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Geometric Inverse Flight Dynamics on SO(3) and Application to Tethered Fixed-Wing Aircraft

Antonio Franchi, Chiara Gabellieri

Abstract

We present a robotics-oriented, coordinate-free formulation of inverse flight dynamics for fixed-wing aircraft on SO(3). Translational force balance is written in the world frame and rotational dynamics in the body frame; aerodynamic directions (drag, lift, side) are defined geometrically, avoiding local attitude coordinates. Enforcing coordinated flight (no sideslip), we derive a closed-form trajectory-to-input map yielding the attitude, angular velocity, and thrust-angle-of-attack pair, and we recover the aerodynamic moment coefficients component-wise. Applying such a map to tethered flight on spherical parallels, we obtain analytic expressions for the required bank angle and identify a specific zero-bank locus where the tether tension exactly balances centrifugal effects, highlighting the decoupling between aerodynamic coordination and the apparent gravity vector. Under a simple lift/drag law, the minimal-thrust angle of attack admits a closed form. These pointwise quasi-steady inversion solutions become steady-flight trim when the trajectory and rotational dynamics are time-invariant. The framework bridges inverse simulation in aeronautics with geometric modeling in robotics, providing a rigorous building block for trajectory design and feasibility checks.

Geometric Inverse Flight Dynamics on SO(3) and Application to Tethered Fixed-Wing Aircraft

Abstract

We present a robotics-oriented, coordinate-free formulation of inverse flight dynamics for fixed-wing aircraft on SO(3). Translational force balance is written in the world frame and rotational dynamics in the body frame; aerodynamic directions (drag, lift, side) are defined geometrically, avoiding local attitude coordinates. Enforcing coordinated flight (no sideslip), we derive a closed-form trajectory-to-input map yielding the attitude, angular velocity, and thrust-angle-of-attack pair, and we recover the aerodynamic moment coefficients component-wise. Applying such a map to tethered flight on spherical parallels, we obtain analytic expressions for the required bank angle and identify a specific zero-bank locus where the tether tension exactly balances centrifugal effects, highlighting the decoupling between aerodynamic coordination and the apparent gravity vector. Under a simple lift/drag law, the minimal-thrust angle of attack admits a closed form. These pointwise quasi-steady inversion solutions become steady-flight trim when the trajectory and rotational dynamics are time-invariant. The framework bridges inverse simulation in aeronautics with geometric modeling in robotics, providing a rigorous building block for trajectory design and feasibility checks.
Paper Structure (17 sections, 53 equations, 2 figures, 4 tables)

This paper contains 17 sections, 53 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Geometric Newton--Euler Model on $\mathrm{SO}(3)$
  3. Geometric Inverse Flight Dynamics
  4. Closed-form construction at each time $t$
  5. Degeneracies and edge cases
  6. Approximate trim solution for simplified aerodynamics
  7. IFD on a Spherical Parallel with External Force Aligned to the Sphere Center
  8. Trim Solution (T,alpha) under a Simple Lift/Drag Law
  9. Bank Dependence on External Force and Geometry
  10. Locus of Parameters for the Efficient Zero Bank Attitude
  11. Sensitivity Analysis at the zero-bank locus
  12. Behavior of the Bank Angle Curves
  13. Numerical Results
  14. Conclusion
  15. Auxiliary Tables
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Behavior of the geometric bank angle $\mu$ required for coordinated tethered flight. (a) Increasing tether tension pulls the aircraft outward (negative bank), competing with centrifugal force. (b) Lowering the trajectory on the sphere (increasing $\theta$) increases the turn radius, reducing centrifugal force and favoring outward banking. (c) The heatmap highlights the zero-bank locus (white contour), separating the regime dominated by centrifugal force (positive bank) from the regime dominated by tether tension (negative bank).
  • Figure 2: Attitude, thrust, and drag forces for the aircraft schematically depicted on the left. The constant tangential speed is $11.7\rm{m/s}$ on a circle of radius $18.544\rm{m}$. The cable tension is increased from $10\rm{N}$ to $16\rm{N}$ with consecutive steps of $1.5\rm{N}$, and the quantities in the displayed plots vary accordingly. For a cable tension of $16\rm{N}$, $\mu$ approaches zero, namely, the cable force compensates the centrifugal effects.