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VeloBins: Learning Velocity and Its Uncertainty via Bins and Error-Conditioned Gaussian Labels for Aerial Inertial Odometry

Maulana Bisyir Azhari, Seungwook Lee, Donghun Han, Sung Jun Park, David Hyunchul Shim

arXiv:2608.29720v1cs.RO

TL;DR

Aerial inertial odometry needs accurate velocity and uncertainty estimates, but regression and separate NLL-trained uncertainty decoders are difficult and can become over-confident. VeloBins predicts a velocity-bin distribution, decodes its expectation and variance, and supervises the variance with error-conditioned Gaussian labels. Across four aerial datasets, it achieves the lowest average errors and improves uncertainty calibration and EKF filter consistency, while requiring a fixed per-dataset velocity range and becoming unreliable when errors are very small.

  • Problem

    Direct velocity regression is difficult, while separate NLL-trained uncertainty decoders complicate learning and may produce over-confident variances.

  • Method

    VeloBins predicts velocity as a discretized-bin distribution, decodes velocity and uncertainty from its expectation and variance, and uses error-conditioned Gaussian labels for explicit uncertainty supervision.

  • Results

    VeloBins achieves the lowest average velocity, relative trajectory, and absolute trajectory errors on four aerial datasets, reducing AVE by 3–27%, RTE by 8–40%, and ATE by 6–53% versus AI-IO.

  • Takeaways & Limitations

    The bin variance provides better-calibrated uncertainty and more consistent EKF fusion without a separate covariance decoder or NLL loss.

  • Takeaways & Limitations

    The bin grid is fixed per dataset with a known operational range, and very small velocity errors can leave too few bins to supervise variance reliably.

Abstract

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Inertial odometry (IO) is critical for aerial robots, where aggressive maneuvers and poor lighting degrade visual sensors. Recent learning-based IO methods improve traditional integration-based approaches by learning motion priors from IMU and platform-specific sensors, then fusing the predictions within an extended Kalman filter. However, learning velocity through regression is difficult, while jointly estimating uncertainty with a separate decoder and negative log-likelihood (NLL) loss further complicates training and can lead to over-confident estimates. We introduce VeloBins, which reformulates velocity regression as classification over discretized velocity bins. We decode both the velocity from the bin distribution's expectation and the uncertainty from its variance, removing the need for a separate uncertainty decoder. We further supervise the uncertainty explicitly using an error-conditioned Gaussian label centered at the ground-truth velocity, with a standard deviation set to the velocity error. We evaluate VeloBins on four aerial datasets, ranging from free-form aggressive flights and a 27 g nano-quadrotor to drone racing at over 21~m/s. VeloBins achieves the lowest average errors on all four datasets, reducing velocity, relative trajectory, and absolute trajectory errors by 3-27%, 8-40%, and 6-53%, respectively, compared with the strongest baseline. Notably, the proposed supervision achieves the lowest NLL and best filter consistency despite never optimizing an NLL loss. The code will be available upon acceptance.

I. INTRODUCTION

Learning-based inertial odometry helps aerial robots estimate motion despite visual degradation, but direct velocity regression and separately learned uncertainty remain difficult. VeloBins addresses these issues with bin-based velocity prediction and error-conditioned uncertainty supervision.

  • Aerial inertial odometry remains important because IMUs work in poor lighting, motion blur, and textureless scenes where vision degrades.
  • Learning-based IO maps IMU windows to motion and fuses predictions with IMU propagation in an extended Kalman filter using predicted uncertainty.
  • Direct velocity regression is difficult for highly dynamic aerial platforms, while separate NLL-trained uncertainty decoders may produce over-confident variances and poor filter consistency.
  • VeloBins reformulates velocity regression as classification over discretized bins and decodes velocity and uncertainty from the distribution's expectation and variance.
  • An error-conditioned Gaussian label sets its standard deviation to the current velocity error, explicitly supervising uncertainty without a separate decoder or NLL loss.
  • Across four aerial datasets, VeloBins achieves the lowest average errors against five baselines and better filter consistency than regression and likelihood-supervised variants.

II. RELATED WORK

Prior learning-based IO methods reduce drift by learning motion priors and fusing them with inertial propagation, but commonly use direct regression with separately decoded covariance. VeloBins instead applies bin-based distribution prediction and error-conditioned supervision to aerial IO.

  • Traditional IO accumulates IMU bias and noise through repeated integration, whereas learning-based methods use temporal networks to reduce long-term drift.
  • UAV-specific methods augment IMU inputs with rotor speeds or thrust and preserve body-frame representations to model highly dynamic flight.
  • Existing methods generally frame velocity estimation as direct regression and obtain measurement covariance from a separate decoder.
  • Bin-based approaches replace continuous regression with classification over discretized bins, with prior applications in depth, pose, and reinforcement learning.
  • Earlier distribution supervision either leaves variance unconstrained, uses fixed-width labels, or learns variance with a separate maximum-likelihood decoder.
  • VeloBins uses one bin distribution for velocity and uncertainty, explicitly trains its variance with error-conditioned Gaussian labels, and passes both quantities to the EKF.

A. Velocity and Uncertainty from Bins

VeloBins discretizes each velocity axis over a fixed operating range and predicts a probability distribution across its bins. It decodes velocity from the distribution's expectation and uncertainty from its variance.

  • Each velocity axis is assigned a fixed operating range [vmin,vmax] and divided into N bins with centers b0 through bN−1.
  • Bin boundaries lie midway between adjacent centers, while the first and last boundaries extend to vmin and vmax.
  • The velocity decoder predicts a per-axis distribution πi over bins and computes the velocity as its expectation over bin centers.
  • The same distribution supplies velocity uncertainty as the variance about the decoded velocity, replacing a separate covariance decoder.

B. VeloBins Network Architecture

VeloBins uses a compact bin-coordinate decoder to predict velocity distributions from multimodal aerial inputs. Error-conditioned Gaussian labels supervise uncertainty, while a KL-based correction addresses discretization bias.

  • Network inputs: The encoder processes accelerometer, gyroscope, and per-rotor actuation signals into a last-step feature for bin decoding.Measured rotor speed is used when available; commanded motor signals are used otherwise.
  • Bin encoding: Each velocity bin is represented by a learnable sine positional encoding that preserves the ordinal structure of the velocity range.The encoding scales a fixed positional basis with learnable per-channel frequencies.
  • Bin decoding: For each velocity axis, a query from the feature is matched against projected bin encodings and normalized with a softmax.The resulting distribution is decoded into velocity and uncertainty.
  • Error-conditioned labels: A Gaussian label centers on ground-truth velocity and sets its standard deviation to the velocity error, with the error detached during label construction.Smaller errors produce sharper labels, whereas larger errors produce broader labels over the bins.
  • Label construction: The continuous Gaussian density is integrated over each bin, preserving ordinal structure through probability mass assigned to bin supports.A KL-nearest tilt then enforces the exact ground-truth expectation and vanishes when the decoded mean is already exact.

D. Training Objective

Training combines KL supervision from an error-conditioned label with a Huber loss on decoded velocity. The KL term explicitly guides the predicted uncertainty while the Huber term directly penalizes mean displacement.

  • Distribution supervision: The predicted bin distribution is supervised with KL divergence from the error-conditioned Gaussian label.The label is normalized over the bins and represents probability mass rather than density values at bin centers.
  • Uncertainty supervision: Matching the label moves the bin variance toward the squared error scale, explicitly supervising uncertainty rather than leaving it as a training by-product.Because the label is detached, the per-input optimum reflects the conditional mean of the labels.
  • Velocity supervision: A Huber loss on decoded velocity complements KL supervision by directly penalizing a displaced mean.The KL term does not give the decoded mean special weight among bins, whereas the Huber term does.

E. Extended Kalman Filter

VeloBins supplies decoded body-frame velocity and bin-derived covariance to the existing error-state EKF. Both measurement quantities come from one distribution and the covariance is used directly.

  • Filter integration: VeloBins is fused with AI-IO’s error-state EKF while retaining its state definition, IMU propagation, and measurement-update formulation.The state includes rotation, global velocity, global position, and accelerometer and gyroscope biases.
  • Measurement: At each network update, the decoded body-frame velocity serves as the EKF measurement.The observation model and covariance are defined for this network-provided measurement.
  • Covariance: The bin variance supplies the covariance, replacing separate velocity and covariance decoders and avoiding rescaling or innovation gating.Both quantities therefore originate from the same bin distribution.

IV. EXPERIMENTS

VeloBins is evaluated on four public UAV datasets spanning lightweight nano-quadrotors to racing platforms, covering 63 held-out sequences totaling 5.3 km and 43 minutes of flight.

  • Datasets: The evaluation uses AI-IO, NanoBench, TII-RATM, and NeuroBEM, spanning a 27 g Crazyflie to a 5-inch racing quadrotor.The platforms provide measured rotor speed or commanded PWM signals, depending on the dataset.
  • Evaluation scope: The four test splits contain 63 sequences totaling 5.3 km and 43 minutes of flight.These held-out sequences form the evaluation scope reported for the datasets.

2) Baselines:

VeloBins is evaluated against five learning-based inertial odometry baselines using EKF-fused trajectory metrics across four aerial datasets. It achieves the lowest average errors, with qualitative trajectories showing less drift than competing methods.

  • Baselines: The comparison includes TLIO, EqNIO, IMO, AirIO, and AI-IO, trained on the same data as VeloBins.TLIO and EqNIO regress pedestrian displacement, IMO uses quadrotor actuation inputs, and AirIO and AI-IO predict body-frame velocity.
  • Quantitative results: VeloBins lowers AVE by 3–27%, RTE by 8–40%, and ATE by 6–53% against AI-IO across the four datasets.These are EKF-fused results, and AI-IO shares VeloBins’s encoder, EKF modules, and training data.
  • Quantitative results: VeloBins achieves the lowest average AVE, RTE, and ATE on all four datasets.AVE is velocity error, while RTE and ATE are relative and absolute trajectory errors; RTE uses Δt = 5 s.
  • Baseline behavior: IMO is competitive on NanoBench and TII-RATM but fails on AI-IO, while AirIO is less accurate than AI-IO and VeloBins on every dataset.The reported differences are associated with IMO’s actuation-driven formulation and AirIO’s sensitivity to orientation errors and online windowed inference.
  • Qualitative comparison: VeloBins drifts least and matches ground truth most closely on one held-out qualitative sequence per dataset.The comparison covers AI-IO, NanoBench, TII-RATM, and NeuroBEM.

V. ABLATION STUDY

The ablations separate the effects of bin-based prediction, uncertainty supervision, and decoder design. Error-conditioned Gaussian supervision with the Huber term provides the strongest overall accuracy and uncertainty behavior without requiring NLL optimization.

  • Ablation setup: The ablation compares bin supervision using Huber, MLE, or error-conditioned Gaussian labels, with optional Huber terms and different uncertainty sources.Uncertainty is taken from bin variance, a covariance decoder, or their sum.
  • A. Regression vs. Classification: Using bins with Huber supervision lowers network AVE on all four datasets, by 2% on AI-IO and up to 25% on NanoBench.This directly compares the bin formulation with Huber supervision against the AI-IO regression baseline.
  • B. Supervision Strategies: Huber-only bin supervision improves accuracy but has far worse NLL than the regression baseline on every dataset because it does not explicitly shape uncertainty.MLE supervision alone or with Huber also exceeds the regression baseline’s NLL, while combining covariance sources only approaches it.
  • B. Supervision Strategies: G+HB attains the lowest NLL on all four datasets, while G+H lowers both network AVE and NLL relative to G.The proposed variants achieve this without optimizing an NLL loss.

C. Filter Accuracy and Consistency

Filter evaluation shows that calibrated uncertainty matters after EKF fusion, not only raw velocity accuracy. G+HB provides the most consistent updates and is closest to ideal NEES, while bin-resolution changes mainly affect NLL.

  • Filter accuracy: After fusion, G and G+H achieve the lowest fused AVE on three of four datasets, with G+H attaining the lowest ATE on AI-IO and NanoBench.The EKF weights each update by its predicted velocity uncertainty, so uncertainty calibration changes fused performance.
  • Filter accuracy: Huber-only supervision predicts accurate velocity but an over-spread bin variance that causes the filter to discard its updates.MLE variants and the regression baseline are over-confident, whereas G+H is better-calibrated and ranks best overall.
  • Filter consistency: G+HB reaches median NEES 3.08 versus the ideal 2.37, with 73% of updates inside the 95% bounds.The corresponding rates are 64% for M+HB+C, 61% for the regression baseline, and 49% for HB.
  • Filter consistency: The G+HB bin variance supplies measurement covariance without a separate covariance decoder or an NLL loss.This connects the filter-consistency result to the proposed uncertainty representation.
  • Bin resolution: Across bin resolutions, accuracy varies little with N, while NLL improves for N > 64 and continues improving until N = 512.The ablation compares network AVE and NLL against the regression baseline.

1) Bin Encoding:

VeloBins combines discretized velocity bins with error-conditioned supervision to estimate velocity and uncertainty, while its bin-resolution study identifies practical accuracy and calibration trade-offs.

  • Bin Encoding: SPE attains the lowest RTE on every dataset and the lowest NLL on three of four datasets.On TII-RATM, SPE is the only encoding that improves NLL over the regression baseline.
  • Bin Encoding: VeloBins maintains lower network AVE than the regression baseline across bin counts on NanoBench and TII-RATM, while requiring no fine grid for velocity estimation.For the other two datasets, its network AVE remains within 0.01 m/s of the baseline.
  • Bin Encoding: NLL generally improves for N = 64 or above and saturates by N = 512, although it remains above the regression baseline on TII-RATM until N = 512.When errors become sufficiently small, the error-conditioned label spans too few bins to supervise variance reliably, producing under-confident uncertainty.
  • Scope: Although evaluated on aerial robots, the formulation and supervision are described as applicable to humanoids, quadrupeds, ground vehicles, and underwater vehicles.These platforms may provide platform-specific modalities or use a cross-platform formulation.
  • Limitations: The fixed per-dataset bin grid requires a known operational velocity range, and sufficiently small velocity errors can undermine reliable variance supervision.These are the two limitations identified for VeloBins.
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