Synergistic Hybrid Feedback for Global Rigid-Body Attitude Tracking on $hbox{ SO }(3)^{ast}$ ${ssr {SO}}(3)^{ast}$

摘要

In this paper, we propose two hybrid feedbacks based on “synergistic” potential functions to achieve robust global asymptotic tracking of rigid-body attitude, a task that is impossible by classical state feedback—be it smooth, nonsmooth, or periodic—due to the topological structure of the special orthogonal group. Both hybrid feedbacks are based upon a proportional-derivative structure where the proportional term is generated from a finite family of control-induced artificial potential energies and the derivative term is due to a damping injection. In a patient-yet-greedy fashion, the hybrid feedback hysteretically switches to the minimum control-induced artificial potential energy in a finite family. The synergy property—which requires that for each undesirable critical point of each potential energy, there exists a lower potential energy in the family—guarantees the globality of robust asymptotic tracking. The first hybrid feedback is the natural extension of existing PD controllers and results in discontinuous jumps in the control signal due to a direct switch the potential energy. Using a backstepping procedure, we design a second hybrid feedback that smooths the jumps in the control torque by dynamically interpolating between the switching potential-energy terms. We show that while the class of “modified trace functions” is not wide enough to generate a “centrally synergistic” potential function, relaxing the centrality assumption allows one to construct a synergistic family and we provide explicit guidelines for doing so.