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Stability Analysis of Geometric Control for a Canonical Class of Underactuated Aerial Vehicles with Spurious Forces

Simone Orelli, Mirko Mizzoni, Antonio Franchi

TL;DR

This work tackles the challenge of stabilizing hovering for a broad class of underactuated aerial vehicles where control moments generate spurious translational forces, violating standard decoupling assumptions. It introduces a canonical SE(3) model with a corresponding geometric controller and provides the first Lyapunov-based proof of local exponential stability for the hovering equilibrium under force–moment coupling, addressing non-minimum-phase effects that invalidate cascade arguments. The analysis yields explicit algebraic conditions on controller gains to certify stability and forward-invariance of a region of attraction, bridging the gap between empirical robustness and formal guarantees. The results offer practical design guidelines for safe static hovering in coupled systems and point to future work on trajectory tracking and reducing conservatism, with potential applicability to fault-tolerant multi-rotor configurations and tilted-propeller platforms.

Abstract

Standard geometric control relies on force-moment decoupling, an assumption that breaks down in many aerial platforms due to spurious forces naturally induced by control moments. While strategies for such coupled systems have been validated experimentally, a rigorous theoretical certification of their stability is currently missing. This work fills this gap by providing the first formal stability analysis for a generic class of floating rigid bodies subject to spurious forces. We introduce a canonical model and construct a Lyapunov-based proof establishing local exponential stability of the hovering equilibrium. Crucially, the analysis explicitly addresses the structural challenges - specifically the induced non-minimum-phase behavior - that prevent the application of standard cascade arguments.

Stability Analysis of Geometric Control for a Canonical Class of Underactuated Aerial Vehicles with Spurious Forces

TL;DR

This work tackles the challenge of stabilizing hovering for a broad class of underactuated aerial vehicles where control moments generate spurious translational forces, violating standard decoupling assumptions. It introduces a canonical SE(3) model with a corresponding geometric controller and provides the first Lyapunov-based proof of local exponential stability for the hovering equilibrium under force–moment coupling, addressing non-minimum-phase effects that invalidate cascade arguments. The analysis yields explicit algebraic conditions on controller gains to certify stability and forward-invariance of a region of attraction, bridging the gap between empirical robustness and formal guarantees. The results offer practical design guidelines for safe static hovering in coupled systems and point to future work on trajectory tracking and reducing conservatism, with potential applicability to fault-tolerant multi-rotor configurations and tilted-propeller platforms.

Abstract

Standard geometric control relies on force-moment decoupling, an assumption that breaks down in many aerial platforms due to spurious forces naturally induced by control moments. While strategies for such coupled systems have been validated experimentally, a rigorous theoretical certification of their stability is currently missing. This work fills this gap by providing the first formal stability analysis for a generic class of floating rigid bodies subject to spurious forces. We introduce a canonical model and construct a Lyapunov-based proof establishing local exponential stability of the hovering equilibrium. Crucially, the analysis explicitly addresses the structural challenges - specifically the induced non-minimum-phase behavior - that prevent the application of standard cascade arguments.
Paper Structure (15 sections, 8 theorems, 81 equations, 1 figure, 1 table)

This paper contains 15 sections, 8 theorems, 81 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Canonical Model and Actuation Properties
  3. Canonical Model of Aerial Rigid Bodies
  4. Actuation Properties
  5. Static Hovering
  6. Control Law
  7. Stability Analysis
  8. Desired Angular Velocity
  9. Equilibrium Point
  10. Candidate Lyapunov Function
  11. Stability of the Error System
  12. Conclusions
  13. Appendix
  14. Time Derivative of the Attitude Tracking Error
  15. Maximum attitude error

Key Result

Lemma III.1

The vector $\bm{\Omega}_d$ admits the following closed-form expression: where $\mathbf{w}_3=\hat{\mathbf{f}}_r$ denotes the third column of the rotation matrix $\mathbf{R}_w$, $\hat{{\mathbf{r}}}_1 \coloneqq \mathbf{R}_r\mathbf{e}_1$ is the constant heading direction, and $\dot{\mathbf{w}}_3 = \dot{\hat{\mathbf{f}}}_r = \frac{1}{\|\mathbf{f}_r\|}\left(\mathbf{I}_3 - \ha

Figures (1)

  • Figure 1: Floating rigid body with frames $\mathcal{F}_W$ and $\mathcal{F}_B$, desired direction $\mathbf d_\star$, total force $\mathbf f$, and its decomposition into clean ($\mathbf{A} \mathbf u_f$) and spurious ($\mathbf{B} \mathbf u_\tau$) components induced by the moment $\boldsymbol{\tau}$.

Theorems & Definitions (18)

  • Lemma III.1
  • Proof 1
  • Lemma III.2
  • Proof 2
  • Proposition 1
  • Proof 3
  • Definition 1
  • Lemma III.3
  • Proof 4
  • Lemma III.4
  • ...and 8 more