In this paper, we address the rigid body pose stabilization problem using dual quaternion formalism. We propose a hybrid control strategy to design a switching control law with hysteresis in such a way that the global asymptotic stability of the closed-loop system is guaranteed and such that the global attractivity of the stabilization pose does not exhibit chattering, a problem that is present in all discontinuous-based feedback controllers. Using numerical simulations, we illustrate the problems that arise from existing results in the literature-as unwinding and chattering-and verify the effectiveness of the proposed controller to solve the robust global pose stability problem.

Unscented Kalman Filters (UKFs) have become popular in the research community. Most UKFs work only with Euclidean systems, but in many scenarios it is advantageous to consider systems with state-variables taking values on Riemannian manifolds. However, we can still find some gaps in the literature's theory of UKFs for Riemannian systems: for instance, the literature has not yet i) developed Riemannian extensions of some fundamental concepts of the UKF theory (e.g., extensions of σ-representation, Unscented Transformation, Additive UKF, Augmented UKF, additive-noise system), ii) proofs of some steps in their UKFs for Riemannian systems (e.g., proof of sigma points parameterization by vectors, state correction equations, noise statistics inclusion), and iii) relations between their UKFs for Riemannian systems. In this work, we attempt to develop a theory capable of filling these gaps. Among other results, we propose Riemannian extensions of the main concepts in the UKF theory (including closed forms), justify all steps of the proposed UKFs, and provide a framework able to relate UKFs for particular manifolds among themselves and with UKFs for Euclidean spaces. Compared with UKFs for Riemannian manifolds of the literature, the proposed filters are more consistent, formallyprincipled, and general. An example of satellite attitude tracking illustrates the proposed theory. Riemannian systems: the works [34], [35] and [36] introduced EKFs for Lie groups; and [5] a UKF for geodesically-complete Riemannian manifolds.geodesically-complete

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