Enhancing Multirotor Drone Efficiency: Exploring Minimum Energy Consumption Rate of Forward Flight under Varying Payload

Ayush Patnaik; Nicolas Michel; Xinfan Lin

REVIEW 3 major objections 3 minor 21 references

Enhancing Multirotor Drone Efficiency: Exploring Minimum Energy Consumption Rate of Forward Flight under Varying Payload

T0 review · 3 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A mass-independent constant governs minimum energy per meter in forward flight.

desk verdict The scaling theorem is sound for mechanical power, but the paper overreaches when it applies it to battery energy and range. read the letter →

arxiv 2501.03102 v1 pith:M2KZDB6W submitted 2025-01-06 cs.RO cs.SYeess.SY

Ayush Patnaik , Nicolas Michel , Xinfan Lin This is my paper · ORCID

classification cs.ROcs.SYeess.SY

keywords multirotorUAVenergyefficiencypermeterpayloadoptimalforwardvelocitypitchangledronedeliveryscalinglaws

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that for a multirotor flying at its energy-optimal forward speed, the minimum energy needed to cover one meter, divided by total vehicle mass, is a constant that does not depend on mass or payload. It derives this from a physics-based model of propeller aerodynamics and rigid-body flight, proving that the optimal pitch angle is mass-independent, that the optimal speed grows as the square root of mass, and that minimum energy per meter grows linearly with mass. This matters because it turns a multidimensional energy-optimization problem into a single constant: measure or compute that constant once, and you can predict best speed, range, and energy requirement for any payload.

What carries the argument

The load-bearing identity is the factorization $\mathrm{EPM} = m \cdot Q'(\Theta)\,\omega'(\Theta)/V'_x(\Theta)$, obtained by expressing horizontal velocity, induced velocity, propeller angular velocity, and torque each as a power of mass times a function of pitch angle alone. The pitch angle $\Theta$ then becomes the sole decision variable, so minimizing EPM with respect to $\Theta$ yields a mass-independent optimum $\Theta^*$ and a single constant $C$. The one-to-one mapping between $\Theta$ and $V_x$ in steady horizontal flight is what lets the paper phrase the result either as "constant $\mathrm{EPM}^*/m$" or as "optimal speed proportional to $\sqrt{m}$."

What would settle it

Measure the energy per meter of a fixed airframe at several payload masses, each flown at its measured energy-optimal horizontal speed; if $\mathrm{EPM}^*/m$ is not the same across masses, the constant relationship fails. A cheaper check is to measure the body drag coefficient with and without a large payload and see whether it stays constant.

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Extended reading notes

Core claim

The central discovery is that at the velocity minimizing energy per meter, the ratio $\mathrm{EPM}^*/m$ equals a constant $C = Q'(\Theta^*)\,\omega'(\Theta^*)/V'_x(\Theta^*)$, where the three factors depend only on the optimal pitch angle $\Theta^*$ and aerodynamic constants, not on mass. The argument works by scaling: steady-level force balance gives $V_x = \sqrt{m}\,V'_x(\Theta)$, the induced-velocity quartic yields $v_i = \sqrt{m}\,v'_i(\Theta)$, the thrust equation gives $\omega = \sqrt{m}\,\omega'(\Theta)$, and torque becomes $Q = m\,Q'(\Theta)$. Substituting into $\mathrm{EPM} = Q\omega/V_x$ leaves a mass factor times a pure function of $\Theta$, and since the optimum of that function is mass-independent, the optimal pitch angle is fixed and the minimal EPM per mass is a constant.

Load-bearing premise

The derivation assumes the body drag coefficient stays the same when payload changes, and it treats energy consumption as the mechanical propeller power, so electrical losses in the motors, speed controllers, and battery must either be negligible or scale in exactly the same way for the constant to hold for real battery-powered drones.

Editorial extensions

If this is right

Optimal cruise speed for a drone carrying payload $m$ is predicted to be $\sqrt{m}$ times the optimal speed of the empty vehicle, so heavier loads should be flown faster, not slower.
The minimum energy needed to fly a fixed distance is $C m L$, giving a ready formula for battery sizing and range estimation.
Delivery-route planning can minimize energy by pairing heavier payloads with shorter route segments, reducing to a linear integer program or a mass-weighted traveling salesman problem.
The energy-optimal pitch angle can be precomputed once and used for all payloads, simplifying flight-control laws for energy-efficient cruising.
The claimed constant is only valid at the optimal speed; at any arbitrary velocity, energy per meter per mass still varies with mass, so the linear mass-dependence formulas do not apply there.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, if $C$ is truly payload-independent, then a drone's range under a fixed battery is inversely proportional to total mass, which would suggest a simple dispatch rule: carry a given payload on the lightest available airframe.
Beyond the paper, the same mass-factorization structure might survive in gentle climb or descent, where the steady-state force balance changes but the scaling of $V_x$, $\omega$, and $Q$ with mass could still hold; the paper does not test this.
Beyond the paper, a testable extension is to check whether $C$ remains mass-independent when the propeller size or blade count changes, since those enter the aerodynamic constants and should change the value of $C$ without destroying its mass-independence.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Punchline: the core scaling theorem is real and the paper is right that the invariant is pitch angle, not velocity. But the paper then uses that theorem to make claims about battery energy and range, and the theorem only holds for mechanical shaft power. That is a serious gap, not a nicety.

What's new: the mass-invariance of the optimal pitch angle and the constancy of EPM*/m are not in the cited literature, and the authors correctly point out that [7] and [18] fix velocity rather than pitch angle. The proof itself is structurally sound: from force balance and the homogeneous BEM equations, V_x, v_i, and omega all scale as sqrt(m), Q scales as m, so EPM/m depends only on pitch angle. The constant C is derived, not fitted, and the one-to-one mapping between V_x and theta is handled properly.

The soft spots are proportionate: (1) Eq. (8) defines power as sum Q_j omega_j, mechanical power only. Real battery-powered drones have motor copper loss ~ I^2 ~ Q^2 ~ m^2, and ESC/battery resistive losses scale similarly. Those add sqrt(m) and m terms to EPM/m, making theta* mass-dependent. Section V then applies Eq. (30) to battery capacity and range in Eqs. (33)-(34). That step is unjustified unless the result is explicitly restricted to lossless actuators. (2) C_BD is assumed constant under payload; if payload changes the drone's drag area, V_x ~ sqrt(m) breaks. (3) Validation is a simulation using the same model, so it is a consistency check, not independent evidence. The quartic and quadratic solutions in Eqs. (20), (22) and the appendix also have typos, but those are fixable.

Who gets value: readers working on multirotor energy-optimal control and drone-delivery routing who want a closed-form payload scaling rule. As a theorem about a mechanical-power model, it holds. As a practical battery-range formula, it needs the loss terms or an explicit scope restriction.

Recommendation: send it to a serious referee. The scaling insight deserves publication, but the authors need either to restrict all claims to mechanical power or to redo the scaling with motor/ESC/battery losses, and experimental validation is essential before the range formula is used.

Referee Report

3 major / 3 minor

Summary. The paper studies steady-state forward flight of a multirotor UAV and argues that there is a mass-invariant energy-efficiency optimum. Using blade-element/momentum-theory propeller equations plus rigid-body force balance, the authors show that at fixed pitch angle the horizontal velocity, induced velocity, and rotor angular velocity all scale as sqrt(m), while torque scales as m; consequently EPM = Q*omega/V_x scales as m times a function of Theta only. Minimizing EPM over Theta then yields a mass-independent optimal pitch angle, so EPM*/m is a constant C. Numerical simulations with the authors' prior physics-based model reproduce the claimed constant minimum, and Section V applies the result to battery sizing, range prediction, and payload routing.

Significance. If restricted to mechanical propeller shaft power and to a payload-independent body drag coefficient, the central derivation is a genuine analytical result: it identifies a dimensionless invariant (EPM*/m), derives the constant explicitly in Eq. (31) with no fitted parameters, and predicts V_x* proportional to sqrt(m). The mass-invariant optimal pitch angle is a concrete, falsifiable prediction that could be tested experimentally. However, the paper's practical headline is about battery-powered drones and onboard energy capacity, and that claim is not established because the minimized quantity in Eq. (8) is mechanical power only, not the electrical power drawn from the battery. The numerical validation also uses the same model that produced the derivation, so it does not independently confirm the load-bearing simplifications. The theoretical result is a useful contribution if clearly scoped; the current manuscript overstates its applicability.

major comments (3)

The power minimized in the derivation is mechanical shaft power P = sum(Q_j omega_j), yet Section V applies the result to battery capacity and maximum range. At fixed Theta, the derivation gives Q ~ m, omega ~ sqrt(m), and V_x ~ sqrt(m), so P_mech ~ m^{3/2} and EPM_mech ~ m. In a real electric drive, motor current is approximately proportional to torque Q, so motor copper loss scales as Q^2 ~ m^2; ESC and battery resistive losses also scale as m^2. The battery-power EPM is therefore (P_mech + P_loss)/V_x ~ m + m^{3/2}, so EPM_batt/m = f(Theta) + sqrt(m) g(Theta), and the minimizer Theta* becomes mass-dependent. The constancy theorem in Eq. (30) does not carry over to the energy actually drawn from the battery. The authors must either restrict all claims to lossless actuators or extend the scaling analysis to the full electro-mechanical model advertised in Section II; otherwise Eqs. (33) and (34) are not supported.
The derivation assumes C_BD is constant regardless of payload. If a payload changes the vehicle's external drag area or orientation, C_BD depends on m, and Eq. (12) becomes V_x = sqrt(m g tanTheta / C_BD(m)), which is not proportional to sqrt(m). The subsequent scaling of v_i, omega, and Q, and the mass-invariance of Theta*, all rely on the exact sqrt(m) form. This assumption is acknowledged in Section III, but it is load-bearing for the headline result, not a minor modeling detail. The paper should state the practical range of validity and ideally test sensitivity to C_BD variation with payload.
The 'validation studies' in Section III are simulations using the same physics-based model from [10] that underlies the analytical derivation. Such numerical experiments confirm the internal consistency of the model and the algebra, but they do not independently validate the two key assumptions: constant C_BD with payload and mechanical-power-only consumption. Without experimental data, the paper cannot claim empirical validation of the mass-invariance result for real battery-powered drones.

minor comments (3)

The notation for Q is ambiguous: Eq. (8) sums over propellers, but Eq. (10) writes EPM = Q omega / V_x without an explicit N_p or a definition of Q as total torque. Please clarify whether Q is per-propeller or total torque.
The quartic solution in the appendix is very hard to follow because several variables (S_1, S_2, S'_2, K_2) are introduced without clear definitions in the main text and the typesetting degrades in places. A concise derivation or a reference to a standard quartic solver would help the reader verify the claimed sqrt(m) scaling of v_i.
The comparison with prior models in Section V is useful, but the equations are introduced without defining the parameters r(V_x), eta, and zeta in the text immediately around them. Please define all symbols or point to the cited sources more explicitly.

Circularity Check

1 steps flagged · score 2.0 of 10

Constant-EPM/m theorem is internally derived, not fitted, but the numerical 'validation' reuses the same prior model; validation circularity only, so score is low.

other [Abstract and Section III (Model-Based Energy Efficiency Simulations); referenced again in Section V]
"The study is based on a previously developed model capturing the first-principle energy dynamics of the multirotor. ... In this section, we present several important relationships regarding multirotor energy efficiency, velocity, and mass obtained from simulation studies using the previously introduced physics-based model."

The numerical results in Figs. 5-7 are generated by the same 'previously introduced physics-based model' whose equations (Section II) are the premises of the analytical derivation in Section IV. The analytical result EPM*/m = C (Eq. 30) is derived from exactly those model equations, so a numerical simulation using the same model cannot independently confirm the constancy; it only verifies algebraic consistency. No parameter is fitted and the derivation is internally self-contained, so this is validation circularity rather than derivation circularity.

full rationale

The analytical chain is self-contained: given the stated assumptions (constant C_BD, steady horizontal flight, P = sum Q_j omega_j mechanical shaft power, and the BEMT equations), Eq. (12) gives V_x ~ sqrt(m), Eq. (19) gives v_i ~ sqrt(m), Eq. (23) gives omega ~ sqrt(m), and Eq. (25) gives Q ~ m. Substituting into Eq. (10) yields EPM/m = Q'(Theta) omega'(Theta)/V'_x(Theta), a function of Theta only. The first-order condition (27)-(28) therefore has a mass-independent minimizer Theta*, giving EPM* = m C. This is a theorem, not a fit or a restatement of the conclusion; Eq. (31) merely names the derived ratio C. The stated assumption C_BD = const and the omission of motor/ESC/battery losses are modeling limitations, not circular reductions. The only circularity-adjacent feature is that the simulation 'validation' uses the authors' own previously developed model [10], so it confirms internal consistency rather than providing independent empirical support. Hence score 2 rather than 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No new free parameters are fit in this paper; the constant C is derived from model coefficients rather than fitted. The proof assumes the BEM model from the authors' prior work, a constant body drag coefficient, and mechanical power as total consumption. The axioms listed above are the load-bearing premises that, if false, would invalidate the real-drone version of the central claim.

assumptions (5)

domain assumption The blade element momentum model in Eqs. 3-7, with small-angle and u_pr << u_pl approximations, accurately represents propeller thrust and torque.
Used to derive T, Q, and v_i; the approximations are validated only in the authors' prior [21], not independently in this paper.
domain assumption Body drag coefficient C_BD is constant across payload and at the optimal pitch angle.
Explicitly assumed in Section III; the V_x ~ sqrt(m) scaling in Eq. 12 depends on this.
domain assumption Total energy consumption equals mechanical propeller power P = sum(Q_j omega_j) in Eq. 8.
The derivation ignores motor, ESC, and battery electrical losses; the abstract references full energy dynamics, but the proof uses only mechanical power.
domain assumption Steady-state horizontal flight with V_z = 0, angular velocity zero, and force balance given by Eq. 11.
Needed to write C_BD V_x^2 = T sin(Theta) and T = mg / cos(Theta), which establish the mass scaling of V_x.
standard math The closed-form quartic solution for v_i in Eq. 20 and the quadratic solution for omega in Eq. 22 are algebraically correct.
The paper says the solution comes after 'a lengthy process' and the appendix contains apparent typos; the scaling result does not depend on the exact closed forms, only on the sqrt(m) scaling.

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Pith. "Pith review of Enhancing Multirotor Drone Efficiency: Exploring Minimum Energy Consumption Rate of Forward Flight under Varying Payload." pith.science (2026). https://pith.science/paper/M2KZDB6W

@misc{pith2026250103102,
  author       = {Pith},
  title        = {Pith review of: Enhancing Multirotor Drone Efficiency: Exploring Minimum Energy Consumption Rate of Forward Flight under Varying Payload},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M2KZDB6W}},
  note         = {Machine review of arXiv:2501.03102}
}

read the original abstract

Multirotor unmanned aerial vehicle is a prevailing type of aircraft with wide real-world applications. Energy efficiency is a critical aspect of its performance, determining the range and duration of the missions that can be performed. In this study, we show both analytically and numerically that the optimum of a key energy efficiency index in forward flight, namely energy per meter traveled per unit mass, is a constant under different vehicle mass (including payload). Note that this relationship is only true under the optimal forward velocity that minimizes the energy consumption (under different mass), but not under arbitrary velocity. The study is based on a previously developed model capturing the first-principle energy dynamics of the multirotor, and a key step is to prove that the pitch angle under optimal velocity is a constant. By employing both analytical derivation and validation studies, the research provides critical insights into the optimization of multirotor energy efficiency, and facilitate the development of flight control strategies to extend mission duration and range.

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Reference graph

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Reviewed August 10, 2026 · model on record in the stance chip above.

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Multi-visit drone routing problem,

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arXiv 2020