  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 $\sigma$-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, formally-principled, and general. An example of satellite attitude
tracking illustrates the proposed theory.
