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

Log-linear Dynamic Inversion Control with Provable Safety Guarantees in Lie Groups