Anlan YANG, Peida HU, Genming LIU, Rong ZHANG, Qiuping WU,Ruikun ZHOU
Department of Precision Instrument, Tsinghua University, Beijing 100084, China
KEYWORDSInertial Navigation System(INS);Attitude reference;Level meter;Turntable;Attitude evaluation;Systematic calibration
AbstractAttitude references are greatly needed for the evaluation and calibration of Inertial Navigation Systems (INSs), which are widely used in gravimeter, marine, and aeronautical navigation.High-accuracy turntable, INS, and Global Navigation Satellite System have been utilized to verify the performance of relatively low-accuracy INS.The accuracy requirement of the attitude reference continuously increases with the rapid improvement of inertial sensors and navigation algorithms.However, the cost of attitude determination system increases rapidly with the increase of attitude accuracy requirement.To solve this limitation, the integration of level meter, INS, and low-cost turntable is proposed to provide level attitude, such as roll and pitch.The turntable is utilized to rotate the INS.An integration model of the level meter and INS is built to estimate the level attitude and reduce the cost of the turntable.The proposed method successfully avoids the dependence on high-accuracy turntables.An observability degree analysis is conducted to improve the level attitude accuracy further.The simulation and turntable test results indicate that the proposed method can provide high-accuracy level attitude without high-accuracy INS or turntable and is applicable to error calibration and attitude evaluation of INS.
Attitude references are crucial in the evaluation1,2and calibration3,4of INSs,which are widely used in gravimeter,5marine,6and aeronautical navigation.7The INS,8Global Navigation Satellite System (GNSS),4and turntable9are widely used attitude references in the calibration and evaluation of INS.INSs with high accuracy have been commonly utilized to evaluate the performance of other INSs.STIM300 has been applied to evaluate the performance of INS on the basis of consumergrade Micro-Electro-Mechanical System (MEMS) sensors.10Fiber Optic Gyroscope (FOG) Strapdown INS (SINS) has been used to examine the performance of six-axis MEMS Inertial Measurement Unit(IMU)and single-axis FOG integrated system.2However, the navigation error of INS diverges with time due to drifts of gyros and accelerometers.Other navigation systems, such as GNSS,11,12Celestial Navigation System(CNS),13and visual navigation system,14are added to suppress the convergence of the navigation error.Yang et al.15used an IMU/GNSS navigation system to evaluate the performance of an attitude determination system.Zhu et al.16applied the highly accurate integrated navigation system SPAN-FSAS from Novatel to evaluate the performance of dual-antenna GNSS and MEMS inertial sensor based attitude determination system.The accuracy requirement of the attitude reference continuously increases with the rapid improvement of inertial sensor and navigation algorithm.17However, the cost of these attitude determination systems increases rapidly with the increase of attitude accuracy requirement.
The attitude can also be determined with the multiantenna carrier phase measurement of GNSS.18The attitude determination accuracy is mainly determined using the baseline length between antennas.19Hence, the attitude accuracy is limited by the vehicle dimension.However, increasing the baseline using two vehicles is complicated and expensive.4The CNS/INS can also output the attitude information for the evaluation of other attitude determination systems.20However, the CNS/INS is more complicated than the GNSS/INS and it is unavailable for most researchers.Although the INS/CNS/GNSS integrated navigation system can provide highprecision and robust attitude information, the complexity of the method is high.21Moreover,the attitude accuracy of these systems should also be evaluated using other attitude determination systems, such as turntable.
The turntable can be utilized to simulate vehicle motion and provide high-accuracy attitude reference.9,22Liang et al.1used the three-axis turntable to evaluate the performance of an attitude determination system consisting of two micromechanical units of the consumer-grade gyro MPU6050.The United States Navy’s Space and Naval Warfare Systems Center Atlantic utilized turntables to evaluate the attitude accuracy of the high-accuracy iXBlue MARINS FOG INS.9Yang et al.23applied a high-precision three-axis turntable to evaluate the performance of an attitude determination system based on MEMS-IMU.The turntable can also be used to calibrate error parameters of the INS.Du et al.3calibrated the error parameters of accelerometers for redundant MEMS INS.Dong et al.24used a turntable to calibrate error parameters of the accelerometer and gyro.However,the attitude accuracy of the turntable is mainly determined by the accuracy of encoders and the mechanically manufactured structure.25Notably, the cost of the turntable is unaffordable for many users because it increases rapidly with the increase of angle accuracy requirement.To date, the attitude accuracy of INSs such as iXBlue MARINS INS,9RINS WSN-7,26and POSAV 310,27has reached the level of tens of arcseconds.That is to say, the attitude reference accuracy should be at the level of several arcseconds, which is unavailable for the majority of researchers.
Many researchers have attempted to decrease the demand of high-precision turntable by utilizing the error characteristic of the INS and turntable rather than increasing the accuracy of the turntable.24,28,29Dong et al.24improved the precision of the turntable by calibrating the constant zero position, installation and orthogonal errors of the turntable using outputs of the turntable and INS.However, installation, orthogonal and angle errors of low-cost turntable change with the rotation angle and time.Xu et al.28and Dai et al.29proposed a calibration method of INS using low-cost turntable, which is only applied to rotate the IMU.However, this method is unable to separate the attitude bias of INS accurately without accurate attitude information.Wei et al.30used the electronic level meter to provide initial level attitude of the high-accuracy INS under static condition.However,this method is inapplicable to both the evaluation and calibration of INS.Wang et al.31and Hu et al.32utilized the electronic level meter to measure the initial attitude of the turntable, which was then applied to simulate the motion of the vehicle and provide attitude reference simultaneously.Hence, the high-accuracy turntable is still needed for the attitude accuracy evaluation of the INS.
Attitude accuracies of these integration methods are mainly determined by the turntable given that the angle of the turntable is used to measure the level attitude.To further improve the accuracy, the attitude error model of the turntable should be modeled accurately.33However, low-cost turntables fail to satisfy the attitude accuracy evaluation and calibration of high-accuracy INS because their installation, orthogonal,and angle errors change with the rotation angle and time.To solve this limitation, the integration of high-accuracy electronic level meter, low-cost turntable, and evaluated INS is proposed to provide level attitude, such as roll and pitch.The roll and pitch are measured by comparing oscillation angles of the level meter when it swings clockwise and counterclockwise.34The electronic level meter is typically utilized to detect the attitude of controllable platforms35and provide level attitude for the high-accuracy INS.30A single-axis turntable is needed at least because the attitude error model of INS is related to its heading.36The heading of INS is provided by the evaluated INS to reduce the cost of turntable further.The heading reference design method is excluded from this study given that the heading reference can be obtained using electronic theodolites and north reference mirror.30.
This paper presents a novel attitude determination method that integrates the electronic level meter, INS, and low-cost turntable for the level attitude evaluation and calibration of INS.The rest of this paper is organized as follows.The attitude error model of the proposed method is described in Section 2.The proposed level attitude determination method is introduced in Section 3.The simulation and turntable tests are conducted to validate the effectiveness of the proposed method for level attitude evaluation and calibration of INS in Sections 4 and 5, respectively.The conclusions are drawn in Section 6.
The attitude of vehicle is obtained from Cnb, which can be expressed using three Euler angles37as follows:
Cnbcan be estimated using an INS with the output of inertial sensors and corresponding attitude determination algorithm.However, unavoidable errors, such as constant gyro drift,accelerometer bias error, accelerometer scale factor error,accelerometer installation error, initial alignment error, and high-frequency measurement error affect the attitude accuracy of INS.38A frequency characteristic-based attitude error model is adopted to build the error model given that the level attitude caused by these error parameters changes periodically.The attitude error at 12 and 24 h,oscillation errors with Schuler and rotation periods,and high-frequency errors39,40should be taken into account in the error model.The attitude error with the Schuler period is ignored because the damping technology is adopted to suppress the Schuler oscillation error of INS.41Given that the rotation periods of different rotational INSs differ with each other38,42and the oscillation attitude error can be compensated accurately,43the residual oscillation attitude is treated as high-frequency measurement error.Thus,the periodical level and down components of the attitude error can be simplified as
Eq.(9)implies that the attitude error at 12 and 24 h can be estimated under the static condition and a rotation test is needed to estimate these parameters given that the attitude error is related to the attitude of the INS.
The evaluated INS is mounted on a single-axis turntable,which is utilized to simulate the motion of the vehicle.The heading is provided by the INS to decrease the accuracy requirement of the turntable.According to Eq.(9), the pitch and roll error can be expressed as
As the level attitude is obtained without using the angle of the turntable, the cost of turntable is reduced greatly.
Observability degrees of the calibration method with different estimation time in Eq.(18) and rotation angle interval in Eq.(21) are further analyzed given that the observability of estimated parameters exerts important effects on the calibration accuracy.44As seen in Fig.1,the estimation time and rotation angle interval of rotation can be optimized to improve the level attitude estimation accuracy in accordance with the observability analysis result.
The observation matrix H in time step k can be written as H(k )in the static test.The local observability matrix can be defined as
The singular value decomposition method is used to estimate the observability degrees.45Observability degrees with different estimation time are shown in Fig.1 (a).The observability degree increases with the estimation time, and the local maximum observability degrees are obtained at integer multiples of 24 h.The least estimation time of 24 h is selected because the observability degree difference in local maximum observability degrees is within 0.14.
The turntable rotates at approximately the same angle and stops at that position continuously in the rotation test.It rotates 360° in the rotation test.The observability analysis with different rotation angle intervals is shown in Fig.1 (b).The observation degree improves with the decrease of the rotation angle interval.The observability degree changes slowly when the rotation angle interval is decreased to 36°.Hence,a rotation angle less than 36° is chosen.With the optimized parameters, a simulation is conducted to verify the proposed method.
Fig.1 Observability degrees with different estimation time and rotation angle intervals.
The periodical level attitude error of INS is mainly caused by the accelerometer bias and installation errors.The magnitude of the accelerometer bias and installation errors determine the values of A01and A11,that is,A01and A11can describe both the magnitude of the level attitude error and the performance of INS.Therefore,the error parameters of level attitude errors in the simulation are listed in Table 1.The periodical level attitude error contains the periodical errors at 24 and 12 h and constant level components of the attitude error and bias.The level attitude error of INS is further added with discrete white noise with a standard deviation of 6.0 arcsec (1 arcsec =(2.8 × 10-4)°).The amplitude of level attitude and highfrequency errors reaches 228.8 and 25.8 arcsec, respectively The constant bias and nonlinear errors of the electronic level meter are set to 1.0 arcsec and 0.02,respectively.The electronic level meter is added with first-order Gauss–Markov bias with a standard deviation of 0.2 arcsec.
Table 1 Designed and estimated values of error parameters.
The estimation result of the roll is analyzed as an example given that the roll and pitch present similar error characteristics.39Fig.2 shows the roll (blue) of INS, estimated roll(red) provided by the proposed method, and true roll (green).The roll estimation error under the static condition is 1.8 arcsec.The roll estimation error in the rotation process is 22.2 arcsec, which is mainly caused by the high-frequency error of INS, as shown in Fig.3.If the standard deviation of discrete white noise of the INS reduces from 6.0 arcsec to 1.0 arcsec,the accuracy of the level attitude error will decrease from 22.2 arcsec to 4.7 arcsec.This finding indicates that the estimation error is mainly caused by the high-frequency error of INS.
The roll provided by the proposed method is utilized to evaluate the performance of the INS.Fig.4 shows the calculated roll error of INS using the roll provided by the proposed method and the true roll value.The effectiveness of the proposed method in evaluating the roll accuracy of INS is verified because the difference is within 1.0 arcsec.Estimated error parameters can be used for the calibration of INS further.The roll error of INS obtained after applying the calibration method of Ref.[46] is presented in Fig.5.The low-frequency roll and pitch error are compensated accurately.The roll error decrease from 146.6 arcsec to 27.6 arcsec and the pitch error decrease from 183.9 arcsec to 28.3 arcsec after calibration.The calibration results verify the effectiveness of the proposed method for the level attitude calibration of the INS.
A turntable test is conducted to evaluate the performance of the proposed method.The evaluated INS is installed on a single-axis turntable, as shown in Fig.6 (a).The INS consists of two dual-axis gyroscopes and three pendulous accelerometers.The gyro drift and accelerometer bias of the RINS are 1×10-3°/h and 2×10-5g,respectively.The single-axis turntable significantly decreases the requirement and cost of the turntable because this turntable is only utilized to rotate the INS and not required to output the attitude.The electronic level meter is placed on the horizontal datum plane of the INS to measure the roll and pitch of the INS directly given that the cost of the turntable increases with the accuracy of the machined surface of the turntable.The precision of the level meter is at the level of several arcseconds.The level meter is placed along the x-axis and y-axis to measure the roll and pitch of the INS, as shown in Fig.6 (b).
Fig.2 Comparison of rolls provided by different methods.
Fig.4 Roll errors obtained using proposed method and true roll.
Fig.5 Roll errors before and after calibration in simulation.
The static and rotation tests conducted are similar to those of the simulation.The rotation interval is 30°.The level meter is utilized to measure the level attitude of INS under the static condition,and the heading output of INS is recorded simultaneously.All attitude data of INS are automatically saved.
Similar to the simulation,the estimation result of the roll is analyzed as an example.The normalized roll values provided by the INS (blue), level meter (green) and proposed method(red) are shown in Fig.7.The roll provided by the INS under the static condition consists of constant, low-frequency, and high-frequency roll errors in addition to the constant roll value.The roll under the rotation condition consists of the true roll value due to rotation and roll errors caused by different error parameters, as shown in Fig.7.This finding is consistent with the simulation result.
Fig.6 Integration of level meter, turntable, and evaluated INS.
Fig.7 Normalized roll in turntable test.
Fig.8 Roll errors before and after calibration in turntable test.
Fig.9 Comparison of proposed method and method using highprecision turntable.
To further verify the performance of the proposed method,a comparative experiment is conducted by using high-accuracy turntable.The same type of INS is installed on a high-accuracy turntable.This turntable is able to simulate motion and provide attitude information simultaneously.The static and rotation tests are also conducted,which are similar to the proposed method.The attitude reference is provided by the turntable instead of the proposed method automatically and consistently.Hence, the operation of the turntable is more convenient than the proposed method.As shown in Fig.9, the error (red) of the method using high-precision turntable is 0.018.The periodical level attitude errors of both methods are suppressed successfully.The level attitude error separated by the proposed method is consistent with that of the highaccuracy turntable.This finding indicates that the proposed method can provide high-accuracy level attitude reference for the evaluation of INS without using high-accuracy turntable.
The attitude reference is crucial in the evaluation and calibration of INS.The expected accuracy of attitude reference increases with the rapid improvement of inertial sensor and attitude determination algorithm.Many methods have been proposed to improve the accuracy of the attitude reference using high-accuracy turntable or integrated GNSS/INS.However,the cost of necessary equipment increases rapidly with the increase in accuracy.The integration of electronic level meter,low-cost turntable, and evaluated INS is proposed to obtain the level attitude reference.The following conclusions can be drawn from this study:
(1) The integration of electronic level meter and the evaluated INS rather than the high-accuracy turntable is adopted to provide high-accuracy level attitude reference and reduce the cost of the turntable significantly.
(2) The attitude estimation model based on the electronic level meter and the evaluated INS is built.Observability degree analysis is conducted to improve the level attitude estimation accuracy.Hence, the proposed method can successfully provide highly accurate level attitude.
(3) The simulation and turntable tests are conducted to verify the performance of the proposed method.The simulation results show that the difference in calculated attitude error values obtained via the proposed method and the true value is within 1.0 arcsec.The turntable result shows that the level attitude accuracy can be further improved by 83.9 % using parameters provided by the proposed method.Hence, the proposed method can provide the roll and pitch reference for the calibration and evaluation of INS without using high-accuracy turntable.
(4) The proposed method can be applied to estimate or calibrate the level attitude error of INS indoor.It is also applicable to the INS installed in car,airplane and other carriers under static condition.
(5) The optimal estimation time of 24 h is selected as local maximum observability degrees are obtained at integer multiples of 24 h.In practical application,the estimation time and rotation angle interval can be further adjusted in accordance with the observability analysis results and practical conditions.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
This work was supported by the National Natural Science Foundation of China (No.62073184).
CHINESE JOURNAL OF AERONAUTICS2023年4期