A Novel GNSS Attitude Determination Method Based on Primary Baseline Switching for A Multi-Antenna Platform

Round 1

Reviewer 1 Report

This reviewer thinks manuscript can be published without changes. Congratulations.

Author Response

Thank you very much. Wish everything goes well with you !

Authors

Reviewer 2 Report

Eq 2 Include descrtiion of the terms not described in Eq 1

4.1: exlpin how the lever arm problem is solved. Specially with regards to the orientation of the antennas.
Figure 7 include IMU/INS while it has been stated IMu is not being employed. indicate it is used for reference and indicate range of costs of the IMU.
Figrue 7: indicate where the SPAN-FSAS is.

Figure 9,10, 11 include name of green line on the figur (basleine switching)

Match table 3 info with Figure 9, 10, 11 results. Not clear the numbers with the graphs.

Lines 370-373: provide more info about the low ratios of the antennas 2 and antenna 3, and how was does measured.

Include graphs with results if baseline is not swithc to see the benefits of the proposed strategy.

Explain why 4 antennas have been chosen and a small anylsis of a lower number of antennas. 

Explain how synchronisation of the 4 receives is done. 

In conclusions it is stated that INS will be included, what are the benefits expected? Quantify them. 

Include references so similar works with multiatenna strategies.

Author Response

Thank you for your review. We will explain specifically to your questions and correct our mistakes seriously. Some editorial errors have been modified. It's really our fault and all of them have been corrected in the revised version.

(1) Eq2 Include descrtiion of the terms not described in Eq 1

[Response]: Thank you for pointing it out. It’s our mistake. In revised manuscript, we will add m and ε in equation 1 and it will be highlighted.

(2) 4.1: explain how the lever arm problem is solved. Specially with regards to the orientation of the antennas.

[Response]: Thanks for your question. We use the commercial IE software to solve this problem, so the calculation of arm is finished by IE software. We only use the position solution of SPAN-FSAS as the reference trajectory. In the revised manuscript, we will explain this part.

(3) Figure 7 include IMU/INS while it has been stated IMu is not being employed. indicate it is used for reference and indicate range of costs of the IMU.

Figrue 7: indicate where the SPAN-FSAS is.

[Response]: SPAN technology combines two different but complementary technologies: GNSS positioning technology and inertial navigation technology. So it includes an IMU (IMU-FSAS-EI) and a high precision GNSS navigation receiver. This set of equipment in our laboratory is worth 400,000 RMB. In revised manuscript, we will indicate it is used for reference and the cost. We will also indicate where is IMU-FSAS-EI.

(4) Figure 9,10, 11 include name of green line on the figure (baseline switching)

[Response]: Thank you for pointing it out. We will follow your advice and add the name of green line. The revised figures will be highlighted.

(5) Match table 3 info with Figure 9, 10, 11 results. Not clear the numbers with the graphs.

[Response]: Thank you for your question. The “switching epochs” in Table 3 are the green dots in figure 9-11. We will change our statement and explain the numbers with the graphs.

Just like this, “In the following figures, green dots represent switching epochs. They are used to indicate the epochs of the corresponding period when primary baseline switching happen.”

(6) Lines 370-373: provide more info about the low ratios of the antennas 2 and antenna 3, and how was does measured. Include graphs with results if baseline is not switched to see the benefits of the proposed strategy.

[Response]: Thank you for your question. In the proposed method, ratio represents the ambiguity ratio test in LAMBDA algorithm. In dynamic test, the ratio of ambiguity fixed resolution is 2. In order to ensure the reliability of the fixed solution, we regard 3 as a ratio threshold. If the ratio of corresponding epoch is lower than 3, we will switch the primary baseline even if the ambiguity can be fixed. Ratio can show the positioning quality to a certain extent, so we use table 4 to further illustrate this problem.

In addition, if baseline is not switched, it corresponds to the blue line in Figure 9-11, but there will be no effective solution at the position of green dots. If we cannot obtain the reliable fixed solutions, the positioning results will be poor and the attitude determination results will not be credible at all. This conclusion is also verified according to Table 4. In this case, the results of attitude determination are far away from the reference value, so it is not suitable to be expressed by graphs.

(7) Explain why 4 antennas have been chosen and a small anylsis of a lower number of antennas.

[Response]: Thank you for your question. Theoretically, the more antennas, the better diversity effect will be obtained. However, the power consumption and calculation will be also increased. So we need to consider a compromise. The hardware platform selected in this paper can support four-way antenna, so the research is mainly focused on four-way antenna. In theory, two antennas can realize the orientation, three antennas can realize the attitude determination, and four antennas can increase the redundancy, which can verify the antenna switching method proposed in this paper when the ambiguity of a single antenna cannot be fixed.

(8) Explain how synchronization of the 4 receives is done.

[Response]: Thank you for your question. The selected M8T module of Ublox Company can directly output the original observations including accurate time information. The clock difference of receivers can be calculated and compensated by the algorithm in M8T according to the pseudorange information of multiple satellites. Therefore, the accuracy of the output time information is nanosecond level, which can meet the needs of research. There is no need for time synchronization between receivers. If an M8T module fails to output time information and original observations due to the insufficient number of satellites, the M8T module is invalid and will not participate in calculation.

(9) In conclusions it is stated that INS will be included, what are the benefits expected? Quantify them.

[Response]: Thank you for your question. The interval of GNSS receiver module is one second, (1Hz), while the sampling rate of INS is much higher than 1Hz, so it can provide data with higher sampling rate for auxiliary positioning. Moreover, the method of antenna switching mentioned in this paper is only applicable to the situation where the ambiguity of one antenna cannot be fixed. If the ambiguities of more antennas cannot be fixed, the attitude information still cannot be provided. At this time, the INS observations can maintain the output of attitude information in a short time, which will be the main work in the next stage. In revised manuscript, we will add this statement in our conclusions and they will be highlighted.

(10) Include references so similar works with multi antenna strategies.

[Response]: Thank you for your question. In fact, references 4, 6, 13-15, 25, 27 in Introduction are all the research based on multi-antenna platform. We will add more references and expand the introduction of multi antenna strategies.

Thanks for your review and wish everything goes well with you !

Author Response File: Author Response.docx

Reviewer 3 Report

The paper “A Novel GNSS Attitude Determination Method Based on Primary Baseline Switching for Multi-Antenna Platform” is devoted to experimental study of the attitude determination algorithm. However, the paper is not clear from several points of view and cannot be published in its present form.

The eq. 1-3 incorrectly written. The indexes of the designations are not explained (for example, “i”, “s”, “k”, “r” etc.) and some of them are changing for the same designation (the ambiguity “N” has different indexes in eq.(1)-(3)). It is unexplained how the double difference “DD” is calculated. The designation “delta_ref” is not explained in the text at all.

Eq. (5) needs clearer explanation. If the angles in eq.(5) are obtained using R from relation X_n=R*X_b then it is unclear why “h”, “p” and “h” do not depend on at least “sin(alpha_0)” from “X_b” vector. Also an explaining figure with these angles will add a clarity.

The paragraph in lines 169-176 is not clear. For attitude determination the measurements from at least two base lines are necessary. If one (of two available) baseline is not available as assumed in the text then the attitude cannot be calculated using direct method explained in the paper. The authors used the third baseline later in the experimental setup, but in Section 2.3 only two baselines are considered and it is unclear how the switching between them solves the problem of the system observability if one baseline is not available.

It is not explained what is the coefficient matrix presented in eq. (11) and how it was obtained?

The authors should add an explanation of the work principle of SPAN-FSAS navigation device which attitude determination results are considered as a true values in the experiments.

The time of the start and of the end of the presented experiment are not correctly presented as “529381” and “530300”. The time of the beginning should be a zero and the end time should be a time from the beginning. Please correct it in all the plots and in the Fig. 6 as well.

Why did the authors consider pairs of the oppositely directed baselines (like “0” and “180” deg)? The errors are just the same for each pair, as it should be if the same measurements are processed. It is strange that the difference between “0” and “180” is 10^-6 deg as presented in Fig. 8 in the first line of the plots while it should be a zero.

The error of the attitude determination is about several degrees using the proposed method. But the authors use too many digits after the dot in all the values like “4.5943 deg”. Such a precision in values is not necessary, two digits would be enough in the considered case.

It is not clear what is “epoch proportion” in the Section 4.3.1. What is “valid epoch”? Please, clarify.

The variance in Table 5 for “delta_r” is not correctly calculated, since 1.16^2=1.34, not 1.23 as presented in the Table.

Author Response

Dear Reviewer,

We would like to thank the anonymous reviewer for providing an opportunity to revise the manuscript. The comments and suggestions of the reviewer are all valuable and very helpful. We have studied them carefully and have made revisions to improve the manuscript. Revised portion are highlighted in the manuscript and the main corrections and additions are given below with a comment followed by a response (in red color).

Best regards,

Authors

Because of the pictures and formulas involved, please download the attachment to review our response, thank you !

Author Response File: Author Response.docx

Reviewer 4 Report

Please consider the attached file.

Comments for author File: Comments.pdf

Author Response

Thank you for your affirmation of our research. We will explain specifically to your questions and correct our mistakes seriously. Some editorial errors have been modified. It's really our fault and all of them have been corrected in the revised version.

Comments and Suggestions for Authors

(1) Lines 118-126 Provide adequate references for this model and equations

[Response]: Thanks for your advice and we will provide references.

Line 118 (Xu, 2007)

[28] Xu G. GPS: theory, algorithms and applications. Springer Science & Business Media Oct 5, 2007.

Line 126

[29] Kouba J (2009) A guide to using International GNSS Service (IGS) products. http://acc.igs.org/UsingIGSProductsVer21.pdf

[30] Shi Junbo, Xiuxiao Yuan*, Yang Cai and Gaojing Wang, “GPS Real-time Precise Point Positioning for Aerial Triangulation”, GPS Solutions, 2017, 21(2):405-414.

[31] Shi J , Gao Y . A comparison of three PPP integer ambiguity resolution methods[J]. GPS Solutions, 2014, 18(4):519-528.

(2) Line 134 m represents carrier phase multipath effect., N is ambiguity and is wavelength.

[Response]: Thanks for pointing it out. It’s really our mistake. In the revised manuscript, we will follow your advice and the changes will be highlighted.

(3) Line 136 Provide reference for the LAMBDA method.

[Response]: Thanks for your advice and we will provide references.

[36] Teunissen, P. J. G. Least-Squares Estimation of the Integer GPS ambiguities. Invited Lecture, Section IV Theory and Methodology, IAG General Meeting, Beijing, China, August, 1993.

[37] Teunissen, P. J. G. The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation. Journal of Geodesy.1995, 70, 65–82.

(4) Lines 140-141 Provide references for the mention methods.

[Response]: Thanks for your question. The mentioned methods in line 140-141 of the previous manuscript has been provided in Introduction, such as reference 6 (direct method), references 7-9 (method based on Wahba problem), reference 10 (least square method) and references 11-15 KF (Kalman filter method).

(5) Lines 195-199 I suggest rephrasing in a shorter statement

[Response]: Thanks for your advice. We will change our statement and they will be highlighted.

“If the selected temporary primary baseline is inconsistent with the starting point of the original primary baseline, the starting point does not need translation first.”

(6) Lines 210-211 “the derivation process of the error will be given:.”

[Response]: Thanks for pointing it out. It’s really our mistake. In the revised manuscript, we will follow your advice and the changes will be highlighted.

(7) Lines 225-226 “the closer pitch angle is to zero, the smaller roll angle error is” Should not this be evident from equation (13)? Please add an equation reporting the direct dependence between p and r or rephrase/include a statement to clarify this point.

[Response]: Thanks for pointing it out. It’s our mistakes. We will change our statement to clarify it and they will be highlighted.

(8) Lines 245-249 If the IMU is not used in this work, I suggest integrating this paragraph with the final one in chapter 5 (conclusions).

[Response]: Thanks for your advice. We will follow your advice and in revised manuscript, it will not be mentioned in chapter 3.

(9) Lines 271-272 “ and heading angle first decreases from 90° to 70°, then increases from 70° to 90° and finally slowly increases to about 140° ” Please indicate how the values of the heading angle were calculated.

[Response]: Thanks for your question. In fact, the change of heading angle is unknown during data collecting. The data given here are approximate values estimated from SPAN-FSAS solution results. According to your advice, we are going to change the way of expression, “the heading angle decreases first and then increases during the whole process”.

(10) Figures 9, 10, 11, 13  The use of the green dot is clearly expressed in the paragraph before Figure 9, nevertheless it is better to indicate it also in the legends or labels of the mentioned figures.

[Response]: Thanks for pointing it out. We will follow your advice and these figures will be highlighted.

(11) Lines 365-366 “the velocity of the car is relatively fast and the lane is changed frequently in this period, which makes the signals of some antennas lose lock in some epochs.” Could the vehicle which the algorithm is addressed to move as the car did during the experiment? What are the velocity and trajectory constraints suggested for the use of this algorithm?

[Response]: Thank you for your question. In this part, we just want to explain that the velocoty of the car is fast and the lane is changing several times. We did not use velocity and trajectory constraints and our hardware platform is mounted on a car. The main reason for losing lock is the multi-path effect of surrounding buildings or the signals attenuation brought by trees, which should not have a great relationship with the velocity and trajectory.

(12) Lines 435-436 “According to GNSS attitude determination theory, a directional precision of 0.6 degrees will be achieved by using a 1-meter long baseline.”

Add a reference to this statement.

[Response]: Thanks for your question. This conclusion is drawn from section 10.2.5 of the following references. We will add it in References part.

[38] Paul D. Groves. Principles of GNSS, Inertial, and Multisensor Integrated Navigation Systems, Second Edition, Artech House, 2013.

(13) Lines 438-439 “Therefore, on one hand, our experimental results show that the proposed method is applicable to small scale mobile platforms, such as UAV or a small driverless car.”

To evaluate if the accuracy of the algorithm is compatible with applications on UAV or driverless car, the performance of the referred vehicle should also be taken into account (i.e. velocity, maneuvers it can perform, ). The results of this research is promising but before applying it further analysis and qualification test are required. I would rephrase the statement to indicate the mentioned applications as a possibility for the future than for the present.

[Response]: Thanks for your question. What we want to express is that our hardware can be equipped on the UAV or driverless car (because of its small size). Indeed, as you said, to evaluate if the accuracy of the algorithm is compatible with applications on UAV or driverless car, the performance of the referred vehicle should also be taken into account. We will adjust our expression.

“Due to its small size, we plan to apply it on the UAV or driverless car. The results of the proposed method is promising but before applying it, analysis and qualification test are required.”

Thanks for your review and wish everything goes well with you!

Author Response File: Author Response.docx

Round 2

Reviewer 3 Report

After the revision, the paper became more clear. But the authors still need to address some of the suggestions that will improve the paper. The new comments are highlighted in green color in the attached file.

Comments for author File: Comments.pdf

Author Response

Thank you for your review. We will explain specifically to your questions and correct our mistakes seriously. Some figures are present in the attached file. Thank you very much.

Comments and Suggestions for Authors

(1) The eq. 1-3 incorrectly written. The indexes of the designations are not explained (for example, “i”, “s”, “k”, “r” etc.) and some of them are changing for the same designation (the ambiguity “N” has different indexes in eq.(1)-(3)). It is unexplained how the double difference “DD” is calculated. The designation “delta_ref” is not explained in the text at all.

[Response]: Thank you for pointing them out. In the revised manuscript, we will correct these mistakes and they will be highlighted.

[New comment]: Now the formulas become clearer.

[New Response]: Thanks for your review.

(2) Eq. (5) needs clearer explanation. If the angles in eq.(5) are obtained using R from relation X_n=R*X_b then it is unclear why “h”, “p” and “h” do not depend on at least “sin(alpha_0)” from “X_b” vector. Also an explaining figure with these angles will add a clarity.

[Response]: Thank you for pointing it out. It is really not clear in the previousmanuscript. The following is a more detailed derivation process:

[New comment]: The mathematical derivations now clear. But the authors steel should add a figure explaining the angles “h”,”r” and “p” for better understanding.

[New Response]: Thanks for your review. We will add a figure to explain angles “h”,“p” and “r”. Just like the following one.

(4) The paragraph in lines 169-176 is not clear. For attitude determination the measurements from at least two base lines are necessary. If one (of two available) baseline is not available as assumed in the text then the attitude cannot be calculated using direct method explained in the paper. The authors used the third baseline later in the experimental setup, but in Section 2.3 only two baselines are considered and it is unclear how the switching between them solves the problem of the system observability if one baseline is not available.

[Response]:Thanks for your question. In Section 2.3, the basic theory of attitude determination is mainly introduced.Only two baseline vectors 1 and 2 are involved, and three antennas are used. In practice, multiple antennas are generally used, taking four antennas as an example.

According to figure 5, three attitude angles can be calculated by taking antenna 4 as the reference, antenna 1 and 4 as the primary baseline, and antenna 3 and 4 as the subsidiary baseline. However,if the ambiguity of antenna 1 cannot be fixed due to the interference, antenna 1 will not participate in attitude determination.At this moment, if the ambiguity can be fixed in the observations of antenna 2, antennas 2 and 4 will be regarded as the primary baseline by the proposed method, and the baseline 3 - 4 will still be used as the subsidiary baseline. The attitude angles can be also obtained by the algorithm in section 2.3. The third baseline mentioned in this paper is essentially a backup baseline or alternate baseline, and only two baselines as explained in Section 2.3 are used during the attitude determination process.

[New comment]: Thank you for the response. Nevertheless some comments about subsidiary baselines that are extra for the two considered baselines are necessary in Section 2.3. Moreover, it is needed because the Figure 3 implies more then 2 baselines for the switching logic.

[New Response]: Thanks for your review. We will add subsidiary baselines in the figures and the revised version will be highlighted.

(5) It is not explained what is the coefficient matrix presented in eq. (11) and how it was obtained?

[Response]: Thanks for your question. It is really not clear in the previousmanuscript. The following is a more detailed derivation process:

[New comment]: Now that part of the text become better. Though, the coefficient matrix should be renamed as “gradient vector” in which direction the error ellipsoid is projected.

[New Response]: Thanks for your review. We will follow your advice and replace “coefficient matrix” as “gradient vector”

(6) The authors should add an explanation of the work principle of SPAN-FSAS navigation device which attitude determination results are considered as a true values in the experiments.

[Response]: Thanks for your question.As SPAN-FSAS is a commercial device, its specific integrated navigation algorithm is not open source. It is generally considered as a standard loose/tight coupling integrated navigation algorithm (including specific problems such as lever arm calculation, alignment, nonlinear Kalman filter, etc.). In this manuscript, only the output position and attitude solution are selected as the reference truth, and no specific integrated navigation algorithm is involved.

[New comment]: This is not very good, that the authors have no idea how the reference “truth” data is obtained. If Kalman filter is involved in SPAN-FSAS then in the beginning during the algorithm converging time the data could be not reliable and be out datasheet accuracy.

[New Response]: Thanks for your review. In fact, SPAN-FSAS is a high-level tactical inertial navigation system, which is built-in with fiber optic gyroscope and works based on tight coupling technology. According to its technical specifications, when the satellite signal is interrupted for 10s, the position accuracy of the integrated navigation output is also better than 2cm, and the attitude accuracy is better than 0.01 degrees. Before our car starts, we first stopped for ten minutes in the open area, giving SPAN-FSAS sufficient time to initialize and complete the initial alignment. During driving, the satellite signal is only interrupted occasionally (not lost for a long time). Therefore, we consider that the positioning and attitude results can be used as truth values. In order to avoid misunderstanding, we will add some notes and they will be highlighted.

(7) The time of the start and of the end of the presented experiment are not correctly presented as “529381” and “530300”. The time of the beginning should be a zero and the end time should be a time from the beginning. Please correct it in all the plots and in the Fig. 6 as well.

[Response]: Thanks for your question. In fact, GPST is used in many places in this paper. In the future, more in-depth research will be carried out. GPST can better correspond to the data and facilitate comparative research. In addition, the trajectory in different GPST is also different.

[New comment]: What is the meaning of the “GPST”? It is not explained in the text. Still, the authors should correct all the time reference to appropriate time in seconds from the beginning of the experiment, in current state the plot axis are not readable.

[New Response]: Thanks for your review. GPST means GPS Time and it can be represent as “GPS week, second”, for example, 2044 530000 means No. 530000 seconds for GPS week 2044. In order to make it clear, we will replace GPST as time series beginning with 0.

(8) Why did the authors consider pairs of the oppositely directed baselines (like “0” and “180” deg)? The errors are just the same for each pair, as it should be if the same measurements are processed. It is strange that the difference between “0” and “180” is 10^-6 deg as presented in Fig. 8 in the first line of the plots while it should be a zero.

[Response]: Thanks for your question. The two baselines with an angle difference of 180°are obtained by the same two antennas. This is to ensure that the difference between the two baselines is only due to the different direction, while the observation noise and multipath error are basically the same, so that the difference between the solution results is only affected by the baseline vector processing algorithm.If the algorithm is correct, the attitude angles of two baselines with an angle difference of 180° should be the same.However, the reference antenna has changed in both two cases. 0° baseline is 4-1 (antenna 4 is the reference antenna) while 180° baseline is 1-4 (antenna 1 is the reference antenna).This is equivalent to that the origin of b-frame has changed, and the attitude determination needs to be carried out again.The coordinates of the antenna calculated from the positioning results are in the ECEF coordinate system, and it should be transformed to ENU coordinate system. There will be a coordinate transformation process here. Therefore, we believe that the error of 10^-6deg should be generated in the specific coordinate conversion calculation (including truncation error).

[New comment]: So, the authors believe that the error of 10^-6 deg is the computational error. This text results of switching the direction of the baselines could be mentioned in the text that there is no difference in the error. But in the table 1 it will be better if the authors leave only one of the directions for each pair.

[New Response]: Thanks for your review. We will follow your advice and mention that there is no difference in the error. In addition, we will leave only one of the directions for each pair in table 1.

(9) The error of the attitude determination is about several degrees using the proposed method. But the authors use too many digits after the dot in all the values like “4.5943 deg”. Such a precision in values is not necessary, two digits would be enough in the considered case.

[Response]: Thank you for pointing it out. In the revised manuscript, we will follow your advice and they will be highlighted.

[New comment]: Now it is better.

[New Response]: Thanks for your review.

(10) It is not clear what is “epoch proportion” in the Section 4.3.1. What is “valid epoch”? Please, clarify.

[Response]: Thank you for pointing it out. In fact, “valid epochs” means the corresponding epochs is available (The reliable fixed solutions can be obtained). In addition, “valid epoch proportion” means the percentage of available epochs in total.We will change our statement and they will be highlighted.

[New comment]: Is it correct that the valid epoch is the time point when the attitude is calculated? If so, please add this simpler explanation in the text.

[New Response]: Thanks for your review. We will follow your advice and add simpler explanation in the revised manuscript.

(11) The variance in Table 5 for “delta_r” is not correctly calculated, since 1.16^2=1.34, not 1.23 as presented in the Table.

[Response]:Thank you for pointing it out. It’s our mistake and we will correct it.

[New comment]: Ok.

[New Response]: Thanks for your review.

Author Response File: Author Response.docx