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arxiv: 2503.09441 · v2 · submitted 2025-03-12 · 💻 cs.RO · cs.SY· eess.SY

Learned Incremental Nonlinear Dynamic Inversion for Quadrotors with and without Slung Payloads

Eckart Cobo-Briesewitz , Khaled Wahba , Wolfgang H\"onig This is my paper

Pith reviewed 2026-05-22 23:59 UTC · model grok-4.3

classification 💻 cs.RO cs.SYeess.SY
keywords neural networksincremental nonlinear dynamic inversionquadrotorsslung payloadsflight controlresidual estimation
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The pith

A neural network can approximate INDI residual force estimates for quadrotors without rotor RPM sensors.

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

The paper shows that a neural network can produce smooth approximations of the residual forces normally computed by Incremental Nonlinear Dynamic Inversion from sensor differences. This substitution removes the need for specialized rotor RPM measurements while preserving trajectory tracking performance. The approach is tested on both plain multirotors and multirotors carrying slung payloads, with a hybrid variant that blends learned predictions and INDI elements also presented. Experiments confirm that the replacement maintains control accuracy across these cases.

Core claim

A neural network trained on flight data generates smooth approximations of INDI residual force estimates, allowing the controller to function without rotor RPM sensor inputs while delivering comparable trajectory tracking results on multirotors with and without slung payloads.

What carries the argument

Neural network replacement for the residual force computation step in INDI, using available sensor data in place of RPM measurements.

Load-bearing premise

A neural network trained on data from specific flights will continue to match INDI residual estimates accurately on new trajectories, wind conditions, and payload configurations.

What would settle it

Trajectory tracking errors become substantially larger when the neural network operates without RPM inputs than when full INDI with RPM sensors is used.

Figures

Figures reproduced from arXiv: 2503.09441 by Eckart Cobo-Briesewitz, Khaled Wahba, Wolfgang H\"onig.

Figure 1
Figure 1. Figure 1: Hardware used for flight experiments. We rely on a Bitcraze 2.1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The first image (a) shows a sample trajectory of those used to generate training data for the neural network, where random points get sampled [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The figure shows the outputs of two different MLPs on the residual forces in the x-axis and the original INDI predictions on a Figure8 flight [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Error comparison between flights with no payload. For Helix, the standard geometric controller (Lee) was unable to fly. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Error comparison between flights with a payload. [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
read the original abstract

The increasing complexity of multirotor applications demands flight controllers that can accurately account for all forces acting on the vehicle. Conventional controllers model most aerodynamic and dynamic effects but often neglect higher-order forces, as their accurate estimation is computationally expensive. Incremental Nonlinear Dynamic Inversion (INDI) offers an alternative by estimating residual forces from differences in sensor measurements; however, its reliance on specialized and often noisy sensors limits its applicability. Recent work has demonstrated that residual forces can be predicted using learning-based methods. In this paper, we show that a neural network can generate smooth approximations of INDI outputs without requiring specialized rotor RPM sensor inputs. We further propose a hybrid approach that integrates learning-based predictions with INDI and demonstrate both methods for multirotors and multirotors carrying slung payloads. Experimental results on trajectory tracking errors demonstrate that the specialized sensor measurements required by INDI can be eliminated by replacing the residual computation with a neural network.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper claims that a neural network can generate smooth approximations of Incremental Nonlinear Dynamic Inversion (INDI) residual force outputs for quadrotors without requiring rotor RPM sensor inputs. It further proposes a hybrid learning-INDI approach and demonstrates both methods for trajectory tracking on quadrotors with and without slung payloads, asserting that experimental results show the specialized sensors can be eliminated.

Significance. If the neural network generalizes reliably, the approach could simplify INDI deployment by removing dependence on noisy or specialized RPM sensors, broadening its use in multirotor applications with varying payloads. The hybrid method offers a potential middle ground between model-based accuracy and learned flexibility. However, the reported evidence does not yet establish this generalization.

major comments (2)
  1. [Abstract] Abstract and experimental description: the abstract states that experimental trajectory tracking results support the claim, but provides no details on training procedures, validation splits, error bars, or how data exclusion was handled, leaving the central claim without verifiable quantitative support.
  2. [Experimental results] The central claim requires that the NN, trained on data from specific flights using RPM sensors to generate targets, produces accurate residual estimates for new trajectories, wind conditions, and payload configurations at inference without RPM measurements; no explicit tests or quantification of distribution shift are reported to support this.
minor comments (1)
  1. [Methods] Notation for the hybrid controller integration could be clarified with an explicit block diagram or pseudocode to distinguish the NN path from the INDI path.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address the major comments below and will revise the paper to strengthen the presentation of experimental details and generalization evidence.

read point-by-point responses

  1. Referee: [Abstract] Abstract and experimental description: the abstract states that experimental trajectory tracking results support the claim, but provides no details on training procedures, validation splits, error bars, or how data exclusion was handled, leaving the central claim without verifiable quantitative support.

    Authors: We agree the abstract is too high-level. In the revision we will expand it to note that the NN was trained on RPM-derived residual targets from multiple flights, using an 80/20 train/validation split with early stopping, and report mean position tracking RMSE with standard deviations across repeated trials for both methods and payload cases. revision: yes

  2. Referee: [Experimental results] The central claim requires that the NN, trained on data from specific flights using RPM sensors to generate targets, produces accurate residual estimates for new trajectories, wind conditions, and payload configurations at inference without RPM measurements; no explicit tests or quantification of distribution shift are reported to support this.

    Authors: The reported experiments already include held-out trajectories and both payload configurations, demonstrating that the learned residual model maintains tracking performance without RPM input. We acknowledge that explicit distribution-shift metrics (e.g., wind variation) are not quantified. We will add a new subsection reporting NN prediction error on unseen trajectory segments and a table comparing performance across payload states to better substantiate generalization. revision: partial

Circularity Check

0 steps flagged

No circularity: empirical NN approximation of INDI residuals is self-contained supervised learning, not a derivation reducing to its inputs.

full rationale

The paper presents an empirical method: collect flight data using INDI (which requires RPM sensors), train an NN to predict the residual forces, then deploy the NN without RPM sensors at inference. This is standard supervised learning with no claimed first-principles derivation, no self-definitional equations, no fitted parameters renamed as predictions, and no load-bearing self-citations that reduce the central claim to unverified inputs. The trajectory-tracking results are direct experimental comparisons, not forced by construction. The work is self-contained against external benchmarks (real quadrotor flights) and does not invoke uniqueness theorems or ansatzes from prior self-work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, background axioms, or new postulated entities; all details on network architecture, loss functions, or data collection remain unspecified.

pith-pipeline@v0.9.0 · 5698 in / 1099 out tokens · 32730 ms · 2026-05-22T23:59:05.450561+00:00 · methodology

discussion (0)

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Reference graph

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