An Unscented Kalman Filter Based on the Adams–Bashforth Method with Applications to the State Estimation of Osprey-Type Drones Composed of Tiltable Rotor Mechanisms

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This article addresses the issue of insufficient computational efficiency of RK-UKF in unmanned aerial vehicle state estimation. It proposes to integrate the Adams Bashforth (AB) multi-step integration method into the sigma point time update stage of UKF. The research topic is in line with engineering practice, The overall structure of the paper is complete, and the experimental design is relatively sufficient. But I still have the following questions and suggestions,

1. The article mentions that the high-order AB method will diverge when the step size is too large (such as when the 4th order AB method h>0.08 diverges), but does not provide the appropriate step size range and engineering selection basis for different order AB methods. Suggest adding simple theoretical explanations (such as stability conditions for multi-step methods) and clarifying the optimal step size range for different scenarios to facilitate the practical application of the method.
2. The literature reviews on UKF should be further refined by considering the recent works such as ‘Dual-event-triggering ANFIS-based unscented Kalman filter for cluster cooperative navigation with measurement anomalies[J]. Chinese Journal of Aeronautics, 2025, in press, DOI: 10.1016/j.cja.2025.103968.’
3. The current computational efficiency analysis is only based on MATLAB simulation, without considering the actual hardware operation of the drone. Can this support the core conclusion of ‘computational efficiency’, based on what?
4. Have the reasons for the differences in key data in Tables 4 and 5 been analyzed? Suggest providing a brief explanation of the differences in core results.
5. The conclusion states that ‘for unmanned aerial vehicle models with more complex state equations, the improvement in computational efficiency is more significant.’ It is recommended to use specific indicators to describe the effectiveness of the method.

Author Response

Thank you very much for your exceptionally detailed and thoughtful review. We confirm that all comments/suggestions mentioned by you have been considered and addressed in the revised manuscript. For more details, please refer to the following point-to-point responses.

 

Comment 1- Reviewer's Comment:

The article mentions that the high-order AB method will diverge when the step size is too large (such as when the 4th order AB method h>0.08 diverges), but does not provide the appropriate step size range and engineering selection basis for different order AB methods. Suggest adding simple theoretical explanations (such as stability conditions for multi-step methods) and clarifying the optimal step size range for different scenarios to facilitate the practical application of the method.

Response:

Thank you for your valuable comment. Following your suggestion, we have added a detailed discussion on the stability limits for each order of the Adams-Bashforth method in the UAV simulation section.

The following explanation has been added to Section 7.6.2 (Comparison of AB Methods with Different Orders of Accuracy):

________________________________________

[Additions and corrections: Added to the end of Section 7.6.2 at line 606 to 623]

Based on these results and additional experiments, the maximum step sizes at which each order can perform stable estimation without divergence are summarized as follows:

%

\begin{itemize}

    \item \textbf{2nd-order AB method}: Stable up to $h = 0.24$ (verified by additional experiments up to $h=0.24$). Divergence occurs at $h \geq 0.25$.

    \item \textbf{3rd-order AB method}: Stable up to $h = 0.14$. Divergence occurs at $h \geq 0.15$.

    \item \textbf{4th-order AB method}: Stable up to $h = 0.07$. Divergence occurs at $h \geq 0.08$.

    \item \textbf{5th-order AB method}: Stable up to $h = 0.04$. Divergence occurs at $h \geq 0.05$.

    \item \textbf{6th-order AB method}: Stable up to $h = 0.02$. Divergence occurs at $h \geq 0.03$.

\end{itemize}

%

These experimentally obtained stability limits are consistent with the theoretical property of linear multistep methods that the absolute stability region shrinks as the order increases \cite{Ab-method_1, Ab-method_2}.

 

In practical applications, the choice of step size involves a trade-off between estimation accuracy and computational efficiency. A larger step size reduces computational load but may degrade accuracy or cause divergence, while a smaller step size improves stability and accuracy at the cost of increased computation time. Users should select the step size based on the specific requirements of their application, referring to the experimentally verified stability limits above as a guideline for the maximum allowable step size.

 

 

Comment 2- Reviewer's Comment:

The literature reviews on UKF should be further refined by considering the recent works such as ‘Dual-event-triggering ANFIS-based unscented Kalman filter for cluster cooperative navigation with measurement anomalies[J]. Chinese Journal of Aeronautics, 2025, in press, DOI: 10.1016/j.cja.2025.103968.’

Response:

Thank you for suggesting this important recent work. We have incorporated the reference into the introduction to strengthen the literature review.

The following sentence has been added to Section 1 (Introduction) within the paragraph discussing recent advances in UKF research:

 

[Additions and corrections: Recent studies at Section 1 at line 53 to 58]

 

Recently, Singh \cite{Singh_2022} has comprehensively reviewed the development of Gaussian filtering, including UKF.

In particular, recent research has been progressing on combining UKF with outlier and event-triggered mechanisms. For example, \cite{Gao_2025_CIAA} proposed a dual-event-triggered ANFIS-based UKF for cluster cooperative navigation with measurement anomalies, demonstrating its applicability to navigation problems in complex environments.

 

【Added to the refence list】

\bibitem{Gao_2025_CIAA} Gao, B.; et al. "Dual-event-triggering ANFIS-based unscented Kalman filter for cluster cooperative navigation with measurement anomalies." Chinese Journal of Aeronautics, in press (2025), DOI: 10.1016/j.cja.2025.103968.

 

 

Comment 3- Reviewer's Comment:

The current computational efficiency analysis is only based on MATLAB simulation, without considering the actual hardware operation of the drone. Can this support the core conclusion of ‘computational efficiency’, based on what?

Response:

We appreciate this important methodological question. To clarify the scope and validity of our computational efficiency analysis, we have added a remark in the experimental setup section.

The following text has been added to Section 7.5.2 (Computational Efficiency Comparison Experiment):

[Additions and corrections: added to the end of Section 7.5.2 at lime 549 to 566]

\begin{remark}

The purpose of this study's computational efficiency evaluation is to compare algorithm-specific computational loads using the MATLAB environment. The validity of this approach is based on the following points:

\begin{itemize}

\item \textbf{Elimination of Hardware Dependence}: MATLAB simulations eliminate hardware-dependent factors such as CPU architecture, memory bandwidth, and cache size, enabling a pure evaluation of the relative computational load differences between the proposed numerical integration method (the AB method) and the conventional method (the RK method).

\item \textbf{Indicator of Computational Amount}: In an onboard environment (flight control computer), the sampling frequency is high and computing resources are limited, making it important to reduce the number of floating-point operations and execution time to achieve the same accuracy. The reduction in computation time measured in MATLAB serves as a basic indicator of the effectiveness of processing speed improvement on an actual aircraft.

\end{itemize}

However, final verification requires implementation on an actual autopilot (e.g., Pixhawk) and benchmarking, which is considered a future challenge. This simulation demonstrates the theoretical and algorithmic advantages prior to actual implementation.

\end{remark}

 

 

Comment 4- Reviewer's Comment:

Have the reasons for the differences in key data in Tables 4 and 5 been analyzed? Suggest providing a brief explanation of the differences in core results.

Response:

Thank you for pointing out the need for a comparative analysis between Table 4 and Table 5. We have added a detailed discussion in Section 7.6.2 to clarify the reasons for the observed differences.

The following analysis has been added to Section 7.6.2 (Comparison of AB Methods with Different Orders of Accuracy) following the presentation of Table 5:

 

[Additions and corrections: Added at the end of Section 7.6.2 (after the stability considerations) at line 624 to 645]

\subsubsection{Comparative Analysis: RK Method vs. AB Method}

Comparing the results in Table \ref{sec04:totalrmse3method} (re-entry vehicle problem) and Table \ref{sec04:totalrmseabmethod} (UAV problem) reveals the following:

\begin{itemize}

\item \textbf{Accuracy Difference Between the RK and AB Methods}: At the same step size (e.g., $h=0.01$), the RMSE of the RK4-UKF (UAV: 0.47646) is nearly identical to the RMSE of the AB4-UKF (UAV: 0.47304), indicating no significant difference in accuracy between the two. This indicates that the RK method achieves high-order accuracy through multiple evaluations within a single step, while the AB method achieves similar accuracy with fewer function evaluations by utilizing past information.

\item \textbf{Relationship between Order and Stability}: For both problems, the higher the order of the AB algorithm, the smaller the upper limit of the step size for stable estimation. This is due to the theoretical property of numerical analysis that the absolute stability region shrinks as the order of the linear multistep algorithm increases.

\item \textbf{Problem Complexity and Computational Efficiency}: For the simple 3D reentry vehicle problem (Table 2), the computational time of the AB4-based UKF increased by approximately 2.9% compared to the Euler-based UKF, while for the complex 12D UAV model (Table 6), it was reduced by approximately 5.1%. This reversal phenomenon is thought to be due to the characteristics of matrix operations in high-dimensional models (regularity of memory access patterns, cache efficiency).

\end{itemize}

These results indicate a trade-off between computational efficiency and numerical stability. When applying this method to real problems, it is necessary to select an appropriate order based on the required estimation accuracy and acceptable step size.

 

 

Comment 5- Reviewer's Comment:

The conclusion states that ‘for unmanned aerial vehicle models with more complex state equations, the improvement in computational efficiency is more significant.’ It is recommended to use specific indicators to describe the effectiveness of the method.

Response:

Following your recommendation, we have revised the conclusion to include specific numerical indicators that clearly demonstrate the computational efficiency improvement.

The following text has been added to the second paragraph of Section 8 (Conclusion):

 

[Additions and Revisions: The second paragraph of Section 8 has been revised as follows, at line 685 to 696]

\item \textbf{Comparison of computational efficiency}: Comparison of total computation time relative to the Euler-based UKF confirmed the effectiveness of the proposed method.

\begin{itemize}

    \item For the simple 3-dimensional reentry vehicle problem (Table 2), the computation time of AB4-based UKF was approximately 2.9\% longer than Euler-based UKF (slower than Euler), whereas RK4-based UKF was approximately 9.4\% longer.

    \item For the complex 12-dimensional UAV model (Table 6), the computation time of AB4-based UKF was approximately 5.1\% shorter than Euler-based UKF (faster than Euler). In contrast, RK4-based UKF exhibited a more pronounced slowdown, with an approximately 20.0\% increase.

\end{itemize}

These results demonstrate that \emph{the computational time reduction effect achieved by the proposed AB method integration becomes more pronounced for models with more complex state equations and higher dimensions}.

 

Reviewer 2 Report

Comments and Suggestions for Authors

REVIEW OF

An Unscented Kalman Filter Based on the Adams-Bashforth Method with Applications to the State Estimation of Osprey-type Drones Composed of Tiltable Rotor Mechanisms

Keigo Watanabe, Soma Takeda and Isaku Nagai

 

The article addresses a typical positioning problem. The logic of the presentation is as follows. The authors formulate the initial thesis that there exists a class of problems for which the best model is a continuous–discrete system. The discrete part consists of observations arriving with a certain frequency. The continuous part corresponds to motion. The intervals between measurements are significant; therefore, during intervals without observations it is important to predict the state of the moving object accurately. For this purpose numerical methods for solving differential equations are used. The authors use the Adams–Bashforth Method (ABM) and demonstrate its advantage. For the actual estimation of position the authors use the Unscented Kalman Filter. This is one of the most well-known and widely used methods of nonlinear filtering. After reminding the reader what UKF and ABM are, the authors describe in detail the Osprey-type Drone model (motion–control–observation). The remaining part of the paper consists of a detailed computational experiment. Before considering the main 12-dimensional Osprey-type Drone model, the authors conduct an experiment “in the sandbox”: planar motion with range measurements on the plane.

The general impression of the authors’ work is very systematic: competence in all the addressed aspects is demonstrated in detail; the filter, the model, and the integration method are described with maximum care and precision. The exposition is consistent and accurate, with graphical and numerical confirmation.

However, in its present form the article cannot be published. It requires substantial revision of the review content and the structure. Specific recommendations are given below. First, the reviewer will attempt to formulate the general reasons behind these recommendations. Let us answer the question: what is this article about, what is its topic, what is its research area?

Let us start with the title: Unscented Kalman Filter and State Estimation. This means that the article concerns filtering, nonlinear filtering, and specifically the UKF filter. This is an area with about 35 years of research, involving leading scientists and producing many fundamental results. Is this the first “violin” in the work? Next, Osprey-type Drones. This refers to aerial vehicles, that is, UAVs, representing a very broad class of navigation problems that have also attracted the attention of many well-known and successful researchers.

In principle, the paper contains some discussion of the UKF — the original formulas from 35 years ago are reproduced. One might expect an answer to the question: what else? No such answer is given. There are other peculiarities. Concerning UAVs, there are no questions: the model is excellent and described in detail. In the sections devoted to this description there are no revolutionary ideas, but everything necessary for an excellent mathematical model is present, and everything is presented carefully and in detail, including a working version of drone control. The problem is that in this presentation there is no clear logic: the reader will not understand why so much attention is paid to the model, because it is “drowned” in details both above and below.

Finally, the third part of the work concerns numerical integration. On the one hand, this is such a typical engineering task that even its presence in the title may be questioned. At the same time, although it is understandable there, it is not entirely clear why so much attention is devoted to obvious and well-known statements. The focus on differential equations is confusing and does not allow the article to be positioned correctly. The authors do not propose anything new regarding the Adams–Bashforth Method; they simply use it and analyze the result. There is no scientific or innovative contribution here. Euler, Runge, and Adams methods are taken, used, and compared. This is correct, simple, and clear. But presenting this as an achievement only diminishes what is actually substantive and interesting in the work.

Despite the above, the study as a whole is not bad. The reviewer therefore attempts to formulate several recommendations on how the text could be revised in order to publish exactly this result. The reviewer believes that if the authors follow these recommendations, the text could reach a form that may be recommended for publication (the suggested references are approximate; the authors may perform their own analysis of the appropriate literature; these are examples and following them exactly is not required; however, it is strongly recommended to preserve the “spirit”).

1. Remove references 1–4 and the mention of SLAM. This is not related to the work.
2. The introduction should start from the problems of navigation of moving objects. There are many such problems.
   Bar-Shalom, Y., X.R. Li, and T. Kirubarajan. 2004. Estimation with Applications to Tracking and Navigation: Theory, Algorithms and Software.
   Determining position is a nonlinear filtering problem. There exists a well-developed theory of optimal filtering:
   Liptser, R.S., and A.N. Shiryaev. 2001. Statistics of Random Processes II. Applications. Berlin: Springer-Verlag. 402 p.
   However, in practice only the linear Kalman filter has practical application:
   Kalman, R.E. 1960. A new approach to linear filtering and prediction problems. J. Basic Eng. — ASME 82(1):35–45.
   because optimal filters cannot be implemented in practice due to computational difficulties. In practice they are replaced by nonlinear suboptimal filters. Among such filters the UKF occupies an important and popular position.
3. The discussion of UKF cannot be limited to classical works. First, many researchers have contributed to the development and improvement of the UKF, including well-known figures in the field: Simon Haykin, Kazufumi Ito, Rudolph van der Merwe, Simo Särkkä, Robert J. Elliott, Grigory Miller. These authors are responsible for the key ideas and for studies of the limitations and instability of the UKF. The problem is that the paper cites secondary articles with minimal contribution, while key researchers are not mentioned. This creates the impression that the structure of the research field is not well understood. Therefore, instead of references 9–13, 7–9 references to the foundational authors listed above should be included.
4. The most important modifications of the UKF cannot be left unmentioned. See how the citations are organized in the review by Abhinoy Kumar Singh, Major Development Under Gaussian Filtering Since Unscented Kalman Filter. In principle, citing this review and/or two or three similar reviews would be sufficient. It is not necessary to review all results. However, mentioning a few minor works and saying nothing about many important ones is very problematic. It is more correct and more academic to avoid situations where one evaluates others’ results as important or unimportant unless there is certainty about their exceptional significance. In this sense the mentioned review is a good example of a correct attitude toward the research field. Instead of citing this review directly, one may use its references and select 10–12 of the most cited works. The reviewer strongly recommends that the authors use scientific databases and choose highly cited works in such matters. In Google Scholar, the citation counts of key authors reach thousands.
5. The issue of numerical methods for solving stochastic differential equations. At present this issue is purely technical. There is no reason to list authors who use Euler, Runge, Adams, and other numerical methods. There exists the classical book Kloeden & Platen — Numerical Solution of SDE, as well as many good analogues. This is a matter of monographs rather than individual papers. Such a reference is sufficient; otherwise the authors will again be forced to review a wide field, which is unnecessary.
6. In addition to the above. One should also avoid providing individual references to justify the continuous–discrete model. There is the classical work Stochastic Processes and Filtering Theory edited by Andrew H. Jazwinski. Despite its age, it was there that such systems were first described and everything necessary regarding the justification of their application was already stated. This is sufficient and academically correct — there is no need to multiply entities.
7. There is also nothing about UAV navigation. Here difficulties with selecting citations may arise as well. However, one may proceed in the correct way: there are recent reviews (A Review of UAV Autonomous Navigation in GPS-Denied Environments (Robotics and Autonomous Systems, 2023), UAV Positioning Using GNSS: A Review of the Current Status (Drones journal, 2026), A Survey of Sensor-Based Autonomous UAV Localization (Springer, 2025)). One should rely on recognized and cited reviews rather than attempting to list everything.
8. Structure of the paper. The introduction, as you understand, must be changed almost completely, as well as the references. Please note that the reviewer strongly recommends limiting the number of references. This is a very niche work, and there is no need to provide many citations. Therefore the recommendations above focus on classical works and reviews.
9. Section 2. Two paragraphs with the notation of the estimates should remain, while the main content should be moved to Appendix 1. You have rewritten other authors’ papers and formulas that have been published in hundreds of works. Why?
10. Section 3. Subsections 3.2–3.3 should be reduced to a reference, and the content moved to an appendix. The formulas of the Runge method should not appear in the main part of a paper published in the journal Sensors. This belongs in a textbook.
11. Section 6 should follow next — the “sandbox” example. Even the reviewer was confused why this example appears between Sections 5 and 7.
12. Sections 4, 5, and 7 should be combined into a single section devoted to the Osprey-Type Drone. Here everything looks reasonable. The authors do not introduce anything fundamentally new, but they construct a coherent model carefully and systematically. Each step individually may be known, but together they form an integrated model, which is important.
13. In all sections with examples, formulas for numerical methods should be moved to an appendix.
14. Remove from the conclusion the sentence about future research “…applying the proposed method to the CD Derivative-Free EKF…”. There exists a very wide spectrum of filters. There are dozens of modifications of the UKF alone. Consider CKF, problems with uncertainties, such filters as the conditional-optimal filter (V.Pugachev), the conditional-minimax filter (A.Pankov), and the filter of linear pseudo-measurements (Lingren A., Bar-Shalom Y., Miller B.). Competing with the Extended Kalman Filter is neither a task nor a challenge.
15. The last and most difficult issue for the authors. Being absorbed by the complex motion model, the authors overlooked the complexity of the observations. The paper concerns nonlinear filtering, yet the measurements in the main example are linear. This can be easily verified: any filter, including the classical Kalman filter, will work perfectly with such observations. Possibly even better than the UKF. For linear measurements the update step is exact; the unscented transformation will not make the filter better because there is no room for improvement. The challenge is not other filters but other navigation problems. Remove telemetry from the observations. Solve the problem not onboard but using external observation with range and azimuth. That would be a challenge. If you can estimate position from observations of one or two radars within your 12-dimensional model, that would indeed be a challenge. Try to explain this to the reader.

Author Response

Response to Reviewer #2

 

Thank you very much for your exceptionally detailed and thoughtful review. We confirm that all comments/suggestions mentioned by you have been considered and addressed in the revised manuscript. For more details, please refer to the following point-to-point responses.

 

Overview

Thank you very much for your detailed and constructive comments. Many of your points are extremely valuable for improving the quality of this paper, and we will respond as follows. At the same time, there may be some misunderstandings regarding several points, so we would like to clarify the factual details.

In particular, we would like to emphasize the following three points, which are the core novelty of this paper:

1. UAV Model:
   The equations of motion, control laws, and control allocation matrix for the Osprey-type drone with a 2-DOF tiltable coaxial counter-rotating rotor derived in Sections 4-5 are all originally derived models in this paper.
2. Formulation of AB-UKF:
   Equations (26)-(31) in Section 3.3 represent a novel formulation for integrating the AB method into the time update of the UKF sigma point matrix. The method of storing and utilizing past sigma point information has not been found in existing research.
3. Quantitative Evaluation of Computational Efficiency:
   As shown in Table 2 and Table 6, we have demonstrated for the first time that the advantage of AB-UKF becomes more pronounced depending on the complexity of the state equation (reduction rate improvement from increase of 2.9% → decrease of 5.1%).

 

Responses to Each Item

Item 1:

[Remove references 1--4 and the mention of SLAM. This is not related to the work.]

Response:

Thank you for your comment. Since SLAM is not the subject of this study, we will delete references [1-4] and remove the mention of SLAM from the introduction.

Revision Plan:

Delete the SLAM-related statements at the beginning of the introduction and start directly with: "The navigation problem for moving objects is a fundamental challenge with important engineering applications \cite{BarShalom_2004}. The problem of position estimation is inherently formulated as a nonlinear filtering problem, and the theory of optimal filtering has been established by researchers such as \cite{Liptser_2001}."

 

Item 2:

[The introduction should start from the problems of navigation of moving objects. There are many such problems.

Bar-Shalom, Y., X.R. Li, and T. Kirubarajan. 2004. Estimation with Applications to Tracking and Navigation: Theory, Algorithms and Software.

Determining position is a nonlinear filtering problem. There exists a well-developed theory of optimal filtering:

Liptser, R.S., and A.N. Shiryaev. 2001. Statistics of Random Processes II. Applications. Berlin: Springer-Verlag. 402 p.

However, in practice only the linear Kalman filter has practical application:

Kalman, R.E. 1960. A new approach to linear filtering and prediction problems. J. Basic Eng. — ASME 82(1):35–45.

because optimal filters cannot be implemented in practice due to computational difficulties. In practice they are replaced by nonlinear suboptimal filters. Among such filters the UKF occupies an important and popular position.]

Response:

Following your suggestion, we will revise the introduction to begin with the navigation problem of moving objects and the importance of nonlinear filtering.

Revision Plan (Beginning of Introduction at line 19 to 24):

"The navigation problem for moving objects is a fundamental challenge with important engineering applications \cite{BarShalom_2004}. The problem of position estimation is inherently formulated as a nonlinear filtering problem, and the theory of optimal filtering has been established by researchers such as \cite{Liptser_2001}. However, in practice, only the linear Kalman filter proposed by Kalman \cite{Kalman_1960} and its nonlinear extensions are feasible for implementation. This is because optimal nonlinear filters are computationally too expensive to implement."

Additional References:

\bibitem{BarShalom_2004} Bar-Shalom, Y.; Li, X.R.; Kirubarajan, T. Estimation with Applications to Tracking and Navigation, Wiley, 2004.

\bibitem{Liptser_2001} Liptser, R.S.; Shiryaev, A.N. Statistics of Random Processes II: Applications, Springer, 2001.

\bibitem{Kalman_1960} Kalman, R.E. "A New Approach to Linear Filtering and Prediction Problems," J. Basic Eng., 1960.

 

Item 3:

[The discussion of UKF cannot be limited to classical works. First, many researchers have contributed to the development and improvement of the UKF, including well-known figures in the field: Simon Haykin, Kazufumi Ito, Rudolph van der Merwe, Simo Särkkä, Robert J. Elliott, Grigory Miller. These authors are responsible for the key ideas and for studies of the limitations and instability of the UKF. The problem is that the paper cites secondary articles with minimal contribution, while key researchers are not mentioned. This creates the impression that the structure of the research field is not well understood. Therefore, instead of references 9--13, 7--9 references to the foundational authors listed above should be included.]

Response:

Thank you for your valuable comment. We failed to properly cite the key researchers who contributed to the development of the UKF. We have added references including the researchers you pointed out (Haykin, Ito, van der Merwe, Särkkä, Elliott) and expanded the description in the introduction as follows.

Regarding Dr. Grigory Miller, we investigated his papers related to UKF but could not identify an appropriate primary source. We judged that the convergence analysis of UKF is covered by Ito & Xiong (2000), so we have refrained from individually citing Dr. Miller.

In the revised version, we have added the following seven references that contributed to the major developments of UKF, and clarified each contribution in the introduction:

Researcher	Contribution	Added Reference
Särkkä (2007)	Extension to continuous-time UKF	Sarkka_2007
van der Merwe (2004)	Square-root UKF	VanDerMerwe_2004
Ito & Xiong (2000)	Convergence analysis of Gaussian filters	Ito_2000
Haykin (2001)	Integration of neural networks and KF	Haykin_2001
Elliott (1995)	Hidden Markov models and stochastic control	Elliott_1995
Julier (2003)	Spherical simplex UT	Julier_2003
Singh (2022)	Review of Gaussian filtering including UKF	Singh_2022

Addition/Revision (Relevant part of the introduction at line 44 to 54)

Since the proposal of the UKF, many researchers have advanced its development and improvement. Särkkä \cite{Sarkka_2007} derived continuous-time and continuous-discrete versions of the UKF and applied them to nonlinear continuous-time filtering and re-entry vehicle tracking problems. van der Merwe \cite{VanDerMerwe_2004} proposed the square-root UKF to improve numerical stability, and Ito \& Xiong \cite{Ito_2000} analyzed the convergence of Gaussian filters, including the UKF. Julier \cite{Julier_2003} proposed the spherical simplex UT to reduce the number of sigma points. Meanwhile, Haykin \cite{Haykin_2001} advanced research on the integration of neural networks and Kalman filtering, demonstrating new possibilities for nonlinear adaptive filtering. Elliott \cite{Elliott_1995} contributed to the foundations of filtering theory through the theory of hidden Markov models and stochastic control. Recently, Singh \cite{Singh_2022} has comprehensively reviewed the development of Gaussian filtering, including the UKF.

Additional References (7 items)

%============================================================================

% 4. Major developments and improvements of UKF (newly added: 7 items)

%============================================================================

\bibitem{Sarkka_2007} Särkkä, S. On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems. \textit{IEEE Trans. Autom. Control} \textbf{2007}, \textit{52}, 1631--1641.

\bibitem{VanDerMerwe_2004} van der Merwe, R.; Wan, E.A. Sigma-Point Kalman Filters for Probabilistic Inference in Dynamic State-Space Models. Ph.D. Thesis, OGI School of Science \& Engineering, 2004.

\bibitem{Ito_2000} Ito, K.; Xiong, K. Gaussian Filters for Nonlinear Filtering Problems. \textit{IEEE Trans. Autom. Control} \textbf{2000}, \textit{45}, 910--927.

\bibitem{Haykin_2001} Haykin, S. (Ed.) \textit{Kalman Filtering and Neural Networks}; Wiley: New York, NY, USA, 2001.

\bibitem{Elliott_1995} Elliott, R.J.; Aggoun, L.; Moore, J.B. \textit{Hidden Markov Models: Estimation and Control}; Springer: New York, NY, USA, 1995.

\bibitem{Julier_2003} Julier, S.J. The Spherical Simplex Unscented Transformation. \textit{Proc. American Control Conf.} \textbf{2003}, 2430--2434.

\bibitem{Singh_2022} Singh, A.K. Major Development Under Gaussian Filtering Since Unscented Kalman Filter. \textit{IEEE/CAA J. Autom. Sinica} \textbf{2022}, \textit{9}, 1304--1325.

Items to be removed (Secondary sources corresponding to references 9–13)

The initial references 9–13 were as follows, but these will be replaced by the major contributors above:

Reference	Judgement
takeno (2011)	Retain (Important as integration of RK method with UT)
Kulikov_2017 (2017)	Retain (High-precision implementation of CD-UKF)
Kulikova_2022 (2022)	Retain (Same as above)
Takeno_2012 (2012)	Retain (Integration of Heun method with UT)
Knudsen_2019 (2019)	Retain (Importance of computational efficiency)

However, these references are retained as they are central to the numerical calculations in this paper.

 

Item 4:

[The most important modifications of the UKF cannot be left unmentioned. See how the citations are organized in the review by Abhinoy Kumar Singh, Major Development Under Gaussian Filtering Since Unscented Kalman Filter. In principle, citing this review and/or two or three similar reviews would be sufficient. It is not necessary to review all results. However, mentioning a few minor works and saying nothing about many important ones is very problematic. It is more correct and more academic to avoid situations where one evaluates others' results as important or unimportant unless there is certainty about their exceptional significance. In this sense the mentioned review is a good example of a correct attitude toward the research field. Instead of citing this review directly, one may use its references and select 10--12 of the most cited works. The reviewer strongly recommends that the authors use scientific databases and choose highly cited works in such matters. In Google Scholar, the citation counts of key authors reach thousands.]

Response:

Thank you for your valuable comment. We will cite Singh's review paper and refer to the major developments of the UKF.

Revision Plan (Addition to Introduction at line 53 to 58):

"Recently, Singh \cite{Singh_2022} has comprehensively reviewed the development of Gaussian filtering, including the UKF. In particular, recent studies have combined UKF with outlier handling and event-triggered mechanisms. For example, \cite{Gao_2025_CIAA} proposed a dual-event-triggering ANFIS-based UKF for cluster cooperative navigation with measurement anomalies, demonstrating potential applications to navigation problems in complex environments."

Additional References:

\bibitem{Singh_2022} Singh, A.K. "Major Development Under Gaussian Filtering Since Unscented Kalman Filter," IEEE/CAA J. Autom. Sinica, 2022.

\bibitem{Gao_2025_CIAA} Gao, B.; Ma, P.; Hu, G.; Zhong, Y.; Liu, Z.

Dual-event-triggering ANFIS-based unscented Kalman filter for cluster cooperative navigation with measurement anomalies.

\textit{Chinese Journal of Aeronautics} \textbf{2025}, in press, doi:10.1016/j.cja.2025.103968.

 

Note that Gao_2025_CIAA is an example of recent research results from Reviewer 1.

 

Item 5:

[The issue of numerical methods for solving stochastic differential equations. At present this issue is purely technical. There is no reason to list authors who use Euler, Runge, Adams, and other numerical methods. There exists the classical book Kloeden & Platen --- Numerical Solution of SDE, as well as many good analogues. This is a matter of monographs rather than individual papers. Such a reference is sufficient; otherwise the authors will again be forced to review a wide field, which is unnecessary.]

Response:

Thank you for your comment. For the fundamental explanation of numerical methods, we will cite the monograph by Kloeden & Platen.

Revision Plan (Added at the end of Section 3.1 at line 183 to 184):

"For the general theory of numerical solutions for stochastic differential equations, please refer to Kloeden & Platen \cite{Kloeden_1999}."

Additional Reference:

\bibitem{Kloeden_1999} Kloeden, P.E.; Platen, E. Numerical Solution of Stochastic Differential Equations, Springer, 1999.

 

Item 6:

[In addition to the above. One should also avoid providing individual references to justify the continuous--discrete model. There is the classical work Stochastic Processes and Filtering Theory edited by Andrew H. Jazwinski. Despite its age, it was there that such systems were first described and everything necessary regarding the justification of their application was already stated. This is sufficient and academically correct --- there is no need to multiply entities.]

Response:

Thank you for your comment. We will cite Jazwinski [1970] as the theoretical foundation for continuous-discrete systems.

Revision Plan (Section 2.1 at line 141 to 142):

"Such systems are called continuous-discrete (CD) systems, and the foundations of their filtering theory are detailed in Jazwinski \cite{Jazwinski_1970}."

Additional Reference:

\bibitem{Jazwinski_1970} Jazwinski, A.H. Stochastic Processes and Filtering Theory, Academic Press, 1970.

 

Item 7:

[There is also nothing about UAV navigation. Here difficulties with selecting citations may arise as well. However, one may proceed in the correct way: there are recent reviews (A Review of UAV Autonomous Navigation in GPS-Denied Environments (Robotics and Autonomous Systems, 2023), UAV Positioning Using GNSS: A Review of the Current Status (Drones journal, 2026), A Survey of Sensor-Based Autonomous UAV Localization (Springer, 2025)). One should rely on recognized and cited reviews rather than attempting to list everything.]

Response:

Thank you for your comment. We will cite recent review papers on UAV navigation.

Revision Plan (Added the following at Introduction at line 89 to 92)

Regarding Unmanned Aerial Vehicle (UAV) navigation, comprehensive reviews exist on topics such as autonomous navigation in GPS-denied environments \cite{UAVReview1}, UAV positioning using GNSS \cite{UAVReview2}, and sensor-based autonomous localization \cite{UAVReview3}. These reviews highlight the importance of filtering techniques in UAV navigation.

Additional References:

\bibitem{UAVReview1} Chang, Y.; Cheng, Y.; Manzoor, U.; Murray, J. A Review of UAV Autonomous Navigation in GPS-Denied Environments. \textit{Robot. Auton. Syst.} \textbf{2023}, \textit{170}, 104533. https://doi.org/10.1016/j.robot.2023.104533

\bibitem{UAVReview2} Jiang, C.; Zhou, X.; Chen, H.; Liu, T. UAV Positioning Using GNSS: A Review of the Current Status. \textit{Drones} \textbf{2026}, \textit{10}, 91. https://doi.org/10.3390/drones10020091

\bibitem{UAVReview3} Liu, H.; Long, Q.; Yi, B.; Jiang, W. A Survey of Sensor Based Autonomous Unmanned Aerial Vehicle (UAV) Localization Techniques. \textit{Complex & Intelligent Systems} 2025, 11, 371. https://doi.org/10.1007/s40747-025-01961-2

 

Item 8:

[Structure of the paper. The introduction, as you understand, must be changed almost completely, as well as the references. Please note that the reviewer strongly recommends limiting the number of references. This is a very niche work, and there is no need to provide many citations. Therefore the recommendations above focus on classical works and reviews.]

Response:

Following your suggestion, we will completely rewrite the introduction and exclude the old references [1]-[4] and [33]-[35]. A new introduction reflecting the revisions from Items 1-7 will be created.

 

Item 9:

[Section 2. Two paragraphs with the notation of the estimates should remain, while the main content should be moved to Appendix 1. You have rewritten other authors' papers and formulas that have been published in hundreds of works. Why?]

Response:

Thank you for your comment. The basic equations of the UKF are well-known, and detailed descriptions in the main text are redundant. We will condense Section 2 and move the detailed derivations to Appendix A.

Revision Plan:

• Compress the current Section 2 (approx. 2 pages) to about half.
• Move detailed equations (3)-(18) to Appendix A.
• Include only the concept of UKF and important equations in the main text.

 

Item 10:

[Section 3. Subsections 3.2--3.3 should be reduced to a reference, and the content moved to an appendix. The formulas of the Runge method should not appear in the main part of a paper published in the journal Sensors. This belongs in a textbook.]

Response:

Thank you for your comment. However, regarding Sections 3.2 and 3.3, there may be one important misunderstanding, so please allow us to explain.

What is described in Section 3.2 (Eqs. (23)-(25)) and Section 3.3 (Eqs. (26)-(31)) are not the general formulas for the ordinary Runge-Kutta method and Adams-Bashforth method. These are time-update formulas derived specifically for the UKF sigma point matrix $\mathcal{X}_k$ (an $n \times (2n+1)$ matrix defined in Eq. (23)).

In particular:

• The ordinary RK method is a time-update formula for a single state vector, but Eqs. (23)-(25) represent the time update for all elements of the sigma point matrix.
• Similarly, the ordinary AB method is a formula for a single state vector, but Eqs. (26)-(31) formulate for the first time a method to update the entire sigma point matrix at once using past sigma point matrices $\mathcal{X}{k-1}, \mathcal{X}{k-2}, ...$ and matrices $\boldsymbol{\mathcal{F}}c(\mathcal{X}{k-1}), \boldsymbol{\mathcal{F}}c(\mathcal{X}{k-2}), ...$ obtained by substituting them into a nonlinear function.

In particular, $\boldsymbol{\mathcal{F}}_c(\mathcal{X}_k)$ in Eq. (26) is a function evaluation matrix for the sigma point matrix, newly defined in this paper, which does not appear in the ordinary AB method.

Therefore, these equations are not mere "textbook restatements" but are original formulations for integrating numerical integration methods into the specific filtering method of UKF, and form the core of the novelty of this paper.

Regarding your point that they "should be moved to an appendix," we will respond with the following policy:

1. Section 3.3 (General formulas for AB-UKF): Keep in the main text as it is central to the novelty (Eqs. (26)-(31)).
2. Section 3.2 (General formulas for RK-UKF): Simplify and move detailed derivations to Appendix B.
3. Application formulas for the specific model (Sections 6.2.1-6.2.3, 7.2.1-7.2.3): Move to Appendices C and D.

Revision Plan (Added at the beginning of Section 3.3 at line 207 to 211):

"This $\boldsymbol{\mathcal{F}}_c(\mathcal{X}_k)$ is a matrix that stores the results of evaluating the nonlinear function for all sigma points at time $k$.

While the ordinary AB method uses function evaluation values for past state vectors, the feature of this method is that it stores function evaluation matrices for past sigma point matrices $\boldsymbol{\mathcal{F}}c(\mathcal{X}{k-1}), \boldsymbol{\mathcal{F}}c(\mathcal{X}{k-2}), \dots$ and uses them as 'past information'."

 

Item 11:

[Section 6 should follow next --- the "sandbox" example. Even the reviewer was confused why this example appears between Sections 5 and 7.]

Response:

Thank you for your comment. We will reorganize the position of Section 6 as follows:

1. Section 6: Falling object model (preliminary experiment)

However, each algorithm including the UT transformation will be moved to Appendix C.

This order creates a natural flow from validation with a simple model to a complex model.

 

Item 12:

[Sections 4, 5, and 7 should be combined into a single section devoted to the Osprey-Type Drone. Here everything looks reasonable. The authors do not introduce anything fundamentally new, but they construct a coherent model carefully and systematically. Each step individually may be known, but together they form an integrated model, which is important.]

Response:

Thank you for your comment. The descriptions related to the UAV model (Section 4) and control (Section 5) remain as they were originally.

This is because the derivation of the dynamics model for the Osprey with a variable tilt mechanism, the controller, and the control allocation problem are only available in our group's Japanese presentations.

However, in Section 7, all concrete UT transformation algorithms have been moved to Appendix D.

 

Item 13:

[In all sections with examples, formulas for numerical methods should be moved to an appendix.]

Response:

Thank you for your comment. We will respond with the following policy:

• Section 3.3 (General formulas for AB-UKF): Keep in main textas it is central to the novelty.
• Section 3.2 (General formulas for RK-UKF): Move to Appendix B and simplify.
• Application formulas for specific models: Move all to appendices.

Appendix Structure:

1. Move Section 2 (Basics of UKF) to Appendix A.
2. Simplify Section 3.2 (General formulas for RK-UKF) (details in Appendix B).
3. Move Section 6.2 (Specific formulas for falling object model) to Appendix C.
4. Move Section 7.2 (Specific formulas for UAV model) to Appendix D.

 

Item 14:

[Remove from the conclusion the sentence about future research "...applying the proposed method to the CD Derivative-Free EKF...". There exists a very wide spectrum of filters. There are dozens of modifications of the UKF alone. Consider CKF, problems with uncertainties, such filters as the conditional-optimal filter (V.Pugachev), the conditional-minimax filter (A.Pankov), and the filter of linear pseudo-measurements (Lingren A., Bar-Shalom Y., Miller B.). Competing with the Extended Kalman Filter is neither a task nor a challenge.]

Response:

Thank you for your important comment. We will remove the mention of "Derivative-Free EKF" and revise it to more realistic research directions.

Revision Plan (Conclusion Section at line 727 to 738):

(Before removal)
"Furthermore, applying the proposed method to the CD Derivative-Free EKF... is also an interesting challenge."

(After removal) As future issues, we will add:

\item \textbf{Reduction of the number of sigma points}: While the standard UKF uses $(2n+1)$ sigma points, the number can be reduced to $n+2$ or $n+1$ by using the spherical simplex UT \cite{Julier_2002, Julier_2003, Lozano_2008, Fu_2017} or the simplex UT \cite{Li_2007}. Combining these methods with the AB method is expected to further improve computational efficiency.

\item \textbf{Implementation on actual hardware}: Implement the proposed algorithm on a microcontroller (e.g., Pixhawk) for the UAV model used in this simulation, and verify the degree of improvement in estimation accuracy and computational efficiency in actual flight environments.

\item \textbf{Adaptive step size control}: Integrate the adaptive step size control based on the degree of nonlinearity proposed by Wang et al. \cite{Wang_2025} into AB-UKF to further optimize estimation accuracy and computational efficiency.

\end{itemize}

 

Item 15:

[The last and most difficult issue for the authors. Being absorbed by the complex motion model, the authors overlooked the complexity of the observations. The paper concerns nonlinear filtering, yet the measurements in the main example are linear. This can be easily verified: any filter, including the classical Kalman filter, will work perfectly with such observations. Possibly even better than the UKF. For linear measurements the update step is exact; the unscented transformation will not make the filter better because there is no room for improvement. The challenge is not other filters but other navigation problems. Remove telemetry from the observations. Solve the problem not onboard but using external observation with range and azimuth. That would be a challenge. If you can estimate position from observations of one or two radars within your 12-dimensional model, that would indeed be a challenge. Try to explain this to the reader.]

Response:

This is the most important point. The fact that the observations in the current UAV model are linear, and the true value of nonlinear filtering is not fully demonstrated, is a limitation of this study.

The "external observation using only range and azimuth" you pointed out is indeed an important challenge that tests the true value of nonlinear filtering. In the revised version, we will take the following actions:

1. State the limitation clearly: Clearly describe the current linear observation setting and acknowledge its limitations.
2. State the extension to nonlinear observations as a future challenge: Make the extension to observations using only range and azimuth the most important future task.

Revision Plan (Added at the end of Section 7.3 at line 469 to 475):

\begin{remark}

The observation setting used in this paper is linear (direct observation of position and attitude angles), and the advantages of nonlinear filtering are not fully utilized. In actual UAV navigation, nonlinear observations such as radar observations in GPS-denied environments are common. While the main focus of this paper is the analysis of the impact of state equation complexity on computational efficiency, and the linearity of observations is not an essential problem, extension to nonlinear observations is an important future task.

\end{remark}

Revision Plan (Added to Conclusion Section):

\item \textbf{Extension to nonlinear observations}: Verify the effectiveness of the proposed method under nonlinear observations using only range and azimuth. Specifically, we will tackle the problem of estimating a 12-dimensional state vector using only range and azimuth information from one or two radars, and verify whether AB-UKF can maintain its advantage in terms of computational efficiency even under nonlinear observations.

 

Emphasis on Novelty (Added to the Overall Response)

Finally, allow us to reiterate the novelty of this paper:

1. Novel Derivation of the UAV Model:
   The equations of motion, control laws, and control allocation matrix for the Osprey-type drone with a 2-DOF tiltable coaxial counter-rotating rotor shown in Sections 4-5 are original derivation results not found in existing literature. In particular, the control allocation matrix A (Eq. 43) is a highly novel achievement that achieves a sufficient and necessary 6-variable representation by introducing the interference efficiency $\eta_{if}$.
2. Novel Formulation of AB-UKF:
   Equations (26)-(31) in Section 3.3 represent a novel formulation for integrating the AB method into the time update of the UKF sigma point matrix. In particular, the method of storing and utilizing the results of function evaluations for past sigma point matrices is an original implementation for UKF, not found in the ordinary AB method.
3. Quantitative Evaluation of Computational Efficiency:
   From the comparison in Table 2 and Table 6, we have demonstrated that the computational efficiency of AB-UKF relative to Euler-UKF depends on the model complexity. For the simple 3D re-entry vehicle problem (Table 2), AB4-UKF was approximately 2.9\% \textit{slower} than Euler-UKF, whereas for the complex 12D UAV model (Table 6), it became approximately 5.1\% \textit{faster} than Euler-UKF. This reversal demonstrates that the advantage of the proposed method becomes more pronounced as the state equation becomes more complex. Furthermore, Table 4 and Table 5 systematically analyze the impact of the AB method's order and step size on estimation accuracy and stability, providing practical guidelines (e.g., for the 4th-order AB method, h < 0.08 is recommended).

 

 

Based on these novelties, we hope you understand that this paper contains theoretically and practically valuable contributions, not merely an "application of existing methods."

 

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Good job