Dynamics and geometric optimal control of a planar air-bearing spacecraft simulator

One of the most popular micro-gravity ground test of space missions is planar air-bearing spacecraft simulator equipped with controllers to perform both pose stabilization and trajectory tracking in a simulated space mission. Previous studies based on the double-integrator equations neglect the geometric structures of the dynamics of the simulator. To address this problem, this paper begins with the continuous dynamic equations formulated on both left and right trivializations of Lie group SE(2), and then presents the intrinsic proportional-derivative controllers derived from these equations. Based on the discrete variational principle, this study presents a Lie group variational integrator derived for the time integration of the dynamic equations of the simulator. Then, the discrete geometric optimal controller is established by including the Lie group variational integrator as equality constraint equations into the optimal control problem. This work also prepares the proposed controllers in simulations in terms of accuracy, convergence rate, and control cost. Numerical results show intrinsic geometric characteristics of the pose trajectories obtained by using different controllers. To further investigate the real-time performances of different controllers, the paper presents a prototype of the planar air-bearing simulator and the experimental tests of pose stabilization and circle tracking. The experimental results are in good agreement with the numerical ones obtained from simulations.