Edited by: Shaowei Zhang, Institute of DeepSea Science and Engineering (CAS), China
Reviewed by: Yongchao Zhu, Hefei University of Technology, China; Yunlong Wu, China University of Geosciences Wuhan, China; Ke Baogui, Chinese Academy of Surveying and Mapping, China; Zhicai Li, China University of Mining and Technology, Beijing, China
*Correspondence: Lin Wu,
This article was submitted to Ocean Observation, a section of the journal Frontiers in Marine Science
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Gravity disturbance compensation is an important technique for improving the positioning accuracy of highprecision inertial navigation systems (INS). Aiming at the current problems of the resolution of gravity compensation background field and the robustness of gravity compensation algorithm are insufficient for gravity compensation. In this study, the error and frequency characteristics of INS caused by gravity disturbances are investigated. The gravity disturbance with a spatial resolution of 1’ × 1’ from a highprecision satellite altimetry marine gravity field model is preliminarily introduced into the initial alignment and pure INS calculation to implement the gravity compensation of the dualaxis rotary modulation INS. Detailed calculation results show that the east gravity disturbance affects the north attitude, and the north gravity disturbance affects the east attitude in the initial alignment. In the pure INS calculation, the horizontal gravity disturbance causes a navigation error in the form of Schuler oscillation. The INS navigation error caused by horizontal gravity disturbance is mainly affected by its amplitude; however, the horizontal gravity disturbance accuracy from the satellite altimetry model for INS gravity compensation can be ignored in practice. In addition, for lowspeed underwater vehicles, the influence of highfrequency gravity disturbance signals on the INS position shows an increasing trend. Finally, the effectiveness of the gravity compensation achieved by the horizontal gravity disturbance from the satellite altimeter model is confirmed by a dynamic shipborne test. The positioning accuracy of the rotary modulation INS is maximally improved by approximately 17.9% after the horizontal gravity disturbance is compensated simultaneously in the pure INS calculation and the initial alignment.
Inertial navigation systems (INS) are widely employed to establish autonomous navigation systems in various fields. This autonavigation feature renders INS stable and reliable for any type of external interference. However, INS is hindered by errors caused by inertial devices, which severely restrict the underwater longtime navigation ability of the submersible. With the development and application of optical inertial devices and rotary modulation technologies, the accuracy of inertial sensors has significantly improved (
Gravity compensation is an important part of gravityaided navigation for INS. Some scholars have studied the influence of gravity disturbances on INS.
The mechanical arrangement of the INS can be improved by measuring the gravity disturbance determined by highdegree GFMs, such as EGM2008 and EIGEN_6C4 (
With the continuous development of satellite altimetry missions, satellite altimetry has become the primary method for determining the global marine gravity field. The marine gravity anomaly and DOV with 1’ × 1’ (~2 km) resolution can be retrieved by integrating multigeneration altimetry satellites (
The velocity kinematical equation of the dualaxis rotary modulation INS is given by (
Where
where
In traditional INS, the gravity vector is usually replaced by the normal gravity by:
where
Where
Compared with the strapdown INS, the accuracy of the rotation modulation INS is improved significantly. Inertial sensor errors are eliminated by periodically rotating the inertial sensor along the rotation axis. At this time, the influence of the gravity disturbance must be considered. The error equation of INS considering gravity disturbance can be written as follows:
where
Based on Eq. (6), the influence of gravity disturbance is not affected by the rotation modulation. First, the INS velocity error is caused by Eq. (6), and the position and attitude of the INS are further affected by Eq. (7) and (8). The gravity disturbance vector is given by:
where
Definition of gravity disturbance vector.
The actual gravity differs from normal gravity owing to the existence of a vertical deflection. As shown in
Because the deflection of the vertical is small, it can be approximated by
The horizontal gravity disturbance components are calculated according to the DOV in the meridian and prime vertical directions. With the continuous development of satellite altimetry, the accuracy of determining the DOV from the altimetry model has been improved. In this study, the horizontal gravity disturbance determined by the altimetry model was used to achieve INS gravity compensation. The calculation process for the horizontal gravity disturbance is presented in section 3.4.
The initial alignment for INS is generally performed under the static base condition, which consists of two steps: coarse alignment and fine alignment. We can only obtain a coarse attitude matrix using coarse alignment. A certain misalignment angle error is observed. Gravity disturbances can be ignored in coarse alignments. However, the influence of horizontal gravity disturbance should be considered in fine alignment. System errors are estimated and corrected using Kalman filtering. According to Eq. (6)–(8), the INS error equation with static base conditions can be expressed as:
The system filtering statespace of fine alignment can be defined as:
According to Eq. (14), the fine alignment statespace model is constructed was follows:
where
where
Taking the velocity under the static base condition as the observation, the measurement equation of the system can be given by:
where
The initial alignment based on Kalman filtering takes the velocity as the measured value; therefore, the values related to velocity (
We can let
Eq. (26) shows that the fine alignment of the INS depends mainly on the ∇^{ n } and horizontal gravity disturbance. The north gravity disturbance caused an east attitude error, and the east gravity disturbance caused a north attitude error. For a dualaxis rotary modulation INS, ∇^{ n } can be weakened by periodically rotating the inertial sensor along the rotation axis. Horizontal gravity disturbance is the main factor affecting fine alignment. Therefore, the horizontal gravity disturbance should be compensated for during the initial alignment.
The gravity compensation of the INS includes compensation in the initial alignment and the pure INS calculation. The gravity compensation in the pure INS calculation means that the gravity disturbance is compensated in the velocity in Eq. (1). For gravity compensation in the initial alignment, Eq. (16) and (23) are used to construct the fine alignment statespace model, and the initial alignment errors caused by gravity disturbances are estimated and corrected. The highresolution horizontal gravity disturbance from the satellite altimetry marine gravity field model is provided to the INS for the mechanical arrangement in this study. The specific gravity compensation process is illustrated in
Gravity compensation procedure of dualaxis rotary modulation INS.
A corresponding staticbased simulation experiment is designed to analyze the impact of gravity disturbance on the initial alignment. The simulation conditions are as follows: gyro drift bias, 0.001°/h; accelerometer drift bias, 10
The initial attitude misalignment angle is set to [0.5°, 0.5°, 0.5°]
Alignment attitude error,
In the pure INS calculation, gravity disturbance first causes a velocity error, then further affects the position and attitude of the INS. In this section, the impact of gravity disturbance on the velocity and position is analyzed by ignoring the errors of the gyro, accelerometer, and attitude. The system error Eq. (6)–(7) can be solved analytically to quantitatively analyze the influence of the gravity disturbance.
The error aroused by different gravity disturbance,
The maximum influence of north gravity disturbance on north position and velocity.
Horizontal gravity disturbance (mGal)  5  24  50  95  143 
North maximum position error (m)  62  311  616  1232  1856 
North maximum velocity error (m/s)  0.038  0.193  0.383  0.765  1.152 
The impact of gravity disturbance on the INS is analyzed above by assuming that the gravity disturbance is constant. In the actual survey line, the gravity disturbance changes from time to time. Therefore, we utilize an actual gravity survey line to study the impact of horizontal gravity disturbance on the INS.
Marine gravity survey trajectory.
Horizontal gravity disturbance in gravity survey trajectory.
Maximum position error influence caused by horizontal gravity disturbance in each Schuler cycle.
Maximum error of INS caused by horizontal gravity disturbance in all Schuler cycles.
Scenarios  Max  Min  Mean 

Position error (nonoise)  1034.6  88.04  532.7 
Position error (withnoise)  1053.9  128.7  538.7 
To analyze and obtain the frequency characteristics of the INS affected by gravity disturbance, the gravity disturbance is modeled as a Markov process (
Root mean square of deflection of the vertical.
Relationship between position error (root mean square value) and spatial wavelength caused by horizontal gravity disturbance at different velocities.
As shown in
The maximum effect of the horizontal gravity disturbance on the INS position is proportional to its amplitude. For areas with obvious topographic fluctuations, such as seamount ranges or trenches, the horizontal gravity disturbance is greater. The distribution of horizontal gravity disturbance in the China Sea and Western Pacific region and its impact on INS are analyzed using a satellite altimetry model. The seafloor topography in the China Sea and Western Pacific region is complex and contains a large number of islands, reefs, submarine plains, and trenches, which can accurately reflect the distribution of horizontal gravity disturbance under different topographies.
The DOV and gravity disturbances from SIO V30.1 model,
Horizontal gravity disturbance and maximum position influence caused by average gravity disturbance on INS in a Schuler period (


C (Sea of Japan)  D (Western Pacific)  


0~237  0~123  0~100.7  0~297.336 

43.05  33.04  20.60  73.27 

0~187.7  0~200.7  0~161.6  0~205.63 

30.09  28.7  21.22  33.3 

682.8  568.9  384.5  1046.3 
From
A marine dynamic shipborne test is conducted in the South China Sea to verify the effectiveness of gravity compensation. A twoaxis rotary modulation laser INS, which contained three laser gyroscopes and three quartz flexible accelerometers, is used. The gyro drift bias is better than 0.001°/h and the accelerometer drift bias is lower than 10
Trajectory of marine experiment.
The horizontal gravity disturbance is calculated using the SIO V30.1 model (section 3.4). For comparative analysis, the highdegree EIGEN_6C4 model is used to determine the horizontal gravity disturbance (
Horizontal gravity disturbance along trajectory of ship,
The horizontal gravity disturbance calculated above is used to achieve gravity compensation for the dualaxis rotation modulation INS. The specific gravity compensation process is illustrated in
For comparison and analysis, two gravity compensation methods are adopted. The first method utilizes the gravity disturbance calculated by the EIGEN_6C4 model, while the second utilizes the gravity disturbance from the SIO V30.1 model. The longitude, latitude, and position errors with and without gravity disturbance compensation are shown in
Positioning accuracy improvement corresponding to different methods.
Improvement of maximum positioning accuracy using two different gravity compensation methods.
Method  Accuracy improvement  Longitude  Latitude  Position 

SIO V30.1  Error reductions (m)  161  334  335 
Improvement ranges (%)  18.5  18.7  17.9  
EIGEN_6C4  Error reductions (m)  152  311  309 
Improvement ranges (%)  17.6  17.5  16.6 
In the gravity compensation of a highprecision INS, the horizontal gravity disturbance determined by the satellite altimetry model could be used to achieve gravity compensation, which can improve the mechanical arrangement of the INS. In particular, for areas with large topographic changes (such as trenches or seamounts), the impact of highfrequency gravity disturbances caused by seafloor topography must be considered. As discussed in section 3.3, the influence of the higherfrequency signal of the horizontal gravity disturbance on the INS position is increasingly significant for lowspeed underwater vehicles. However, the highdegree gravity field model (EGM2008 or EIGEN_6C4) could not effectively represent higherfrequency gravity field signals because the spatial wavelength of these models is limited. The satellite altimetry model (SIO V30.1) can effectively represent the highfrequency gravity field signal, which meets the requirements of gravity compensation.
Position error caused by different compensation strategies.
Gravity disturbance compensation is essential for further improving the positioning accuracy of the INS. In this study, the error and frequency characteristics of INS caused by gravity disturbances are investigated. The results show that the east gravity disturbance affects the north attitude, and the north gravity disturbance affects the east attitude in the initial alignment. In the pure INS calculation, the horizontal gravity disturbance causes a navigation error in the form of Schuler oscillation. The position error is mainly caused by the mediumlong wavelength (mediumlow frequency) gravity disturbance when the underwater vehicle velocity is larger. With a decrease in underwater vehicle velocity, the influence of the shortwavelength (highfrequency) of gravity disturbance become increasingly significant.
The distribution of horizontal gravity disturbance in the China Sea and Western Pacific region and its impact on INS are analyzed based on a satellite altimetry model (SIO V30.1). Four regions are divided to analyze the distribution of horizontal gravity disturbances on four important channels and their influence on the INS. The results show that the average gravity disturbance in the South China Sea, the East China Sea, and the Sea of Japan have a maximum impact on the INS position of several hundred meters in the Schuler period, and the average gravity disturbance in the western Pacific has a maximum impact on the INS position of more than 1000 m in the Schuler period. In particular, in areas with rugged topography, such as trenches and seamounts, gravity disturbance is found to have a maximum impact on the INS position error of several thousand meters in a Schuler period.
Finally, the dynamic shipmounted experiment verified the effectiveness of the satellite altimetry model (SIO V30.1) in achieving gravity compensation. After gravity compensation using the altimeter model, the positioning accuracy has been improved by 18%. The satellite altimetry model is able to effectively represent the highfrequency gravity field signal, which meets gravity compensation requirements. In particular, for regions with large topographic changes (e.g. trenches or seamounts), the impact of highfrequency gravity disturbances caused by seafloor topography must be considered.
The original contributions presented in the study are included in the article/supplementary material. Further inquiries can be directed to the corresponding authors.
LW, LB and YW conceived the study. The data processing and collection were conducted by QL, HL and BW. The analysis of the results was implemented by PZ, LB, and LW. And the initial draft of the manuscript was written by PZ with improvements and substantial edits from all authors. All authors contributed to the article and approved the submitted version.
This work was supported by the National Natural Science Foundation of China (grant nos. 42192535, 41931076, 42274116, and 42174102), and the Basic Frontier Science Research Program of the Chinese Academy of Sciences (grant no. ZDBSLYDQC028).
We acknowledge David Sandwell, University of California, San Diego, for providing radar altimeterderived gravity data used in this study. We also acknowledge the International Centre for Global Earth Models (ICGEM) for providing us with the gravity field model.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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