  The rigid body attitude estimation problem is treated using the discrete-time
Lagrange-d'Alembert principle. Three different possibilities are considered for
the multi-rate relation between angular velocity measurements and direction
vector measurements for attitude: 1) integer relation between sampling rates,
2) time-varying sampling rates, 3) non-integer relation between sampling rates.
In all cases, it is assumed that angular velocity measurements are sampled at a
higher rate compared to the inertial vectors. The attitude determination
problem from two or more vector measurements in the body-fixed frame is
formulated as Wahba's problem. At instants when direction vector measurements
are absent, a discrete-time model for attitude kinematics is used to propagate
past measurements. A discrete-time Lagrangian is constructed as the difference
between a kinetic energy-like term that is quadratic in the angular velocity
estimation error and an artificial potential energy-like term obtained from
Wahba's cost function. An additional dissipation term is introduced and the
discrete-time Lagrange-d'Alembert principle is applied to the Lagrangian with
this dissipation to obtain an optimal filtering scheme. A discrete-time
Lyapunov analysis is carried out to show that the optimal filtering scheme is
asymptotically stable in the absence of measurement noise and the domain of
convergence is almost global. For a realistic evaluation of the scheme,
numerical experiments are conducted with inputs corrupted by bounded
measurement noise. These numerical simulations exhibit convergence of the
estimated states to a bounded neighborhood of the actual states.
