In order to achieve fast and high precision force/position control of on-orbit mission, the impedance control problem of dual-arm space robot on-orbit auxiliary docking operation is studied. Firstly, the dynamics equation of the closed chain hybrid system formed after the capture operation of the dual-arm space robot is established by using the Lagrange method. A second-order linear impedance model and a second-order approximate environment model are established based on impedance control theory. Then, theradial basis function neural networks(RBFNNs) are used for blocking approximation of the system uncertainties. Considering the control requirement of convergence in finite time, the fast nonlinear sliding mode surface(FNSMS) is introduced, and the buffeting caused by the introduced sliding mode surface is suppressed by designing the boundary layer function. Finally, the stability of the system is verified by Lyapunov stability determination. The simulation results show that the proposed control algorithm has fast convergence speed, good stability and strong robustness, and can simultaneously achieve high-precision force/position control. The attitude control accuracy is better than 0.5°, the position control accuracy is better than 0.001 m, and the output force control accuracy is better than 0.5 N, which can meet the task requirements of the dual-arm space robot on-orbit auxiliary docking operation.