force position control of free-floating space robot with output constraints is an important research direction, not to mention the unmodeled error and improving computer efficiency. The issue is actually open due to few people pay attention to it. The main contributions of this manuscript are: (1). The system dynamics model and the end second-order impedance model are established for the force position control problem of the on-orbit insertion and extraction operation of the free-floating space robot. (2). The reconstruction avoids the use of acceleration signal by introducing filtering operation. It improves the calculation efficiency and is conducive to the application of robot actual control. (3). A simple and reliable estimator requiring only the system inertia matrix is designed to compensate the modeling uncertainty, external bounded disturbance and impact effect in the process of inserting and extracting. There are only one control parameter needs to be adjusted. This also greatly reduces the amount of calculation. (4).