With the rapid development of inertial technology, the rotational inertial system has been widely used for highaccuracy navigation. In addition, the singleaxis rotational inertial navigation system is one of the most popular navigation systems due to its low cost. However, the fact which cannot be ignored is that although single axis rotational inertial navigation system can restrain divergence of attitude error, it cannot eliminate velocity error, which is mainly caused by the gyro misalignment and scale factor error related to rotational axis. A novel calibration method established on filter has been proposed. Position, velocity, and attitude are chosen to be the state variables. In order to estimate the gyro errors related to rotational axis, these errors must be included in the state equation. Correspondingly, zero velocity and rate of turntable are chosen to be measured. Simulation has been carried out to verify the correctness of the theory, and the real test has been performed to further demonstrate the validity of the method. By compensating the main error estimated by filter, it can be found that the accumulated errors in velocity and attitude are decreased. So, the precision of navigation is greatly improved with the proposed method.
Inertial navigation system (INS) can obtain position, attitude, and velocity of the craft by resolving the data sampled by its inertial measurement unit, which is IMU and contains three orthogonal gyros and accelerometers [
Device error is affected by the precision of the navigation. Meanwhile, velocity and attitude errors will accumulate with time quickly [
In rotational inertial navigation system (RINS), IMU is placed on the indexing mechanization which is fixed on the craft [
The advantage of rotation modulation is that it can realize selfcompensation of sensor errors during navigation process [
In triaxis RINS, instrument errors, such as accelerometer nonlinearity and inner leverarm, can be accurately compensated through optimal estimation with velocity and position error measurements. Meanwhile, proper rotation scheme design can help modulate and estimate error parameters of IMU. Taking commercial cost into consideration, triaxis and dualaxis RINSs are more expensive than singleaxis [
In singleaxis RINS, gyro drift is the main concern. Shang et al. [
Besides the gyro drift, error propagation of the single axis rotational INS showed that scale factor error and misalignment errors related to the rotational axis cannot be modulated [
Compared with other methods, the method proposed in this paper is suitable for singleaxis RINS to calibrate main device errors by using zero velocity and attitude errors. Thus, compensation can be made to reduce the error accumulation during the navigation process. Based on this, the precision of the singleaxis RINS can be promoted.
The advantages of the proposed method are shown as follows: according to singleaxis rotational inertial navigation system, this paper provides the deduction of error propagation in terms of misalignment and scale factor error related to rotational axis. Besides, it points out that in singleaxis RINS, attitude error caused by gyro misalignment related to rotational axis can be modulated, but velocity error still exists. Simulation has been carried out to verify this thesis. By observing the velocity and rate of turntable, gyro misalignment and scale factor errors of the rotational axis can be estimated under stationary base.
The paper is organized as follows. Section
The denotation of frames is shown in Figure
Wander azimuth frame.
The wander frame is widely used in navigation system, and it can be obtained by rotating with respect to the local geodetic frame
In this paper, considering the fact calibration is performed on stationary base,
To further demonstrate the error mechanism in this paper, simplification and assumption are given.
The IMU can be rotated about the vertical body axis with constant angular velocity
The gyro angular velocity error equation is shown above, where
Gyro error model is given by equation (
The simplified gyro error model shows that the most important error sources are
The attitude error can be defined as (concrete deduction is shown in Appendix)
Besides, scale factor error of the rotational axis gyro will affect yaw angle. The velocity error dynamic equations demonstrate the relationship between attitude error and velocity error as follows:
Velocity errors can be obtained by the integral of equation (
From the final expansion of equation (
Similarly, the error velocity
Obviously, the main angular velocity error above, which is induced by scale factor error and misalignment error related to body spin axis, cannot be modulated during the rotation. The navigation attitude error will accumulate quickly because of this. Thus, the precision of the INS navigation system can be improved by using the selfcalibration method to estimate the errors related to body spin axis.
According to the error mechanism, the proposed method should estimate the instrument errors including scale factor errors and misalignment of the related rotation axis. Besides, the navigation information, position, velocity, and attitude should also be contained in the measurement matrix. Correspondingly, the measurement model of the proposed method should be established by parameters that can be obtained precisely.
The state model chosen in this paper is listed as follows:
Instrument error equations
The gyro errors are modelled as constant during the filtering process:
Attitude error equations:
where
Velocity error equations:
where
Position error equations:
where
Gyro error model:
where
The angular velocity provided by rate turntable is much larger than that of the earth, so the errors are mainly caused by scale factor errors and misalignment errors related to rotational axis. Thus, the error of gyro output and the velocity during navigation process can be utilized as observations to estimate the corresponding errors.
Given that the IMU is tested on stationary base, the velocity of IMU should be zero. The output velocity of IMU can be defined as the measurement
The measurement equation of sensed velocity of tested IMU can be described as
where the measurement matrix is
The attitude error always exists and the computed attitude matrix translating from navigation frame to body frame is represented as
Flow chart of selfcalibration filtering is given in Figure
The principle of the proposed calibration method in this paper to estimate the unknown errors is shown in Figure
Flow chart of selfcalibration filter in the proposed method.
To support the theory discussed above and evaluate the performance of the proposed method, simulations are carried out in Section
The IMU errors in this simulation are defined as follows: gyro bias is 0.1°/h with a white noise of 0.01°/h/√Hz, accelerometers bias is 100
Main error sources of accelerators.
Bias ( 
Misalignment (sec)  Scale factor error (ppm)  Noise (  














100  100  100  20  20  20  20  20  20  100  100  100  30 
In order to validate the theory above, the initial attitude angles of roll, pitch, and yaw are 20°, 30°, and 40°, respectively. The tested IMU is under stationary base. The tested IMU rotates with the angular velocity of 10°/s, and the length of the test period
Assume that there is no attitude error at the beginning of navigation and the whole navigation process lasts 360 s, including 10 whole periods
Main error sources of gyros.
Bias (°/h)  Misalignment (sec)  Scale factor error (ppm)  Noise (°/h/√Hz)  














0.05  0.05  0.05  20  60  20  60  20  20  100  100  300  0.02 
Calibration result.
Parameter 




Estimate  59.96  59.96  300.02 
Figure
Error covariance rootmeansquare curve of estimated device error.
Figure
Estimation curve of device error related to rotational axis.
Pure navigation results.
Parameter  Before compensation  After compensation  Theory value 


2.982  0.0447  0 

17.548  0.1184  0 

0  0  0 
Pitch (°)  20.467  19.954  20 
Roll (°)  29.697  30.030  30 
Yaw (°)  40.911  39.910  40 
Figure
Velocity error curve resolved by single axis rotational INS without compensation.
Table
Thus, simulation results show that velocity error caused by scale factor errors and misalignment errors related to rotation axis cannot be modulated by the singleaxis rotational inertial navigation system. With the proposed method, the corresponding errors can be calibrated precisely. Simulation results show that navigation performance is greatly enhanced by calibration and compensation.
To further validate the proposed method, the real calibration test is established. The test IMU consists of three orthogonal gyros whose accuracy is 0.1°/h and accelerometers whose accuracy is 100
Data are sampled by the frequency of 100 Hz. The turntable shown in Figure
Velocity error of pure navigation before compensation.
In the beginning, the turntable remains stationary for 140 s to collect data for initial alignment. Then, the IMU rotates at the angular velocity of 10°/s around its
Gyro errors of rotation axis are estimated by the designed selfcalibration filter. Estimated values of
Calibration result.
Parameter 





Estimated  154.2511  −107.7430  84.0126 
Figure
Errors estimation curve related to rotational axis gyro.
Velocity errors of pure navigation after compensation.
Navigation results about velocity and attitude are listed in Table
Pure navigation results in experiment.
Parameter  Theory value  Before compensation  After compensation 


0  4.9222  0.2950 

0  −1.9773  0.5888 

0  0  0 
Pitch (deg)  0.0456  0.0043  0.0467 
Roll (deg)  0.0279  −0.0758  −0.0455 
Yaw (deg)  0.0197  0.8809  0.0968 
Thus, the proposed method with filter can estimate the device error related to rotational axis effectively. The precision of the single axis rotational INS is improved by compensating the estimated steady value of corresponding errors.
In the singleaxis rotational inertial navigation system, instrument errors related to rotational axis cannot be modulated and these errors will have great impact on the navigation precision. Error mechanism and equations of velocity and attitude have been shown in this paper, and a calibration method with filter has been proposed. The state model should contain device errors which cannot be modulated by rotation. In the stationary base, zero velocity and the known turntable angular velocity can be chosen as the measurement. IMU errors can be estimated and compensated online to reduce the navigation error. Simulation has been carried out to verify the correctness of error mechanism and the feasibility of the proposed method, the experiment results further illustrate the validation of the proposed method, and the precision of the singleaxis rotational INS can be improved.
It is shown that the precision of navigation is enhanced at least 20%. The proposed method using filter can estimate the device error related to rotational axis effectively. The precision of the single axis rotational INS has been improved by compensating the estimated steady value of corresponding errors.
Deduction of equation (
In the equation,
Deduction of equation (
Normally, accelerometer error
Combining with equation (
Similarly, the error velocity
Due to project team constraints, other data materials cannot be provided for the time being.
The authors declare that they have no conflicts of interest.