A Generalized Thrust Estimation and Control Approach for Multirotors Micro Aerial Vehicles

Davi Santos; Martin Saska; Tiago Nascimento

REVIEW 4 major objections 5 minor 23 references

A Generalized Thrust Estimation and Control Approach for Multirotors Micro Aerial Vehicles

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims that a BEMT-based closed-loop thrust estimator, calibrated by a single static bench-test scaling per rotor, can replace the standard quadratic thrust-to-speed map and improve tracking robustness under wind on small…

desk verdict A practical single-scaling BEMT thrust controller with real two-platform flight tests; the thrust-tracking metric is circular, but the trajectory data and open-source implementation make it worth engaging. read the letter →

arxiv 2412.02874 v1 pith:YNQHTIBA submitted 2024-12-03 cs.RO

Davi Santos , Martin Saska , Tiago Nascimento This is my paper · ORCID

classification cs.RO

keywords thrustestimationcontrolmultirotorUAVquadrotorbladeelementmomentumtheoryfeedforwardPIDwinddisturbancestaticbenchcalibration

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

Multirotor autopilots usually convert rotor speed to thrust with a quadratic curve fitted on a test bench, a map that degrades in forward flight and wind. This paper argues the right place to fix that is at the rotor level, and proposes replacing the open-loop map with a closed-loop thrust estimator and controller built on blade element momentum theory (BEMT). The estimator reads each ESC's voltage, current, and speed, solves the rotor's inflow equations online, and needs only a single scaling constant per rotor from a simple static bench test. In 32 outdoor flights per platform across two very different quadrotors, the closed-loop thrust control produced lower and more consistent tracking errors in position, velocity, acceleration, and thrust than the quadratic map under wind. The payoff, if the result holds, is a low-calibration upgrade path for small multirotors that improves robustness to aerodynamic disturbances without per-platform aerodynamic identification.

What carries the argument

The carrying object is the power-balance equation of each rotor: the mechanical power from the motor, $P_m = K_q i_a \omega$, minus the shaft friction and inertia power $P_r = I_r \omega \dot{\omega}$, gives the aerodynamic power $P_{am}$, whose coefficient $C_{Pam} = P_{am}/\omega^3$ feeds an iterative Newton–Secant solver. The solver recovers the stream inflow ratio $\lambda_s$ by matching $C_{Pam}$ computed from ESC telemetry to the BEMT expression, then obtains the induced inflow $\lambda_i$ from a quadratic relation, the thrust coefficient $C_T = c_1(c_2 - \lambda)$, and hence thrust $T = C_T \omega^2$. The generalization trick is that the $c$ and $d$ coefficients are taken from one published rotor dataset as a baseline, and a static bench test establishes that the estimate is a linear scaling of true thrust (correlation above 0.99), so one scalar per rotor calibrates the whole estimator. The control side is a feedforward PID: $u = K_{ff}T_{sp} + K_p e_T + K_i \int e_T + K_d \frac{de_T}{dt}$, with $e_T = T_{sp} - \hat{T}$, running at 500 Hz.

What would settle it

Mount a thrust load cell on one rotor arm and fly the same outdoor waypoint path in wind; if the BEMT estimate calibrated by the static scaling diverges from the measured thrust as airspeed grows past the small-advance-ratio regime, the central robustness claim would not hold.

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Extended reading notes

Core claim

The paper's central claim is that replacing the standard static quadratic thrust-to-speed mapping with a BEMT-based closed-loop thrust estimator and feedforward PID controller improves robustness under aerodynamically varying flight conditions. The key move is to show that the BEMT model's seven aerodynamic coefficients do not need to be re-identified for each platform: reusing one published set of baseline coefficients and fitting a single scale factor from one static thrust experiment yields thrust estimates with Pearson correlations above 0.99 to ground truth for two different motor–propeller combinations. The estimated thrust is fed back at 500 Hz into a per-rotor PID with feedforward, which tracks the thrust setpoint instead of merely commanding a motor speed. Across 32 outdoor flights per platform, the thrust control gives lower mean, median, and standard deviation of RMSE for acceleration, velocity, position, and thrust than the quadratic map, with the only exceptions in some x/y acceleration axes for the smaller 250 mm platform. The result is stated as better robustness to wind, not as a general optimality claim.

Load-bearing premise

The single per-rotor scaling constant fitted on a static bench is assumed to stay valid in dynamic flight, including forward speeds up to 12.7 m/s and gusts, and the derivation also assumes small advance ratio and small horizontal speed to drop the horizontal-force terms.

Editorial extensions

If this is right

Standard autopilots can swap the quadratic thrust-to-speed map for this estimator using only propeller mass and radius plus one static bench test per rotor, removing the need for per-platform aerodynamic identification.
Tracking errors under lateral wind decrease for both a 250 mm and a 500 mm quadrotor, and the variance across runs also drops, meaning more consistent flight in gusts.
The estimator averages about 25.23 \mu s per call and the controller about 5.29 \mu s, so the 500 Hz closed loop runs in real time on a Pixhawk 6C.
The more aggressive rotor commands raise battery drain by 15.30% for the 250 mm platform and 20.60% for the 500 mm platform, so the benefit is a deliberate trade: precision in wind costs endurance.
On two platforms of very different size and thrust-to-weight ratio, the same calibration procedure applies unchanged, supporting the generalization claim.

Reading between the lines

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

If the single static scaling is carried into aggressive forward flight, the small-advance-ratio assumption that decouples horizontal force will eventually break; an instrumented rotor with an in-flight load cell could map where the scaling drifts.
The paper's own future direction — estimating the advance ratio and horizontal force directly from IMU acceleration — would remove the main modeling assumption while keeping the one-scalar calibration, and is a natural next experiment.
The measured power increase suggests the controller gains favor tracking accuracy; scheduling gains by flight phase or wind level could recover some endurance without losing the wind benefit.
The portability claim is strongest for rotors in the tested 5-inch and 13-inch range; extending to other sizes, blade counts, or coaxial layouts would test how far the single-scaling generalization reaches.

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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

Short version: this is a genuine practical contribution—reusing Bangura–Mahony's BEMT estimator with one static bench scaling per rotor, plus a feedforward PID thrust loop in PX4, validated on two quadrotors outdoors. If the single-scaling generalization holds, it removes a real calibration barrier. The implementation is open source and the computational load numbers (25 µs per estimate at 500 Hz) are useful.

The static bench results show Pearson correlations of 0.997/0.995 between estimated and measured thrust across throttle, which supports the claim that the baseline coefficients give a linearly scaled estimate for these rotors. That is the strongest evidence in the paper. The outdoor trials are real: 16 per method per platform, alternating batteries, wind perpendicular to the path.

Where it gets soft: the thrust-RMSE comparison is circular. The controller minimizes error against the same estimator that produces the reported thrust output, so lower estimated-thrust RMSE is guaranteed by construction, not proof of actual thrust tracking. There is no in-flight load cell or independent thrust measurement. The single scaling constant is fitted at zero airspeed, then extrapolated to flights at 11–12.7 m/s with 3–5 m/s wind, while the model's small-advance-ratio and µ²≈0 assumptions are invoked. A single multiplicative constant can correct a static scaling error, but it is not obvious it corrects dynamic errors in inflow states. The trajectory-level RMSE improvements (position/velocity) are non-circular, but the paper reports no significance tests; the 250mm acceleration x/y axes actually favor the baseline, which the authors explain but does not strengthen the case. Power consumption rises 15–20%, which the paper discloses.

Overall: the central claim—better robustness than the quadratic map—is plausible but not settled. The paper is a solid engineering report, not a definitive aerodynamic validation.

Who it's for: someone building a multirotor controller who wants a drop-in thrust loop with minimal calibration, or someone working on rotor-level control allocation. It deserves a serious referee: the question of whether one static scaling suffices in dynamic flight is worth a careful experimental test, and the authors have shipped the code and data structure to enable it.

Referee Report

4 major / 5 minor

Summary. The paper proposes replacing the standard quadratic rotor thrust-to-speed map with a BEMT-based thrust estimator and a feedforward PID thrust controller for multirotor UAVs. The estimator reuses aerodynamic coefficients from the literature as a baseline and applies a per-rotor scaling factor obtained from static bench tests, avoiding per-platform aerodynamic identification. The method is implemented in PX4 and evaluated outdoors on two quadrotors (250 mm and 500 mm) by comparing trajectory and thrust RMSE against the quadratic model under wind. The authors claim that the proposed thrust estimation and control improves robustness under aerodynamically varying flight conditions.

Significance. If the claims hold, the practical contribution is substantial: a low-calibration, cross-platform thrust estimation and control scheme that runs in real time at 500 Hz, with an open-source PX4 implementation and experiments on two different rotor sizes. The paper also reports useful computational-load measurements. However, the significance is conditional on resolving serious validation issues: the thrust-tracking evidence is partly circular, the static scaling is extrapolated to flight conditions that violate the model assumptions, and the trajectory-level statistical claims lack significance testing.

major comments (4)

[Section V, Eq. (14), Fig. 10] The thrust RMSE comparison is circular. The control law in Eq. (14) feeds back the estimated thrust via e_T = T_sp - \hat T, so the controller is explicitly designed to drive the estimated thrust to the setpoint. The thrust RMSE reported in Figs. 10a and 10b therefore measures how well the controller tracks its own feedback signal, not how accurately the actual rotor thrust follows the setpoint. No independent in-flight thrust measurement (e.g., load cell, accelerometer-derived total thrust, or wind-tunnel validation) is provided. This metric cannot support the central claim that real thrust is controlled more accurately.
[Section IV-A, Table III and Section V] The static bench scaling is extrapolated to flight regimes that violate the model assumptions. The estimator assumes a small advance ratio and sets mu^2 approximately to zero in Section IV to decouple the horizontal force, yet the reported maximum flight speeds are 11.3 m/s for the 250 mm platform and 12.7 m/s for the 500 mm platform, with wind speeds of 3.05 m/s and 4.8 m/s respectively. A single multiplicative constant fitted at zero airspeed cannot correct for dynamic errors in lambda_i, lambda_s, or kappa if the baseline aerodynamic coefficients from Table I, identified for a different motor and propeller, do not transfer to the two test rotors. The paper provides no in-flight thrust ground truth to validate this extrapolation, so the estimator's dynamic accuracy is unestablished.
[Section V, Figs. 8-9] The claimed statistical improvement over the quadratic model is not supported by significance tests. The paper reports only means, medians, and standard deviations of RMSE distributions over 16 trials per method per platform. Moreover, for the 250 mm platform the mean and median acceleration RMSE in the x and y axes are slightly better with the thrust-to-speed map than with thrust control, which is acknowledged in the text but nonetheless contradicts the blanket statement that thrust control provides better robustness. Without p-values, confidence intervals, or effect-size statistics, the trajectory-level evidence is suggestive rather than conclusive.
[Section IV-A, Table III] The claim that the thrust estimate is 'simply a scaled version' of the ground-truth thrust is not consistent with Table III. The ratio of estimated to measured thrust varies with throttle: for the 250 mm rotor it ranges from about 12.3 at 10% throttle to about 8.9 at 100% throttle, and for the 500 mm rotor from about 0.75 to 0.59. A high Pearson correlation is compatible with a non-constant affine or nonlinear relation, so the single-scaling-value interpretation is not uniquely supported. In addition, the implementation described in Section IV-A uses the maximum estimated thrust during flight for normalization, whereas the abstract and introduction state that a single scaling value from the bench test is used; the relationship between these two calibration procedures should be clarified.

minor comments (5)

[Section V] The text refers to the standard thrust-to-speed map as 'Equation (5a)', but Eq. (5a) defines the aerodynamic power coefficient C_Pam; the quadratic thrust model is Eq. (1). Please correct the citation.
[Algorithm 1] The secant update in line 13 is missing a division sign in the typeset equation; the formula should read lambda_s^{k+1} = lambda_s^k - f(lambda_s^k) * (lambda_s^k - lambda_s^{k-1}) / (f(lambda_s^k) - f(lambda_s^{k-1})).
[Table III] Please state explicitly that the 'Throttle' column is in percent and specify whether the estimated thrust values are raw estimates from the baseline coefficients or already scaled values.
[Section IV-A] The calibration description is inconsistent: the introduction and Section IV-A mention a test-bench experiment producing a scaling value, while Section IV-A later states that the user need only perform an in-flight calibration by inputting maximum throttle and storing maximum estimated thrust. Please unify the description of the calibration procedure.
[Fig. 7] The caption uses 'f-PID control' without defining the term; please spell out 'feedforward PID' in the caption or in the text.

Circularity Check

2 steps flagged · score 6.0 of 10

The thrust-level evidence is circular: the static bench agreement is the calibration, and the thrust RMSE compares the setpoint with the estimator that the control law is designed to null; only the trajectory-level comparisons are independent, and they are weakly supported.

fitted input called prediction [Abstract; Section IV-A, Table III and following paragraph]
"This experiment shows that, with the coefficients presented in Table I, the method overestimates the thrust for the 250mm and underestimates the thrust for the 500mm, and in both cases, the Pearson correlation between the estimated thrust and the ground truth are 0.997 and 0.995 respectively."

The abstract promises to 'fit the thrust estimate to values closest to the real ones with a simple test bench experiment, resulting in a single scaling value,' and Section IV-A performs that fit with the static bench data in Table III, then presents the resulting linear relationship as an indication that the estimate is 'simply a scaled version' of measured thrust. The high correlation and the scaled relationship are the calibration outcome, not an independent validation of the estimator. Because this same scaled estimate is later used as flight feedback, the static agreement cannot support claims about dynamic thrust accuracy under wind.
self definitional [Section IV-B, Eq. (14); Section V, Fig. 7 and Fig. 10]
"This controller is an augmented PID with feedforward of the setpoint to increase convergence speed. The control law can then be formulated as u = KffTsp + KpeT + Ki ∫ eT + Kd deT/dt, (14) ... eT = Tsp − T̂ . As shown in Fig. 7, the thrust control acts so that the thrust estimate closely follows the thrust setpoint."

The control law (14) is designed to drive eT = Tsp − T̂ to zero, and the thrust RMSE results in Fig. 10 compare exactly the same Tsp with the same T̂ produced by the estimator. Thus, the reported thrust-tracking improvement is a direct consequence of closing the loop around the estimator's own output, not an independent measurement of real rotor thrust. Since no in-flight load cell or other thrust ground truth is provided, the thrust-level robustness evidence reduces to a self-consistency check of the estimator/controller pair rather than a validation of actual thrust tracking.

full rationale

The BEMT estimator derivation itself is not circular: equations (11a)-(11c) and (12) form a standard underdetermined system closed by the measured aerodynamic power, and the Newton-Secant iteration is an internal numerical solve. However, the paper's central robustness evidence splits into two parts. The static-thrust agreement in Table III is not an independent check; it is the calibration of the promised single scaling value, since the raw BEMT output with Table I coefficients is declared to be 'simply a scaled version' of the measured thrust after the bench experiment. The thrust-tracking comparison is circular by construction: Eq. (14) feeds back eT = Tsp − T̂, and Fig. 10 measures RMSE between the same Tsp and the same estimator output T̂, so lower estimated-thrust RMSE is a built-in consequence of the controller, not evidence about actual thrust. No in-flight thrust ground truth is reported. The position, velocity, and acceleration RMSE comparisons in Figs. 8-9 are not circular because they use independent navigation state estimates, but they are presented without significance tests and under alternating battery and wind conditions, so they cannot single-handedly support the strong robustness claim. The self-citations in the reference list are not load-bearing. Overall, part of the central claim reduces to a fit or to a control loop acting on its own estimate, giving a partial circularity score of 6.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The estimation core is reused from Bangura and Mahony [15]; the paper's own added free parameters are the per-rotor scaling factor and control/tuning constants, plus ad hoc assumptions about coefficient transferability and dynamic validity of the static scaling.

free parameters (3)

Per-rotor scaling factor s = Not quoted directly; implied from Table III (about 0.11 for 250mm and 1.7 for 500mm)
Fit by linear regression of measured static thrust on raw BEMT estimate; this is the paper's 'single scaling value'.
PID gains Kp, Ki, Kd, Kff = 250mm: 1.0, 0.3, 0, 0.8; 500mm: 2.15, 0.5, 0, 0.9
Tuned through flight experiments; no tuning procedure is specified.
Algorithm 1 parameters N, delta, epsilon = N=20, delta=0.1, epsilon=1e-5
Chosen constants for the iterative Newton-Secant thrust estimation.

assumptions (5)

domain assumption BEMT power balance and inflow equations (2)-(9) from Bangura and Mahony and Bramwell
Standard rotor aerodynamics model adopted as background.
domain assumption Small advance ratio mu^2 approximately 0 and small horizontal speeds, used to decouple horizontal force CH
Simplifies equations (8b), (8a), (8c); contradicts flight speeds of 11 to 12 m/s and is flagged in the conclusion as a limitation.
ad hoc to paper Aerodynamic coefficients in Table I, identified for a D2836/9 950KV motor and 10-inch propeller, apply to both test platforms up to a constant scaling factor
Core generalization premise; no physical justification is presented.
domain assumption ESC telemetry of current, voltage, and speed is accurate enough to compute aerodynamic power Pam
Needed for the estimator and not independently validated.
ad hoc to paper Linear static scaling extrapolates to dynamic flight conditions
Load-bearing assumption for the flight results; no in-flight thrust ground truth.

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Cite this review

Pith. "Pith review of A Generalized Thrust Estimation and Control Approach for Multirotors Micro Aerial Vehicles." pith.science (2026). https://pith.science/paper/YNQHTIBA

@misc{pith2026241202874,
  author       = {Pith},
  title        = {Pith review of: A Generalized Thrust Estimation and Control Approach for Multirotors Micro Aerial Vehicles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YNQHTIBA}},
  note         = {Machine review of arXiv:2412.02874}
}

read the original abstract

This paper addresses the problem of thrust estimation and control for the rotors of small-sized multirotors Uncrewed Aerial Vehicles (UAVs). Accurate control of the thrust generated by each rotor during flight is one of the main challenges for robust control of quadrotors. The most common approach is to approximate the mapping of rotor speed to thrust with a simple quadratic model. This model is known to fail under non-hovering flight conditions, introducing errors into the control pipeline. One of the approaches to modeling the aerodynamics around the propellers is the Blade Element Momentum Theory (BEMT). Here, we propose a novel BEMT-based closed-loop thrust estimator and control to eliminate the laborious calibration step of finding several aerodynamic coefficients. We aim to reuse known values as a baseline and fit the thrust estimate to values closest to the real ones with a simple test bench experiment, resulting in a single scaling value. A feedforward PID thrust control was implemented for each rotor, and the methods were validated by outdoor experiments with two multirotor UAV platforms: 250mm and 500mm. A statistical analysis of the results showed that the thrust estimation and control provided better robustness under aerodynamically varying flight conditions compared to the quadratic model.

Figures

Figures reproduced from arXiv: 2412.02874 by the authors.

**Figure 2.** Quadrotors body frame and rotor thrust. for the quadrotor frame is in the X configuration ( [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** PX4 control pipeline. We removed the red block and added the green blocks. The thrust control and estimator blocks [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Different forces, torques, and powers on a rotor. [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: Setup used for the thrust test bench experiment. [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: Thrust control diagram. The tuning of the thrust controller was performed through several flight experiments and observations of the thrust curves. In the following section, we discuss the results using two types of UAVs, the 250mm and the 500mm, where 250mm and 500mm …

**Figure 7.** Figure 7: Thrust setpoint tracking with f-PID control. [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

**Figure 8.** Figure 8: Acceleration (x, y, z), Velocity (x, y, z), and Position (x, y, z) RMSE distribution for the 32 experiments with the [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

**Figure 9.** Figure 9: Acceleration, Velocity, and Position RMSE distribution for the 32 experiments with the 500mm quadrotor using the [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]

**Figure 10.** Figure 10: Thrust RMSE distribution using the thrust-to-speed map and thrust estimation and control for the 32 experiments with [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]

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Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Position and attitude control of multi- rotor aerial vehicles: A survey,

T. P. Nascimento and M. Saska, “Position and attitude control of multi- rotor aerial vehicles: A survey,” Annu Rev Control, vol. 48, pp. 129–146, 2019

work page 2019

[2] [2]

Influence of the ground, ceiling, and sidewall on micro-quadrotors,

D. J. Carter, L. Bouchard, and D. B. Quinn, “Influence of the ground, ceiling, and sidewall on micro-quadrotors,” AIAA Journal, vol. 59, no. 4, pp. 1398–1405, 2021

work page 2021

[3] [3]

Px4: A node-based mul- tithreaded open source robotics framework for deeply embedded plat- forms,

L. Meier, D. Honegger, and M. Pollefeys, “Px4: A node-based mul- tithreaded open source robotics framework for deeply embedded plat- forms,” in 2015 IEEE ICRA . IEEE, 2015, pp. 6235–6240

work page 2015

[4] [4]

The mrs uav system: Pushing the frontiers of reproducible research, real-world deployment, and education with autonomous un- manned aerial vehicles,

T. Baca, M. Petrlik, M. Vrba, V . Spurny, R. Penicka, D. Hert, and M. Saska, “The mrs uav system: Pushing the frontiers of reproducible research, real-world deployment, and education with autonomous un- manned aerial vehicles,” J Intell Robot Syst , vol. 102, no. 1, p. 26, 2021

work page 2021

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work page 2006

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