Nested Power Models for Multirotor Propulsion: From Aerodynamic Drag to Electrical Losses

Antonio Franchi; Aaron Saini; Ahmed Ali; Chiara Gabellieri

cs.ROOct 6, 2026

Nested Power Models for Multirotor Propulsion: From Aerodynamic Drag to Electrical Losses

Authors: Antonio Franchi, Aaron Saini, Ahmed Ali, Chiara Gabellieri

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

Abstract

Speed-only aerodynamic power models for multirotor propulsion cannot represent acceleration-dependent effects. This work develops a nested sequence of propulsion-power models that starts from aerodynamic power dissipation and progressively introduces a reversible kinetic-energy rate, torque-dependent electromechanical dissipation, and lumped speed-proportional dissipation. The models are identified using one subset of experiments and validated using the other on a motor-drive-propeller unit. Independent estimates of rotational inertia and aerodynamic drag complement predictive validation by assessing whether the models correctly attribute the measured power to reversible kinetic-energy exchange and irreversible dissipation and, within the latter, to aerodynamic and electromechanical losses. The results show that the reversible kinetic-energy rate is necessary but insufficient for accurate dynamic power prediction. Dissipation proportional to the squared motor torque provides the main additional improvement, while speed-proportional dissipation further prevents irreversible losses from being attributed to reversible kinetic-energy exchange. The resulting methodology provides a reusable and experimentally verifiable basis for developing and selecting dynamic propulsion-power models for multirotor systems.

Figures & tables

Nested model	Model parameters and input variables	Reversible power contribution	Irreversible power loss	Parameters and variables contributing to dissipation
$P_{1}(v)=c_{\tau}\|v\|^{3}$	$c_{\tau}$ ; $v$	Absent	Aerodynamic: $c_{\tau}\|v\|^{3}$	$c_{\tau}$ , $v$
$P_{2}(v,\dot{v})=P_{1}(v)+Jv\dot{v}$	$c_{\tau}$ , $J$ ; $v$ , $\dot{v}$	Kinetic-energy rate: $Jv\dot{v}$	Aerodynamic: $c_{\tau}\|v\|^{3}$	$c_{\tau}$ , $v$
$P_{3}(v,\dot{v})=P_{2}(v,\dot{v})+k_{\mathrm{tl}}\tau_{\mathrm{m}}^{2}$	$c_{\tau}$ , $J$ , $k_{\mathrm{tl}}$ ; $v$ , $\dot{v}$	Kinetic-energy rate: $Jv\dot{v}$	Aerodynamic and electromechanical: $c_{\tau}\|v\|^{3}+k_{\mathrm{tl}}\tau_{\mathrm{m}}^{2}$	$c_{\tau}$ , $J$ , $k_{\mathrm{tl}}$ ; $v$ , $\dot{v}$
$P_{4}(v,\dot{v})=P_{3}(v,\dot{v})+q_{1}\|v\|$	$c_{\tau}$ , $J$ , $k_{\mathrm{tl}}$ , $q_{1}$ ; $v$ , $\dot{v}$	Kinetic-energy rate: $Jv\dot{v}$	Aerodynamic, electromechanical, and lumped: $c_{\tau}\|v\|^{3}+k_{\mathrm{tl}}\tau_{\mathrm{m}}^{2}+q_{1}\|v\|$	$c_{\tau}$ , $J$ , $k_{\mathrm{tl}}$ , $q_{1}$ ; $v$ , $\dot{v}$

TABLE I: Nested electrical-power models and their energetic structure.

Panel of Figure 2 — Fig. 1: Port-Hamiltonian-inspired representation of the energetic structure underlying the nested models. The electrical source supplies $P$ through a conceptual power-preserving interconnection. The kinetic-energy storage element exchanges reversible power, whereas the other branches represent aerodynamic, torque-dependent electromechanical, and lumped speed-proportional dissipation. Accordingly, each model combines the contributions indicated by its labels. The diagram provides an energetic interpretation rather than a complete bond graph of the motor-drive–propeller system.

Component	Device	Relevant characteristic
Motor	Xnova Lightning 2208 V2 1500KV	1500 rpm/V; listed resistance 0.078 $\Omega$ ; mass 36.5 g
Propeller	Graupner 3D-Prop 8x4.5	8-in diameter; 4.5-in pitch; measured mass 7.81 g
Electronic speed controller	Aikon AKC 70A 8S 8-in-1	BLHeli_32; 70-A continuous rating
Current sensing	Mateksys Hall Current Sensor 150A	20 mV/A sensitivity; 5% stated current-output error
Acquisition and logging	D1 Mini, Chimera board, and Raspberry Pi	Timestamped current, voltage, and rotor-frequency acquisition

TABLE II: Experimental hardware and instrumentation.

Panel of Figure 4 — Fig. 2: Propulsion and measurement components used in the experiments: motor-propeller unit, electronic speed controller, current sensor and acquisition electronics, voltage acquisition, and logging computer.

Panel of Figure 5 — Fig. 3: Ramp-level quality control. Transition regions around commanded frequency reversals are excluded, and complete ramp interiors are retained or rejected according to measured-motion and processing-consistency criteria established before fitting any power model.

Fig. 4: Independent determination of the rotational inertia: (a) separation of one propeller blade from the hub; (b) division of the blade into eight segments for radial mass-distribution measurement; and (c) disassembly of the motor to characterize the outer rotor and shaft.

Model	RMSE	MAE	$\boldsymbol{R^{2}}$	MAPE
	[W]	[W]		[%]
$P_{1}$	103.320	61.785	0.18044	53.945
$P_{2}$	42.457	26.602	0.86161	16.579
$P_{3}$	19.799	12.465	0.96991	14.687
$P_{4}$	17.388	9.651	0.97679	8.739

TABLE III: Held-out prediction metrics along the nested model hierarchy.

Fig. 5: Measured source-side electrical power and held-out predictions of the nested models over the sampled angular-speed and acceleration domain. The common color scale shows the progressive recovery of acceleration-dependent power, with $P_{4}$ providing the closest overall prediction among the nested models.

Fig. 6: Held-out prediction diagnostics for the nested models: prediction parity, cumulative absolute-error distributions, residual dependence on angular acceleration, and MAPE for measured source-side power satisfying $P\geq 20$ W. The principal improvements occur through $P_{3}$ , while $P_{4}$ provides a smaller refinement and therefore overlaps closely with $P_{3}$ in some panels.

Fig. 7: Held-out prediction residuals, defined as predicted minus measured source-side power. The strong acceleration-organized structure observed for $P_{1}$ is progressively reduced by the dynamic models, with $P_{3}$ providing the principal correction and $P_{4}$ a smaller additional refinement. Each panel uses a separate symmetric color scale based on the 99th percentile of the corresponding absolute residuals; color magnitudes must therefore be interpreted using the panel-specific colorbar.

Fig. 8: Local held-out RMSE improvement across the nested hierarchy. The panels show the effects of introducing reversible kinetic-energy rate, torque-dependent electromechanical dissipation, and lumped speed-proportional dissipation. Positive values indicate a lower local RMSE for the richer model. Uncolored cells contain insufficient validation samples.

Model	$\widehat{J}$	$\widehat{c_{\tau}}$	$\widehat{k_{\mathrm{tl}}}$	$\widehat{q_{1}}$	$\delta_{J}$	$\delta_{c_{\tau}}$
	$[10^{-5}]$	$[10^{-8}]$	$[10^{3}]$	$[10^{-2}]$	$[\%]$	$[\%]$
$P_{1}$	–	14.01	–	–	–	$+96.00$
$P_{2}$	3.570	13.99	–	–	$+72.03$	$+95.73$
$P_{3}$	1.584	7.396	7.820	–	$-23.68$	$+3.47$
$P_{4}$	1.902	7.155	5.229	2.176	$-8.36$	$+0.10$
Independent reference	2.075	7.148	–	–	–	–

TABLE IV: Identified coefficients and agreement with the independent aeromechanical estimates. Dashes: coefficients absent from a model.

Fig. 9: Held-out prediction and aeromechanical parameter recovery along the nested hierarchy. Prediction metrics quantify total-power accuracy, whereas the signed parameter discrepancies quantify agreement with the independent inertia and aerodynamic-drag estimates.

Fig. 10: Power decomposition obtained with $P_{3}$ and $P_{4}$ . Both models separate reversible kinetic-energy rate from aerodynamic and torque-dependent electromechanical dissipation, while $P_{4}$ additionally represents lumped speed-proportional dissipation. Component-specific robust color scales are used to reveal contributions of different magnitudes; quantitative color comparisons must therefore use the corresponding colorbars.

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