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

TL;DR

This paper introduces a novel control approach for Lie group systems that guarantees safety and robustness by using log-linear dynamic inversion, with applications demonstrated in urban air mobility scenarios.

Contribution

It develops a log-linear dynamic inversion control law with provable safety guarantees for systems on Lie groups, incorporating LMI-based error bounds and flow pipes.

Findings

Effective error bounding using LMIs under disturbances

Application to UAM with improved tracking robustness

Flow pipe construction for safety assurance

Abstract

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.