Abstract— This work documents a case study in the application
of Mortensen’s nonlinear filtering approach to invariant
systems on general Lie groups. In this paper, we consider the
special orthogonal group SO(3) of all rotation matrices. We
identify the exact form of the kinematics of the minimumenergy
(optimal) observer on SO(3) and note that it depends
on the Hessian of the value function of the associated optimal
control problem. We derive a second order approximation of
the dynamics of the Hessian by neglecting third order terms in
the expansion of the dynamics. This yields a Riccati equation
that together with the optimal observer equation form a second
order minimum-energy filter on SO(3). The proposed filter is
compared to the multiplicative extended Kalman filter (MEKF),
arguably the industry standard for attitude estimation, by
means of simulations. Our studies indicate superior transient
and asymptotic tracking performance of the proposed filter as
compared to the MEKF
