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arxiv: 2606.30804 · v1 · pith:K6FAWFESnew · submitted 2026-06-29 · 💻 cs.RO

Wind and State Estimation on SE(3): Comparative Evaluation of EKF and UKF with Continuous and Discrete Quadrotor Models

Hiranya Udagedara , Adam Bigsby , Mahdis Bisheban This is my paper

Pith reviewed 2026-07-01 01:38 UTC · model grok-4.3

classification 💻 cs.RO
keywords quadrotorwind estimationSE(3)Lie group variational integratorUKFEKFdiscrete dynamics
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The pith

Discrete SE(3) quadrotor dynamics paired with UKF yield higher wind-velocity estimation accuracy than continuous models or EKF.

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

The paper compares continuous and discrete formulations of quadrotor rigid-body dynamics on the special Euclidean group SE(3) inside both extended and unscented Kalman filters for simultaneous state and wind-velocity estimation. It replaces the usual discretized continuous equations with an exact discrete model obtained from a Lie-group variational integrator that preserves the geometry without approximation. Numerical simulations of hovering and trajectory tracking plus outdoor free-flight experiments show that the discrete model inside the unscented filter produces the smallest estimation errors while still allowing the vehicle to follow commanded paths when only low-cost onboard sensors are used.

Core claim

When the quadrotor dynamics are expressed exactly on SE(3) by a Lie-group variational integrator and the resulting discrete-time model is inserted into an unscented Kalman filter, the combined estimator recovers both vehicle state and wind velocity with measurably lower error than the same filter driven by the continuous dynamics or than an extended Kalman filter driven by either model, and this accuracy holds during both hovering and trajectory-tracking maneuvers even when sensor noise levels match those of inexpensive hardware.

What carries the argument

Lie Group Variational Integrator that supplies an exact discrete-time representation of the quadrotor equations of motion on SE(3) without linearization or discretization error.

If this is right

  • The same discrete-UKF combination can be used for other disturbance estimation tasks that require geometric fidelity on SE(3).
  • Low-cost sensor suites become sufficient for accurate wind-aware trajectory tracking once the discrete model is adopted.
  • Filter tuning effort can be reduced because the unscented transform handles the nonlinear discrete map without Jacobian computation.
  • Real-time implementation on embedded hardware is feasible because the variational integrator evaluates in closed form at each step.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result suggests that geometric integrators may improve estimator performance in other rigid-body control problems where continuous-time approximations currently dominate.
  • If the accuracy gain persists under wind gusts or payload changes, the discrete formulation could become the default model for online disturbance observers on SE(3).
  • Extending the comparison to particle filters or moving-horizon estimators would test whether the benefit is specific to the unscented transform.

Load-bearing premise

The variational integrator gives a discrete dynamics whose integration error remains small enough that it does not increase filter estimation error relative to the continuous model.

What would settle it

An outdoor flight in which independent anemometer measurements of wind velocity are recorded simultaneously with the onboard estimator outputs; if the discrete-UKF errors exceed those of the continuous-EKF case by more than the reported margin, the claimed superiority is falsified.

Figures

Figures reproduced from arXiv: 2606.30804 by Adam Bigsby, Hiranya Udagedara, Mahdis Bisheban.

Figure 1
Figure 1. Figure 1: Experimental Setup for Operating the Holybro X500 V2 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Performance of EKF and UKF when propagated using the [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Performance of the 4 algorithms compared to the [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
read the original abstract

Use of quadrotor UAVs for wind velocity estimation is gaining popularity in recent studies, leveraging their maneuverability, compact size and low cost. Among available approaches, model-based wind velocity estimation is most commonly used, since it relies only on onboard sensors. However, as the quadrotor is a highly nonlinear system, thus making this task challenging. This study evaluate the use of both discrete and continuous dynamic equations of the quadrotor UAV for wind velocity estimation on SE(3), rather than commonly adapted continuous or discretized form. Lie Group Variational Integrator, developed on discrete Lagrangian is used as the discrete model without any approximation or discritization. The study assess both the discrete and continuous form of the quadrotor dynamics on SE(3) using Extended Kalman filter (EKF), and Unscented Kalman filter (UKF). The quadrotor UAV performance is evaluated in both MATLAB-based numerical simulations and free outdoor flight. The numerical simulations are conducted during both hovering and trajectory-tracking flights. Results demonstrate that, by using discrete SE(3) dynamics coupled with UKF, the quadrotor achieves higher estimation accuracy while maintaining trajectory tracking, even with low-cost sensors. These findings highlight the potential of discrete quadrotor models with UKF not only for wind velocity estimation but also for other high-accuracy tasks, even when relying on low-cost onboard sensors.

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 / 2 minor

Summary. The paper evaluates wind and state estimation for quadrotor UAVs on SE(3) by comparing EKF and UKF applied to both continuous dynamics and a discrete model obtained via Lie Group Variational Integrator (LGVI). It claims that the discrete SE(3) formulation paired with UKF yields superior estimation accuracy during hovering and trajectory tracking, even with low-cost sensors, as demonstrated in MATLAB simulations and outdoor flights, while preserving trajectory tracking performance.

Significance. If the accuracy advantage is robust to integrator choice and noise tuning, the work would indicate that structure-preserving discrete Lagrangian integrators can measurably improve filter performance on Lie groups for UAV wind estimation tasks, offering a practical route to higher accuracy without additional hardware.

major comments (2)
  1. [Methods / Continuous-model propagation] The manuscript does not specify the numerical integration scheme (Euler, RK4, etc.) or step-size/tolerance settings used to propagate the continuous SE(3) dynamics inside the EKF and UKF prediction steps. Because the central claim attributes performance gains to the discrete LGVI formulation rather than to integration accuracy, this omission directly affects whether the reported superiority of discrete+UKF survives a fair comparison; the stress-test concern therefore lands as a load-bearing issue.
  2. [Numerical results / Simulation setup] No quantitative comparison of integration error (e.g., local truncation error or conserved quantities) between the continuous integrator and the LGVI is provided. Without such metrics, it remains unclear whether the discrete model’s reported advantage is intrinsic to the SE(3) discretization or an artifact of differing numerical fidelity.
minor comments (2)
  1. [Abstract] The abstract states that the LGVI is used “without any approximation or discretization,” yet the continuous model necessarily requires discretization for implementation; a brief clarification of this terminology would avoid reader confusion.
  2. [Figures and tables] Figure captions and table headings should explicitly state the sensor noise levels and wind gust profiles used in each experiment so that reproducibility does not rely solely on the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments that highlight important aspects of methodological clarity. We address each major comment below and will incorporate the requested details in the revised manuscript.

read point-by-point responses

  1. Referee: [Methods / Continuous-model propagation] The manuscript does not specify the numerical integration scheme (Euler, RK4, etc.) or step-size/tolerance settings used to propagate the continuous SE(3) dynamics inside the EKF and UKF prediction steps. Because the central claim attributes performance gains to the discrete LGVI formulation rather than to integration accuracy, this omission directly affects whether the reported superiority of discrete+UKF survives a fair comparison; the stress-test concern therefore lands as a load-bearing issue.

    Authors: We agree that the integration scheme for the continuous model must be stated explicitly to support a fair comparison. In the revised manuscript we will add to the Methods section that the continuous SE(3) dynamics were propagated with the classical fourth-order Runge-Kutta (RK4) integrator using a fixed step size of 1 ms (identical to the discrete time step of the LGVI). This specification will allow readers to verify that the reported accuracy advantage of the discrete+UKF combination is not an artifact of unequal numerical integration fidelity. revision: yes

  2. Referee: [Numerical results / Simulation setup] No quantitative comparison of integration error (e.g., local truncation error or conserved quantities) between the continuous integrator and the LGVI is provided. Without such metrics, it remains unclear whether the discrete model’s reported advantage is intrinsic to the SE(3) discretization or an artifact of differing numerical fidelity.

    Authors: We accept that a direct quantitative comparison of integration accuracy is needed to strengthen the central claim. In the revision we will insert a new subsection (or supplementary table) that reports local truncation error norms and drift in conserved quantities (kinetic plus potential energy, angular momentum) for both the RK4 continuous integrator and the LGVI over the same simulation horizons and initial conditions used in the filter evaluations. These metrics will demonstrate that both integrators maintain comparable fidelity, thereby supporting the conclusion that the performance difference arises from the structure-preserving discrete formulation. revision: yes

Circularity Check

0 steps flagged

No circularity: comparative evaluation uses independent simulation and flight data

full rationale

The paper is a comparative study of EKF/UKF filters on continuous vs. discrete SE(3) quadrotor models (LGVI). Reported accuracy metrics come from separate MATLAB simulations (hovering and trajectory tracking) and outdoor flights; no parameters are fitted on a subset and then re-predicted as the performance metric. The LGVI is invoked as an existing structure-preserving integrator without the paper deriving its properties from the present results or via load-bearing self-citation. The central claim therefore rests on external data rather than reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit list of fitted parameters, background axioms, or new postulated entities; standard Kalman filter noise covariances and sensor models are presumed but not enumerated.

pith-pipeline@v0.9.1-grok · 5796 in / 1153 out tokens · 34076 ms · 2026-07-01T01:38:15.715342+00:00 · methodology

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Reference graph

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