  Nonlinear observer design for systems whose state space evolves on Lie groups
is considered. The proposed method is similar to previously developed nonlinear
observers in that it involves propagating the state estimate using a process
model and corrects the propagated state estimate using an innovation term on
the tangent space of the Lie group. In the proposed method, the innovation term
is constructed by passing the gradient of an invariant cost function, resolved
in a basis of the tangent space, through a linear time-invariant system. The
introduction of the linear system completes the extension of linear
complementary filters to nonlinear Lie group observers by allowing higher-order
filtering. In practice, the proposed method allows for greater design freedom
and, with the appropriate selection of the linear filter, the ability to filter
bias and noise over specific bandwidths. A disturbance observer that accounts
for constant and harmonic disturbances in group velocity measurements is also
considered. Local asymptotic stability about the desired equilibrium point is
demonstrated. A numerical example that demonstrates the desirable properties of
the observer is presented in the context of pose estimation.
