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

Antonio Franchi; Chiara Gabellieri

cs.ROFeb 19, 2026

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

Authors: Antonio Franchi, Chiara Gabellieri

Organizations: Robotics and Mechatronics lab, Faculty of Electrical Engineering, Mathematics & Computer Science, University of Twente, Enschede, The Netherlands. · Department of Computer, Control and Management Engineering, Sapienza University of Rome, 00185 Rome, Italy.

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.

Figures & tables

Panel of Figure 1 — Fig. 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).

Symbol & Parameter	Sign of $\frac{\partial\mu}{\partial(\cdot)}$	Physical Interpretation
$\eta$ : Normalized External Force	(-)	Higher tether tension requires the aircraft to bank outward.
$\kappa$ : Dimensionless Speed Squared	(+)	Higher speed increases centrifugal force, requiring inward bank.
$\theta$ : Colatitude	(-)	Increasing $\theta$ (moving down) increases radius, reducing centrifugal force.
$L$ : Tether Length	(-)	Longer tether reduces angular rate (for fixed $v_{0}$ ), thus centrifugal force.

TABLE I : Sensitivity of Bank Angle at the Zero-Bank Equilibrium

Panel of Figure 3 — Fig. 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 increases from $10\rm{N}$ to $16\rm{N}$ with 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.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Description	Units	Frame
$\mathbf{p}^{W}\in\mathbb{R}^{3}$	CoM position	m	$W$
$\mathbf{v}^{W}\in\mathbb{R}^{3}$	CoM velocity ( $\dot{\mathbf{p}}^{W}$ )	m/s	$W$
$R\in\mathrm{SO}(3)$	Rotation matrix ( $B\to W$ )	–	–
$\bm{\omega}^{B}\in\mathbb{R}^{3}$	Ang. vel. of $B$ w.r.t $W$	rad/s	$B$
$m>0$	Mass	kg	–
$I_{B}\in\mathbb{R}^{3\times 3}$	Inertia (pos-def, const.)	kg·m 2	$B$

Appendix

TABLE II: Symbols, units, and frames used throughout the geometric

\mathrm{SO}(3)

flight-dynamics model. The frame in which vectors are expressed is specified in the corresponding column.

Kinematics	$\dot{\mathbf{p}}^{W}=\mathbf{v}^{W},\hskip 9.24994pt\dot{R}=R\,[\bm{\omega}^{B}]_{\times}$
Relative wind	$\mathbf{v}_{a}^{W}=\mathbf{v}^{W}-\mathbf{w}^{W},\hskip 9.24994pt\mathbf{v}_{a}^{B}=R^{\top}\mathbf{v}_{a}^{W},\hskip 9.24994pt\mathbf{e}_{a}^{B}=\frac{\mathbf{v}_{a}^{B}}{\\|\mathbf{v}_{a}^{B}\\|},\hskip 9.24994ptq=\tfrac{1}{2}\rho\\|\mathbf{v}_{a}^{B}\\|^{2}$ Regularization: $\\|\mathbf{v}_{a}^{B}\\|_{\epsilon}=\max(\\|\mathbf{v}_{a}^{B}\\|,\epsilon),\hskip 9.24994pt\tilde{\mathbf{e}}_{a}^{B}=\frac{\mathbf{v}_{a}^{B}}{\\|\mathbf{v}_{a}^{B}\\|_{\epsilon}},\hskip 9.24994pt\tilde{q}=\tfrac{1}{2}\rho\\|\mathbf{v}_{a}^{B}\\|_{\epsilon}^{2}$
Aerodynamic directions (Geometric)	$\mathbf{e}_{D}^{B}=-\,\mathbf{e}_{a}^{B},\hskip 9.24994pt\mathbf{e}_{L}^{B}=\frac{\mathbf{n}^{B}-(\mathbf{n}^{B}\cdot\mathbf{e}_{a}^{B})\,\mathbf{e}_{a}^{B}}{\\|\mathbf{n}^{B}-(\mathbf{n}^{B}\cdot\mathbf{e}_{a}^{B})\,\mathbf{e}_{a}^{B}\\|},\hskip 9.24994pt\mathbf{e}_{Y}^{B}=\mathbf{e}_{a}^{B}\times\mathbf{e}_{L}^{B}$
Aerodynamic Angles	$\alpha=\operatorname{atan2}\left(-\,\mathbf{e}_{a}^{B}\cdot\mathbf{n}^{B},\ \mathbf{e}_{a}^{B}\cdot\mathbf{c}^{B}\right),\hskip 9.24994pt\beta=\operatorname{atan2}\left(-\mathbf{e}_{a}^{B}\cdot\mathbf{s}^{B},\ \sqrt{1-(\mathbf{e}_{a}^{B}\cdot\mathbf{s}^{B})^{2}}\right)$
Aero Loads	$\mathbf{F}_{\text{aero}}^{B}=q\,S\left(C_{D}\,\mathbf{e}_{D}^{B}+C_{L}\,\mathbf{e}_{L}^{B}+C_{Y}\,\mathbf{e}_{Y}^{B}\right),\hskip 9.24994pt\mathbf{F}_{\text{aero}}^{W}=R\,\mathbf{F}_{\text{aero}}^{B}$ $\bm{\tau}_{\text{aero}}^{B}=q\,S\left(C_{\ell}\,b\,\mathbf{c}^{B}+C_{m}\,\bar{c}\,\mathbf{s}^{B}+C_{n}\,b\,\mathbf{n}^{B}\right)+D_{\omega}\,\bm{\omega}^{B}$
Propulsion & External	$\mathbf{F}_{\text{prop}}^{B}=T\,\mathbf{u}_{t}^{B},\hskip 9.24994pt\bm{\tau}_{\text{prop}}^{B}=\bm{\tau}_{\text{prop}}^{B}(\mathbf{u})$ $\mathbf{f}_{\text{ext}}^{W}\text{ (Force at }r^{B}\text{)},\hskip 9.24994pt\bm{\tau}_{\text{ext,net}}^{B}=\bm{\tau}_{\text{ext}}^{B}+r^{B}\times(R^{\top}\mathbf{f}_{\text{ext}}^{W})$
Rigid-Body Dynamics	$m\,\dot{\mathbf{v}}^{W}=m\,\mathbf{g}^{W}+R(\mathbf{F}_{\text{prop}}^{B}+\mathbf{F}_{\text{aero}}^{B})+\mathbf{f}_{\text{ext}}^{W}$ $I_{B}\,\dot{\bm{\omega}}^{B}+\bm{\omega}^{B}\times(I_{B}\bm{\omega}^{B})=\bm{\tau}_{\text{prop}}^{B}+\bm{\tau}_{\text{aero}}^{B}+\bm{\tau}_{\text{ext,net}}^{B}$

Appendix

TABLE III: Model Summary: Kinematics, Aerodynamics, and Dynamics

Parameter	Symbol	Class A (1–3 m)	Class B (4–6 m)	Sources
Span	$b$ [m]	$1$ – $3$ (e.g., $2.12$ )	$4$ – $6$ (e.g., $5.00$ )	[ 53 , 54 , 55 , 56 ]
Reference area	$S$ [m 2 ]	$0.2$ – $0.9$ (e.g., $0.80$ )	$1.3$ – $4.5$ (e.g., $2.615$ )	[ 53 , 56 ]
Aspect ratio	$\mathrm{AR}$ [-]	$\approx 5$ – $7$ (e.g., $5.62$ )	$\approx 8$ – $12$ (e.g., $9.56$ )	[ 57 , 56 ]
Oswald eff.	$e$ [-]	$0.75$ – $0.85$ (nom. $0.80$ )	$0.85$ – $0.92$ (nom. $0.90$ )	[ 58 , 59 ]
Lift slope	$C_{L\alpha}$ [rad -1 ]	$4.2$ – $4.7$ (nom. $4.35$ )	$4.9$ – $5.3$ (nom. $5.10$ )	[ 58 , 59 ]
Zero-lift drag	$C_{D0}$ [-]	$0.03$ – $0.05$ (nom. $0.035$ )	$0.02$ – $0.04$ (nom. $0.030$ )	[ 60 ]

Appendix

TABLE IV : Recommended parameter ranges and nominal values for two fixed-wing UAV classes. See the rightmost column for the sources supporting each number.

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