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Application of Log-Linear Dynamic Inversion Control to a Multi-rotor

Li-Yu Lin, James Goppert, Inseok Hwang

TL;DR

This work develops a safety-verification framework for multi-rotors by embedding the dynamics in the $SE_2(3)$ Lie group and deriving exact log-linear error dynamics in the associated Lie algebra $\mathfrak{se}_2(3)$. A log-linear dynamic inversion controller, with $K_\zeta$ designed by $LQR$ in the Lie algebra, yields a linearized error model; angular-velocity error is governed by a separate linear-in-input law. Linear Matrix Inequalities (LMIs) are then used to compute invariant sets for the log-linear system and the angular-velocity subsystem, providing provable boundedness of the tracking error under bounded disturbances, with the invariant set mapped back to the nonlinear $SE_2(3)$ dynamics via the exponential map. Simulation on a reference-trajectory task demonstrates robust safety guarantees and the practical viability of the approach for UAM-like scenarios.

Abstract

This paper presents an approach that employs log-linearization in Lie group theory and the Newton-Euler equations to derive exact linear error dynamics for a multi-rotor model, and applies this model with a novel log-linear dynamic inversion controller to simplify the nonlinear distortion and enhance the robustness of the log-linearized system. In addition, we utilize Linear Matrix Inequalities (LMIs) to bound the tracking error for the log-linearization in the presence of bounded disturbance input and use the exponential map to compute the invariant set of the nonlinear system in the Lie group. We demonstrate the effectiveness of our method via an illustrative example of a multi-rotor system with a reference trajectory, and the result validates the safety guarantees of the tracking error in the presence of bounded disturbance.

Application of Log-Linear Dynamic Inversion Control to a Multi-rotor

TL;DR

This work develops a safety-verification framework for multi-rotors by embedding the dynamics in the Lie group and deriving exact log-linear error dynamics in the associated Lie algebra . A log-linear dynamic inversion controller, with designed by in the Lie algebra, yields a linearized error model; angular-velocity error is governed by a separate linear-in-input law. Linear Matrix Inequalities (LMIs) are then used to compute invariant sets for the log-linear system and the angular-velocity subsystem, providing provable boundedness of the tracking error under bounded disturbances, with the invariant set mapped back to the nonlinear dynamics via the exponential map. Simulation on a reference-trajectory task demonstrates robust safety guarantees and the practical viability of the approach for UAM-like scenarios.

Abstract

This paper presents an approach that employs log-linearization in Lie group theory and the Newton-Euler equations to derive exact linear error dynamics for a multi-rotor model, and applies this model with a novel log-linear dynamic inversion controller to simplify the nonlinear distortion and enhance the robustness of the log-linearized system. In addition, we utilize Linear Matrix Inequalities (LMIs) to bound the tracking error for the log-linearization in the presence of bounded disturbance input and use the exponential map to compute the invariant set of the nonlinear system in the Lie group. We demonstrate the effectiveness of our method via an illustrative example of a multi-rotor system with a reference trajectory, and the result validates the safety guarantees of the tracking error in the presence of bounded disturbance.
Paper Structure (7 sections, 17 equations, 3 figures)

This paper contains 7 sections, 17 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Vehicle Dynamics Embedded in $SE_2(3)$ Lie group
  3. Control Design for Error Dynamics
  4. Log-Linearization of Error Dynamics in the $SE_2(3)$ Lie Group
  5. Error Dynamics of Angular Velocity
  6. Application of Invariant Set
  7. Conclusion

Figures (3)

  • Figure 1: Projected 2D Invariant Sets Comparison with Dynamic Inversion, Left: Invariant Set in the Lie Algebra, Right: Invariant Sets in the Lie Group, Top: Small Disturbance, Bottom: Large Disturbance
  • Figure 2: Projected 3D Invariant Sets Comparison with Dynamic Inversion, Left: Invariant Set in the Lie Algebra, Right: Invariant Sets in the Lie Group, Top: Small Disturbance, Bottom: Large Disturbance
  • Figure 3: The Bound of the Invariant Set with Simulated Trajectories, Left: Small Disturbance, Right: Large Disturbance