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Abstract
In a fiber optic gyroscope rotational inertial navigation system (RINS), attitude errors may change after vibration due to the change of misalignment angles. There are two kinds of misalignment angles which can cause the same attitude errors: the one is misalignment angles of gyroscopes, and the other is misalignment angles between input axis of gyroscope and rotating gimbal axis. Thus, it is difficult to calibrate any kind of misalignment angles by attitude errors alone. Self-calibration methods can separate and calibrate the two kinds of misalignment angles. But single-axis RINSs rely on a turntable to realize the rotation scheme. And misalignment angles may change during repeated removal. Therefore, it is necessary to study an efficient and convenient method to analyze which kind of misalignment angles leads to the change of attitude errors and calibrate these misalignment angles. According to the different influences of two kinds of misalignment angles on navigation errors and fine alignment errors, this paper proposes a calibration method based on fine alignment algorithm to calibrate the gyroscopes’ misalignment angles. Its accuracy is proven by simulations and experiments. From experimental results, position errors have decreased at least 21.4% with the proposed method.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Taking advantages of low angle random walk, simple structure, low cost and high reliability [1–3], fiber optic gyroscopes (FOGs) have been widely used in rotational inertial navigation systems (RINSs). According to the number of gimbals, RINS is divided into single-axis RINS, dual-axis RINS and tri-axis RINS. Inertial measurement unit (IMU) is driven to rotate around gimbal axis. Constant inertial sensors’ errors can be modulated into sine or cosine signals whose mean value is zero [4]. As a result, the navigation precision will be improved [5,6].
However, a shortcoming of FOGs is its misalignment angle is influenced by working environment [7–9]. Although the misalignment angles can be calibrated by discrete calibration methods and self-calibration methods [10–17], they may change after the INS is vibrated significantly. Under such circumstance, there will be residual misalignment angles if they are compensated with laboratory calibration results. Due to the misalignment angle, a FOG which is perpendicular to the rotating gimbal axis will incorrectly measure the rotational angular velocity. The angle measurement error will lead to attitude errors, velocity errors and position errors. Meanwhile, there will be misalignment angles between gimbal axis and input axis of FOGs because of fabrication error [12,14,18]. This kind of misalignment angles also result in measurement errors for FOGs whose input axis is perpendicular to the rotating gimbal axis. Misalignment angles between gimbal axis and input axis of FOG only lead to attitude errors in b-frame and have no influence on velocity errors and position errors.
In traditional methods [10,14], output of FOGs and attitude errors are always used to calibrate the residual misalignment angles. But such methods cannot distinguish the misalignment angles of FOGs and the misalignment angle between gyroscope input axis and gimbal axis. The two kinds of misalignment angle can cause the same output errors, and hence the same attitude errors in b-frame. Therefore, output of FOGs or attitude errors are not appropriate for calibrating misalignment angles in a RINS.
According to self-calibration methods [10–13,15,17], error observability is promoted by different rotation schemes. Misalignment angles of FOGs can be estimated from velocity errors and position errors. However, the rotation scheme in navigation is generally different from what it is in calibration. If attitude errors change during navigation, self-calibration methods are hardly used to analyze which kind of misalignment angles result in the attitude errors. Moreover, single-axis RINSs rely on turntable to realize rotation scheme, which is inconvenient and not always available. Misalignment angles may change during repeated removal.
Therefore, it is necessary to put forward an efficient solution to separate and calibrate the two kinds of misalignment angles for single-axis RINSs. According to the different influences of two kinds of misalignment angles on navigation errors and fine alignment errors, this paper proposes a calibration method based on fine alignment algorithm to calibrate the gyroscopes’ misalignment angles. Attitude errors in b-frame and velocity errors are both used as measurement to separate two kinds of misalignment angles. Simulations and experiments are conducted to verify the proposed method.
The rest of this paper is organized as follows. Section 2 defines coordinate systems. System configuration of a self-researched single-axis RINS is also introduced in this section. Section 3 introduces the calculation process in RINSs and analyzes the influences of FOG’s misalignment angles on navigation errors and fine alignment errors. Simulation results are shown to verify the analysis. In Section 4, experiments are conducted using the single-axis RINS. Experimental results and analysis are also shown in this section. Finally, the conclusion is given in Section 5.
2. System configuration and coordinate definitions
2.1 Single-axis RINS
In this paper, experiments are conducted to verify the proposed method with a single-axis RINS. The system mainly consists of one gimbal (inner gimbal), one IMU and data processing circuits. IMU is equipped inside of the inner gimbal. Three FOGs (x-gyro, y-gyro, z-gyro) and three quartz flexible accelerometers (x-acc, y-acc, z-acc) make up of the IMU. The inner gimbal axis is coincident with the input axis of z-gyro. A rotary optical encoder and a brushless servo motor is mounted on the two ends of inner gimbal axis respectively. The motor drives the IMU rotating around the inner gimbal axis.
The configuration of single-axis RINS is shown in Fig. 1.
2.2 Coordinate definitions
Besides inertial frame (i-frame) and navigation frame (n-frame, defined as east-north-up), several coordinate systems are defined in Section 2.
The frame is an orthogonal frame defined as right-forward-upward. The relationship between b-frame and n-frame is described by transformation matrix $C_b^n$.${\theta _b}$, ${\gamma _b}$, ${\psi _b}$ is pitch, roll and azimuth in b-frame.
The frame is an orthogonal frame. $O{Z_{{A_1}}}$ directs to the inner gimbal axis. $O{Y_{{A_1}}}$ directs to the projection of $O{Y_p}$ on the normal plane of $O{Z_{{A_1}}}$. $O{X_{{A_1}}}$ directs to the projection of $O{X_p}$ on the normal plane of $O{Z_{{A_1}}}$.
If the rotation angle of inner gimbal is zero, A1-frame is coincident with b-frame. The relationship between A1-frame and b-frame is described by transformation matrix $C_b^{{A_1}}$, as shown in Eq. (2).
The relationship between p-frame and n-frame is described by transformation matrix $C_p^n$. ${\theta _p}$, ${\gamma _p}$, ${\psi _p}$ is pitch, roll and azimuth in p-frame.
The relationship between p-frame and A1-frame is described by transformation matrix $C_p^{{A_1}}$.
The definitions of ${\beta _{ZY}}$ and ${\beta _{ZX}}$ are shown in Fig. 2.
Gyroscope measurement frame (mg-frame) and accelerometer measurement frame (ma-frame) is defined by gyroscope input axes and accelerometer input axes, respectively. The two frames are nonorthogonal frames.
The installation errors of gyroscopes and accelerometers are shown in Fig. 3. $C_{mg}^p$ and $C_{ma}^p$, shown in Eq. (5), describe the rotation matrix from mg-frame and ma-frame to p-frame, respectively.
Fig. 3. The definition of installation errors. (a) Gyroscope installation errors; (b) Accelerometer installation errors.
3. Calibration method based on the fine alignment algorithm
3.1 Calculation process in the single-axis RINS
In the single-axis RINS, IMU rotates around inner gimbal axis clockwise and counter-clockwise. The rotational angular velocity is shown in Eq. (6).
The typical calculation process is shown in Fig. 4. ${\vec{\omega }^{mg}}$, ${\vec{f}^{mg}}$ and ${\varphi ^{\prime}_z}$ is original data received from FOGs, accelerometers and the optical encoder, respectively.
Before beginning the calculation process, the original output of FOGs and accelerometers should be compensated as shown in Eq. (7).
Therefore, misalignment angles of FOGs will cause both attitude errors, velocity errors and position errors. However, misalignment angles between input axes of FOGs and gimbal axis are only used to calculate attitude in b-frame. It is excluded from the process of updating velocity and position. Attitude in b-frame is updated as shown in Eq. (8) and Eq. (9).
From Eq. (8), misalignment angles of FOGs can cause attitude errors in b-frame through attitude matrix $C_p^n$, even if there are no misalignment angles between input axes of FOGs and gimbal axis. As a result, attitude errors alone cannot be used to separate the two kinds of misalignment angles.
3.2 Influence of misalignment angles on navigation error
In single-axis RINSs, gimbal axis is perpendicular to the input axis of x-gyro and y-gyro. Therefore, among the misalignment angles, the change of $\beta _{gx}^Y$, $\beta _{gy}^X$, ${\beta _{ZY}}$ and ${\beta _{ZX}}$ is the main source of attitude errors. Other misalignment angles have little effect on basic horizontal motion.
Set pitch, roll and yaw in b-frame is zero and carrier’s velocity is zero. If IMU rotates around inner gimbal axis, equivalent gyroscope drift in n-frame is shown in Eq. (10) and Eq. (11).
On the basis of Eq. (10) and Eq. (11), equivalent gyroscope drift in n-frame caused by $\beta _{gx}^Y$ is shown in Eq. (12).
By integrating Eq. (12), we can obtain Eq. (13).
According to Eq. (13), $\Delta {\phi _E}$ is a sine signal whose mean value is zero. $\Delta {\phi _N}$ is a cosine signal whose mean value is $- \frac{{\beta _{gx}^Y}}{{{\omega _r}}}({{\omega_{ie}}\sin L + {\omega_r}} )$. Similarly, if there is $\beta _{gy}^X$, $\Delta {\phi _E}$ is a cosine signal whose mean value is $- \frac{{\beta _{gy}^X}}{{{\omega _r}}}({{\omega_{ie}}\sin L + {\omega_r}} )$. $\Delta {\phi _N}$ is a sine signal whose mean value is zero.
If the carrier’s attitude is not zero and both $\beta _{gy}^X$ and $\beta _{gy}^X$ exist, $\Delta {\phi _E}$ and $\Delta {\phi _N}$ are shown in Eq. (14) and Eq. (15), respectively.
Angular velocity error in b-frame caused by ${\beta _{ZY}}$ is shown in Eq. (16).
By integrating Eq. (16), attitude errors in b-frame are obtained in Eq. (17).
From Eq. (17) and Eq. (13), if ${\beta _{ZY}}$ is equal to $\beta _{gx}^Y$, they will result in the same attitude errors in b-frame. It can be proven that ${\beta _{ZX}}$ will also lead to the same attitude errors in b-frame as $\beta _{gy}^X$ does.
3.3 Reflection of the FOG misalignment angle in the fine alignment model
Fine alignment model in this paper is shown in Eq. (18) with state variables are chosen as $X = {[{\Delta {\phi_{E0}},\Delta {\phi_{N0}},\Delta {\phi_{U0}},\Delta {\varepsilon_N},\Delta {\varepsilon_U},\Delta {K_{gz}},{\psi_0}} ]^T}$[19,20].
The measurement is derived from Eq. (19) [19].
Recursive least square (RLS) algorithm is applied to estimate state variables as shown in Eq. (20). The initial value of P is $1000 \cdot {I_{7 \times 7}}$.
Since ${\omega _{ie}}\sin L \le {\omega _r}$, Eq. (13) can be simplified as Eq. (21).
Mean value of $\beta _{gx}^Y\sin {\varphi _z}$ is zero and its period is ${{2\pi } / {{\omega _r}}}$. Mean value of $- \beta _{gx}^Y({1 - \cos {\varphi_z}} )$ is $- \beta _{gx}^Y$ and its period is ${{2\pi } / {{\omega _r}}}$.
Applying Eq. (21) to Eq. (18), we can obtain Eq. (22).
According to the characteristics of fine alignment errors caused by state variables in Eq. (22), the period of $\sin {\omega _{ie}}t$ and $\cos {\omega _{ie}}t$ is 24 hours. Such errors cannot be estimated from the measurement whose period is ${{2\pi } / {{\omega _r}}}$ and their estimation values are zero. Fine alignment error caused by $\beta _{gx}^Y$ is shown in Eq. (23).
Similarly, $\beta _{gy}^X$ leads to fine alignment error shown in Eq. (24).
If carrier’s attitude is not zero and fine alignment does not last long time, Eq. (23) and Eq. (24) can be written as Eq. (25) and Eq. (26).
From Eq. (25) and Eq. (26), if $\beta _{gx}^Y$ or $\beta _{gy}^X$ changes during navigation, it will lead to fine alignment errors. Therefore, the method can be used to calibrate the misalignment angles of FOGs.
In the analysis above, $\Delta {\phi _{E0}}$ and $\Delta {\phi _{N0}}$ are assumed only influenced by misalignment angles. But, in reality, turntable tilt and the change of accelerometer’s bias will also change $\Delta {\phi _{E0}}$ and $\Delta {\phi _{N0}}$.
The accelerometer outputs are used to monitor system’s horizontal angular motion. They are transformed from p-frame to b-frame and averaged every rotation period as shown in Eq. (27). Pitch angle and roll angle in b-frame are calculated as shown in Eq. (28).
Set the change of pitch angle and roll angle is $\Delta {\theta _b}$ and $\Delta {\gamma _b}$, Eq. (25) and Eq. (26) are corrected as Eq. (29) and Eq. (30).
3.4 Simulations and analysis
Simulations are conducted under static base to verify the influences of FOG misalignment angles on fine alignment errors. The velocity is zero. Fine alignment lasts for 20 minutes. The simulation frequency is 200Hz. Attitude and misalignment errors of FOGs are set as Table 1. Other errors are set as zero.
Table 1. Attitude and misalignment errors of FOGs in simulations
Simulations are conducted eight times. The fine alignment results and estimation values of misalignment angles are shown in Table 2.
Table 2. Fine alignment results
From Table 2, the misalignment angles of FOGs are reflected from fine alignment errors and simulation errors are within ±1″.Therefore, fine alignment method can be used to calibrate misalignment angles of FOGs.
4. Experiments and analysis
Experiments were conducted three times under stationary base with a self-researched single-axis RINS. The RINS was mounted on a turntable. Angular velocity of inner gimbal axis is 3°/s. Navigation lasted for 120 minutes. In the middle of navigation, there was vibration lasted for about 15 seconds. The navigation process is shown in Fig. 5.
The attitude in b-frame is shown in Table 3.
Table 3. Attitude in b-frame of three experiments
During Stage1, fine alignment method was applied to the data from 4min to 24min. During Stage2, fine alignment method was applied to the data from 64min to 84min. The estimation results of fine alignment method before calibration misalignment angles are shown in Table 4.
Table 4. Fine alignment results in experiments before calibration misalignment angles
According to Table 4, the change of $\Delta {\phi _{N0}}$ is about +1.1″. The change of $\Delta {\phi _{E0}}$ is within ±0.2″. Compared with the change of $\Delta {\phi _{N0}}$, the change of $\Delta {\phi _{E0}}$ is small enough to be ignored.
The attitude in b-frame obtained by accelerometer is shown in Fig. 6.
Table 5 lists the change of attitude before vibration and after vibration calculated by accelerometer.
Table 5. The change of attitude in b-frame calculated by accelerometer
From Table 4 and Table 5, the change of $\Delta {\phi _{N0}}$ should be +1.7″, instead of +1.1″.
Therefore, the change of misalignment angles is calculated by Eq. (31).
After compensated with calibration results in Eq. (35), pitch angle error and roll angle error in 1st experiment is shown in Fig. 7.
Two-hour navigation errors (CEP) are shown in Table 6.
Table 6. Two-hour navigation errors (CEP, n mile/h)
From Fig. 7 and Table 6, after compensation, attitude errors have been decreased significantly and navigation accuracy has increased 21.4% at least. However, in traditional methods, attitude errors in sine or cosine form are treated as misalignment angles between input axis of gyroscope and rotating gimbal axis. These traditional methods calibrate and compensate this kind of misalignment angles. As a result, navigation errors will not be decreased.
Therefore, compared with traditional methods, the proposed method is effective in separating and calibrating misalignment angles of FOGs.
5. Conclusion
In this paper, a calibration method for FOG misalignment angles is proposed. Misalignment angles of FOG and misalignment angles between input axis of FOG and rotating gimbal axis are separated quickly and precisely, which improves the system performance in the case of vibration. According to experimental results, after compensated with proposed method, position errors (CEP) have decreased at least 21.4%, while traditional methods will only promote attitude accuracy in b-frame. Therefore, the proposed method has an advantage in separating and calibrating the misalignment angles of FOGs over traditional methods.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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