Incremental Nonlinear Dynamics Inversion and Incremental Backstepping: Experimental Attitude Control of a Tail-Sitter UAV

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

1. A conclusion section should be added to the manuscript.

2, The manuscript should show a picture of the structural layout of the studied Tail-sitter UAVs, because the dynamics are different for different layouts.

3, The data in Table 3 of the manuscript is all zeroes and these results should be explained.

Author Response

Dear Reviewer,

The authors would like to thank your careful reading and review of the manuscript. Unfortunately, when submitting the manuscript, a small lapse took place, and the submitted version corresponded to an outdated document. In the correct version, which has already been uploaded, the remarks that were kindly provided by you have been addressed.

• Remark 1: A conclusion section should be added to the manuscript.
  • A conclusion was incorporated into the document, as the sixth section, providing the major conclusions of the research work, as well as some insight about future development.

• Remark 2: The manuscript should show a picture of the structural layout of the studied Tail-sitter UAVs, because the dynamics are different for different layouts.
  • In Figure 6, a picture of the experimental tail-sitter prototype is provided, allowing to understand the layout of the tail-sitter.

• Remark 3: The data in Table 3 of the manuscript is all zeroes and these results should be explained.
  • This issue has been corrected, and the adequate values were introduced.

Reviewer 2 Report

Comments and Suggestions for Authors

This paper presents an incremental control strategy for tail-sitter UAV systems. The contribution is unclear. There are multiple clear mistakes in the analysis.

 

Major issues are as follows.

1. In Lines 135 - 136, the authors mentioned, "… the sonar would sometimes be partially blocked by the tail …". This sounds like a design problem or could be fixed by redesign. While this problem could be partially solved through sensing and estimation, a redesign sounds more reasonable. In the meantime, the error introduced by the sonar sensor or barometer could be reduced by some other sensors.
2. In Eq. (24), the authors used the inversion of G_{kin, err}. Please provide proof that this matrix is always non-singular.
3. In Eq. (25), the authors introduced a positive-definite matrix K_1. Please show why it is always possible to find this PD matrix. A similar issue exists in Eq. (30).
4. The equation in Line 243 is incorrect. It doesn't equal the results obtained from Eqs. (30) and (29).
5. Does the saturated signals in Figure 2(b) cause problems?
6. In Figure 9(a), why are the signals shifted? Any analysis for the potential reasons?
7. The error in Figure 11(b) seems too large.

Author Response

Dear Reviewer,

The authors would like to thank your careful reading and review of the manuscript, and hope that it is now improved by addressing your remarks. Regarding the contribution of the paper, a small note has been included in the Introduction in order to make said contribution clearer.

• Remark 1: In Lines 135 - 136, the authors mentioned, "… the sonar would sometimes be partially blocked by the tail …". This sounds like a design problem or could be fixed by redesign. While this problem could be partially solved through sensing and estimation, a redesign sounds more reasonable. In the meantime, the error introduced by the sonar sensor or barometer could be reduced by some other sensors.
  • The authors understand that the option of redesigning the UAV to avoid or minimize the sonar-related issues would be reasonable, but, since a commercial platform was to be used (the X-Vert VTOL), the barometer solution was favoured to a redesign. Furthermore, the central issue is that the HC-SR04 sonar has an angle of effect of 15 degrees, meaning that readings it provides may become unreliable for angles larger than this, which are to be expected in tail-sitters. The partial block by the tail of the UAV only exacerbated this issue at smaller angles. Nevertheless, the authors understand the possible confusion regarding how these aspects were phrased, and the beginning of subsection 2.2 has been rewritten.

• Remark 2: In Eq. (24), the authors used the inversion of G_{kin, err}. Please provide proof that this matrix is always non-singular.
  • This was an aspect that has been overlooked, and it is extremely relevant. A note has been introduced after Eq. (25) that provides the analytical expression for the determinant of G_{kin,err} and states that this matrix will always be invertible for the expected conditions of controlled flight.

• Remark 3: In Eq. (25), the authors introduced a positive-definite matrix K_1. Please show why it is always possible to find this PD matrix. A similar issue exists in Eq. (30).
  • Whilst the authors understand the relevance of the demonstration of K_1 and K_2, we believe that would be out of place in this article. Therefore, a suitable reference (Nonlinear Systems, by Hassan K. Khalil) has been introduced that accounts for the usage of gain constants in the backstepping design approach.

• Remark 4: The equation in Line 243 is incorrect. It doesn't equal the results obtained from Eqs. (30) and (29).
  • From what the authors noticed, the inclusion of the Input Scaling Gain (\lambda^{IBKS}) caused the aforementioned inequality, and therefore was removed from Eq. (30). The remark after Eq. (31) was rewritten to include said IGS.

• Remark 5: Does the saturated signals in Figure 2(b) cause problems?
  • The time where the signals in Figure 2(b) saturated was during take-off and did not introduce additional problems. This can be attributed to the absence of filtering in the command-filter during the Hardware-in-the-loop simulations, which was justified by the introduced delay, as described in the text. Furthermore, the HITL simulations were taken as a qualitative validation of the hardware implementation, and no exhaustive effort was taken to tune these parameters, which could, eventually, minimize the moments when the signals become saturated during take-off.

• Remark 6: In Figure 9(a), why are the signals shifted? Any analysis for the potential reasons?
  • The difference between the blue and black curves in Fig. 9(a) is related with the difference in the initial orientation, as explained in the text: the FC assumes that q1 and q3 start as zero (blue lines), but the MCS accounts for a small initial rotation (black lines).

• Remark 7: The error in Figure 11(b) seems too large.
  • Regarding altitude control, the experimental trials for both the INDI (Figure 11(a)) and IBKS (Figure 11(b)) show that the altitude estimation requires further work, but in (Figure 11(b)) the discrepancy between the estimated altitude (blue) and the one assumed as true (black) is larger. Nevertheless, as stated in the text, it allowed for stabilization on a given altitude without significant oscillation, which in turn enabled the testing of different attitude control solutions, which was the focus of this article. We understand that the altitude estimation and control require significant future work, and we intend to tackle this topic in the future.

Reviewer 3 Report

Comments and Suggestions for Authors

This paper aims to develop two nonlinear control strategies for a tail-sitter UAV. Specifically, an incremental nonlinear dynamics inversion and a backstepping control are implemented in the studied UAV. The paper shows the implementation of the control strategies in a microcontroller unit and validates it in a hardware-in-the-loop environment.  However, the paper is incomplete from the fourth section onwards. According to what is understood from the writing, it seems that the only novelty is the implementation of two existing techniques in a hardware-in-the-loop environment. That is, there is no theoretical contribution regarding control algorithms. There is no compelling justification for the simulator to demonstrate its relevance.

Comments to improve the paper:

* Since there is no explicit statement of the paper's contribution, it appears that the only novelty is the implementation of two existing techniques in a hardware-in-the-loop environment. Then, please state explicitly the contribution of the paper.

* Throughout the article there is an abuse in the writing of what has been previously done by the authors and other works of literature. There seems to be no theoretical contribution in the article.

* Most of the literature review in the introduction is dedicated to the motivation of the topic and linear control techniques. From there the writing goes directly to the nonlinear techniques used in the manuscript (NDI and BKS), mentioning that they are widely used. However, an analysis is not presented where the research gap can be seen in which the research problem is contextualized. Please improve this analysis and present the problem tackled.

* It is suggested to expand the analysis of the literature by mentioning other recent nonlinear techniques applied to hybrid aircrafts, in this sense, a work that is worth mentioning is: modeling and passivity-based control for a convertible fixed-wing vtol, amc.

* Include the paper organization at the end of the introduction to improve the presentation.

* Ensure all acronyms and variables are defined in the first use.

* The relevance of the aircraft simulator should be emphasized and not only mention that it is published in a previous work. Was this simulator experimentally validated? Do you have any certification that guarantees the validity of the simulation? How accurate is the simulator with respect to the original aircraft? This point is very important if the contribution is only the implementation of the strategies found in the literature.

* For the model presented in section 2, it is necessary to specify the units of the variables. Since angles are normally expressed in radians. However, in the results the angles are shown in degrees. If the units differ it is necessary to place a note highlighting this point.

* Are q_1, q_2 and q_3 the orientation with respect to the North-East-Down frame? why they do not have units in figures 2 and 3?

* What is \mu in tables 1 and 3? what is \mu and sigma in Table 2? Include the definitions in the manuscript.

* Add names and units to the axis of the plot in Figure 4.

* The novelty of the manuscript is not clear, everything is taken from the literature and previous works of the authors, including the controllers, the simulator and experimental platform.

Author Response

Dear Reviewer,

The authors would like to thank your careful reading and review of the manuscript. Unfortunately, when submitting the manuscript, a small lapse took place, and the submitted version corresponded to an outdated document. We have tried reaching to the Editorial Office about this, but it seems that the document had already been sent. In the correct version, which has already been uploaded, many of the remarks that were kindly provided by you have been addressed and are explained next. We hope that you will recognize the scientific contribution of this paper, as we believe that it has been further improved by your attentive remarks.

• Remark 1: Since there is no explicit statement of the paper's contribution, it appears that the only novelty is the implementation of two existing techniques in a hardware-in-the-loop environment. Then, please state explicitly the contribution of the paper.
  • A small note has been included in the fourth paragraph of the Introduction in order to make said contribution clearer.

• Remark 2: Throughout the article there is an abuse in the writing of what has been previously done by the authors and other works of literature. There seems to be no theoretical contribution in the article.
  • The authors believe that this remark is addressed by the inclusion of the note described in the response to your first remark.

• Remark 3: Most of the literature review in the introduction is dedicated to the motivation of the topic and linear control techniques. From there the writing goes directly to the nonlinear techniques used in the manuscript (NDI and BKS), mentioning that they are widely used. However, an analysis is not presented where the research gap can be seen in which the research problem is contextualized. Please improve this analysis and present the problem tackled.
  • The literature review begins with a generic description of UAVs, then focusing on tail-sitters and their control challenges due to the complex aerodynamics phenomena and transitions (first paragraph). The linear control techniques are then described, and so are their limitations, which leads to the introduction of nonlinear techniques, NDI and BKS (second paragraph). Lastly, it is explained that these two methods still rely greatly on the model of the UAV, and their incremental versions, INDI and IBKS, are introduced (third paragraph). The third paragraph ends with “The limited model dependency makes them ideal candidates for the challenges involved in controlling tail-sitters, due to the difficulty in modelling the complex aerodynamics during transitions and the propeller/control surface interaction.”, thus linking the application of the INDI and IBKS as possible effective control strategies for tail-sitters, capable of addressing the control challenges described in the first paragraph.
  • Nevertheless, a note in the end of the third paragraph has been introduced, evidencing the research gap and how the proposed article intends to address it.

• Remark 4: It is suggested to expand the analysis of the literature by mentioning other recent nonlinear techniques applied to hybrid aircrafts, in this sense, a work that is worth mentioning is: modeling and passivity-based control for a convertible fixed-wing vtol, amc
  • We understand the relevance of the suggested article, but we believe that it would create a divergent path from the analysis of the literature. There are a number of different control techniques suitable for UAV stabilization and control, and it would not be possible to cover all of them, and therefore the introduction conducts the reader from conventional flight control techniques (linear control) to incremental/sensor-based methods (INDI/IBKS). Nevertheless, we will take this work in consideration when expanding our research in the future.

• Remark 5: Include the paper organization at the end of the introduction to improve the presentation.
  • The last paragraph of the introduction has been rewritten in order to include the references to the different sections, thus explaining how the paper is organizer.

• Remark 6: Ensure all acronyms and variables are defined in the first use.
  • The document has been reviewed extensively to ensure the definitions of the acronyms and variables, and hope that is now correct.
• Remark 7: The relevance of the aircraft simulator should be emphasized and not only mention that it is published in a previous work. Was this simulator experimentally validated? Do you have any certification that guarantees the validity of the simulation? How accurate is the simulator with respect to the original aircraft? This point is very important if the contribution is only the implementation of the strategies found in the literature.
  • The simulator has not been experimentally validated. The aircraft model which was published in the previous work relies greatly on a research paper that modelled the X-Vert with experimental identification (Chiappinelli, R.; Nahon, M. Modeling and Control of a Tailsitter UAV.), specially for the aerodynamics, and we have adapted it to include additional aspects such as a BLDC motor model to portray its effects in the actuation. We did not conduct extensive work to validate this aircraft model, instead opting to direct efforts to develop control strategies that are robust to possible (and probable) mismatches that may occur between the UAV model and the real aircraft. A note has been introduced at the end of section 2.1 (before Eq. (5)) explaining these aspects.
  • The authors understand that this remark may be due to the lack of experimental validation of the solutions, which was absent from the first submitted PDF file, for which we once again apologize.

• Remark 8: For the model presented in section 2, it is necessary to specify the units of the variables. Since angles are normally expressed in radians. However, in the results the angles are shown in degrees. If the units differ it is necessary to place a note highlighting this point.
  • The units for the different variables have been introduced in the beginning of section 2. A remark about the plots for delta_a and delta_e having been converted to degrees was introduced before figure 2.

• Remark 9: Are q_1, q_2 and q_3 the orientation with respect to the North-East-Down frame? why they do not have units in figures 2 and 3.
  • Yes, the three vectorial components represent the orientation of the UAV with respect to the NED frame, as stated in the beginning of section 2: “quaternion-expressed orientation in relation to this same frame, q^{NED} = [q_0, q_1, q_2, q_3]^{T}”.
  • From the available literature, the quaternion attitude representation was taken as unitless/dimensionless, similarly to rotation matrices. This is stated in the description of the variables in the beginning of section 2, as a response to your previous remark.

• Remark 10: What is \mu in tables 1 and 3? what is \mu and sigma in Table 2? Include the definitions in the manuscript.
  • These definitions have been included in the manuscript before Table 1, and are analogous to Table 3. The definitions for Table 2 have also been included.

• Remark 11: Add names and units to the axis of the plot in Figure 4.
  • The names of the variables and their units have been added to Figure 4.

• Remark 12: The novelty of the manuscript is not clear, everything is taken from the literature and previous works of the authors, including the controllers, the simulator and experimental platform.
  • The paper contributions were highlighted in the end of paragraph four of the Introduction.

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have addressed all my previous comments. Even though some responses are not perfectly as expected, they are acceptable at this point.  I have no further questions.

Author Response

Dear Reviewer,

The authors would like to thank you again for all your commentaries and suggestions in the review of the manuscript. We would like to underline the relevance, detail and objectivity in your analysis, which we believe was fundamental in improving our article.

Reviewer 3 Report

Comments and Suggestions for Authors

I thank the authors for their efforts in answering my questions. In my view, the contribution is marginal since, as mentioned in my first review report, the main work is about two control strategies taken from the literature applied to a tail-sitter UAV. In the modifications presented by the authors,  it is acknowledged that there are applications in the case of INDI for these uavs, and the main novelty argument is that the IBKS has not been applied to a tail-sitter UAV. The abstract of the article is equally weak in this sense, providing a very general description of the work without clarifying the contribution. It only points out that the main work is the application of INDI and IBKS in the tail-sitter UAV, it is applied to a simulator and applied experimentally. In this form, I think the manuscript might be more suitable for a conference, but it is not enough to justify a contribution to a journal.

Perhaps the most redeeming part would be the experimental validation of the algorithm. In this case, I suggest emphasising this part in the literature review and abstract. In fact, this is what the new title suggests is the main work: an experimental validation of two strategies. Consider including details about the experimental implementation of the algorithms and the scientific-technological challenges that arise when implementing the algorithms. On the other hand, given there is no theoretical contribution and the described simulator is not guaranteed to represent a real tail-sitter (it is not validated), I consider that this section of the manuscript does not contribute anything and can be omitted.

Author Response

Dear Reviewer,

The authors would like to thank you for the commentaries from your previous review report, and once again apologize for the confusion regarding the PDF version of the manuscript. Given the extension of your commentary in this review round, we took the liberty of dividing it into several components (shown in blue), and answer separately to each of them, as presented next.

 

I thank the authors for their efforts in answering my questions. In my view, the contribution is marginal since, as mentioned in my first review report, the main work is about two control strategies taken from the literature applied to a tail-sitter UAV. In the modifications presented by the authors,  it is acknowledged that there are applications in the case of INDI for these uavs, and the main novelty argument is that the IBKS has not been applied to a tail-sitter UAV. The abstract of the article is equally weak in this sense, providing a very general description of the work without clarifying the contribution. It only points out that the main work is the application of INDI and IBKS in the tail-sitter UAV, it is applied to a simulator and applied experimentally. In this form, I think the manuscript might be more suitable for a conference, but it is not enough to justify a contribution to a journal.

Whilst we understand your point of view, we believe that the contribution of our proposed article meets the criteria for the journal, given its threefold contribution components: 1) theoretically, we propose a quaternion-based Incremental Backstepping solution for controlling UAVs, original in the literature; 2) as an experimental contribution, the quaternion-based IBKS controller is then applied to a tail-sitter, with experimental validation in flight tests, together with the INDI controller; 3) the last component of the contribution is related to the analysis done, focused on evaluating the advantages and limitations of each incremental solution, effectively comparing the INDI and IBKS implementation for the same UAV, which is also absent from the literature.

To make these several contributions more evident, the third paragraph in the Introduction section has been modified to include the relevance of quaternion-based versions of the INDI and IBKS controllers, and more specifically the absence of quaternion-based Incremental Backstepping applications from the available literature, by stating that:

“a quaternion-based form of IBKS is absent from the current literature, and so are implementations of this control strategy to tail-sitters with experimental validation”.

Furthermore, the fourth paragraph of the Introduction was rewritten aiming to clearly state the aforementioned three contributions and their nature:

“Within this research article, three separate scientific contributions are offered: firstly, the deduction of a quaternion-based Incremental Backstepping is made, representing the theoretical contribution of this article; as a second contribution, this IBKS controller, together with the INDI solution that results from a previous research work [23], is implemented in the tail-sitter prototype and experimentally validated; and the third and final contribution is a systematic analysis of the INDI and IBKS when applied to the X-Vert VTOL tail-sitter, focusing on the aspects that differ, such as global stability proof and computational requirements, as well as the tracking results obtained from the experimental flight trials.”

We hope that this clarification will be sufficient to make the contributions of our manuscript more evident, and how these address the research gaps that exist in the current literature.

 

Perhaps the most redeeming part would be the experimental validation of the algorithm. In this case, I suggest emphasising this part in the literature review and abstract. In fact, this is what the new title suggests is the main work: an experimental validation of two strategies. Consider including details about the experimental implementation of the algorithms and the scientific-technological challenges that arise when implementing the algorithms.

The abstract has been reviewed, focusing on the experimental implementation and posterior validation of the IBKS and INDI controllers in a tail-sitter, as was the literature review as explained in the reply to the first part of your commentary. It can now be read in the abstract the following text, which we hope underlines the different scientific contributions, focusing on the experimental validation of both INDI and IBKS controllers:

“In this research article, a quaternion-based form of IBKS is originally deduced and applied to the stabilization of a tail-sitter in vertical flight, which is then implemented in a flight controller and validated in a Hardware-in-the-Loop simulation, which is also made for the INDI controller. Experimental validation with indoor flight tests of both INDI and IBKS controllers follows, evaluating their performance in stabilizing the tail-sitter prototype in vertical flight. Lastly, the tracking results obtained from the experimental trials are analysed allowing to draw an objective comparison between these controllers, evaluating their respective advantages and limitations.”

Regarding the details of the implementation, these are provided in Section 4, which already addresses the challenge related with the computational requirements of these controllers when implemented in a microcontroller. Furthermore, section 5.1 explains in detail the experimental setup, namely the utilized flight controller (section 5.1.1), the ground station (section 5.1.2), and the tools used for tracking the movement of the tail-sitter during the experiments (section 5.1.3). Lastly, some considerations about additional challenges are addressed within the results section 5.2.

 

On the other hand, given there is no theoretical contribution and the described simulator is not guaranteed to represent a real tail-sitter (it is not validated), I consider that this section of the manuscript does not contribute anything and can be omitted.

Although we understand that the simulator in itself does not represent a significant contribution of this scientific article, we believe that Section 2 should be kept in the manuscript as it introduces the relevant variables, which are crucial not only for simulation purposes but also for the experimental trials:

• in section 2.1, the simulator is briefly introduced, focusing on describing the state (x) and input (u) vectors, which are fundamental in understanding the behaviour of both the simulator and real UAV.
• in section 2.2, the output (z) and tracking (y) vectors are described, as these play a fundamental role in the experimental trials. Additionally, the usage of a barometer for altitude estimation is an aspect that differs from the previous work, which is explained in this subsection.

In order to underline the importance of these vectors in the experimental trials, a note has been included in the end of section 2.2.

Finally, in section 2.3, the affine form of the attitude kinematics (eq. (16)) and attitude dynamics (eq. (11)) are introduced, as so is the incremental form of the latter (eq. (14)), and the functions or constants necessary for these (eqs. (12), (13), (15), (17)). We believe that this section should also be kept in the manuscript as it introduces the different equations that are imperative for describing the application of the incremental controllers described in Section 3.

 

 

 

Round 3

Reviewer 3 Report

Comments and Suggestions for Authors

I have no more comments