Error Modeling, Calibration, and Nonlinear Interpolation Compensation Method of Ring Laser Gyroscope Inertial Navigation System

Abstract

In order to improve the precision of Strapdown Inertial Navigation System (SINS) and reduce the complexity of the traditional calibration method, a novel calibration and compensation scheme is proposed. An optimization calibration method with four-direction rotations is designed to calculate all error coefficients of Ring Laser Gyroscope (RLG) SINS in a series of constant temperatures. According to the actual working environment, the temperature errors of RLG SINS are compensated by a nonlinear interpolation compensation algorithm. The experimental results show that the inertial navigation errors of the proposed method are reduced.

1. Introduction

Motion error compensation based on a high precision Position and Orientation System (POS) is an effective avenue to improve the imaging quality of the interferometric Synthetic Aperture Radar (SAR) [1, 2]. The Strapdown Inertial Navigation System (SINS) of the POS is fixed on the antenna of the SAR outside the aircraft cabin; so, the measurement results are easily affected by the extreme temperature environment. Therefore, an accurate calibration and compensation method for SINS is necessary to improve the measurement precision and environmental adaptability of SINS [3].

The core sensors of SINS are the gyroscopes and accelerometers; the gyroscopes are sensitive to the rotation rate, and the accelerometers are sensitive to the acceleration. So, the precision of SINS depends on the precision of the gyroscopes and accelerometers. The traditional calibration methods for SINS include static multiposition calibration method [4], rotation rate calibration method [5], and dynamic-static hybrid calibration method [6], and so on. Among all the methods, the static multiposition calibration method can accurately calibrate the error coefficients of acceleration channel, while the accuracy of the error coefficients in angular rate channel is very low. On the other hand, the traditional rotation rate calibration can improve the precision of the angular rate channel error coefficients, but it reduces the precision of acceleration channel. Reference [7] proposed a 24-point dynamic-static hybrid calibration method. The method improves the precision of the calibrated coefficients in both of the two channels, but increasing the complexity of the calibration. So, it is another key technology for SINS to find a better method to reduce the complexity and error of the calibration at the same time.

Furthermore, the change of the working environmental temperature also decreases the precision of the SINS significantly [8, 9]. The linear fitting method is often applied to compensate for the temperature errors of inertial devices biases and scale factors. The method is easy to be implemented, while the precision of compensation is low. Other nonlinear methods, such as the high order least-squares approximation method and the Radial Basis Function (RBF) neural network method, improve the precision of the SINS, but they cannot meet the requirement of real-time processing because of the large amount of calculation. So, it is another key problem for SINS to resolve the previous conflicts. Based on the theoretical analysis and experimental data, the calibration and compensation methods of the temperature and dynamic error for SINS are studied in this paper, and an optimization calibration method with four-direction rotations and nonlinear interpolation compensation method is proposed for SINS in full temperature ranges.

2. Error Modeling of RLG SINS

2.1. Error Modeling of the Angular Rate Channel

Dithered Ring Laser Gyroscope (RLG) is not sensitive to the gravity acceleration. Considering the scale factors, biases, and misalignment errors, the error model of angular rate channel is written as 𝐺_𝑥𝐺_𝑦𝐺_𝑧=𝐺_𝑏𝑥𝐺_𝑏𝑦𝐺_𝑏𝑧+𝑆_𝑔𝑥𝐸_𝑥𝑦𝐸_𝑥𝑧𝐸_𝑦𝑥𝑆_𝑔𝑦𝐸_𝑦𝑧𝐸_𝑧𝑥𝐸_𝑧𝑦𝑆_𝑔𝑧𝜔_𝑥𝜔_𝑦𝜔_𝑧, (1) where 𝐺_𝑖 and 𝐺_𝑏𝑖 are gyroscope output and bias in 𝑖 axis, 𝜔_𝑖 is inputting rational rate, 𝑆_𝑔𝑖 is gyroscope scale factor, and 𝐸_𝑖𝑗 is the misalignment error of angular rate channel, where 𝑖=𝑥,𝑦,𝑧; 𝑗=𝑥,𝑦,𝑧.

2.2. Error Modeling of the Acceleration Channel

According to the characteristic of the Quartz Mechanical Accelerometer (QMA), the acceleration channel error model of the SINS can be given by 𝐴_𝑥𝐴_𝑦𝐴_𝑧=𝐴_𝑏𝑥𝐴_𝑏𝑦𝐴_𝑏𝑧+𝑆_𝑎𝑥𝑀_𝑥𝑦𝑀_𝑥𝑧𝑀_𝑦𝑥𝑆_𝑎𝑦𝑀_𝑦𝑧𝑀_𝑧𝑥𝑀_𝑧𝑦𝑆_𝑎𝑧𝑎_𝑥𝑎_𝑦𝑎_𝑧, (2) where 𝐴_𝑖 is the accelerometer output of RLG SINS in 𝑖 axis, 𝑎_𝑖 is the accelerometer input, 𝐴_𝑏𝑖 is the accelerometer bias, 𝑆_𝑎𝑖 is the acceleration scale factor, and 𝑀_𝑖𝑗 is the misalignment error of the acceleration channel, where 𝑖=𝑥,𝑦,𝑧; 𝑗=𝑥,𝑦,𝑧.

3. The Optimization Calibration Method with Four-Direction Rotations

The measurement precision of RLG SINS is mostly affected by the temperature variation, the change rate, and the gradient of the temperature. The work temperature of airborne SAR varies a lot with the flight altitude. Compared with the temperature variation, the gradient and change rate of the temperature do not change much during everyprocession of airborne imaging. The effects of different temperature on the precision of RLG SINS are mainly studied in this paper. The range of calibration temperature can be set according to the actual working environment and the accuracy requirement.

According to the error model of RLG SINS, an optimization calibration method with four-direction rotations is designed for RLG SINS based on a single-axis turntable with temperature control and a high-precision hexahedral fixture. In the course of calibration, the SINS fixed in hexahedral fixture will be overturn four times at each temperature point, making the 𝑋, 𝑌, and 𝑍 axes of SINS superposed with 𝑍_𝑡 axis and −𝑍_𝑡 axis of the turntable (𝑍_𝑡 axis points up in Geographic Coordinate System). At each direction, the turntable will be pivoted the angle of 2𝜋𝑛 with 𝑛 being the cycle number. When 𝑥-axis directs up as the first direction as shown in Figure 1, the inputting angular rates of the three sensing axes can be given by 𝜔_𝑥(𝑡)𝜔_𝑦(𝑡)𝜔_𝑧(𝑡)=𝜔_𝑖𝑒(𝑡)𝜔cos𝜙sin𝜑_𝑖𝑒cos𝜙cos𝜑(𝑡)Ω+𝜔_𝑖𝑒sin𝜙, (3) where 𝜔_𝑖𝑒 is rotation rate of earth. 𝜙 is local latitude. Ω is the inputting rotation rate of turntable. 𝜑(𝑡)is head of RLG IMU. The inputting acceleration of the three sensing axes can be given by 𝑎_𝑥(𝑡)𝑎_𝑦(𝑡)𝑎_𝑧(𝑡)=00𝑔. (4)

Figure 1 

An optimization four-direction rotation rate calibration method using a single-axis turntable with temperature control.

The mean value of inputting angular rates and acceleration at the first direction can be obtained as 𝜔_𝑥𝜔_𝑦𝜔_𝑧=Ω∫2𝜋𝑛^{02𝜋𝑛/Ω}𝜔_𝑥(𝑡)∫𝑑𝑡_{02𝜋𝑛/Ω}𝜔_𝑦(𝑡)∫𝑑𝑡_{02𝜋𝑛/Ω}𝜔_𝑧(𝑡)=00𝑑𝑡Ω+𝜔_𝑖𝑒=00sin𝜙𝜔,𝑎_𝑥𝑎_𝑦𝑎_𝑧=Ω∫2𝜋𝑛^{02𝜋𝑛/Ω}𝑎_𝑥(𝑡)∫𝑑𝑡_{02𝜋𝑛/Ω}𝑎_𝑦(𝑡)∫𝑑𝑡_{02𝜋𝑛/Ω}𝑎_𝑧(𝑡)=00𝑔.𝑑𝑡 (5)

The system states equation of angular rate channels of RLG SINS during the optimization calibration can be expressed as 𝐺_𝑥1𝐺_𝑦1𝐺_𝑧1𝐺_𝑥2𝐺_𝑦2𝐺_𝑧2𝐺_𝑥3𝐺_𝑦3𝐺_𝑧3𝐺_𝑥4𝐺_𝑦4𝐺_𝑧4=1𝜔0010𝜔0100𝜔100−𝜔𝐺_𝑏𝑥𝐺_𝑏𝑦𝐺_𝑏𝑧𝑆_𝑔𝑥𝐸_𝑦𝑥𝐸_𝑧𝑥𝐸_𝑥𝑦𝑆_𝑔𝑦𝐸_𝑧𝑦𝐸_𝑥𝑧𝐸_𝑦𝑧𝑆_𝑔𝑧, (6) where 𝜔 is inputting angular rate; 𝜔=Ω+𝜔_𝑖𝑒sin𝜙. 𝐺_𝑖𝑚 is gyroscope output in 𝑖 axis for the 𝑚th direction, where 𝑖=𝑥,𝑦,𝑧; 𝑚=1,2,3,4. The system status equation of acceleration channel of RLG SINS can be given by 𝐴_𝑥1𝐴_𝑦1𝐴_𝑧1𝐴_𝑥2𝐴_𝑦2𝐴_𝑧2𝐴_𝑥3𝐴_𝑦3𝐴_𝑧3𝐴_𝑥4𝐴_𝑦4𝐴_𝑧4=1𝑔00𝐴10𝑔0100𝑔100−𝑔^𝑏𝑥𝐴_𝑏𝑦𝐴_𝑏𝑧𝑆_𝑎𝑥𝑀_𝑦𝑥𝑀_𝑧𝑥𝑀_𝑥𝑦𝑆_𝑎𝑦𝑀_𝑧𝑦𝑀_𝑥𝑧𝑀_𝑦𝑧𝑆_𝑎𝑧, (7) where 𝐴_𝑖𝑚 is the output of the i axis accelerometer on the 𝑚th direction. According to (6) and (7), the error coefficients of angular rate and acceleration channels of RLG SINS can be calculated by 𝐺_𝑏𝑥𝐺_𝑏𝑦𝐺_𝑏𝑧𝑆_𝑔𝑥𝐸_𝑦𝑥𝐸_𝑧𝑥𝐸_𝑥𝑦𝑆_𝑔𝑦𝐸_𝑧𝑦𝐸_𝑥𝑧𝐸_𝑦𝑧𝑆_𝑔𝑧=12𝜔𝜔𝐺_𝑥3+𝐺_𝑥4𝜔𝐺_𝑦3+𝐺_𝑦4𝜔𝐺_𝑧3+𝐺_𝑧42𝐺_𝑥1−𝐺_𝑏𝑥2𝐺_𝑦1−𝐺_𝑏𝑥2𝐺_𝑧1−𝐺_𝑏𝑥2𝐺_𝑥2−𝐺_𝑏𝑥2𝐺_𝑦2−𝐺_𝑏𝑥2𝐺_𝑧2−𝐺_𝑏𝑥𝐺_𝑥3−𝐺_𝑥4𝐺_𝑦3−𝐺_𝑦4𝐺_𝑧3−𝐺_𝑧4,𝐴_𝑏𝑥𝐴_𝑏𝑦𝐴_𝑏𝑧𝑆_𝑎𝑥𝑀_𝑦𝑥𝑀_𝑧𝑥𝑀_𝑥𝑦𝑆_𝑎𝑦𝑀_𝑧𝑦𝑀_𝑥𝑧𝑀_𝑦𝑧𝑆_𝑎𝑧=1𝑔𝐴2𝑔^𝑥3+𝐴_𝑥4𝑔𝐴_𝑦3+𝐴_𝑦4𝑔𝐴_𝑧3+𝐴_𝑧42𝐴_𝑥1−𝐴_𝑏𝑥2𝐴_𝑦1−𝐴_𝑏𝑦2𝐴_𝑧1−𝐴_𝑏𝑧2𝐴_𝑥2−𝐴_𝑏𝑥2𝐴_𝑦2−𝐴_𝑏𝑦2𝐴_𝑧2−𝐴_𝑏𝑧𝐴_𝑥3−𝐴_𝑥4𝐴_𝑦3−𝐴_𝑦4𝐴_𝑧3−𝐴_𝑧4. (8)

4. Nonlinear Interpolation Compensation of Temperature Errors

Both the RLG and QMA are sensitive to the temperature, and the precision of RLG SINS is limited by the temperature errors. Two measures are usually taken to solve the problem: (1) adopting temperature control technology to reduce the change of temperature [10]; (2) establishing the mathematical model of the temperature errors based on the analysis of the test data and using compensation algorithm to reduce the temperature errors [11]. The first method is disadvantageous in the high cost and an increase of the size and weight of RLG SINS. Due to the easy implementation and validity, the second method draws lots of the researcher’s attention. The second method is implemented by analyzing the test data sampled by the RLG SINS to find the hidden relation between the outputs of RLG SINS and the external temperature.

The conventional methods of the errors compensation include the linear fitting method and RBF neural network method. The linear fitting method is commonly used to find the hidden law of data. Though the method has the least amount of calculation, it does not meet the precision demand. Other nonlinear approximation methods, such as the high order least-squares approximation method and the RBF neural network method, can improve the precision of the SINS, but they increase the amount of calculation. Due to the great complexity of the algorithm, these methods are not suitable for real-time processing in RLG POS.

Considering the precision and calculational amount, a nonlinear low-order interpolation method is used to compensate for the temperature errors of RLG SINS. The method divides the temperature into several small intervals and uses piecewise linear interpolation in each interval. Due to the closed-loop control of RLG, the error of scale factor is very small. The scale factor temperature errors of RLG SINS are unnecessary to be compensated. A series of calibration temperature points are supposed as 𝑇=[𝑇₁𝑇₂⋯𝑇_𝑘⋯𝑇_𝑛]. If the working temperature is 𝑇∈[𝑇_𝑘−1,𝑇_𝑘] (𝑘=1,2,…,𝑛), the piecewise linear interpolation algorithm about the temperature errors of RLG biases can be expressed as follows: 𝐺_𝑏𝑥(𝑇)𝐺_𝑏𝑦(𝑇)𝐺_𝑏𝑧(𝑇)=𝑇−𝑇_𝑘𝑇_𝑘−1−𝑇_𝑘𝐺_{𝑏𝑥,𝑘−1}+𝑇−𝑇_𝑘−1𝑇_𝑘−𝑇_𝑘−1𝐺_{𝑏𝑥,𝑘}𝑇−𝑇_𝑘𝑇_𝑘−1−𝑇_𝑘𝐺_{𝑏𝑦,𝑘−1}+𝑇−𝑇_𝑘−1𝑇_𝑘−𝑇_𝑘−1𝐺_{𝑏𝑦,𝑘}𝑇−𝑇_𝑘𝑇_𝑘−1−𝑇_𝑘𝐺_{𝑏𝑧,𝑘−1}+𝑇−𝑇_𝑘−1𝑇_𝑘−𝑇_𝑘−1𝐺_{𝑏𝑧,𝑘}. (9)

The temperature errors of accelerometer biases and scale factors are given by 𝐴_𝑏𝑥(𝑇)𝐴_𝑏𝑦(𝑇)𝐴_𝑏𝑧(𝑇)=𝑇−𝑇_𝑘𝑇_𝑘−1−𝑇_𝑘𝐴_{𝑏𝑥,𝑘−1}+𝑇−𝑇_𝑘−1𝑇_𝑘−𝑇_𝑘−1𝐴_{𝑏𝑥,𝑘}𝑇−𝑇_𝑘𝑇_𝑘−1−𝑇_𝑘𝐴_{𝑏𝑦,𝑘−1}+𝑇−𝑇_𝑘−1𝑇_𝑘−𝑇_𝑘−1𝐴_{𝑏𝑦,𝑘}𝑇−𝑇_𝑘𝑇_𝑘−1−𝑇_𝑘𝐴_{𝑏𝑧,𝑘−1}+𝑇−𝑇_𝑘−1𝑇_𝑘−𝑇_𝑘−1𝐴_{𝑏𝑧,𝑘},𝑆_𝑎𝑥(𝑇)𝑆_𝑎𝑦(𝑇)𝑆_𝑎𝑍(𝑇)=𝑇−𝑇_𝑘𝑇_𝑘−1−𝑇_𝑘𝑆_{𝑎𝑥,𝑘−1}+𝑇−𝑇_𝑘−1𝑇_𝑘−𝑇_𝑘−1𝑆_{𝑎𝑥,𝑘}𝑇−𝑇_𝑘𝑇_𝑘−1−𝑇_𝑘𝑆_{𝑎𝑦,𝑘−1}+𝑇−𝑇_𝑘−1𝑇_𝑘−𝑇_𝑘−1𝑆_{𝑎𝑦,𝑘}𝑇−𝑇_𝑘𝑇_𝑘−1−𝑇_𝑘𝑆_{𝑎𝑧,𝑘−1}+𝑇−𝑇_𝑘−1𝑇_𝑘−𝑇_𝑘−1𝑆_{𝑎𝑧,𝑘}. (10)

5. Experiment and Result

5.1. Calibration Experiment of RLG SINS

The RLG SINS hardware includes an Inertial Sensing Assembly (ISA), a power module, eight absorbers, an I/F conversion circuit, and a data sampling and processing module as shown in Figure 2. The ISA includes three RLGs and three QMAs assembled in a supporting structure. The data sampling and processing module completes the data sampling from the RLG SINS, data filtering, synchronization, errors compensation, and real-time inertial navigation arithmetic. The sampling data are sent to the integrated navigation computer by RS422 serial port, and the Kalman filter is used in SINS/ Global Position System (GPS) integrated navigation. The integrated navigation data are sent to airborne SAR to realize real-time motion error compensation. In addition, the original data of SINS and GPS can be restored for postprocessing to improve the precision and smoothness of integrated navigation results and get a higher image resolution of SAR.

Figure 2 

The hardware components and operation principle of RLG POS.

According to the actual working environment and precision requirement of RLG POS, the calibration experiment is carried on in the temperature points of −25°C, −10°C, 5°C, 20°C, 30°C, and 40°C. The RLG SINS is fixed in the center of the hexahedral fixture, and the 𝑥, 𝑦, and 𝑧 axes are perpendicular to the datum plane of the turntable separately as shown in Figure 1. The calibration experiment is implemented by turning over the hexahedral fixture, which is placed on the single-axis turntable with temperature control as shown in Figure 3.

Figure 3 

The single-axis turntable with temperature control for the calibration experiment.

According to (6)–(8), the error coefficients of acceleration and angular rate channels can be calibrated. The curves of gyroscope bias changing with temperature are shown in Figure 4, and the curves of accelerometer biases and scale factors changing with temperature are shown in Figure 5. Compared to the traditional 24-point hybrid calibration method without regard to temperature effect, the calibration time of the proposed method is declining from about 3 hours to 1.5 hours. There is no necessity to calibrate the temperature transducers. Since the heating of POS, the temperatures of transducers are higher than the environmental temperature as shown in Figures 4 and 5. The gyroscope biases, acceleration biases, and scale factor are nonlinear with respect to the temperature.

Figure 4 

Curves of gyroscope bias changing with temperature.

Figure 5 

Curves of accelerometer bias and scale factor changing with temperature.

5.2. Performance Test of RLG SINS

In order to validate the method proposed in this paper, the proposed method and the traditional 24-point dynamic-static hybrid calibration method are used to calibrate the coefficients separately, and then vehicle experiments and airborne experiments are conducted. The RLG POS was fixed on an immovability marble platform in the experimental vehicle. The GPS antenna wasfitted on the top of the vehicle, and the GPS receiver was placed in the vehicle. The POS power was switched on. Firstly, the vehicle was parked in a location for 300 seconds, and the data were sampled for initial alignment. Then, the vehicle was driven along layout route for more than 3600 seconds. Two groups of tests were conducted with each method.

Comparing the vehicle experiment navigation results with the GPS positioning results, the precisions of inertial navigation in the two methods are shown in Table 1. The experimental results show that the inertial navigation average error of the proposed method is reduced to 0.594 nautical mile/h from 1.850 nautical mile/h for vehicle experiment. The experiments validate the practicability and validity of the proposed method.

Table 1 

The 3600 seconds position errors of inertial navigation in vehicle experiment.

6. Conclusion

This paper proposed an optimization calibration method with four-direction rotations for RLG SINS based on a single-axis turntable with temperature control and a high-precision hexahedral fixture in full temperature. Compared to the traditional 24-point dynamic-static hybrid calibration and linear fitting method, the inertial navigation average errors of the proposed method are reduced to 0.594 nautical mile/h from 1.850 nautical mile/h for vehicle experiments. The experimental results validate the practicability and validity of the proposed method.

Acknowledgments

The work was sponsored in part by the National Program on Key Basic Research Projects of China under Grant 2009CB724000, the CAST Innovation Foundation CAST201205, and in part by the National Natural Science Foundation of China under Grant nos. 60904093, 50975049, 60905056, 61004140, and 61273033.