Analytic Method for Estimating Aircraft Fix Displacement From Gyroscope’s Allan-Deviation Parameters

Abstract

The noise and drift requirements for a navigation-grade gyroscope are widely known, yet there is no simple analytic model of how the noise and drift of a gyroscope influence the fix displacement error (FDE) of an inertial navigation system (INS). This work derives simple analytical expressions for the cross-track and along-track errors of an aircraft whose INS consists solely of a three-axis gyroscope system with perfect knowledge of the vertical direction. The error signal of each gyroscope is Gaussian white noise and drift modeled as a first-order Markov random walk. These expressions provide a straightforward mean of calculating the FDE of an aircraft as a function of the flight duration, velocity, noise amplitude, drift amplitude, and drift’s time constant. These expressions are validated with Monte-Carlo simulations of long flights. This model quantifies the noise-versus-drift trade-off for a gyroscope in an inertial navigation system. It can save time when estimating the noise and drift that a gyroscope must exhibit to satisfy a given position-error requirement, or vice versa. They are used in particular to confirm the values, often cited without demonstration, of the noise and drift required to meet the maximum position error of an aircraft imposed by the Federal Aviation Administration’s required navigation performance 10 specification. Finally, it demonstrates that using the minimum in the measured Allan deviation of a gyroscope as a metric of the drift is incorrect, because it fails to capture the drift’s time constant. The proper metric is the maximum in the Allan deviation.