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arxiv: 1907.09450 · v1 · pith:EGZ5XN75new · submitted 2019-07-22 · 📡 eess.SY · cs.SY· eess.SP

A New computation reduction based nonlinear Kalman filter

Milad Behvandi , Mohammad Azam Khosravi , Amir Abolfazl Suratgar This is my paper

Pith reviewed 2026-05-24 17:55 UTC · model grok-4.3

classification 📡 eess.SY cs.SYeess.SP
keywords nonlinear Kalman filterunscented Kalman filterextended Kalman filtersigma pointscomputation reductionstate estimationreal-time filtering
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The pith

A hybrid Kalman filter uses sigma-point propagation with EKF linearization to reach third-order accuracy at half the cost of the unscented filter.

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

The paper introduces a nonlinear state estimator that propagates the state mean and covariance using deterministic sigma points while applying EKF-style linearization to form the prior. This combination is claimed to deliver third-order Taylor series accuracy, cutting the unscented Kalman filter's computation load by about 50 percent and exceeding the extended Kalman filter's accuracy. The authors position the method as simpler and more numerically stable than prior reduced-cost UKF variants, making it practical for real-time use. Examples in the paper are presented to show the claimed gains in accuracy and efficiency.

Core claim

The central claim is that a Kalman filter variant using deterministic sigma-point sampling to propagate mean and covariance while employing EKF linearization for the prior achieves third-order accuracy and reduces UKF computation cost by roughly 50 percent, with better accuracy than the EKF.

What carries the argument

The hybrid framework that applies deterministic sigma-point propagation for the state mean and covariance together with EKF linearization for the prior.

If this is right

  • The filter maintains numerical stability for real-time applications.
  • Accuracy exceeds the EKF because of third-order propagation of mean and covariance.
  • Computation cost is reduced by about 50 percent relative to the UKF.
  • The method is simpler than other reduced-cost UKF variants previously proposed.

Where Pith is reading between the lines

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

  • The same mixing of sigma-point and linearization steps might be tried in other nonlinear estimators to trade accuracy against cost.
  • Verification on systems with stronger nonlinearity or higher state dimension would test whether the claimed savings and accuracy hold beyond the presented examples.
  • If the third-order property is robust, the filter could serve as a drop-in replacement in applications where EKF accuracy is insufficient but full UKF cost is prohibitive.

Load-bearing premise

The assumption that mixing sigma-point propagation with EKF linearization simultaneously produces third-order accuracy, halves computation, and avoids new approximation errors or instability.

What would settle it

A benchmark test on a standard nonlinear system where the method either fails to cut computation by 50 percent or shows accuracy no better than the EKF would falsify the claim.

Figures

Figures reproduced from arXiv: 1907.09450 by Amir Abolfazl Suratgar, Milad Behvandi, Mohammad Azam Khosravi.

Figure 4
Figure 4. Figure 4: x1 error of EKF, UKF and HUKF Fig.5. State covariance For this example, the computation reduction of new algorithm according to (26) (27) is ukf nkf ukf tt t  =65% for n=1, m=1, j=5 In table below the Time requirement and MSE of estimation of different filter method were investigated for 1000 simulation. TABLE 1: filters mean MSE and execution time for 1000 independent simulations. FILTER Execution Time (… view at source ↗
Figure 5
Figure 5. Figure 5: Fig.5. State covariance [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
read the original abstract

This article introduces a new algorithm for nonlinear state estimation based on deterministic sigma point and EKF linearized framework for priori mean and covariance respectively. This method reduces the computation cost of UKF about 50% and has better accuracy compared to EKF due to propagating mean and Covariance of state to 3rd order Taylor series. Several types of Kalman filter have been presented before to reduce the computation cost of UKF, however, this new KF is a better choice because of its simplicity, numerical stability, and accuracy for real-time implementation. Examples verify the effectiveness of the proposed method.

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 proposes a hybrid nonlinear Kalman filter that combines deterministic sigma-point propagation with an EKF-linearized framework specifically for the prior mean and covariance. It claims this yields a roughly 50% reduction in computation relative to the UKF while delivering third-order Taylor-series accuracy for both mean and covariance (hence superior accuracy to the EKF), together with simplicity and numerical stability suitable for real-time use; effectiveness is asserted via examples.

Significance. A correctly derived and validated hybrid filter of this type would be of practical interest for real-time nonlinear estimation, offering a potential middle ground between EKF speed and UKF accuracy. The significance is currently limited by the absence of an explicit derivation showing how EKF linearization on the prior can still recover third-order moments.

major comments (2)
  1. [Abstract] Abstract: the central claim that the method propagates mean and covariance 'to 3rd order Taylor series' while employing an 'EKF linearized framework for priori mean and covariance respectively' is internally inconsistent; standard first-order EKF linearization cannot restore the third-order terms of the UKF sigma-point expansion without an additional, unstated correction step.
  2. [Abstract] Abstract: the asserted 50% computation reduction relative to UKF is presented without any operation-count table, flop comparison, or reference to the specific sigma-point set and linearization steps that would produce that factor; this figure is load-bearing for the paper's contribution yet unsupported by derivation or data.
minor comments (1)
  1. [Abstract] Abstract: the sentence 'Several types of Kalman filter have been presented before to reduce the computation cost of UKF' lacks citations to the relevant prior art.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on the abstract. We will revise the manuscript to improve clarity on the hybrid approach and provide the requested supporting analysis and derivation.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the method propagates mean and covariance 'to 3rd order Taylor series' while employing an 'EKF linearized framework for priori mean and covariance respectively' is internally inconsistent; standard first-order EKF linearization cannot restore the third-order terms of the UKF sigma-point expansion without an additional, unstated correction step.

    Authors: The abstract phrasing is ambiguous and we agree it requires clarification. The proposed method is a hybrid: deterministic sigma-point sampling is used to propagate the state and achieve third-order Taylor accuracy for the updated mean and covariance, while an EKF-style linearization is applied only to the prior mean and covariance computation to reduce the number of nonlinear function evaluations. We will revise the abstract for precision and add an explicit derivation (in the main text) showing how the hybrid retains the higher-order moments without requiring an extra correction term beyond the sigma-point mechanism. revision: yes

  2. Referee: [Abstract] Abstract: the asserted 50% computation reduction relative to UKF is presented without any operation-count table, flop comparison, or reference to the specific sigma-point set and linearization steps that would produce that factor; this figure is load-bearing for the paper's contribution yet unsupported by derivation or data.

    Authors: We agree the 50% reduction claim needs quantitative support. In the revision we will insert a computational complexity analysis, including an operation-count table. This will reference the specific sigma-point set (2n+1 points) and demonstrate that applying EKF linearization solely to the prior step halves the nonlinear propagations relative to a full UKF, yielding the stated reduction factor. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims presented as outcomes of proposed hybrid method

full rationale

The paper proposes a hybrid nonlinear Kalman filter using deterministic sigma-point propagation combined with EKF linearization for the prior mean and covariance. Performance assertions (approximately 50% UKF cost reduction and third-order Taylor accuracy) are stated as consequences of the algorithm rather than inputs or self-referential definitions. No equations, fitted parameters, or self-citations appear in the provided text that would reduce any central claim to its own construction. The derivation is self-contained against external references such as standard UKF and EKF.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Because only the abstract is available, the ledger records the minimal set of unverified premises required for the central claim to hold; no free parameters, invented entities, or additional axioms are extractable from the given text.

axioms (1)
  • domain assumption The hybrid sigma-point plus EKF-linearization construction propagates statistics to third-order Taylor accuracy while halving UKF cost
    This premise is asserted in the abstract but not derived or evidenced within the provided text.

pith-pipeline@v0.9.0 · 5630 in / 1265 out tokens · 26108 ms · 2026-05-24T17:55:54.431237+00:00 · methodology

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Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

  1. [1]

    A new method for the nonlinear transfo rmation of means and covariances in filters and estimators

    Julier, Simon, Jeffrey Uhlmann, and Hugh F. Durrant -Whyte. "A new method for the nonlinear transfo rmation of means and covariances in filters and estimators." IEEE Transactions on automatic control 45.3 (2000): 477-482

    work page 2000

  2. [2]

    A novel a priori state computation strategy for the unscented Kalman filter to imp rove computational efficiency

    Biswas, Sanat K., Li Qiao, and Andrew G. Dempster. "A novel a priori state computation strategy for the unscented Kalman filter to imp rove computational efficiency." IEEE transactions on Automatic Control 62.4 (2017): 1852-1864.W.-K. Chen, Linear Networks and Systems. Belmont, CA: Wadsworth, 1993, pp. 123–135

    work page 2017

  3. [3]

    Infrared navigation —Part I: An assessment of feasibility,

    J. U. Duncombe, “Infrared navigation —Part I: An assessment of feasibility,” IEEE Trans. Electron Devices, vol. ED-11, no. 1, pp. 34–39, Jan. 1959

    work page 1959

  4. [4]

    Computationally efficient simplex unscented Kalman filter based on numerical integration

    Hu, B. Q., et al. "Computationally efficient simplex unscented Kalman filter based on numerical integration." Asian Journal of Control 16.3 (2014): 915-921.E. H. Miller, “A note on reflec tor arrays,” IEEE Trans. Antennas Propagat., to be published

    work page 2014

  5. [5]

    Reduced sigma point filters for the propagation of means and covariances through nonlinear transformations

    Julier, Simon J., and Jeffrey K. Uhlmann. "Reduced sigma point filters for the propagation of means and covariances through nonlinear transformations." American Control Conference, 2002. Proceed ings of the 2002. Vol. 2. IEEE, 2002

    work page 2002

  6. [6]

    The spherical simplex unscented transformation

    Julier, Simon J. "The spherical simplex unscented transformation." American Control Conference, 2003. Proceedings of the 2003 . Vol. 3. IEEE, 2003. E. E. Reber, R. L. Michell, and C. J. Carter, “Oxygen absorption in the earth’s atmosphere,” Aerospace Corp., Los Angeles, CA, Tech. Rep. TR-0200 (4230-46)-3, Nov. 1988

    work page 2003

  7. [7]

    Julier, Simon J., and Jeffrey K. Uhlmann. A general method for approximating nonlinear transformations of probability dis tributions. Technical report, Robotics Research Group, Department of Engineering Science, University of Oxford, 1996

    work page 1996

  8. [8]

    A numerical -integration perspective on Gaussian filters,

    Y. Wu, D. Hu, M. Wu, and X. Hu, “A numerical -integration perspective on Gaussian filters,” IEEE Transactions on Signal Processing, vol. 54 , no.8, pp. 2910–2921, 2006

    work page 2006

  9. [9]

    Cubature Kalman filters,

    I. Arasaratnam and S. Haykin, “Cubature Kalman filters,” IEEE Transactions on Automatic Control, vol. 54, no.6, pp. 1254 –1269, 2009

    work page 2009

  10. [10]

    Gaussian filters for nonlinear filtering problems,

    K. Ito and K. Xiong, “Gaussian filters for nonlinear filtering problems,” IEEE Transaction s on Automatic Control, vol. 45, no.5, pp. 910–927, 2000

    work page 2000

  11. [11]

    Loosely Coupled INS/GPS Integration with Constant Lever Arm using Marginal Unscented Kalman Filter,

    G. Chang, “Loosely Coupled INS/GPS Integration with Constant Lever Arm using Marginal Unscented Kalman Filter,” Journal of Navigation, vol. 67, pp. 419–436, 2013

    work page 2013

  12. [12]

    Evaluation of matrix square root operations for UKF within a UAV GPS/INS sensor fusion application

    Rhudy, Matthew, et al. "Evaluation of matrix square root operations for UKF within a UAV GPS/INS sensor fusion application." International Journal of Navigation and Observation2011 (2012)

    work page 2012

  13. [13]

    On unscented Kalman filtering for state estimation of continuous-time nonlinear systems

    Sarkka, Simo. "On unscented Kalman filtering for state estimation of continuous-time nonlinear systems ." IEEE Transactions on automatic control 52.9 (2007): 1631-1641

    work page 2007

  14. [14]

    Characteristics Optimization of the Maglev Train Hybrid Suspension System Using Genetic Algorithm

    Safaei, Farhad, et al. "Characteristics Optimization of the Maglev Train Hybrid Suspension System Using Genetic Algorithm." IEEE Transactions on Energy Conversion 30.3 (2015): 1163-1170

    work page 2015

  15. [15]

    Unscented Filtering and Nonlinear Estimation,

    S. Julier and J. Uhlmann, “Unscented Filtering and Nonlinear Estimation,” Proceedings of the IEEE, vol. 92, pp. 401–422, Mar. 2004

    work page 2004

  16. [16]

    Maximum correntropy Kalman filter

    Chen, Badong, et al. "Maximum correntropy Kalman filter." Automatica 76 (2017): 70-77

    work page 2017

  17. [17]

    Computational complexity

    Papadimitriou, Christos H. Computational complexity. John Wiley and Sons Ltd., 2003

    work page 2003

  18. [18]

    SVD-based Kalman Filter Derivative Computation

    Tsyganova, Julia V., and Maria V. Kulikova. "SVD-based Kalman Filter Derivative Computation." IEEE Transactions on Automatic Control (2017)

    work page 2017

  19. [19]

    The unscented Kalman filter for nonlinear estimation

    Wan, Eric A., and Rudolph Van Der Merwe. "The unscented Kalman filter for nonlinear estimation." Adaptive Systems for Signal Processing, Communications, and Control Symposium 2000. AS -SPCC. The IEEE

    work page 2000

  20. [20]

    Industrial applications of the Kalman filter: A review

    Auger, François, et al. "Industrial applications of the Kalman filter: A review." IEEE Transactions on Indust rial Electronics 60.12 (2013): 5458-5471

    work page 2013

  21. [21]

    A sensor fault detection and isolation method in interior permanent - magnet synchronous motor drives based on an extended Kalman filter

    Foo, Gilbert Hock Beng, Xinan Zhang, and D. Mahinda Vilathgamuwa. "A sensor fault detection and isolation method in interior permanent - magnet synchronous motor drives based on an extended Kalman filter." IEEE Transactions on Industrial Electronics 60.8 (2013): 3485 - 3495

    work page 2013

  22. [22]

    Accurate numerical implementation of the continuous -discrete extended Kalman filter

    Kulikov, Gennady Yu, and Maria V. Kulikova. "Accurate numerical implementation of the continuous -discrete extended Kalman filter." IEEE Transactions on Automatic Control 59.1 (2014): 273-279

    work page 2014