  This paper presents solutions to the following two common quaternion attitude
estimation problems: (i) estimation of attitude using measurement of two
reference vectors, and (ii) estimation of attitude using rate measurement and
measurement of a single reference vector. Both these problems yield to a direct
geometric analysis and solution. The former problem already has a well
established analytic solution in literature using linear algebraic methods.
This note shows how the solution may also be obtained using geometric methods,
which are not only more intuitive, but also amenable to unconventional
extensions. With respect to the latter problem, existing solutions typically
involve filters and observers and use a mix of differential-geometric and
control systems methods. However, no analytic solution has yet been reported to
this problem. In this note, both the problems are formulated as optimization
problems, which can be solved analytically to obtain a unique closed-form
solution. The analytic attitude estimates are (i) instantaneous with respect to
the measurements, thus overcoming the latency inherent in solutions based upon
negative feedback upon an error, which can at best show asymptotic convergence,
(ii) exact, thus overcoming errors in solutions based upon linear methods, and
(iii) geometry-based, thus enabling imposition of geometric inequality
constraints.
