MAO Yu-liang,CHEN Jia-bin,SONG Chun-lei

(School of Automation,Beijing Institute of Technology,Beijing 100081,China)

Optimal axis selection scheme of SINS single-axis rotation modulation

MAO Yu-liang,CHEN Jia-bin,SONG Chun-lei

(School of Automation,Beijing Institute of Technology,Beijing 100081,China)

The rotation modulation technology of strapdown inertial navigation system(SINS) is based on a self-calibration method,which modulates the constant drift of gyroscope and bias of accelerometer into periodic signals.And these periodic signals can be eliminated through integral calculation,so that the inertial navigation system(INS) can automatically compensate the navigation error caused by gyro drift and accelerometer bias without using external information,thereby improving the system’s accuracy.Based on the single-axis rotation modulation principle,this paper also analyzes the manifestations of constant drift,installation errors,and scale factor error of gyro and accelerometer when IMU uniaxially rotating about an arbitrary axis.The optimal axis selection scheme is given to minimize the gyro and accelerometer constant drifts.The simulation and experiments are carried out,which verify the effectiveness of the proposed scheme.The conclusions can also assist the design of the optimal single-axis rotational SINS to minimize the adverse effects of gyro and accelerometer constant drifts.

error compensation; gyro drift; scale factor error; single-axis rotation; strapdown inertial navigation system

For longtime working INS,the drift of inertial measurement unit(IMU) will cause positioning error accumulated over time,which is an important factor to affect the INS performance.There are two main methods to improve the accuracy of INS,one is the device-level method,i.e.,developing high-precision inertial devices,mainly high-precision gyroscopes,the other is systemlevel approach,including the system calibration,zero velocity update,and rotation modulation method[1-4].Nowadays,an increasing number of researches have been focused on the latter.In 1880s,Levinson put forward the idea of rotation modulation[5],since then,the research of the rotary INS has drawn extensive attention.Currently,there are two commonly implemented rotation schemes,i.e.,single-axis rotation[6-9]and dual-axis rotation.From the perspective of engineering implementation,the advantage of single-axis rotation scheme is simplicity and high reliability.A simple single-axis rotary INS can be built by installing a SINS on the turntable.Although dual-axis rotation can compensate all the errors but random walk,accurate rotation scheme design,higherorder Kalman filtering techniques and more rational design of the IMU structure make it difficult to implement and the cost is greatly increased compared to the single-axis rotation system[10].

The single-axis rotation modulation is a technique that by rotating IMU around an axis periodically according to some rule,makes the inertial device mean error as close to zero as possible in a rotating cycle,thereby reducing the accumulation of systematic errors,and improving navigation accuracy[11-12].Starting from the general SINS error equation,the rotary INS error propagation equation is analogized according to the relationship between the rotating coordinate system and the carrier coordinate system.So far,numerous work has been done with regard to uniaxial rotation around the vertical axis,and clear conclusions have been drawn as to the modulation of gyro constant drift and accelerometer bias[13-14].In addition,the influence of scale factor error and installation error is analyzed in the situation that the rotating axis doesn’t coincide with either the carrier coordinate system or the IMU coordinate system.

1 Quaternions and rotation vector

Quaternion algebra was introduced by Hamilton in 1843.As a set,the quaternions are equal toR4,a four-dimensional vector space over the real numbers.,.i,j,andkare mutually orthogonal unit vectors,which satisfy the following equations[15]:

The angular positional relationship between two coordinate systems can be expressed as the fixed-point rotation of a rigid body.Coordinate system A rotates around an axis to coordinate system B with a certain angle and this rotation can be represented by rotation vectorΦ.denotes the rotation angle and the unit vectordenotes the direction of rotation.The relationship between quaternions and rotation vector is:

The corresponding direction cosine matrix(DCM)can also be obtained as

2 Selection of optimal axis

Fig.1 Schematic of IMU rotation

The initial coordinates and the rotating coordinates of IMU are defined asPandR,respectively.In general situation,the IMU rotates around axisat the speed of,where,Uis the unit vector which satisfies the formula.CoordinatesPandRcoincide with each other in the initial moment,and aftertseconds,the rotating angle betweenPandRis.Using (1) and (2),we obtain:

Specially,when IMU rotates around axis,,

Set the initial IMU coordinatesPcoincide with the vehicle coordinates b,and obviously.

The IMU errors projected to b frame are:

Here we take gyro errors as an example for analysis,and the results are identical to that of accelerometer errors.

2.1 Modulation of gyro drift and accelerometer bias

The first term of the right part of (7) is:

It shows that,in three axial directions of b frame,the constant drift of gyro is modulated into two parts:one is the sinusoidal signal; the other is the constant component.The sinusoidal signal can be eliminated through integral operation,while the constant component causes system error accumulated over time.The purpose of the paper is to minimize the constant component error by choosing appropriate combinations of l,m ,and n.

The constant components of gyro drift in three axial directions of b frame are,.

In order to completely modulate the constant drift of gyro into sinusoidal signals,we need to set all the constant components to zero:

Noting the symmetrical form ofl,m, andn, any two variables have the same relationship shown in Fig.2.

Fig.2 Solutions of m and n

The range ofl,m, andnis.

In this scheme, the rotating axis lies in theplane,θ=45°.

Fig.3 IMU rotating around an optimal axis

Solution of (10) is similar to that of (9), except thatl ,m,n no longer has a completely symmetrical form. Givenεx=0.003 (°)/h,εy=0.004 (°)/h,εz=0.005 (°)/h, the relationship of m and n is shown in Fig.4.Specially, whenm=0,n=±0.514,.

Fig.4 Solutions of m and n

2.2 Modulation of scale factor error

The second and third terms of the right part of (7) are:

Where, (×)is periodic term. If we want to eliminate the equivalent gyro drift caused by the gyro scale factor error in three directions, the non-periodic term of (11) and (12) must be set zero.

In order to get real number solution ofl,m, andn,the following conditions need to be met:

Or,equivalently,

The second Inequality of (15) is met as long asare not exactly equal to each other.However,it’s very harsh to satisfy the first inequality.The influence of scale factor error on system includes two parts: one is brought by rotating angular velocity,the other is introduced by,i.e.,body movement and rotation of the earth.We notice from (11) and (12) that,the installation error is modulated automatically,because it is contained in the periodic terms.Compared with the constant drift of gyro,the scale factor error has a much smaller impact on system.However,as rotating angular velocity gets larger,the influence of scale factor error becomes significant and the system performance degenerates faster.

3 Simulation study

This section is devoted to numerical verification of analytical results,using extensive simulations.The strapdown INS(gyroscope drift of 0.005 deg/h,accelerometer bias of 50 μg) is assumed to be located at latitude 39.9461 deg,longitude 140 deg,and.The initial attitude of strapdown INS is known precisely and simulations of 72-hours-static navigation are carried out.

3.1 Without rotation

The simulation of static navigation without rotation is to examine the influence of gyroscope constant drift and accelerometer bias on positioning error of strapdown INS.The latitude error oscillates within 3 arcmin,whereas the longitude error diverges to 27 arcmin after 72 hours,leading to position error divergence up to 40 km.

3.2 Single-axis rotation aroundp

The strapdown INS is rotated alongat 5 deg/s,i.e.,.Gyro drift and accelerometer bias perpendicular to the rotating axis can be modulated into sinusoidal terms,improving the system performance.Latitude error,longitude error,and position error are reduced to 1.2′,6′,and 14 km respectively,compared to those of navigation result without rotation.However,the longitude error still diverges over time in spite of accuracy improvement.That is because the gyro drift alongcan not be modulated,causing longitude error accumulated over time.

Fig.5 Latitude error

Fig.6 Longitude error

Fig.7 Position error

3.3 Optimal single-axis rotation

Fig.8 Latitude error

Fig.9 Longitude error

Fig.10 Position error

Fig.11 Latitude error

Fig.12 Longitude error

Fig.13 Position error

3.4 Scale factor error and rotating angular speed

The positioning error caused by scale factor error (5×10-6) is shown in Fig.14-16.

The scale factor error has a much smaller impact on system compared with the gyro drift and accelerometer bias,with latitude error 0.003 arcmin,longitude error 0.4 arcmin,and position error 900 m during 72 h staticnavigation.When IMU rotates consecutively around,the error caused by scale factor grows much larger,as shown is Fig.17-19.

Fig.14 Latitude error

Fig.15 Longitude error

Fig.16 Position error

Fig.17 Latitude error

Fig.18 Longitude error

Fig.19 Position error

Due to the influence of scale factor error on system positioning accuracy,continuous single-axis rotation can not meet the accuracy requirement of long-time navigation.Positive-negative rotation is applied to reduce the positioning error.The IMU rotates aroundat 1 deg/s for one cycle,and then counter-rotates for another cycle,and so forth.The navigation results of continuous rotation and positive- negative rotation at 1 deg/s is shown in Fig.20-22.

Fig.20 Latitude error

Fig.21 Longitude error

Fig.22 Position error

4 Conclusions

Rotation modulation technology is an effective way to improve the strapdown INS accuracy.This paper proposes an optimization-based axis selection scheme based on minimizing positioning error caused by gyro drift and accelerometer bias.This optimal axis selection scheme can modulate all the gyro drift and accelerometer bias into sinusoidal signals in three axial directions.With continuous single-axis rotation,the positioning error caused by scale factor error is enlarged through rotating angular rate.A large number of simulation tests are carried out to evaluate the single-axis rotation modulation scheme proposed in this paper.The result shows that the system performance improves dramatically by rotating IMU around optimal axis,and with positive-negative rotation,the error brought by rotating angular velocity is reduced significantly.

[1]Yang Y,Miao LJ.Fiber optic strapdown inertial system with sensing cluster continuous rotation[J].IEEE Transactions on Aerospace and Electronic Systems 2004;40(4): 1173-8.

[2]Ishibashi S,Tsukioka S,Yoshida H.Accuracy improvement of an inertial navigation system Brought about by the Rotational Motion[C]//OCEANS.2007: 1-5.

[3]Yi G X,Xie Y G,Wang C H,Zhao H L.A new type of rotating modulation SINS[J].Journal of Chinese Inertial Technology,2011,19(3): 261-6.

[4]Zha F,Xu J N,Qin F J.Influence of Rotation Schemes on Error of SINS[J].Electronics Optics & Control 2012,19(3): 47-51.

[5]Emaneul L,Robert M.Accuracy enhancement techniques applied to the marine ring laser inertial navigator(MARLIN)[J].Navigation,1987,34(1): 64-87.

[6]Shang S T,Deng Z H,Fu M Y,et al.Analysis of single-axial rotation modulation influence on FOG INS precision[C]//Proceedings of the 29th Chinese Control Conference.Beijing,China,2010: 5469-72.

[7]Long X W,Yu X D,Zhang P F,Wang Y,Tang J X.Single-rotating inertial navigation system with ring laser gyroscope[J].Journal of Chinese Inertial Technology,2010,18(2): 149-53.

[8]Ben Y Y,Wu X,Chai Y L,Li Q.Research on error modulating of SINS based on single-axis rotation[C]//Proceedings of the 2011 IEEEIICME.Harbin,China,2011.

[9]Yu X D.Research on some key technologies for single-axis rotation inertial navigation system with mechanically dithered ring laser gyroscope[D].Changsha: National University of Defense Technology,2011.

[10]Wang Z H,Chen X J,Zeng Q S.Ship’s inertial navigation system rotating modulation based on navigation coordinates[J].Journal of Harbin Engineering University,2011,32(7): 921-6.

[11]Lai J Z,Yu M Q,Lv P,et al.Analysis on rotation SINS precision modulated in different directions[J].Journal of Chinese Inertial Technology,2012,20(1): 7-11.

[12]Sun F,Sun W.Research on auto-compensation by rotation in strapdown inertial navigation systems[J].System Engineering and Electronics,2010,32(1): 122-5.

[13]Titterton D H,Weston J L.Strapdown inertial navigation technology[M].2nd ed.United Kingdom: Institution of Electrical Engineers,2004.

1005-6734(2014)02-0149-08

單軸旋轉(zhuǎn)捷聯(lián)慣導系統(tǒng)最優(yōu)轉(zhuǎn)軸選取方案

毛玉良，陳家斌，宋春雷
（北京理工大學 自動化學院，北京 100081）

捷聯(lián)慣性導航系統(tǒng)的旋轉(zhuǎn)調(diào)制技術(shù)是一種自校正方法，它能將慣性測量單元中陀螺儀的常值漂移和加速度計的零偏調(diào)制成周期性的信號，通過積分運算消除這些周期信號對系統(tǒng)的影響。從而使得慣導系統(tǒng)在不使用外部信息的條件下，自動補償由陀螺漂移和加速度計零偏引起的導航誤差，提高系統(tǒng)精度。從單軸旋轉(zhuǎn)調(diào)制原理入手，詳細推導分析了IMU繞任意轉(zhuǎn)軸做單軸旋轉(zhuǎn)時，陀螺和加速度計常值漂移、安裝誤差、刻度系數(shù)誤差在單軸旋轉(zhuǎn)下的誤差表現(xiàn)形式，基于最大限度消除陀螺和加速度計常值漂移的原則，給出了最優(yōu)的轉(zhuǎn)軸選取方案。進行了大量仿真和實驗，證明了提出的旋轉(zhuǎn)方案的有效性。

誤差補償；陀螺漂移；刻度系數(shù)誤差；單軸旋轉(zhuǎn)；捷聯(lián)慣導系統(tǒng)

U666.12

：A

2013-11-13;

：2014-03-11

國家自然科學基金（90920304）

毛玉良（1985—），男，博士研究生，從事定位定向技術(shù)研究。E-mail：maoyulaingready@163.com

聯(lián) 系 人：陳家斌（1965—），男，教授，博士生導師。E-mail：chenjiabin@bit.edu.cn

10.13695/j.cnki.12-1222/o3.2014.02.002