Breaking Topological Obstructions in Rigid Body Attitude Control using Time-varying Localized Feedback Perturbations

Rigid body attitude control is a nonlinear problem, as the configuration space of three-dimensional orientations is the compact, non-contractible Lie group SO(3). The largest achievable domain of convergence using smooth state feedback is given by feedback control laws based on a Morse function on SO(3), which is a measure of attitude stabilization or tracking error. A polar Morse function on SO(3) has four isolated critical points, with a unique minimum. When used as part of a Morse-Lyapunov function on the tangent bundle TSO(3) comprising the attitude and angular velocity states, it yields a globally continuous, state-feedback proportional-derivative (PD) type control law, with the critical points of the Morse function giving rise to four equilibria of the feedback system. The equilibrium state corresponding to the minimum with zero angular velocity is rendered almost globally asymptotically stable (AGAS), while the other critical points give hyperbolic equilibria on TSO(3). In our recent research, we used time-varying gains in the Morse function to improve the domain of convergence of the desired equilibrium state on TSO(3), by periodically changing the indices of the critical points corresponding to the undesired equilibria. In this work, we design compactly supported bump functions to generate continuously time-varying and localized perturbation torques near the three undesired critical points of the Morse function. These perturbations act as deliberate, state and time-dependent torque injections added to the PD-type nominal torque given by the Morse-Lyapunov function, so that the undesired equilibria of the nominal system do not remain equilibria of the perturbed feedback system on TSO(3). The resulting state and time-varying continuous feedback controller is shown to be globally asymptotically stable (GAS) at the single remaining equilibrium of the feedback dynamics. In addition, the proposed perturbed control approach is also extended to two-agent attitude synchronization relevant to spacecraft autonomous rendezvous and proximity operations (ARPO), highlighting the important fundamental result on breaking topological limitations on GAS. Numerical simulations demonstrate that the proposed time-varying localized feedback perturbations expel trajectories initialized at undesired equilibria of the nominal feedback system associated with a Morse function employing constant or time-varying gains. As a result, the trajectories of both the single rigid body attitude system and the two-body relative attitude synchronization system converge to the desired equilibrium.