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.
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
Fig. 1 : Behavior of the geometric bank angle μ 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 θ ) 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 ∂(⋅)∂μ
Physical Interpretation
η : Normalized External Force
(-)
Higher tether tension requires the aircraft to bank outward.
TABLE I : Sensitivity of Bank Angle at the Zero-Bank Equilibrium
Fig. 2 : Attitude, thrust, and drag forces for the aircraft schematically depicted on the left. The constant tangential speed is 11.7m/s on a circle of radius 18.544m . The cable tension increases from 10N to 16N with steps of 1.5N , and the quantities in the displayed plots vary accordingly. For a cable tension of 16N , μ 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
pW∈R3
CoM position
m
W
vW∈R3
CoM velocity ( p˙W )
m/s
W
R∈SO(3)
Rotation matrix ( B→W )
–
–
ωB∈R3
Ang. vel. of B w.r.t W
rad/s
B
m>0
Mass
kg
–
IB∈R3×3
Inertia (pos-def, const.)
kg·m 2
B
Appendix
TABLE II: Symbols, units, and frames used throughout the geometric SO(3) flight-dynamics model. The frame in which vectors are expressed is specified in the corresponding column.
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
AR [-]
≈5 – 7 (e.g., 5.62 )
≈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
CLα [rad -1 ]
4.2 – 4.7 (nom. 4.35 )
4.9 – 5.3 (nom. 5.10 )
[ 58 , 59 ]
Zero-lift drag
CD0 [-]
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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