**3.3.3 Influence on inertial navigation**

**Introduction**

Inertial navigation relies on Newton=s law, and integrates position and velocity from measured accelerations, either in a physically mechanized reference frame (gimbal systems) or mathematically computed reference frame (strapdown INS navigation). Commercial strapdown INS systems are used widely as primary navigation tool for long-range civilian aircraft, as well as military aircraft and missiles. For a review of INS principes, see Farrell (1976). The systems typically assume the gravity field to be the normal (ellipsoidal) field, i.e. deflections of the vertical are ignored. This gives rise to errors, which for military systems is known to be the primary error source in navigation accuracy.

INS errors show a different error behaviour in the horizontal and vertical channels. Due to the double integration involved, errors due to gravity field and system noise quickly integrates vertical accelerations into excessive height errors (10=s of km), unless stablized by other means (barometric altitude or GPS). Horizontal navigation error growth are, on the other hand, bound due to feed-back between computation system angular errors and position B giving rise to the characteristic "Schuler period" of 84 min for terrestrial INS navigation. In practice horizontal errors grow in a random walk fashion, with typical drift values of 0.5-1.0 nm/hr for commercial systems (and probably one order of magnitude better for the classified military systems).

To limit error growth, some military systems incorporate active gravity field compensation, i.e. gravity field quantities are computed in real time from models, and applied in the position computations. This has been a prime motivation for the military interests (and classifications) of gravity and d.o.v. surveys globally for decades.

**Simulation study of INS navigation errors for GOCE **

To make a simple order-of-magnitude simulation experiment, INS errors were propagated using the "one-dimensional" error equation of INS. The integration solves the Schuler loop error propagation equation by the approximation

where Mg is the gravity vector component, x the position error and R the earth radius. The solutions to this equation gives rise to the characteristic Schuler period (84 min) of INS errors.

Two types of comparisons were made assuming reference fields, complete to degree 360, perturbed by either the EGM96 model or the assumed GOCE errors. 10 or 20 error grid realizations were used, and the statistics were expressed as the r.m.s. drift and maximum errors encountered for a particular flight.

The simulation errors were either high-altitude flights over the Atlantic region (30-60 N, 90 W-30 E), i.e. a 10 hr flight from Russia to the US, or short low-altitude flights over the Alps (46-48 N, 6- 16 E). In both cases results were accumulated on assumed E-W flights, flying along parallels of 3 deg spacing (North Atlantic) or 1 deg spacing (Alps).

Computations are done stepwise with up to 6000 integration steps pr. flight. The gravity field is assumed compensated at initial point using an observed deflection value, that is during alignment INS system do align to actual vertical (this gives smaller errors).

Performance numbers are given as random walk drift in nm/hr and max error in nm (nautical miles) for the assemblage of error realizations.

There are two different cases of error simulations:

1. Propagate the full field with errors (either GOCE or EGM) B this corresponds to the full INS errors encountered for a system which assumes a normal gravity field model, or

2. Propagate errors *only* B this corresponds to a system which compensates for a 360-degree reference model, assumed to be perfect.

Additionally the EGM96 gravity field errors are propagated deterministically without errors.

Results.

North Atlantic: Results at 240 m/s (10 realizations, 10 latitude parallels):__<TBODY>__

Type of integration |
Drift (nm/hr) |
Maximal error (nm) |

GOCE errors alone |
0.002 |
0.006 |

EGM96+GOCE errors |
0.065 |
0.136 |

EGM96 errors alone |
0.011 |
0.024 |

EGM96 + EGM96 errors |
0.066 |
0.142 |

EGM96 without errors |
0.065 |
0.131 |

North Atlantic at supersonic speed (600 m/s, i.e. Concorde):__<TBODY>__

Type of integration |
Drift (nm/hr) |
Maximal error (nm) |

GOCE errors alone |
0.003 |
0.008 |

EGM96+GOCE errors |
0.094 |
0.211 |

EGM96 errors alone |
0.010 |
0.019 |

EGM96+EGM96 errors |
0.095 |
0.217 |

EGM96 without errors |
0.094 |
0.203 |

Alps: Results at 240 m/s (20 realizations, 3 latitude parallels):__<TBODY>__

Type of integration |
Drift (nm/hr) |
Maximal error (nm) |

GOCE errors alone |
0.006 |
0.021 |

EGM96+GOCE errors |
0.277 |
0.442 |

EGM96 errors alone |
0.028 |
0.069 |

EGM96 + EGM96 errors |
0.278 |
0.471 |

EGM96 without errors |
0.276 |
0.430 |

**Conclusions**

The incurred drifts from the satellite fields are quite small, 0.1-0.2 nm/hr, largest over the Alps due to the more rough gravity field. This is because the INS errors are to a large degree caused by the higher-wavelength gravity field variations, not taken into account in these simulations. The overall performance for a normal gravity field compensated INS will be very small once the gravity field is improved after GOCE mission, because the dominant error of the integrated anomalous field itself.

For INS systems which incorporate gravity compensation, the simulated errors of GOCE versus EGM shows that an improvement of about a factor 3 may be obtained for a perfectly compensated system. Therefore GOCE would be beneficial to aid navigation accuracy for the most accurate INS sytems, especially at higher (supersonic) speeds.