pith. sign in

arxiv: 2604.14545 · v1 · submitted 2026-04-16 · 💻 cs.RO

CT-VIR: Continuous-Time Visual-Inertial-Ranging Fusion for Indoor Localization with Sparse Anchors

Yu-An Liu , Li Zhang This is my paper

Pith reviewed 2026-05-10 11:42 UTC · model grok-4.3

classification 💻 cs.RO
keywords continuous-time state estimationvisual-inertial odometryUWB rangingindoor localizationB-splinesensor fusionvirtual anchorssparse anchors
0
0 comments X

The pith

A B-spline continuous-time framework fuses visual, inertial and ranging data for indoor localization using sparse anchors.

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

This paper develops a spline-based continuous-time state estimation technique for combining visual-inertial odometry with ultra-wideband ranging. Virtual anchors are built during preprocessing from motion priors and range measurements to counter geometric problems when anchors are few. The trajectory is then represented as a B-spline and optimized in a sliding window graph incorporating all sensor constraints. Readers interested in robot navigation would care if this reduces the need for extensive anchor setups while maintaining accuracy and handling unsynchronized data.

Core claim

We propose a spline-based continuous-time state estimation method for VIR fusion localization. In preprocessing, VIO motion priors and UWB ranging measurements construct virtual anchors and reject outliers. In estimation, the pose trajectory is parameterized using a B-spline, with constraints formulated as factors in a sliding-window graph, and control points are jointly optimized.

What carries the argument

B-spline parameterization of the pose trajectory in continuous time, with virtual anchors constructed from VIO priors and UWB measurements to support the sliding-window factor graph optimization.

If this is right

  • Localization accuracy improves in environments with sparse UWB anchors by alleviating geometric degeneration.
  • Continuous-time B-spline modeling maintains trajectory consistency better under asynchronous sensor sampling than discrete-time methods.
  • Virtual anchors constructed from VIO priors enhance range reliability without requiring dense real anchor deployment.
  • The sliding-window graph optimization balances positioning accuracy, trajectory smoothness, and computational efficiency.

Where Pith is reading between the lines

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

  • This continuous-time formulation could support larger-scale indoor operations where anchor placement is constrained by power or space limits.
  • The virtual-anchor construction step might generalize to other sensor pairs that combine predictive motion models with occasional absolute measurements.

Load-bearing premise

VIO motion priors combined with UWB ranges can reliably construct virtual anchors that alleviate geometric degeneration without introducing new biases.

What would settle it

A controlled experiment in which position errors with the virtual-anchor construction exceed those obtained from the same sparse real anchors without virtual anchors, or where the outlier rejection step fails to remove consistent range biases visible in ground-truth comparison.

Figures

Figures reproduced from arXiv: 2604.14545 by Li Zhang, Yu-An Liu.

Figure 1
Figure 1. Figure 1: System overview of the proposed continuous-time VIR localization framework. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Overall sensing configuration of the VIR system. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of physical/virtual anchors and range inliers/outliers in 3D [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of the continuous-time trajectory estimation. [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Trajectory estimation on the Euroc sequences. [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Trajectory estimation on the UZH-FPV indoor 45/forward sequences. [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Experimental setup and hardware configuration. [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Trajectory estimation on the real world sequences. [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
read the original abstract

Visual-inertial odometry (VIO) is widely used for mobile robot localization, but its long-term accuracy degrades without global constraints. Incorporating ranging sensors such as ultra-wideband (UWB) can mitigate drift; however, high-accuracy ranging usually requires well-deployed anchors, which is difficult to ensure in narrow or low-power environments. Moreover, most existing visual-inertial-ranging (VIR) fusion methods rely on discrete time-based filtering or optimization, making it difficult to balance positioning accuracy, trajectory consistency, and fusion efficiency under asynchronous multi-sensor sampling. To address these issues, we propose a spline-based continuous-time state estimation method for VIR fusion localization. In the preprocessing stage, VIO motion priors and UWB ranging measurements are used to construct virtual anchors and reject outliers, thereby alleviating geometric degeneration and improving range reliability. In the estimation stage, the pose trajectory is parameterized in continuous time using a B-spline, while inertial, visual, and ranging constraints are formulated as factors in a sliding-window graph. The spline control points, together with a small set of auxiliary parameters, are then jointly optimized to obtain a continuous-time trajectory estimate. Evaluations on public datasets and real-world experiments demonstrate the effectiveness and practical potential of the proposed approach.

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

1 major / 2 minor

Summary. The manuscript proposes CT-VIR, a continuous-time spline-based state estimation method for visual-inertial-ranging (VIR) fusion localization with sparse UWB anchors. In preprocessing, VIO motion priors combined with UWB ranges are used to construct virtual anchors and reject outliers to mitigate geometric degeneration. The trajectory is then parameterized as a B-spline and optimized jointly with inertial, visual, and ranging factors inside a sliding-window factor graph.

Significance. If the virtual-anchor construction can be shown to introduce negligible bias relative to UWB noise, the continuous-time formulation offers a principled way to fuse asynchronous sensors while maintaining trajectory smoothness, potentially enabling reliable indoor localization with far fewer physical anchors than conventional discrete-time VIR methods. This would be a practical contribution to robotics applications in power- or space-constrained environments.

major comments (1)
  1. [Preprocessing stage] Preprocessing stage: the claim that virtual anchors constructed from drifting VIO priors alleviate geometric degeneration 'without introducing new biases' lacks any error-propagation analysis, uncertainty bounds on the derived anchor positions, or ablation that isolates virtual-anchor factors from real UWB factors. Because VIO drift is correlated with the trajectory being estimated, biased range residuals treated as hard factors in the subsequent B-spline optimization risk producing locally consistent but globally drifted solutions; this assumption is load-bearing for the sparse-anchor promise.
minor comments (2)
  1. [Abstract] The abstract states that evaluations on public datasets and real-world experiments 'demonstrate the effectiveness' yet supplies no quantitative error metrics, baselines, or ablation tables; the full manuscript should include these in the evaluation section for reproducibility.
  2. [Estimation stage] Notation for the B-spline control points and auxiliary parameters is introduced without an explicit list of free parameters or their initialization strategy, which would aid readers in reproducing the sliding-window optimization.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive critique of our manuscript. The concern regarding the preprocessing stage and potential biases from virtual anchors is well-taken and highlights an area where additional analysis will strengthen the paper. We address the comment below.

read point-by-point responses

  1. Referee: [Preprocessing stage] Preprocessing stage: the claim that virtual anchors constructed from drifting VIO priors alleviate geometric degeneration 'without introducing new biases' lacks any error-propagation analysis, uncertainty bounds on the derived anchor positions, or ablation that isolates virtual-anchor factors from real UWB factors. Because VIO drift is correlated with the trajectory being estimated, biased range residuals treated as hard factors in the subsequent B-spline optimization risk producing locally consistent but globally drifted solutions; this assumption is load-bearing for the sparse-anchor promise.

    Authors: We agree that the manuscript does not provide a formal error-propagation analysis, explicit uncertainty bounds on virtual-anchor positions, or an ablation isolating the contribution of virtual-anchor-based outlier rejection from the real UWB range factors. Virtual anchors are generated in preprocessing from VIO motion priors and available UWB ranges solely to improve geometric conditioning and reject outliers before the main optimization; the subsequent B-spline factor graph uses the original UWB range measurements as ranging factors together with inertial and visual factors. Nevertheless, the referee correctly identifies that any residual bias in the virtual-anchor positions could correlate with the trajectory being estimated and affect global consistency. In the revised manuscript we will add (i) a short error-propagation derivation that propagates VIO covariance and UWB ranging noise into approximate position uncertainty for each virtual anchor, (ii) a quantitative bound showing that this uncertainty remains smaller than typical UWB noise under the operating conditions considered, and (iii) an ablation that runs the full pipeline with and without the virtual-anchor outlier-rejection step, reporting both local and global trajectory metrics. These additions will directly test whether the introduced bias is negligible relative to the benefit in sparse-anchor regimes. revision: yes

Circularity Check

0 steps flagged

No circularity: independent formulation and external validation

full rationale

The paper's derivation chain consists of a preprocessing step that builds virtual anchors from separate VIO motion priors plus UWB ranges, followed by an independent B-spline parameterization whose control points are optimized against inertial, visual, and ranging factors in a sliding-window graph. No equation reduces the final trajectory estimate to a redefinition or direct fit of the preprocessing inputs; the ranging constraints to virtual anchors are treated as additional factors rather than tautological re-encodings. Public-dataset evaluations supply external benchmarks outside the fitted values, satisfying the criteria for a self-contained, non-circular derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Abstract-only review; ledger populated from stated components only. Virtual anchors are constructed rather than measured. B-spline is treated as a standard approximation tool.

axioms (1)
  • domain assumption B-spline can accurately represent robot trajectories under the given sensor constraints
    Invoked when parameterizing the pose trajectory in continuous time
invented entities (1)
  • virtual anchors no independent evidence
    purpose: To provide additional ranging constraints and reject outliers when real anchors are sparse
    Constructed in preprocessing from VIO motion priors and UWB measurements

pith-pipeline@v0.9.0 · 5520 in / 1160 out tokens · 33809 ms · 2026-05-10T11:42:00.979093+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages

  1. [1]

    Z. Chen, Y . Hou, R. Huang, Q. Cheng, Neural network compensator- based robust iterative learning control scheme for mobile robots non- linear systems with disturbances and uncertain parameters, Appl. Math. Comput. 469 (2024) 128549

    work page 2024

  2. [2]

    Huang, M

    Z. Huang, M. Chen, P. Shi, H. Shen, Adaptive neural network control for fixed-wing UA V with disturbance observer under switching disturbance, IEEE Trans. Netw. Learn. Syst. 36 (6) (2025) 11520 – 11533

    work page 2025

  3. [3]

    C. Wang, W. Zhao, S. Lv, H. Shen, J. H. Park, Predefined-time optimized tracking control for QUA Vs with extended state observer and global prescribed performance, IEEE Trans. Automation Sci. Eng. 22 (2025) 19961 – 19973

    work page 2025

  4. [4]

    H. Shen, Y . Song, J. Wang, J. H. Park,H ∞ state estimation for PDT-switched coupled neural networks under round-robin protocol: A cooperation-competition-based mechanism, IEEE Trans. Neural Sci. Eng. 10 (2) (2023) 911–921

    work page 2023

  5. [5]

    J. Wang, X. Hu, J. Cao, J. H. Park, H. Shen,H ∞ state estimation for switched inertial neural networks with time-varying delays: A persistent dwell-time scheme, IEEE Trans. Syst. Man, Cybern. Syst. 52 (5) (2022) 2994–3004

    work page 2022

  6. [6]

    F. Li, Y . Cai, L. Su, H. Shen, S. Xu,H ∞ state estimation for two-time- scale Markov jump complex networks under analog fading channels: A hidden-markov-model-based method, IEEE Trans. Circuits Syst. I, Reg. Papers 71 (8) (2024) 3833–3842

    work page 2024

  7. [7]

    T. Qin, P. Li, S. Shen, Vins-mono: A robust and versatile monocular visual-inertial state estimator, IEEE Trans. Robot. 34 (4) (2018) 1004– 1020

    work page 2018

  8. [8]

    M. K. Paul, K. Wu, J. A. Hesch, E. D. Nerurkar, S. I. Roumeliotis, A comparative analysis of tightly-coupled monocular, binocular, and stereo VINS, in: Proc. IEEE Int. Conf. Robot. Autom. (ICRA), IEEE, 2017, pp. 165–172. 9

    work page 2017

  9. [9]

    Geneva, K

    P. Geneva, K. Eckenhoff, W. Lee, Y . Yang, G. Huang, Openvins: A research platform for visual-inertial estimation, in: Proc. IEEE Int. Conf. Robot. Autom. (ICRA), IEEE, 2020, pp. 4666–4672

    work page 2020

  10. [10]

    Felicioni, B

    S. Felicioni, B. M. Rizzo, C. Tortorici, G. Costante, Vision-based topological localization for MA Vs, IEEE Robot. Autom. Lett. 9 (2) (2023) 1158–1165

    work page 2023

  11. [11]

    J. Xu, T. Li, H. Wang, Z. Wang, T. Bai, X. Hou, Intermittent VIO- assisted LiDAR SLAM against degeneracy: Recognition and mitigation, IEEE Trans. Instrum. Meas. 74 (2024) 8500613

    work page 2024

  12. [12]

    Cheng, M

    Y . Cheng, M. Jiang, Y . Liu, MS-VRO: a multistage visual-millimeter wave radar fusion odometry, IEEE Trans. Robot. 40 (2024) 3004–3023

    work page 2024

  13. [13]

    W. Li, G. Wang, Q. Zhang, J. Liu, LG-VIWO: Visual-inertial-wheel odometry leveraging the depth-aided local ground constraints for mobile robots, IEEE Trans. Instrum. Meas. 74 (2025) 8507911

    work page 2025

  14. [14]

    T. Qin, S. Cao, J. Pan, S. Shen, A general optimization-based frame- work for global pose estimation with multiple sensors, arXiv preprint arXiv:1901.03642 (2019)

    work page Pith review arXiv 1901

  15. [15]

    Z. Gong, P. Liu, F. Wen, R. Ying, X. Ji, R. Miao, W. Xue, Graph-based adaptive fusion of GNSS and VIO under intermittent GNSS-degraded environment, IEEE Trans. Instrum. Meas. 70 (2020) 1–16

    work page 2020

  16. [16]

    T. H. Nguyen, T.-M. Nguyen, M. Cao, L. Xie, Loosely-coupled ultra- wideband-aided scale correction for monocular visual odometry, Un- manned Syst. 8 (02) (2020) 179–190

    work page 2020

  17. [17]

    Y . Gao, G. Li, A GNSS instrumentation covert directional spoofing algorithm for UA V equipped with tightly-coupled GNSS/IMU, IEEE Trans. Instrum. Meas. 72 (2023) 1–13

    work page 2023

  18. [18]

    T. Liu, B. Li, G. Chen, L. Yang, J. Qiao, W. Chen, Tightly coupled integration of GNSS/UWB/VIO for reliable and seamless positioning, IEEE Trans. Intell. Transp. Syst. 25 (2) (2023) 2116–2128

    work page 2023

  19. [19]

    Zhang, P

    J. Zhang, P. V . Orlik, Z. Sahinoglu, A. F. Molisch, P. Kinney, UWB systems for wireless sensor networks, in Proc. IEEE 97 (2) (2009) 313– 331

    work page 2009

  20. [20]

    J. P. Queralta, C. M. Almansa, F. Schiano, D. Floreano, T. Westerlund, Uwb-based system for uav localization in gnss-denied environments: Characterization and dataset, in: Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst. (IROS), IEEE, 2020, pp. 4521–4528

    work page 2020

  21. [21]

    Taponecco, A

    L. Taponecco, A. A. D’Amico, U. Mengali, Joint TOA and AOA estima- tion for UWB localization applications, IEEE Trans. Wirel. Commun. 10 (7) (2011) 2207–2217

    work page 2011

  22. [22]

    R. Liu, Y . He, C. Yuen, B. P. L. Lau, R. Ali, W. Fu, Z. Cao, Cost- effective mapping of mobile robot based on the fusion of UWB and short-range 2-D LiDAR, IEEE/ASME Trans. Mechatron. 27 (3) (2021) 1321–1331

    work page 2021

  23. [23]

    L. Tan, X. Song, Y . Zhang, Maximum correntropy unscented filter for simultaneous unknown input and state estimation under non-gaussian disturbances, Appl. Math. Comput. 516 (2026) 129868

    work page 2026

  24. [24]

    Bloesch, S

    M. Bloesch, S. Omari, M. Hutter, R. Siegwart, Robust visual inertial odometry using a direct EKF-based approach, in: Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst. (IROS), IEEE, 2015, pp. 298–304

    work page 2015

  25. [25]

    M. Li, A. I. Mourikis, High-precision, consistent EKF-based visual- inertial odometry, Int. J. Robot. Res. 32 (6) (2013) 690–711

    work page 2013

  26. [26]

    Leutenegger, P

    S. Leutenegger, P. Furgale, V . Rabaud, M. Chli, K. Konolige, R. Sieg- wart, Keyframe-based visual-inertial slam using nonlinear optimization, Proc. Robotics Sci. Syst. (RSS) (2013)

    work page 2013

  27. [27]

    J. Wang, Z. Huang, H. Shen, J. H. Park, A parallel homotopic optimized control scheme of uncertain nonlinear Markov jump systems and its applications, IEEE Trans. Automation Sci. Eng. 22 (2025) 19403 – 19414

    work page 2025

  28. [28]

    Y . Zhou, X. Zhang, F. Ding, Partially-coupled nonlinear parameter optimization algorithm for a class of multivariate hybrid models, Appl. Math. Comput. 414 (2022) 126663

    work page 2022

  29. [29]

    B. Yang, J. Li, H. Zhang, Resilient indoor localization system based on UWB and visual–inertial sensors for complex environments, IEEE Trans. Instrum. Meas. 70 (2021) 1–14

    work page 2021

  30. [30]

    H. Chen, L. Zhang, D. Li, J. Xu, W. Yang, Z. Yang, Refloc: A resilient evolutionary fusion framework for robust indoor localization, IEEE Trans. Instrum. Meas. 73 (2024) 1–13

    work page 2024

  31. [31]

    Zhang, X

    L. Zhang, X. Zhou, D. Li, Z. Yang, HCCNet: Hybrid coupled cooper- ative network for robust indoor localization, ACM Trans. Sens. Netw. 20 (4) (2024) 1–22

    work page 2024

  32. [32]

    Y . Fan, X. Huang, Z. Wang, J. Xia, H. Shen, Discontinuous event- triggered control for local stabilization of memristive neural networks with actuator saturation: Discrete-and continuous-time lyapunov meth- ods, IEEE Trans. Netw. Learn. Syst. 34 (4) (2021) 1988–2000

    work page 2021

  33. [33]

    Cioffi, T

    G. Cioffi, T. Cieslewski, D. Scaramuzza, Continuous-time vs. discrete- time vision-based SLAM: A comparative study, IEEE Robot. Autom. Lett. 7 (2) (2022) 2399–2406

    work page 2022

  34. [34]

    T. Liu, X. Zhang, X. Gao, Stability analysis for continuous-time and discrete-time genetic regulatory networks with delays, Appl. Math. Comput. 274 (2016) 628–643

    work page 2016

  35. [35]

    T ¨or¨ok, J

    C. T ¨or¨ok, J. Hud´ak, V . Pristaˇs, L. Antoni, Explicit forms of interpolating cubic splines and data smoothing, Appl. Math. Comput. 500 (2025) 129411

    work page 2025

  36. [36]

    Furgale, C

    P. Furgale, C. H. Tong, T. D. Barfoot, G. Sibley, Continuous-time batch trajectory estimation using temporal basis functions, Int. J. Robot. Res. 34 (14) (2015) 1688–1710

    work page 2015

  37. [37]

    Talbot, J

    W. Talbot, J. Nubert, T. Tuna, C. Cadena, F. D ¨umbgen, J. Tordesillas, T. D. Barfoot, M. Hutter, Continuous-time state estimation methods in robotics: A survey, IEEE Trans. Rob. 41 (2025) 4975 – 4999

    work page 2025

  38. [38]

    X. Lang, C. Chen, K. Tang, Y . Ma, J. Lv, Y . Liu, X. Zuo, Coco-lic: continuous-time tightly-coupled lidar-inertial-camera odometry using non-uniform b-spline, IEEE Rob. Autom. Lett. 8 (11) (2023) 7074– 7081

    work page 2023

  39. [39]

    B. He, W. Dai, Z. Wan, H. Zhang, Y . Zhang, Continuous-time LiDAR- inertial-vehicle odometry method with lateral acceleration constraint, in: 2023 IEEE Int. Conf. Robot. Autom. (ICRA), IEEE, 2023, pp. 3997– 4003

    work page 2023

  40. [40]

    Dellenbach, J.-E

    P. Dellenbach, J.-E. Deschaud, B. Jacquet, F. Goulette, Ct-icp: Real- time elastic lidar odometry with loop closure, in: 2022 IEEE Int. Conf. Robot. Autom. (ICRA), IEEE, 2022, pp. 5580–5586

    work page 2022

  41. [41]

    K. Li, Z. Cao, U. D. Hanebeck, Continuous-time ultra-wideband-inertial fusion, IEEE Rob. Autom. Lett. 8 (7) (2023) 4338–4345

    work page 2023

  42. [42]

    X. Lang, J. Lv, J. Huang, Y . Ma, Y . Liu, X. Zuo, Ctrl-VIO: Continuous- time visual-inertial odometry for rolling shutter cameras, IEEE Rob. Autom. Lett. 7 (4) (2022) 11537–11544

    work page 2022

  43. [43]

    J. Sun, W. Sun, G. Zhang, K. Yang, S. Li, X. Meng, N. Deng, C. Tan, CT- UIO: Continuous-time UWB-inertial-odometer localization using non- uniform B-spline with fewer anchors, arXiv preprint arXiv:2502.06287 (2025)

    work page arXiv 2025