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

Antonio Franchi, Aaron Saini, Ahmed Ali, Chiara Gabellieri

arXiv:2610.08177v1cs.ROcs.SYeess.SY

TL;DR

Speed-only aerodynamic models cannot represent acceleration-dependent propulsion power or its physical attribution. This work develops nested models and evaluates them on held-out experiments against independent aeromechanical estimates. The kinetic-energy rate is necessary but insufficient; torque-dependent dissipation provides the main additional improvement, and speed-proportional dissipation yields the best combined prediction and attribution among the models considered.

  • Problem

    Speed-only models cannot represent acceleration-dependent effects, and total-power prediction alone does not establish whether power is attributed to reversible exchange or to aerodynamic and electromechanical losses.

    Source: §I · p. 2, G. Position of the present work · p. 4

  • Method

    The study identifies nested propulsion-power models on one group of dynamic experiments and evaluates them on held-out experiments, with independent inertia and aerodynamic-drag estimates for attribution assessment.

    Source: §I · p. 2

  • Results

    RMSE falls 58.9% from P1 to P2 and a further 53.4% from P2 to P3; P4 gives the lowest errors and the best combined prediction and power attribution among the considered models.

    Source: A. Held-out prediction along the… · p. 9, §VII · p. 11

  • Takeaways & Limitations

    Held-out prediction and independent aeromechanical references assess complementary properties and together provide a physically grounded basis for selecting dynamic propulsion-power models.

    Source: B. Implications and conclusion · p. 12

    The independent inertia and aerodynamic-drag references are point estimates, so parameter discrepancies show relative proximity rather than statistical compatibility.

    Source: A. Scope and limitations · p. 11

Abstract

from arXiv · show

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.

I. Introduction

Existing aerial-power models support design, energy estimation, and planning, but typically predict total power rather than its physical attribution. This work develops and experimentally assesses dynamic propulsion-power models that separate reversible kinetic-energy exchange from aerodynamic and electromechanical losses.

II. Related Work

  • Flight-level, trajectory-based, and design-oriented approaches estimate total power, energy, or performance, while component-level models represent interconnected electrical, mechanical, and aerodynamic subsystems.The cited literature spans vehicle consumption and mission planning as well as propulsion-chain selection, diagnosis, simulation, and control.Source: §I · p. 1, B. Electric propulsion-chain modeling… · p. 2

I. Introduction

  • The proposed models progress from aerodynamic dissipation through reversible kinetic-energy rate and torque-dependent loss to lumped speed-proportional dissipation.Assessment combines held-out dynamic experiments with independent inertia and aerodynamic-drag estimates.Source: §I · p. 2

A. Rotor aerodynamics and speed-only power models

  • With constant aerodynamic coefficients, quadratic thrust and resisting-torque laws yield a cubic speed-only power model that compactly represents aerodynamic dissipation.Such models cannot express reversible kinetic-energy exchange or additional acceleration-associated losses.Source: A. Rotor aerodynamics and speed-only… · p. 2

B. Electric propulsion-chain modeling and design

  • Component-resolved propulsion models predict quantities such as thrust, rotor speed, and current, but this work instead tests compact acceleration-dependent terms and their physical attribution.Prior approaches combine source, controller, motor, and propeller descriptions, including dynamic and physics-informed formulations.Source: B. Electric propulsion-chain modeling… · p. 2

C. Battery, range, endurance, and vehicle-level power

Related work spans vehicle energy estimation, planning, diverse aircraft, and physical interaction, while the paper targets rotor-level attribution of transient electrical power. Its nested models distinguish stored energy from aerodynamic and electromechanical dissipation.

C. Battery, range, endurance, and vehicle-level power

  • Battery-aware and vehicle-level models estimate discharge, degradation, endurance, range, and energy from flight or aircraft characteristics.The reviewed methods include empirical flight-energy models, analytical flight-condition models, and integrated aerodynamic, motor, and battery models.Source: C. Battery, range, endurance, and… · pp. 2–3

D. Energy-aware trajectory, allocation, and mission planning

  • Trajectory, coverage, communication, and actuator-allocation planners use energy models to optimize missions, speeds, trajectories, or redundant actuator choices.The cited work also describes planning under battery, sensing-resolution, return-safety, payload, and obstacle constraints.Source: D. Energy-aware trajectory,… · p. 3

E. Fixed-wing, flapping-wing, and convertible MAVs

  • Fixed-wing, flapping-wing, convertible, and reconfigurable aircraft involve energetic and actuation mechanisms that differ from hovering multirotors.These platforms motivate models that remain meaningful when rotor states change rapidly, even when vehicle motion changes comparatively little.Source: E. Fixed-wing, flapping-wing, and… · p. 3

F. Aerodynamic interaction and physically interactive flight

  • Ground effect, obstacle proximity, rotor interference, body interaction, payload aerodynamics, and wake recirculation can alter rotor thrust, resisting torque, and efficiency.These effects matter in close-to-environment operation and aerial interaction with limited translation.Source: F. Aerodynamic interaction and… · p. 3

G. Position of the present work

  • The reviewed literature commonly predicts thrust, current, total power, range, endurance, or mission energy, whereas this work addresses attribution of source-side power among reversible exchange and dissipation.It uses nested models, held-out dynamic experiments, and independent aeromechanical reference estimates.Source: G. Position of the present work · pp. 3–4

III. Nested Propulsion-Power Models

  • The speed-only model represents nonnegative aerodynamic dissipation but cannot capture acceleration-dependent power; adding the inertia term separates reversible kinetic-energy exchange from aerodynamic loss.The inertial contribution is reversible, while the aerodynamic contribution is nonnegative and irreversible.Source: §III · p. 4
  • Torque-dependent loss lets the acceleration-producing torque contribute to irreversible electromechanical dissipation; a further speed-proportional term can represent losses otherwise partly assigned to reversible power.The speed-proportional coefficient represents lumped loss over the investigated operating domain.Source: §III · pp. 4–5

IV. Model-Adequacy Criteria and Methodology for Dynamic Identification and Validation

The adequacy framework asks whether each added model term is justified for prediction and physical attribution, not merely whether a richer model fits better. It evaluates held-out accuracy alongside independent aeromechanical references.

IV. Model-Adequacy Criteria and Methodology for Dynamic Identification and Validation

  • Because each richer model recovers its predecessor when its added coefficient is zero, minimum fitting error cannot increase under the same data and criterion.The relevant question is whether added complexity is justified for the intended task.Source: §IV · p. 5
  • The hierarchy asks whether speed-only power suffices, whether reversible kinetic-energy rate is enough, and whether torque-dependent or speed-proportional dissipation is also required.It also tests separation of reversible and dissipated power, and of aerodynamic and electromechanical loss where modeled.Source: §IV · p. 5
  • Models are compared on complete experiments excluded from parameter identification, while identified inertia and aerodynamic coefficients are compared with independent estimates.Together these checks assess attribution rather than only accurate prediction of the total power sum.Source: §IV · p. 5

A. Data acquisition methodology

The acquisition method samples dynamically realizable rotor-speed and acceleration conditions, then uses realized motion rather than command references for identification and validation.

1) Realizable operating-domain sampling:

A. Data acquisition methodology

  • Reference trajectories repeatedly ramp commanded frequency up and down across a frequency interval to sample positive and negative acceleration over the relevant speed range.The internal speed controller converts commanded frequency to commanded angular speed.Source: 1) Realizable operating-domain sampling · p. 5

1) Realizable operating-domain sampling:

  • Experiments are organized by nominal frequency-rate magnitude, with each condition comprising repeated ramps in both directions.The nominal rate defines the reference trajectory, while measured rotor frequency and its reconstructed derivative describe realized motion.Source: 1) Realizable operating-domain sampling · p. 6

2) Grouped identification and held-out validation:

Complete experiments are divided between identification and held-out validation, with validation using nominal acceleration rates excluded from fitting. The measured signals are synchronized and processed before model parameters are estimated from identification data.

2) Grouped identification and held-out validation:

  • Every complete experiment, including increasing and decreasing ramps, belongs wholly to identification or validation, and validation uses nominal rates excluded from identification.Parameters are estimated exclusively from the processed identification samples; held-out predictions assess complete experiments at those rates.Source: 2) Grouped identification and held-out… · p. 6

3) Signal synchronization and processing:

  • Voltage, current, and rotor-frequency measurements are synchronized on a common time grid and filtered identically for identification and validation.Zero-phase filtering avoids relative phase shifts among reconstructed power, angular speed, and angular acceleration; transition regions and failed ramp interiors are excluded before fitting.Source: 3) Signal synchronization and processing · p. 6

B. Parameter estimation and prediction metrics

1) Fitting algorithms:

The fitting procedures match each model's parameter structure, and performance is assessed on held-out data using global, experiment-level, and local measures. Relative error is restricted to positive-power samples, while resampling preserves dependence within experiments.

1) Fitting algorithms:

  • Models P1 and P2 use scaled nonnegative linear least squares, whereas P3 and P4 use constrained nonlinear least squares with positive parameterization.The latter models are nonlinear because c_tau and J also enter the squared motor-torque term; external estimates of J and c_tau are not fitting targets.Source: 1) Fitting algorithms · p. 6

2) Prediction metrics:

  • Relative prediction error is evaluated only where measured power is at least P_min, and MAPE does not bound individual prediction errors.This metric averages absolute relative error over the selected positive-power region.Source: 2) Prediction metrics · p. 6

3) Experiment-level statistical and local assessment:

  • Experiment-level bootstrap resampling preserves within-trajectory dependence, while local relative RMSE improvement compares consecutive models in sufficiently populated validation-domain cells.A positive local improvement means the richer model has lower RMSE than its predecessor in that cell.Source: 3) Experiment-level statistical and… · pp. 6–7

C. Independent aeromechanical validation

Independent aeromechanical references test whether fitted models attribute power consistently, beyond merely predicting total electrical power. The joint criterion combines held-out predictive accuracy with discrepancies in inertial and aerodynamic parameters.

C. Independent aeromechanical validation

  • Held-out prediction alone cannot establish correct power attribution because different inertia, aerodynamic-drag, and electrical-loss values may compensate while yielding similar total power.The study therefore obtains independent references for J and c_tau from sources other than the electrical-power measurements used for identification.Source: C. Independent aeromechanical validation · p. 7

1) Independent inertia determination:

  • The independent inertia estimate combines propeller inertia from segmented mass and radius with motor-rotor and other rotating-component inertias.This estimate is obtained independently of the electrical-power data used to identify the models.Source: 1) Independent inertia determination · p. 7

2) Independent aerodynamic-drag determination:

  • An independent torque measurement identifies the quasi-steady aerodynamic resisting-torque relation, providing an external reference for each model's aerodynamic contribution.Signed parameter discrepancies indicate overestimation when positive and underestimation when negative relative to the independent value.Source: 2) Independent aerodynamic-drag… · p. 7, 3) Joint validation criterion · p. 7

3) Joint validation criterion:

  • Prediction metrics assess total-power accuracy, while δJ and δc_tau assess inertial and aerodynamic attribution against independent estimates; neither criterion replaces the other.The criteria are complementary because accurate total-power prediction does not imply physically accurate attribution.Source: 3) Joint validation criterion · p. 7

V. Experimental Realization and Dataset

The experimental realization applies the methodology to a specified motor-propeller setup and a set of dynamic ramp experiments. Signal processing and quality control define which ramp interiors enter identification and validation.

A. Propulsion unit and data acquisition

  • The tested propulsion unit comprised an Xnova Lightning 2208 V2 1500KV motor, a Graupner 3D-Prop 8x4.5 propeller, and an Aikon AKC 70A 8S electronic speed controller.Source: 1) Hardware and instrumentation · p. 7

1) Hardware and instrumentation:

  • The commanded rotor-frequency interval was [35, 220] Hz, with the lower bound avoiding reduced sensorless-speed accuracy and the upper bound avoiding current saturation.Identification and validation rates were interleaved, with extreme rates included in identification to assess predictions at unseen rates within the sampled domain.Source: 2) Trajectory family and dataset · p. 7

2) Trajectory family and dataset:

  • Thirteen complete experiments were used for identification and thirteen distinct experiments for validation; held-out nominal rates included 400, 500, 600, 800, 1000, and 1500 Hz/s.After exclusions and quality control, 26 experiments yielded 278,627 retained samples, including 140,508 validation samples.Source: 2) Trajectory family and dataset · p. 8
  • Third-order noncausal zero-phase Savitzky–Golay filters processed rotor frequency, voltage, and current, with a 25-Hz margin around each commanded reversal excluded.The rotor-frequency filter used 101 samples and the voltage and current filters used 31-sample windows.Source: 3) Signal processing and quality control · p. 8

3) Signal processing and quality control:

  • Quality control rejected complete ramp interiors when measured-versus-filtered rotor frequency differed by more than 10 Hz or the realized-rate criterion failed.The speed-controller gains remained fixed throughout the campaign.Source: 3) Signal processing and quality control · p. 8

B. Independent references and validation settings

Independent mechanical references and pre-fit ramp screening anchor the held-out evaluation of the nested models.

1) Rotational-inertia reference:

  • The rotating assembly’s inertia was independently estimated by measuring segmented propeller blades and characterizing the motor rotor and shaft.Blade segments accounted for nonuniform radial mass distribution; the rotor and shaft were approximated as cylinders.Source: 1) Rotational-inertia reference · p. 8, Figure 4 · p. 9

B. Independent references and validation settings

  • Before fitting, transition regions around commanded frequency reversals were excluded, and complete ramp interiors were retained or rejected using motion and processing-consistency criteria.Source: Figure 3 · p. 8

2) Aerodynamic-drag reference and validation threshold:

A. Held-out prediction along the nested hierarchy

The nested models improve held-out prediction as they add dynamic power contributions, while independent references show how those additions affect physical attribution.

C. Identified coefficients and aeromechanical parameter recovery

  • P3 closely recovers c_tau but retains an inertia discrepancy, whereas P4 gives the closest joint recovery of the independent estimates.The complete parameter array is re-estimated at each hierarchy level; P1 cannot estimate J, and P2 closely recovers neither reference.Source: Table IV · p. 9, C. Identified coefficients and… · p. 9

A. Held-out prediction along the nested hierarchy

  • Negative-acceleration regions show less residual dispersion than corresponding positive-acceleration regions, where measured power varies more broadly with acceleration.Residual color magnitudes use separate panel-specific scales.Source: A. Held-out prediction along the… · p. 9, Figure 7 · p. 10

B. Incremental improvement over the operating domain

  • Across the accelerated operating domain, P1-to-P2 gives the dominant correction, P2-to-P3 improves broadly where torque is large, and P3-to-P4 adds a smaller localized refinement.Positive local values indicate lower RMSE for the richer model; uncolored cells have insufficient validation samples.Source: B. Incremental improvement over the… · p. 9

A. Held-out prediction along the nested hierarchy

D. Decomposition of reversible and dissipative power

  • P3 and P4 separate reversible kinetic-energy rate from irreversible dissipation, while P4 additionally assigns speed-proportional loss to q1|v|.That contribution is smaller than the principal aerodynamic and torque-dependent terms, yet its inclusion moves both J and c_tau closer to independent estimates.Source: D. Decomposition of reversible and… · p. 10

VII. Discussion and Conclusions

Across the nested hierarchy, the kinetic-energy rate is necessary but insufficient, torque-dependent dissipation provides the main further predictive improvement, and P4 gives the best combined prediction and power attribution. These conclusions apply to interpolation within the tested operating domain and motor-drive–propeller unit.

VII. Discussion and Conclusions

  • P4 gives the best combined prediction and power attribution, while P3 already adds substantial predictive improvement but retains an inertia discrepancy.P3 closely recovers the aerodynamic resisting-torque coefficient; P4 gives the closest joint recovery of independent aeromechanical point estimates.Source: §VII · pp. 10–11

A. Scope and limitations

  • The experiments span rotor frequencies of 60.28–195.05 Hz and frequency rates of −1489.83–2093.45 Hz/s, so conclusions concern interpolation within this domain and the tested unit.The results are specific to the motor-drive–propeller unit considered.Source: A. Scope and limitations · p. 11
  • Identified parameter discrepancies show proximity to independent point estimates, not statistical compatibility.The reference inertia and aerodynamic coefficient are point estimates from segmented inertia determination and force–torque measurements.Source: A. Scope and limitations · p. 11
  • Residuals indicate the four modeled contributions do not capture all power variations, and the lumped k_tl and q_1 coefficients are not isolated microscopic loss parameters.Possible omitted effects include braking and DC-bus energy return, temperature dependence, controller switching and conduction losses, iron losses, voltage dependence, and synchronization or differentiation errors.Source: A. Scope and limitations · pp. 11–12

B. Implications and conclusion

  • P3 and P4 remain compact for energy-aware trajectory generation, control allocation, and propulsion analysis while separating temporary rotor-energy storage from aerodynamic and electromechanical losses.This distinction supports transient-energy estimation, efficiency assessment, and interpretation of identified propulsion parameters.Source: B. Implications and conclusion · p. 12
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