Analytical and experimental stability investigation of a hardware-in-the-loop satellite docking simulator

The European Proximity Operation Simulator (EPOS) of the DLR-German Aerospace Center is a robotics-based simulator that aims at validating and verifying a satellite docking phase. The generic concept features a robotics tracking system working in closed loop with a force/torque feedback signal. Inherent delays in the tracking system combined with typical high stiffness at contact challenge the stability of the closed-loop system. The proposed concept of operations is hybrid: the feedback signal is a superposition of a measured value and of a virtual value that can be tuned in order to guarantee a desired behavior. This paper is concerned with an analytical study of the system's closed-loop stability, and with an experimental validation of the hybrid concept of operations in one dimension (1D). The robotics simulator is modeled as a second-order loop-delay system and closed-form expressions for the critical delay and associated frequency are derived as a function of the satellites' mass and the contact dynamics stiffness and damping parameters. A numerical illustration sheds light on the impact of the parameters on the stability regions. A first-order Pade approximation provides additional means of stability investigation. Experiments were performed and tests results are described for varying values of the mass and the damping coefficients. The empirical determination of instability is based on the coefficient of restitution and on the observed energy. There is a very good agreement between the critical damping values predicted by the analysis and observed during the tests. The contact duration shows also a very good fit between analysis and experiment. In addition, results from a 1D contact experiment carried on an air-floating testbed are successfully emulated using the proposed hybrid docking simulator. This illustrates the flexibility of the hybrid simulator, where various contact dynamics can be emulated without changing any hardware elements.