Angular Misalignment Calibration for Dual-Antenna GNSS/IMU Navigation Sensor

Round 1

Reviewer 1 Report

I have only one short comment/question. It seams that the authors consider the IMU including gyroscope measuring Euler rotations directly. Since the three obtained rotations are simultaneous and not sequential, considering them as Euler rotations will introduce an error. The angles of the measured rotations are in the presented study small, this error is aslo expected to be small. Nevertheless, do you have an estimate for this error?  

Author Response

Speaking of sequential rotations, we believe you most probably refer to Euler rotations angles. 
However, in strapdown inertial navigation the concept of the Euler vector, also called the rotation vector, is generally used, as opposed to Euler angles inherent to older gimbaled systems. 
The Bortz equation describes the behaviour of the Euler vector analytically, and it is precise. 
You are very correct that numerical integration on a finite time grid, surely, introduces some error. 
For the classic 4-th order Runge-Kutta (RK4) integration method, total accumulated error is estimated to be of the order of O(h^4), where h is the time step.
In our approximated method similar to RK4, we expect errors to be at or below O(h^3) ≈ (ω x h)^3 which for our sampling frequency of 250 Hz and rotations slower than 1 rad/sec yields less than 6 x 10^-8 rad accumulated error per each second of integration.
Moreover, the way the Bortz equation is constructed, guarantees the numerical error to be zero under constant rotation rate, so what really matters is angular acceleration in our case, so the error should stay negligible in our calibration procedure.
We hope this answers your question. To clear up the ambiguity in describing rotations, we have also added the term "strapdown" to our text on pages 1 and 5 to emphasize the terminology.

Thank you very much for your time and comments.

Reviewer 2 Report

Is a good manuscript, deserves to be published

Author Response

We are very grateful for your kind words and for the time taken to read through our manuscript.

Reviewer 3 Report

In this paper, the research have reduced the problem of angular misalignment calibration between the instrumental reference frame associated with an IMU, and the carrier body reference frame with known locations of two GNSS antennas in it, to a conventional linear stochastic estimation problem. The research is relevant to all applications aimed at tracking orientation using a low-grade IMU and dual-antenna GNSS within a sub-degree level of precision. The research has certain practical value in engineering. Some suggestions are provided to consider as follows: the format of the references is not uniform, for example reference 9,11, 12 and 21, et al.

Author Response

We are very thankful for your suggestions and we totally agree with you regarding the style uniformity in References. 
We have fixed some flaws after their revision as you had kindly noted:
1. To reference 21 we have applied the same "Thesis" style as reference 11 have had.
2. For references 6, 9, 12, and 15, we have changed the format from the "Journal" style to the "Proceedings" one.
3. We have added a DOI to the 9-th reference, and made sure that all DOIs are correct too.
4. Couple of minor typos in references 17 and 24 have been fixed.

The differences between the initial and current versions of the manuscript are highlighted in yellow, so you can easily check it out.

Reviewer 4 Report

The study is interesting since it presents an approach to reducing the problem of angular misalignment calibration between the instrumental reference frame associated with an IMU, and the carrier body reference frame with known locations of two GNSS antennas in it to a conventional linear stochastic estimation problem. The methodology is explained in detail. Simulation and experiment are conducted to confirm the validity of the proposed method.

In general, the paper is well written. My concerns include:

1. The proposed method is expected to compared with conventional calibration methods. Although Fig.5 presents a comparison between Euler attitude integration and the proposed algorithm, it is not a comparison of calibration method.

2. Fig.7 presents the estimates within 2-sigma confidence intervals while in Table 4 1-confidence interval is used. It is expected to explain the increase of accuracy or the different use of confidence interval.

3. The limitation of the proposed method is expected to be remarked in the conlcusion section.

4. The full name of GSNN should be given after appearance.

Author Response

Thanks a lot for your guidance. 
We highly appreciate your feedback to make our research better, and we have tried to address all your points of concern below.

1. You are absolutely correct that Fig. 5 does not provide any comparison of calibration methods. 
As the Introduction says, we have been unable to find any published methods for the calibration of angular misalignment despite there exist papers limited to high-and mid-grade INS, without using GNSS information. 
From recent communication (after the manuscript submission) with an expert in IMU/GNSS/odometer integration, this problem appears to be very often overlooked. 
In particular, the lack of conventional approaches in literature has motivated us to tackle the problem.

2. 2-σ intervals in Fig. 7 demonstrate the degree of consistency between two realizations of the estimates for ϰ1, ϰ2 in two different experiments. 
2-σ value corresponds to approximately 95% confidence interval (CI), which satisfy the following condition: CIs of two estimates for ϰ2 do not overlap before conical motions and overlap after they end. 
We believed that this fact would be less visible when using 1-σ confidence level in Fig. 7. 
In addition, Table 4 demonstrates that two final (after conical motions) estimates for ϰ1 correspond to each other within 1-σ CIs. 
So, Table 4 contains all results as they stand, and Fig. 7 just visually complements them.

3. The main and possibly the single limitation of the suggested calibration method is the requirement for specific (conical) motions. 
The Abstract, Introduction, and Sections 3, 4 contain this information explicitly, that is why we decided to avoid apparent repetition in Discussion. 
Please let us know if you still feel it should appear in the Discussion as well.
Also, if we deal with linear estimation problem, there appears a natural question about knowing the a priori moments of the stochastic terms, 
but it is well-known that problems of such sort are eventually solved in engineering practice usually by trial and error.

4. We have expanded the abbreviation at its first appearance (see the 1-st paragraph of the Introduction).