  This paper conveys attitude and rate estimation without rate sensors by
performing a critical comparison, validated by extensive simulations. The two
dominant approaches to facilitate attitude estimation are based on stochastic
and set-membership reasoning. The first one mostly utilizes the commonly known
Gaussian-approximate filters, namely the EKF and UKF. Although more
conservative, the latter seems to be more promising as it considers the
inherent geometric characteristics of the underline compact state space and
accounts -- from first principles -- for large model errors. We address the
set-theoretic approach from a control point of view, and we show that it can
overcome reported deficiencies of the Bayesian architectures related to this
problem, leading to coordinate-free optimal filters. Lastly, as an example, we
derive a modified predictive filter on the tangent bundle of the special
orthogonal group $\mathbb{TSO}(3)$.
