Novel On-Line North-Seeking Method Based on a Three-Axis MEMS Gyroscope

Yu Liu, Gaojun Xiang, Junqi Guo, Min Zhou and Hongzhi Liu

(Chongqing Municipal Level Key Laboratory of Photoelectronic Information Sensing and Transmitting Technology, Chongqing University of Post and Telecommunications, Chongqing 400065, China)

Abstract: A novel on-line north-seeking method based on a three-axis micro-electro-mechanical system (MEMS) gyroscope is designed. This system processes data by using a Kalman filter to calibrate the installation error of the three-axis MEMS gyroscope in complex environment. The attitude angle updating for quaternion, based on which the attitude instrument will be rotated in real-time and the true north will be found. Our experimental platform constitutes the dual-axis electric rotary table and the attitude instrument, which is developed independently by our scientific research team. The experimental results show that the accuracy of north-seeking is higher than 1°, while the maximum root mean square error and the maximum mean absolute error are 0.906 7 and 0.910 0, respectively. The accuracy of north-seeking is much higher than the traditional method.

Key words: micro-electro-mechanical system (MEMS) gyroscope; quaternion; rotating system in real-time; north-seeking

Inertial technology, combined with classical mechanics and physics, can control the trajectory and movement posture that promotes the automation and precision of the north-finding system[1]. The gyro-based north-finding system adopts inertial technology and can result in a rapid realization of high-precision positioning with a longer period of continuous work. These advantages allow this north finding system to be used in a wide range of applications[2-3].The micro-electro-mechanical system (MEMS) gyroscope is a new type of all-solid-state gyroscope. Compared with the traditional mechanical gyroscope and optical gyroscope, MEMS gyroscope has many advantages such as small size, light weight, low cost, good reliability, large measuring range, easy to digitize and intelligent[4].At present, a north-seeking method based on single-axis MEMS gyroscope has been reported[5]with an accuracy of 2° through multiple turntable tests. The same method has achieved an accuracy of 1° approximately[6], but it still relies more on the turntable. Both methods are based on the two-position north-seeking required latitude information to find the north, so they are not suitable for rapid north-seeking at the location of an unknown latitude.

In this paper, a novel on-line north-seeking method based on a three-axis MEMS gyroscope is proposed. First, the output data is compensated and calibrated, then the attitude angles are updated by the quaternion, and north-seeking is realized by operating the rotation control system. Compared with the two-position north-seeking, the latitude information is undesired, and the turntable is used only once, while the accuracy is higher than 1°and the seeking time is less than 1 min.

1 System Introduction

Fig.1 System architecture

2 Data Filtering and Installation Error Calibrating

2.1 Data filtering

The Kalman filter is an optimal estimation method under the minimum covariance error with advantages of small computation and high real-time performance. Taking real-time and stability into account can improve the estimation accuracy of the future gyroscope continuously, by using the variance parameters measured[7].

When the three-axis MEMS gyroscope raw data is processed with the Kalman filter, (0,0,0) is used as the initial value of the state variable, and the square of the original zero bias stability of the three-axis MEMS gyroscope acts as the measured noise variance. We collected about 10 minutes’ static data from the three-axis MEMS gyroscope and then processed them by Kalman filter. The comparisons between before and after filtering data are shown in Fig.2.

The analysis of the raw data of the three-axis MEMS gyroscope processed by Kalman filter shows that the filtering effect is obvious. Therefore, the Kalman filter can effectively reduce the random noise and improve the measurement accuracy of the three-axis MEMS gyroscope, improve the accuracy of the updated attitude instrument, and provide higher accuracy of north-seeking for the program.

Fig.2 Comparison between before and after filtering data

2.2 Installation error calibration

There is an installation error between the attitude instrument and the three-axis MEMS gyroscope. The carrier coordinate system can be overlapped with the coordinate system of the three-axis MEMS gyroscope after three rotations[8]. The calibration model of the installation error of the three-axis MEMS gyroscope is

(1)

The installation error of the three-axis MEMS gyroscope can be calibrated by Eq.(1), whereωxg,ωyg, andωzgare the measurements of the three-axis MEMS gyroscope, andωxb,ωyb, andωzbare the input angular rate.Kij(i,j=x,y,z) ifi=j,Kijis the scale factor, if not,kijis the installation error factor.ωx0,ωy0, andωz0represent the zero bias of the three axes of the gyroscope respectively, which are considered as fixed zero bias when we calibrate the installation error because they are very small[9].The calibration matrix is

(2)

The -100°/s-+100°/s data of the MEMS gyroscopeZaxis were used to test Eq.(2) to get the value after calibration; the calibrated and uncalibrated data are shown in Tab.1.

As seen from Tab.1, at the angular rate of -100°/s-+100°/s, the absolute error of the Z-axis data of the MEMS gyroscope increases with the increase of the angular rate, and the installation error reaches a maximum at the angular rate of ± 100°/s. The maximum absolute error of theZ-axis data of the uncalibrated MEMS gyroscope is 1.068 61°/s, and the absolute error is higher than 0.069 99°/s after calibrating. This is improved by 1-2 orders of magnitude by improving the measurement accuracy of the MEMS gyroscope and enhancing the accuracy of the attitude instrument.

Tab.1 Comparison between before and after

3 Attitude Angle Algorithm

The quaternion and its differential equation are used to solve the conversion matrix. The attitude angle information can be updated in real time through the angular rate of the three axes of the MEMS gyroscope and the four state quantities, and that makes it convenient to operate, easy to implement and extensively applicable[10-11].

The differential equation of the quaternion is

(3)

where Q is the attitude quaternion, and Q=q0+q1i+q2j+q3k;q0,q1,q2,q3are real numbers,t0is the initial moment of movement of the attitude instrument, and Q0is the quaternion of the initial moment of the attitude angle. We set the initial attitude angles of the attitude instrument as (γ,β,α),W is the angular rate quaternion of the attitude instrument in the carrier coordinate system, and ? is the multiplication sign of quaternion.The matrix form of the differential equation is

(4)

whereωi(i=1,2,3) is the angular rate component of the three-axis MEMS gyroscope on thex,y, andzaxis of the carrier coordinate system, andΩ(ω) is an anti-symmetric matrix of 4×4.

Normally, we assume that the values of the MEMS gyroscope are constant during the sampling timeT; the discretized quaternion attitude updated formula becomes

(5)

(6)

Therefore, the updated quaternion can be obtained by knowingq0,q1,q2,q3, and the angular rate of the MEMS gyroscope, and the attitude matrix can be gotten from the relationship between the attitude quaternion and the conversion matrix

(7)

(8)

4 On-Line North-Seeking Algorithm of the Three-Axis MEMS Gyroscope

4.1 Rotary calibration algorithm

Through the process of the above data, including the installation of error calibration and the quaternion solution of attitude angles, we proposed a rotary calibration algorithm to achieve on-line north-seeking based on three-axis MEMS gyroscope. In the process of operation, we first filter the data of the three-axis MEMS gyroscope and calibrate the installation error when we ensure the rotation control system stays at the stop state. Next, according to the data of the three-axis MEMS gyroscope, the attitude angle is updated and sent to the host computer monitor by the interface RS232 in real-time. The realization process of the calibration algorithm of the on-line north-seeking based on three-axis MEMS gyroscope is shown in Fig.3.

Fig.3 Rotary calibration algorithm on-line

The north-seeking system based on the three-axis MEMS gyroscope waits for a north-seeking instruction of the host after initialization. When the bottom of the system receives the north-seeking command of the host computer, the north-seeking system would stop rotating, and the pitch and the roll of the attitude instrument can be judged by the system. According to this result, the pitch and the roll can be controlled by the rotation control system. The system sends the stop signal of the rotation control system to the host computer when the pitch and roll is 0°, and then the rotation control system will stop working. At this point, the system sends the data of the attitude instrument to the host computer where we see that the heading is the magnetic north direction. We can obtain the true north when the magnetic declination is taken into consideration in the host computer. Thus, the algorithm of the on-line rotation calibration is realized, and the calibration process of the entire algorithm is embodied.

4.2 Determination of the north-seeking

The initial state of the attitude instrument is (0,0,α), as shown in Fig.4. In the process of north-seeking, we can get the attitude information (γ′,β′,α′) in real-time through the updated attitude angles by the quaternion method, as shown in Fig.5.

When receiving the north-seeking command, the system will judge if the rollγ′ and the pitchβ′ of the attitude instrument are zero. If not, the system will be rotated to make the pitch and the roll becoming zero, as shown in Fig.6. At this point, the heading of the attitude instrument can be expressed as the angle between the attitude instrument and the magnetic north.

Fig.4 Initial state of the attitude instrument

Fig.5 Process of the attitude instrument

Fig.6 Find the north

Because there is declination between magnetic north and true north, the angle between the attitude instrument and the true north can be expressed as

Φ=α′+Δδ

(9)

whereα′ is the heading of the attitude instrument, Δδis the local magnetic declination, andΦis the angle between the attitude instrument and the true north.

5 Experimental Results

The physical status of the system is shown in Fig.7. We manually turn the roll and pitch of the

attitude instrument to ensure they display on the host computer as (-90°,-60°), (-45°,-45°), (30°,10°), (45°,45°), and (90°,60°) respectively. Then, in 30 min, six experiments were carried out for each angular position, and the heading angle was obtained by combining the local magnetic declination. The experimental results are shown in Tab.2. From Tab.2 we can see that the mean values of the five locations are -111.995 0°, -111.743 3°, -111.576 7°, -111.301 7°, and -110.440 0° respectively, and its accuracy meets the general requirements of north-seeking.

Fig.7 System of north-seeking experiment

Tab.2 Finding results at several locations

6 Conclusion

A novel on-line north-seeking method based on three-axis MEMS gyroscope has been proposed in this paper. The heading angle information of the attitude instrument is updated in real-time by using the three-axis MEMS gyroscope data which has been filtered and installation error calibrated. The true heading information of the attitude instrument can be obtained by combining the local magnetic declination when the roll and pitch of the attitude instrument are zero. This is ensured by the rotation control system, in order to realize the north-finding. The accuracy of the north-seeking is better than 1°,and the root mean square error ranged from 0.244 3 to 0.906 7, which indicated that the north-seeking method is stable, repeatable and has high accuracy.