Target capturing is an essential and key mission for tethered space robot (TSR) in future on-orbit servicing, and it is quite meaningful to investigate the stabilization method for TSR during capture impact with target. In this paper, the stabilization control of TSR during target capturing is studied. The space tether is described by the lumped mass model, and the impact dynamic model for target capturing is derived using the Lagrange method with the consideration of space tether length, in/out-plane angles, and gripper attitude. Given the structure of the TSR's gripper, a position-based impedance control method is proposed for target capturing operation. The neural network is used to estimate and compensate the uncertainties in the dynamic model of TSR, and an adaptive robust controller is designed to overcome the influence of the space tether and track the desired position generated by impedance controller. Numerical simulations suggest that the proposed controller can realize the stabilization of TSR during target capturing; besides, the uncertainties of the TSR can effectively be compensated via adaptive law and the influence of the space tether can be suppressed via the robust control strategy, which lead to smaller overshoot, less convergence time, and higher control accuracy during capturing operation.