Robust Trajectory-Tracking for a Bi-Copter Drone Using INDI: A Gain Tuning Multi-Objective Approach

Round 1

Reviewer 1 Report

Comments to the authors

Manuscript ID: robotics-1826098

Title: Robust trajectory-tracking for a Bi-copter Drone using INDI a gain tuning multi-objective approach

1) What is the contribution(s) of the manuscript? Provide a list of contributions at the end of Section 1 and give sufficient evidences to support your claim.

2) Literature review is just poor. There are many recent and relevant publications that combine a PID/PD controllers with trajectory planning schemes to control drones/UAVs. Consider discussing the following publications to enhance the literature review:

[R1]. “Constrained Control of UAVs in Geofencing Applications”, 2018. [https://doi.org/10.1109/MED.2018.8443035]

[R2]. “MIMO PID Controller Tuning Method for Quadrotor Based on LQR/LQG Theory”, 2019. [https://doi.org/10.3390/robotics8020036]

3) I failed to find the definitions of functions f(x), G(x), and h(x) in Equations (6) and (7).

4) Equation (9) is derived based upon the assumption that the lie derivative of the function h(x) is invertible. Is this a plausible assumption?

5) Equation (11): what do you mean by \dot{x}_0? Does that mean f evaluated at x_0 and u_0?

6) Assuming a small enough sampling time ΔT is used, the state variation between samples can 226 be considered negligible. This assumption can be restrictive in practice. More elaborations are required.

7) Equation (25) is confusing. G is a function of x in Equation (24), but a function of x and u in Equation (25).

8) Figure 9 provides objective values before and after optimization. However, I could not find the optimization problem.

Author Response

Thank you for your comments on our submitted manuscript. Below we provide brief statements on how the revised paper has been modified to address each and every one of your comments or concerns.

COMMENT #1:

The paper has been revised and a new paragraph has been added to the end of Section I with a detailed description of the contributions of the paper. The added list of contributions includes references to prior papers which provide readers with prior work from which they can better understand the contributions of the paper. Ten additional references have been added to the revised manuscript complementing and enhancing the previously cited works.

COMMENT #2:

The list of papers used as part of the literature review in the revised manuscript has been extended to include additional relevant papers as suggested by the reviewer. Furthermore, ten additional references have been added to the revised manuscript complementing and enhancing the previously cited works.

Some of the papers that the reviewer suggested being included are somewhat out of the scope of the paper’s discussion. New sentences and paragraphs have been added to the revised manuscript to make such points more evident so that readers will be able to better understand the presented work. The added referenced papers do include the other papers suggested by the reviewer.

COMMENT #3:

The revised paper provides a clear definition of the mathematical formulations used. Where appropriate a reference has been added from where the readers refer to, to fully understand the formulations presented in the paper. For this the following new reference has been added from where similar formulations are described:

Lu, Q., Sun, Y., Mei, S. (2001). Design Principles of Multi-Input Multi-Output Nonlinear Control Systems. In: Nonlinear Control Systems and Power System Dynamics. The Springer International Series on Asian Studies in Computer and Information Science, vol 10. Springer, Boston, MA.

COMMENT #4:

The revised paper provides an adequate description to deal with this concern. The revised paper clearly justifies the provided assumption that one can define diverse constraints to eliminate the associated problems or use diverse tools to handle the challenges of obtaining the inverse function.  

COMMENT # 5:

A proper definition and enhanced explanation of the mathematical formulation used were added to the revised paper.

COMMENT # 6:

For more clarification, minor changes were applied to the paper. The revised manuscript better justifies such an assumption by specifying that the state variable can be considered negligible when using high dynamic actuators.

COMMENT # 7:

The mentioned equations were indeed confusing. In the revised manuscript such formulations were revised making them more clearer.

COMMENT # 8:

To clarify the reviewer’s concern, an improved explanation has been added to the revised paper where the revised text makes explicit reference to Section 4 of the paper where the optimization problem and the associated cost functions defining the problem are described in detail.

Reviewer 2 Report

1. Overview and general recommendation:

The authors of the article have undertaken an interesting topic involving a nonlinear trajectory controller based on nonlinear dynamic inversion  including an outer NDI loop controller for position and an inner INDI loop controller for  orientation, is proposed. The proposed control architecture is improved with the introduction of a NSGA-II-PID controller. In this controller, the linear gains were optimized in an off-line process via the NSGA-II algorithm.

This paper  presents an optimized robust trajectory control system for an autonomous  tiltrotor Bi-copter based on an incremental nonlinear dynamic inversion (INDI) strategy combined  with a set of PID/PD controllers. The methodology includes two coupled controllers, one for position and one for attitude enabling the control of the coupled options of the bi-copter.

The performance of the control system was  tested over diverse flight paths including the  sever circular maneuver reported in this  paper.

The research problem was formulated correctly, and the researcher's methodology was also precisely presented. I have no comments on the scientific part.

The presented material corresponds to the profile of the Journal " Robotics". The scientific value of the submitted material qualifies the article for publication in this Journal.

2. Minor comments:

Please consider changing the layout of the manuscript by introducing the main points:

1.     Introduction

2.     Materials and methods

3.     Results and discussion

4. Conclusions

Author Response

Thank you for your comments on our submitted manuscript. Even though you had positive comments on our submitted paper below we provide brief statements on how the revised paper has been enhanced.

The paper has been revised and a new paragraph has been added to the end of Section I with a detailed description of the contributions of the paper. The added list of contributions includes references to prior papers which provide readers with prior work from which they can better understand the contributions of the paper. Ten additional references have been added to the revised manuscript complementing and enhancing the previously cited works.

The list of papers used as part of the literature review in the revised manuscript has been extended to include additional relevant papers as suggested by the reviewer. Furthermore, ten additional references have been added to the revised manuscript complementing and enhancing the previously cited works.

The revised paper provides a clear definition of the mathematical formulations used. Where appropriate a reference has been added from where the readers refer to, to fully understand the formulations presented in the paper.

Reviewer 3 Report

Comments to the Author:

  In this manuscript, the so called Robust trajectory-tracking for a Bi-copter Drone using INDI: a gain tuning multi-objective approach is analyzed and modelled.

Nevertheless, the comented key issues must be re emphasized to the authors, due to the low content quality of this eidtion.

  (1) No numerical performance gain is explicitly included in the abstract of this manuscript. Obviously, there is too obvious similarity between this manuscript and   the ref (10) which is also the work of the authors, including the mathematical architecure and illustration resluts. The authors must provide sufficient novelty in this  manuscript.

  (2) What is meaning of symbol C in equ(1)? The complete Drone system block diagram must be included in this work, which must inlclded all function modules of the UAV, the wireless link subsystem, the   user controlling the UAV at the far end. Moreover, the input and output signals of all modules must be included in this block diagram. The mathematical expression of these inputs and outputs must be  explicitly presented.

  (3) Obviously, NDI and INDI is not firstly proposed in this work, the references must be added and labeled to this work.

  (4) The wind and outdoor atmosphere effct must be considered in this work to identify and test the applicability and robustness of the proposed scheme.

  (5) The meaning of all varibles must be added to this work. In addition, the variables must be labeled in Fig 2 and addition scenrio descriptions.

  (6) Why the unit is missing in Fig 6, and Fig. 7, 9, 10, 11? The most parameters are missing in this work, the authors must provide all paramters setting in table and the    references of all parameters must be labeled and the relevant actual modules or devices datasheet must be included as well for proving the implementation of the proposed scheme. 

  (7)  The performance comparison must be made the proposed scheme and the work in ref (10) and the work in ref: modeling and attitude control of bi-copter to illustrate the superiority of this work,

  (8)  The majority of the references is published more than 5 years ago. The authors must cover the relevant works in  latest 3 years and the superiority of this work must be discussed compared to the latest works.

Before all above modifications are made, I cannot recommend this manuscript to publish. 

Author Response

Thank you for your comments on our submitted manuscript. Below we provide brief statements on how the revised paper has been modified to address every one of your comments or concerns.

COMMENT # 1:

An additional paragraph was added to the Introductory section where the difference and superiority of this work compared to prior work has been added to the revised manuscript. The added statements clearly state the advances of the proposed controller with respect to prior work.

COMMENT # 2:

Equation 1, as well as all other equations, thought the manuscript, were rewritten to reduce the use of shorthand notations typically used in the robotics literature which might be confusing for readers not familiar with the use of the “c” and “s” to denote the typical cosine and sin functions use when representing rotational matrices and homogeneous transformations. All parameters used in the manuscript are now clearly defined in the revised paper.

A complete block system diagram of the drone being used in the paper was redrawn showing the aspects relevant to the topics presented and discussed in the paper. Specifically, Figure 2 was redrawn which now better complements Figure 1 in the revised manuscript.

COMMENT # 3:

A number of new references were added to the revised manuscript and control sections increasing the references by 10 from the previous version of the paper. The added references provide an improved state-of-the-art background related to the areas discussed in the paper. 

COMMENT # 4:

In the revised paper a new section (Section 5.3) was added where new results are presented showing the effects of wind disturbances on the performance of the aircraft and its controller. The new section provides relevant information showing the robustness of the proposed controller against external disturbances.

COMMENT # 5:

Definitions and labels have been added throughout the revised paper making the equations, figures, and plots easier to understand.

COMMENT # 6:

All figures showing the results of the work have been revised to include the units used. The revised figures also provide improved labels enabling readers to clearly identify the diverse curves shown in the plots.

COMMENT # 7:

Additional explanation defining the superiority and contributions of the presented work has been added to the revised paper. The revised paper better described the contributions and improvements from prior work by the authors of the paper and others.

COMMENT # 8:

The list of papers used as part of the literature review in the revised manuscript has been extended to include relevant and newer papers published within the past 5 years as suggested by the reviewer.  Now the citations used throughout the revised paper have been updated with more recent papers published in top conferences and journals

Round 2

Reviewer 1 Report

No further comment.

Author Response

Thank you for your comments on our submitted manuscript. Even though you had positive comments on our submitted paper below we provide brief statements on how the revised paper has been further enhanced.

The abstract and the conclusion section of the paper have been enhanced showing the comparative gain metrics showing the superiority of the proposed controller with respect to previously developed controllers for the same aircraft.

In addition, a new section (i.e., Section 5.3.1) has been added where a comparison between the results obtained with the proposed controllers and the results obtained with the previously developed Backstepping controller are presented. In such a section, results showing the robustness of the proposed controller to external disturbances (i.e., wind gusts) are presented. 

Reviewer 3 Report

The numerical and comparative gain metrics must be included in the Abstract and Conclusion of this manuscript to identify the superiority.

Author Response

Thank you for your comments on our submitted manuscript. Below we provide brief statements on how the revised paper has been modified to address every one of your comments or concerns.

COMMENT # 1:

As per your suggestion, the numerical and comparative gain metrics have been included in the Conclusion of this manuscript to identify the superiority of the proposed controller with respect to previously developed controllers for the same aircraft.

Furthermore, the abstract has been enhanced with a set of sentences detailing the comparison between the proposed and previously developed controllers.

In addition to the above-mentioned changes, a new section (i.e., Section 5.3.1) has been added where a comparison between the results obtained with the proposed controllers and the results obtained with the previously developed Backstepping controller are presented. In such a section, results showing the robustness of the proposed controller to external disturbances (i.e., wind gusts) are presented.