Quasi-Stationary North Finding with a FOG-Based IMU

Theory, Sensor Requirements, and a Quaternion Error-State Kalman Filter

Abstract

North finding by gyrocompassing determines the azimuth of an instrument from the Earth’s rotation rate sensed by its own gyroscopes, without any external reference. Because the horizontal component of the Earth rate is only 15.04◦/hcos 𝐿 at latitude 𝐿, the achievable accuracy is governed by the bias stability and the noise of the gyroscopes. In addition, practical installations are rarely perfectly still. This paper gives a self-contained treatment of quasi-stationary north finding. Starting from the reference frames and the measurement model of a stationary inertial measurement unit (IMU), the analytic leveling and gyrocompassing equations are derived together with a closed-form heading error budget, from which the sensor requirements for a target accuracy and alignment time are obtained. A noise identification procedure based on the Allan variance and the power spectral density links the measured sensor noise to the process noise of the estimator. Fine alignment is performed by a quaternion error-state Kalman filter that processes zero-velocity pseudo-measurements and models the vibration of the support as a Gauss-Markov disturbance. An observability analysis explains why the attainable accuracy is bounded by the east gyro bias. The method is evaluated on a 60s record of a fiber-optic gyroscope (FOG) IMU at 40.92◦ N. The filter estimates the heading as 342.70◦ with a 1𝜎 uncertainty of 0.88◦, in agreement with the error budget, in which the 0.1◦/h gyro bias sets a floor of 0.50◦. A Monte Carlo simulation confirms that this uncertainty is realistic. The analytic gyrocompass gives 343.13◦, and the difference is traced to a slow drift of the horizontal specific force, to which the filter is sensitive and plain averaging is not.