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Global Attitude Control via Contraction on Manifolds with Reference Trajectory and Optimization

In this paper, we present a simple geometric attitude controller that is globally, exponentially stable. To overcome the topological restriction, the controller is designed to follow a reference trajectory that in turn converges to the desired equilibrium (making it discontinuous in the initial conditions, but continuous in time). The system and reference dynamics are studied as a single augmented system that can be analyzed and tuned simultaneously. The controller’s stability is proved using contraction analysis (on the manifold), and the bounds on the convergence rate can be found via a semi-definite program with linear matrix inequalities. Additionally, our approach allows the use of the Nelder-Mead algorithm to automatically select controller gains and reference trajectory parameters by optimizing the aforementioned bounds. The resulting controller is verified through simulations.