Analysis and calibration of the gyro bias caused by geomagnetic field in a dual-axis rotational inertial navigation system

A rotational inertial navigation system (RINS) has been wildly used in long term marine navigation. In a dual-axis RINS, with all constant biases averaged out, the errors which can not be averaged out become the main error source. In this paper, the gyro geomagnetic biases of a dual-axis RINS are modelled, analysed and calibrated. The gyro geomagnetic biases are proved unable to be averaged out, but can be modulated to be a constant value in the navigation frame. A slope error term of longitude error is found to be caused by gyro geomagnetic biases in north and upward directions, which increases linearly with time and is remarkable in long term navigation. Thus, a calibration method based on least square regression is proposed to compensate the slope error term. Laboratory and sailing experimental results show that the divergence speed of longitude error can be effectively slowed down by the compensation of gyro geomagnetic biases. In long term independent navigation, the position accuracy of dual-axis RINS is improved about 50% by the calibration method proposed in this paper.


Introduction
A rotational inertial navigation system (RINS) has been wildly used in marine navigation, as it can bring a much better performance in long term navigation without using higherlevel inertial sensors. By rotating an inertial measurement unit (IMU) using a single-axis indexing mechanism every few minutes, the constant biases of the inertial sensors perpendicular to the axis can be averaged out in a single-axis RINS, such as MK39 and WSN-7B (Lahham et al 2000, Tucher andLevinson 2000). Moreover, all constant biases of inertial sensors can be averaged out in a dual-axis RINS, such as MK49 and WSN-7A Brazell 1992, Levinson et al 1994).
With the benefit of having high precision without using costly inertial sensors, the RINS has become hot issue in recent years. Not only a ring laser gyro (RLG) (Ishibashi et al 2006(Ishibashi et al , 2007, but also a fibre optic gyro (FOG) (Yin 2012) and micro-electromechanical systems (MEMS) (Renkoski 2008 were all attempted to be used in RINS by researchers. More and more rotation schemes for single-axis (Ishibashi 2006(Ishibashi , 2007 or dual-axis RINS (Yuan et al 2012, Measurement Science and Technology Analysis and calibration of the gyro bias caused by geomagnetic field in a dual-axis rotational inertial navigation system Qingzhong Cai 1 , Gongliu Yang 1 , Ningfang Song 1 , Hongliang Yin 2 and Yiliang Liu 3 A rotational inertial navigation system (RINS) has been wildly used in long term marine navigation. In a dual-axis RINS, with all constant biases averaged out, the errors which can not be averaged out become the main error source. In this paper, the gyro geomagnetic biases of a dual-axis RINS are modelled, analysed and calibrated. The gyro geomagnetic biases are proved unable to be averaged out, but can be modulated to be a constant value in the navigation frame. A slope error term of longitude error is found to be caused by gyro geomagnetic biases in north and upward directions, which increases linearly with time and is remarkable in long term navigation. Thus, a calibration method based on least square regression is proposed to compensate the slope error term. Laboratory and sailing experimental results show that the divergence speed of longitude error can be effectively slowed down by the compensation of gyro geomagnetic biases. In long term independent navigation, the position accuracy of dual-axis RINS is improved about 50% by the calibration method proposed in this paper.
Keywords: inertial navigation, rotational initial navigation system (RINS), gyro geomagnetic bias, error modelling, calibration method (Some figures may appear in colour only in the online journal) Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Zhang et al 2012) are studied in both theory and experiments. To improve the accuracy, various errors in RINS are analysed, calibrated and compensated. The stochastic errors which can't be averaged out by rotational technology are analysed by simulation (Lv et al 2014) and compensated by a twice positionfix reset method (Zheng et al 2016). Some IMU errors are self-calibrated by rotating the indexing. A total least squares (TLS) method is utilized for identification of the error parameters in self-calibration for dual-axis RINS (Ren et al 2014). A backward working calibration method based on velocity error observation is proposed by Zhang et al (2015). A gyro g-dependent bias can be calibrated by introducing two more positions into a conventional six-position method (Zheng et al 2015). Some additional errors caused by the indexing are also investigated. An optical angle encoder calibration method using accelerometers are proposed by Liu et al (2013). The mounting errors between the IMU and the indexing in dualaxis RINS is analysed and calibrated by Song et al (2013). A self-calibration method for nonorthogonal angles between gimbals of RINS is proposed by Wang et al (2015).
When all constant biases of inertial sensors are averaged out in a dual-axis RINS, the errors which can't be averaged out became the main error source, such as the gyro g-dependent bias (Zheng et al 2015) and the stochastic errors (Lv et al 2014, Zheng et al 2016. The calibration and compensation of these errors becomes a key issue. The gyro bias caused by geomagnetic field (Wang et al 2010 andZhang et al 2013), named gyro geomagnetic bias in this paper, is also an unable averaged out error, whose effect on RINS has not been analysed in existing researches. The gyro geomagnetic bias is generated from the Faraday Effect (Saida 1999), and its range of optical gyro can still reach 0.0001-0.001° h −1 G −1 s −1 after double-layer magnetic shielding. In strapdown INS, the gyro geomagnetic bias is difficult to model or compensate because the directions of the three gyros are changed randomly with the maneuver of the carrier. A commonly used method for decreasing the errors caused by the gyro geomagnetic bias is magnetic shielding (Rong 2015). The gyro geomagnetic bias after magnetic shielding is much smaller, thus it is usually ignored in strapdown INS. However, in dual-axis RINS, the const gyro biases are averaged out except the gyro geomagnetic bias because the magnetic induction intensity detected by each gyro changes with the rotation of the IMU in geomagn etic field with an intensity of about 0.5 Gs. The navigation errors caused by gyro geomagnetic bias can not be ignored in dual-axis RINSs especially during longterm navigation, which has not been discussed in previous studies. In this paper, the gyro geomagnetic bias of dualaxis RINS is model led, then by analysing the effect on the long term navigation precision, a calibration method based on least square regression is proposed.
The present paper is organized as follows. In section 2, the gyro geomagnetic bias of the dual-axis RINS is modelled and analysed. In section 3, the effect of the gyro geomagnetic bias on long term navigation is deduced and simulated. In section 4, a calibration method for the gyro geomagnetic bias in dual-axis RINS is proposed and elaborated. In section 5, the calibration and compensation results of the gyro geomagnetic bias in laboratory and sailing experiments are presented. Finally, section 6 is the conclusions.

Error model of gyro geomagnetic bias
In the IMU frame, which is defined by three gyro axes and denoted by m (Song et al 2013), gyro geomagnetic bias can be written as Since common-used north-slaved local-level INSs always work in mid and low latitudes, where the geomagnetic field can be regarded as a parallel magnetic field pointing to north, the geomagnetic field intensity can be simplified in the geographic frame (defined by East, North and Upward and denoted by n) as In dual-axis RINS, because the rotation of the vehicle is isolated by the indexing mechanism, the IMU only rotates according to the designed rotation modulation scheme in n-frame. The indexing mechanism rotation matrix is equivalent to the IMU attitude matrix C n m , and the gyro geomagnetic bias of RINS is obtained as

Analysis of averaging results
Based on the gyro geomagnetic bias model of RINS, the averaging results of gyro geomagnetic bias in dual-axis RINS can be analysed in Z-axis rotation type and X-axis rotation type.
In the Z-axis rotation type with an angle of ϕ, the indexing mechanism rotation matrix can be written as When the Z-axis rotates 360°, the averaging result of gyro geomagnetic bias can be written as T is the vector of equivalent gyro geomagnetic bias in three axis directions of n-frame.
The calculation result of equation (5) is Similarly, the averaging result of gyro geomagnetic bias in the X-axis rotation type is The results show that the gyro geomagnetic bias can't be averaged out by dual-axis rotation, but it can be modulated to be a constant value in n-frame when the vehicle rotation is isolated by indexing mechanism. It should be pointed out that, the result is appropriate for all dual-axis rotating modulation schemes because all schemes are designed based on strict symmetrical rotation periodically around two axes.

The solution of error equation
By introducing gyro geomagnetic bias into the error equation of traditional strapdown INS, a complete error equation of a RINS can be written as where, C m n is the direction cosine matrix for transforming the measured specific force vector from the IMU frame to the navigation frame; [ ] ψ δα δβ δγ = T is the attitude error vector on the mathematical platform; δα, δβ and δγ are the three attitude error elements, which can be replaced by the roll, pitch and yaw Euler errors for small angle misalignments; v n represents the vector defined by the velocity directions of east, north and upward in the navigation frame; ] is the velocity error vector; L, λ and h are the latitude, longitude and height; L δ , δλ and h δ stand for the latitude, longitude and height error, respectively; R N and R E are the meridian and transverse radius of curvature, respectively; im m ω represents the angular velocity vector with respect to the inertial space measured by the gyro triad in the IMU frame; f m is the specific force vector measured by the accelerometer triad; ie ω and g are the self-rotation angular velocity and the gravity acceleration of the earth; In order to analyse the error characteristic of the dual-axis RINS in long term navigation, the traditional error equation is simplified. The earth model is simplified as a sphere with R R R N M = = . Due to the divergence and the damping of the altitude by altimeter, altitude error, vertical velocity error and other altitude coupling error terms are ignored. Because a ship's speed is slow and the angle velocity caused by a ship sailing on the surface of earth ( en n ω ) is much smaller than that of the earth's rotation ( ie n ω ), the terms of v n and en n ω in velocity errors are ignored. Also because of the low dynamics, the sailing acceleration is much smaller than the gravitational acceleration, the specific force can be simplified    When the Earth oscillation and Schuler oscillation are also ignored, the time-domain solution of position error can be further simplified as It can be concluded that a slope error term of longitude error is caused by gyro geomagnetic biases in north and upward directions, which grows linearly with time.

Mathematical simulation
To verify the theoretical analysis in this section, a mathematical simulation test is carried out. The simulation condition is given as follow: the biases of the gyros are set as 0.01° h −1 with a noise of 0.1° h −1 Hz −1/2 (sampled at 100 Hz); the scale factor errors of the gyros are set as 3 ppm; the misalignments of the gyros are set as 5″. After a 10 min initial static alignment and a 30 min multi-position dynamic alignment, the system worked for 5 d in a pure inertial navigation mode. The IMU rotates using a 16-sequence rotation scheme (Yuan 2012) for averaging out most of the constant errors. Three tests are performed with a gyro geomagnetic bias of 0.0005° h −1 in east, north and upward directions, respectively. We also assume an accelerometer geomagnetic bias of 2ug in three directions of all the three tests. The test results of latitude and longitude error are shown in figures 1(a) and (b), respectively.
It can be seen that the longitude error increases to about 2 nmiles during a 5 d pure inertial navigation when the gyro geomagnetic biases in north and upward directions are 0.0005° h −1 , which is much bigger than the oscillating errors caused by the other error sources, including east gyro geomagnetic bias, accelerometer geomagnetic biases, alignment errors and random errors. Thus, the gyro geomagnetic biases in north and upward directions are the main error sources of dual-axis RINS in long term navigation, which should be calibrated and compensated.

Calibration method
The time-domain solution of position error caused by accelerometer geomagnetic biases and alignment errors are all  oscillating or constant by the same analysis method in section 3, which is also verified by the simulation test results in section 3. Based on this, the sum of gyro geomagnetic biases in north and upward directions can be calibrated by regression analysis on the longitude error of a long term navigation result according to equation (13).
Defining the regressive equation of longitude error is t k t 0 ( ) ( ) δλ δλ = + , the traditional least square method can be used to fit the parameters as However, the north and upward gyro geomagnetic biases can not be decoupled by only longitude error. The latitude error can be used to evaluate the contribution of the two errors according to equation (12). This method needs more than one time offline calculation, classical dichotomy or other iterative algorithm can be used to get the contribution of the two errors close to the real situation. In practice, because the contribution of the two errors has little effect on long term navigation, the north and upward gyro geomagnetic biases are decoupled by a rule of not affecting the latitude error. The calibration equation can be written as To sum up, the gyro geomagnetic biases in north and upward directions can be calculated by equations (15) and (16) with k fitted by equation (14). It should be pointed out that, because the period of Earth oscillation is 24 h and the period of Schuler oscillation is 84.4 min, the calibration method need more than a 48 h navigation data with a sampling interval no larger than 42.2 min according to Shannon sampling theory.

Experiments
To verify the analysis and calibration result of gyro geomagnetic bias, laboratory and sailing experiments are carried out using a dual-axis RINS. The IMU of the dual-axis RINS contains three RLGs with an accuracy of 0.008° h −1 and three     quartz flexible accelerometers with an accuracy of 10 µg. The precision of the indexing mechanism is 5″.

Laboratory experiment
In the laboratory experiment, two groups of static tests are carried out in different latitude regions to evaluate the applicability of the calibration method. The raw data of gyros and accelerometers are stored. One group of data is used for calibration and the other is used for verification.
The first test is carried out at 40° north latitude with a 4 h alignment and a 5 d static pure inertial navigation. The position errors before the compensation of the gyro geomagn etic bias are shown in figure 2(a). The regression curve is also shown in figure 2(a) with a result: k = 0.000 347° h −1 , δλ(0) = 0.023 nmile. The calibration result of north and upward gyro geomagnetic biases are calculated as 0.000 270° h −1 and 0.000 226° h −1 by equations (12) and (13). The calibration result lies in the range of optical gyro geomagnetic bias after double-layer magnetic shielding. The position errors after the compensation of the gyro geomagnetic bias are shown in figure 2 The second test is carried out at 21° north latitude, which also contains 4 h alignment and a 5 d static pure inertial navigation. The position errors before and after the compensation of the gyro geomagnetic bias are shown in figures 3(a) and (b).
It is shown from the figures that the compensation of the gyro geomagnetic bias can slow down the divergence speed of longitude error effectively. The max longitude error is reduced from 2.7 nmiles to 1.3 nmiles, which verifies the theoretical analysis results in sections 2 and 3. The applicability of the calibration method in mid or low latitude regions is also proved by the compensation result of test 2 using the parameters calculated by test 1.

Sailing experiment
In the sailing experiment, the dual-axis RINS is mounted in a cabin of an experimental ship and the Display and Control Device is mounted on the wall (shown in figure 4).
Two groups of sailing tests are carried out at 10-20° north latitude. Both tests contain a 6 h alignment and a 5 d horizontal-damped navigation, and the velocity information of damping is provided by the on-board log. It has to point out that Schuler oscillation error can be restrained in horizontaldamping mode, but the other errors, including the slope error caused by gyro geomagnetic bias, has the same characteristic as in pure inertial navigation. In each sailing, no fewer than three times of 360° rotation, acceleration and deceleration, '∞ -type' maneuver and 'S-type' maneuver are carried out every day. Sailing trajectories of the two tests are shown in figures 5(a) and (b), respectively.
Using the high-accuracy position information of a differential Global Position System (GPS) as the true value, the position error of the dual-axis RINS before and after compensation the gyro geomagnetic bias in both sailings are shown in figures 6 and 7, respectively.
The sailing test results show that, the divergence speed of longitude error can be slowed down effectively by the compensation of the gyro geomagnetic bias. The max longitude error is restrained from 2.5 nmiles to 1.2 nmiles and from 3 nmiles to 1.5 nmiles in twice sailings, respectively. There is a little overcompensation in the result of sailing test 1, which is because the gyro geomagnetic bias drifted with the environ ment changed. Even then, the compensation result is acceptable because the drift is much smaller than the compensated error. It can be concluded from the four tests that the calibration and compensation of gyro geomagnetic bias can effectively retrain the divergence speed of longitude error, and improve the position accuracy of a dual-axis RINS about 50% in long term independent navigation.

Conclusions
The gyro geomagnetic bias is one of the main error sources in the dual-axis RINS with all constant biases averaged out. By the analysis of the gyro geomagnetic bias in this paper, we found that it can not be averaged out, but can be modulated to be a constant value in the navigation frame. Theoretical derivation and simulation indicate that a slope error term of longitude error is caused by gyro geomagnetic biases in north and upward directions, which increases linearly with time and is remarkable in long term navigation. Based on this, a calibration method based on least square regression is proposed. Laboratory and sailing experimental results show that the divergence speed of longitude error can be slowed down effectively by the compensation of gyro geomagnetic bias. In a 5 d long term independent navigation, the position accuracy of the dual-axis RINS is improved about 50% by the proposed calibration method. The analysis, calibration and compensation of the gyro geomagnetic bias in this paper apply to the long term navigation of all kinds of RINS or other high acc uracy INS.