  In this paper, we use the derivative of the exponential map to derive the
exact evolution of the logarithm of the tracking error for mixed-invariant
systems, a class of systems capable of describing rigid body tracking problems
in Lie groups. Additionally, we design a log-linear dynamic inversion-based
control law to remove the nonlinearities due to spatial curvature and enhance
the robustness of the controller. We apply Linear Matrix Inequalities (LMIs) to
bound the tracking error given a bounded disturbance amplified by the
distortion matrix and leverage the tracking error bound to create flow pipes.
To demonstrate the usefulness of our method, we show its application with Urban
Air Mobility (UAM) scenarios using a simplified kinematic aircraft model and
polynomial-based path planning methods.
