Separation property for the rigid-body attitude tracking control problem

Quaternion-based proportional-derivative controllers for rigid-body attitude dynamics provide globally stabilizing solutions to both set-point regulation and trajectory tracking problems. Because the quaternion vector, or for that matter, any other attitude representation, can never be directly exactly measured, proportional-derivative controllers are invariably implemented under the assumption that the attitude error is available from suitable observers whose estimates converge sufficiently fast to the corresponding true attitude. To compound the situation, given the nonlinearities within the governing dynamics, most existing attitude observers can at best be proven to provide only asymptotic (i.e., nonexponential) convergence for the attitude estimation errors. This has the serious consequence that closed-loop stability assurances provided by classical proportional-derivative control laws no longer remain valid when the true attitude errors are replaced by their corresponding estimates. In this paper, we present a new quaternion-based attitude tracking controller that guarantees global asymptotic stability for the closed-loop dynamics while adopting an observer to generate the quaternion-based attitude estimates. We show that the state feedback control law and the estimator can be independently designed so that closed-loop stability is maintained even when they are combined. Accordingly, a separation property is established for the rigid-body attitude tracking problem, the first such result to our best knowledge. The crucial step in our stability analysis involves introduction ofa novel class ofstrict Lyapunov functions whose time derivatives contain additional negative terms that help dominate the error terms arising due to the attitude observer implementation. Detailed proofs and numerical simulation examples are presented to help illustrate all the technical aspects of this work.