# Gravity Compensation Using EGM2008 for High-Precision Long-Term Inertial Navigation Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Gravity Disturbance Vector Induced Position Errors

#### 2.1. Error Propagation

#### 2.2. Gravity Disturbance Vector and Its Induced Position Errors

**f**is the combination of the kinematic acceleration vector

**a**and the gravitational acceleration vector

**G**, as:

**g**is the gravity vector and ${\omega}_{ie}$ is the Earth’s rotation vector. The radius vector

**r**defines the position to the Earth’s center of mass. Obviously it can be seen from Equations (2) and (3) that INSs need gravitational information to extract the kinematic acceleration of the vehicle. The normal gravity model is frequently employed because it can meet the accuracy requirement in most cases and is both simple and convenient to be calculated. This model is based on an ellipsoid of revolution having the same mass and rotation rate with the Earth, namely the so-called reference ellipsoid. As the normal gravity vector $\gamma $ is perpendicular to the surface of the reference ellipsoid, its vertical component equals its magnitude $\gamma $, as:

## 3. Gravity Compensation Using a Spherical Harmonic Model

## 4. Real-Time Gravity Compensation

#### 4.1. Time Requirements for Real-Time Compensation

#### 4.2. Improvement of Computation Efficiency

#### 4.3. Compromise between Accuracy and Computing Efficiency

## 5. The Sea Test of a Shipborne INS

## 6. Results and Discussion

#### 6.1. The Static and the Round-Trip Experiment

#### 6.2. Dynamic Experiments after a Long-Time Navigation

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Levine, S.A.; Gelb, A. Effect of reflections of the vertical on the performance of a terrestrial inertial navigation system. J. Spacecr. Rockets
**1969**, 6, 978–984. [Google Scholar] - Schwarz, K.P. Gravity Induced Position Errors in Airborne Inertial Navigation; The Ohio State University: Columbus, OH, USA, 1981. [Google Scholar]
- Harriman, D.W.; Harrisont, J.C. Gravity-induced errors in airborne inertial navigation. J. Guid. Control Dyn.
**1986**, 9, 419–426. [Google Scholar] - Vanderwerf, K. Schuler pumping of inertial velocity errors due to gravity anomalies along a popular North Pacific airway. In Proceedings of the IEEE Position Location and Navigation Symposium, Atlanta, GA, USA, 22–25 April 1996; pp. 642–648.
- Dai, D.; Wang, X.; Zhan, D.; Huang, Z.; Xiong, H. An improved method for Gravity disturbances compensation in INS/GPS integrated navigation. In Proceedings of the 12th International Conference on Signal Processing (ICSP), Hangzhou, China, 19–23 October 2014; pp. 148–153.
- Hu, P.; Gao, Z.; She, Y.; Cai, L.; Han, F. Shipborne heading determination and error compensation based on a dynamic baseline. GPS Solut.
**2015**, 19, 403–410. [Google Scholar] [CrossRef] - Grejner-Brzezinska, D.; Yi, Y.; Toth, C. On improved gravity modelingg supporting direct georeferenceing of multisensor systems. Proc. Int. Soc. Photogramm. Remote Sens.
**2004**, XXXV, 908–913. [Google Scholar] - Jekeli, C. Airborne vector gravimetry using precise, position-aided inertial measurement units. Bull. Geod.
**1994**, 69, 1–11. [Google Scholar] [CrossRef] - Leonard, J.M.; Nievinski, F.G.; Born, G.H. Gravity error compensation using second-order Gauss-Markov processes. J. Spacecr. Rockets
**2013**, 50, 217–229. [Google Scholar] [CrossRef] - Jordan, S.K. Self-Consistent statistical models for the gravity anomaly, vertical deflections, and undulation of the geoid. J. Geophys. Res.
**1972**, 77, 3660–3670. [Google Scholar] [CrossRef] - Jekeli, C. Precision free-inertial navigation with gravity compensation by an onboard gradiometer. J. Guid. Control Dyn.
**2006**, 29, 704–713. [Google Scholar] [CrossRef] - Heller, W.G.; Jordant, S.K. Error analysis of two new gradiometer-aided inertial navigation systems. J. Spacecr. Rockets
**1976**, 13, 340–347. [Google Scholar] [CrossRef] - Kwon, J.H.; Jekeli, C. Gravity requirements for compensation of ultra-precise inertial navigation. J. Navig.
**2005**, 58, 479–492. [Google Scholar] [CrossRef] - Hirt, C.; Claessens, S.; Fecher, T.; Kuhn, M.; Pail, R.; Rexer, M. New ultrahigh-resolution picture of Earth’s gravity field. Geophys. Res. Lett.
**2013**, 40, 4279–4283. [Google Scholar] [CrossRef] - Jekeli, C.; Lee, J.K.; Kwon, J.H. Modeling errors in upward continuation for INS gravity compensation. J. Geod.
**2007**, 81, 297–309. [Google Scholar] [CrossRef] - Pavlis, N.K.; Holmes, S.A.; Kenyon, S.C.; Factor, J.K. The development and evaluation of the Earth Gravitational Model 2008 (EGM2008). J. Geophys. Res. Solid Earth
**2012**, 117, 1–38. [Google Scholar] [CrossRef] - Hirt, C.; Marti, U.; Bürki, B.; Featherstone, W.E. Assessment of EGM2008 in Europe using accurate astrogeodetic vertical deflections and omission error estimates from SRTM/DTM2006.0 residual terrain model data. J. Geophys. Res.
**2010**, 115. [Google Scholar] [CrossRef] - Fu, G.Y.; Zhu, Y.Q.; Gao, S.H.; Liang, W.F.; Jin, H.L.; Yang, G.L.; Zhou, X.; Guo, S.S.; Xu, Y.M.; Du, W.J. Discrepancies between free air gravity anomalies from EGM2008 and the ones from dense gravity/GPS observations at west Sichuan Basin. Chin. J. Geophys.
**2013**, 56, 3761–3769. (In Chinese) [Google Scholar] - Abeho, D.R.; Hipkin, R.; Tulu, B.B. Evaluation of EGM2008 by means of GPS levelling Uganda. S. Afr. J. Geomat.
**2014**, 3, 272–284. [Google Scholar] [CrossRef] - Abeyratne, P.G.V.; Featherstone, W.E.; Tantrigoda, D.A. Assessment of EGM2008 over Sri Lanka, an area where “fill-in” data were used in EGM2008. Newton Bull.
**2009**, 4, 284–316. [Google Scholar] - Dawod, G.; Mohamed, H.; Ismail, S. Evaluation and adaptation of the EGM2008 geopotential model along the Northern Nile Valley, Egypt: Case study. J. Surv. Eng.
**2009**, 136, 36–40. [Google Scholar] [CrossRef] - Jekeli, C. Gravity on precise, short-term, 3-D free-inertial navigation. Navigation
**1997**, 44, 347–357. [Google Scholar] [CrossRef] - Wang, J.; Yang, G.; Li, X.; Cheng, J. Research on time interval of gravity compensation for airborne INS. In Proceedings of the 2015 34th Chinese Control Conference (CCC), Hangzhou, China, 28–30 July 2015; pp. 5442–5446.
- Wang, J.; Yang, G.; Li, J.; Zhou, X. An Online Gravity Modeling Method Applied for High Precision Free-INS. Sensors
**2016**, 16. [Google Scholar] [CrossRef] [PubMed] - Wang, J.; Yang, G.; Li, X.; Zhou, X. Application of the spherical harmonic gravity model in high precision inertial navigation systems. Meas. Sci. Technol.
**2016**, 27, 95103. [Google Scholar] [CrossRef] - He, Q.; Gao, Z.; Wu, Q.; Fu, W. Design of horizontal damping network for INS based on complementary filtering. J. Chin. Inert. Technol.
**2012**, 20, 157–161. (In Chinese) [Google Scholar] - Hofmann-Wellenhof, B.; Moritz, H. Physical Geodesy; Springer Science & Business Media: Berlin, Germany, 2006. [Google Scholar]
- Pavlis, N.K.; Holmes, S.A.; Kenyon, S.C.; Factor, J.K. An Earth Gravitational Model to Degree 2160: EGM2008. In Proceedings of the European Geosciences Union General Assembly 2008, Vienna, Austria, 13–18 April 2008; Volume 84.
- Holmes, S.A.; Featherstone, W.E. A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions. J. Geod.
**2002**, 76, 279–299. [Google Scholar] [CrossRef] - Wang, J.; Yang, G.; Li, X.; Zhou, X. Error indicator analysis for gravity disturbing vector’s influence on inertial navigation system. J. Chin. Inert. Technol.
**2016**, 24, 285–290. (In Chinese) [Google Scholar]

**Figure 2.**Pole plot of $I\left(s\right)\text{}(\mu \text{}=\text{}0.5,\text{}\zeta \text{}=\text{}1.296)$.

**Figure 3.**Bode plot of $I\left(s\right)\text{}(\mu \text{}=\text{}0.5,\text{}\zeta \text{}=\text{}1.296)$.

**Figure 5.**Simulation and theoretical values of the RMS difference between ${x}_{c}\left(t\right)$ and ${x}_{r}\left(t\right)$ under the common experimental flight condition.

**Figure 6.**RMS compensation errors vs. update interval and computing time delay under the shipborne condition.

**Figure 8.**(

**a**) Sizes of spherical harmonic coefficient files; and (

**b**) Average computing time vs. maximum degree of the used spherical harmonic model.

**Figure 10.**Results of the static experiment: (

**a**) Latitude errors before (line 1) and after (line 2) compensation; (

**b**) Longitude errors before (line 1) and after (line 2) compensation.

**Figure 11.**Latitudinal results of the round-trip experiment: (

**a**) $\xi $; (

**b**) Latitude errors before (line 1) and after (line 2) compensation; (

**c**) Compensated latitude error; (

**d**) Difference between $\xi $ and the compensated latitude error.

**Figure 12.**Longitudinal results of the round-trip experiment: (

**a**) $\eta $; (

**b**) Longitude errors before (line 1) and after (line 2) compensation; (

**c**) Compensated longitude error; (

**d**) Difference between $\eta $ and the compensated longitude error.

**Figure 13.**Compensation results of Segment II-5: (

**a**) $\xi $; (

**b**) Latitude errors before (line 1) and after (line 2) compensation; (

**c**) $\eta $; (

**d**) Longitude errors before (line 1) and after (line 2) compensation.

**Table 1.**SSEs of fittings of the normalized horizontal position errors before and after compensation.

No. | II-1 | II-2 | II-3 | II-4 | II-5 | II-6 | II-7 |
---|---|---|---|---|---|---|---|

Time span (h) | 7.22 | 10.28 | 10.55 | 6.66 | 13.88 | 7.22 | 7.50 |

Latitude scope ($\xb0$) | 13.15~14.65 | 10.07~10.44 | 9.80~10.21 | 9.60~9.85 | 13.41~15.51 | 13.96~15.10 | 14.05~15.16 |

Longitude scope ($\xb0$) | 115.52~115.61 | 114.21~115.34 | 114.23~115.54 | 114.64~115.53 | 112.33~112.52 | 112.35~112.44 | 112.30~112.43 |

Uncompensated $\delta L$ SSE | 62.01 | 65.95 | 92.88 | 33.06 | 78.33 | 36.52 | 39.25 |

Compensated $\delta L$ SSE | 12.41 | 26.57 | 49.09 | 13.49 | 34.27 | 9.66 | 19.19 |

Uncompensated $\delta l$ SSE | 34.89 | 75.69 | 367.66 | 384.84 | 187.18 | 236.06 | 58.58 |

Compensated $\delta l$ SSE | 20.65 | 33.21 | 157.40 | 99.80 | 66.22 | 81.99 | 15.88 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wu, R.; Wu, Q.; Han, F.; Liu, T.; Hu, P.; Li, H.
Gravity Compensation Using EGM2008 for High-Precision Long-Term Inertial Navigation Systems. *Sensors* **2016**, *16*, 2177.
https://doi.org/10.3390/s16122177

**AMA Style**

Wu R, Wu Q, Han F, Liu T, Hu P, Li H.
Gravity Compensation Using EGM2008 for High-Precision Long-Term Inertial Navigation Systems. *Sensors*. 2016; 16(12):2177.
https://doi.org/10.3390/s16122177

**Chicago/Turabian Style**

Wu, Ruonan, Qiuping Wu, Fengtian Han, Tianyi Liu, Peida Hu, and Haixia Li.
2016. "Gravity Compensation Using EGM2008 for High-Precision Long-Term Inertial Navigation Systems" *Sensors* 16, no. 12: 2177.
https://doi.org/10.3390/s16122177