Analysis and Suppression of Nonlinear Error of Pendulous Integrating Gyroscopic Accelerometer at Instrument Level.

The error coefficients of the pendulous integrating gyroscopic accelerometer (PIGA) mainly include the bias, scale factor, and nonlinear error. Previous works have fully studied and suppressed the bias and scale factor of PIGAs. At present, the nonlinear error is the most critical factor restricting the measurement accuracy of PIGAs. To address this barrier, a study on the analysis and suppression of the nonlinear error of PIGAs at the instrument level was carried out. Firstly, the error model of a PIGA is established by kinematics and dynamics analyses. Then, nonlinear error is analyzed based on the established model. Finally, a suppression method for the nonlinear error is proposed based on the analysis results. The nonlinear error analysis found that (1) the nonlinear error includes a quadratic term error caused by unequal inertia and the inertia product, cross-coupling error is caused by lateral accelerations, and error is caused by unequal stiffness; (2) unequal inertia and the inertia product were the most critical factors resulting in nonlinear error. Based on the results in the nonlinear error analysis, the suppression method for error focuses on unequal inertia and the inertia product. The proposed method of analysis and suppression was validated experimentally as the quadratic term coefficient was reduced by an order of magnitude from 1.9 × 10-6/g0 to 1.91 × 10-7/g0.

Precise Measurement and Compensation of the Micro Product of Inertia for Float Assembly in Pendulous Integrating Gyroscopic Accelerometers.

Nonlinear error has become the most critical factor restricting the measurement accuracy of pendulous integrating gyroscopic accelerometers (PIGA) during their improvement. The key to nonlinear error suppression for PIGA is the precise measurement and compensation of the micro product of inertia (MPOI) of the float assembly. However, the existing equipment and procedure for product of inertia (POI) measurement and compensation do not meet the accuracy requirements for MPOI. To solve this problem, novel equipment and procedures are proposed for the measurement and compensation of MPOI. The principle of the proposed measurement method is to simulate the error produced by MPOI in PIGA by using a single-axis turntable to rotate the float assembly along the eccentric axis to generate a centrifugal moment due to MPOI. The principle of the proposed compensation method is to remove the asymmetric mass to reduce the MPOI to zero. Through experimental validation, it is concluded that: (1) the measurement and compensation accuracy of the proposed method are better than 1 × 10-10 kg·m2 and 3 × 10-10 kg·m2, respectively; (2) the proposed method is validated as the MPOI is reduced from 7.3 × 10-9 kg·m2 to 3 × 10-10 kg·m2 for a real float assembly in PIGA, and the quadratic error of PIGA is reduced from 10-5/g0 to 3 × 10-7/g0.

An Improved Online Self-Calibration Method Utilizing Angular Velocity Observation for Ultra High Accuracy PIGA-Based IMU.

In the field of ultra high accuracy inertial measurement unit (IMU), pendulous integrating gyroscopic accelerometer (PIGA) has become a research hot spot due to its high-end performance. However, PIGA is sensitive to angular velocity, and the calibration process of PIGA-based IMU will be very complicated, which makes online self-calibration difficult to implement. To solve the above problems, we proposed an online self-calibration method utilizing angular velocity observation. The main contributions of this study are twofold: (1) An error analysis of PIGA is conducted in this paper, and the error model has also been simplified to suit the self-calibration model. (2) An improved online self-calibration method utilizing angular observation based on a simplified PIGA error model is proposed in this study. Experimental results show that the self-calibration method proposed in this study can improve the PIGA online calibration accuracy effectively (with the accuracy within 0.02 m/s/pulse), which can improve the dynamic accuracy of the PIGA.

An Alignment Method Based on KF-ASMUKF Hybrid Filtering for Ship's SINS under Mooring Conditions.

To solve the problem that the ship's strapdown inertial navigation system (SINS) alignment accuracy decreases with the increase of the nonlinear filtering state dimension under mooring conditions, a method based on Kalman filter (KF) and Adaptive scale mini-skewness single line sampling Unscented Kalman Filter (ASMUKF) hybrid filtering algorithm is proposed in this paper. Three improvements are made as the following: (1) adopt a new sampling strategy. To obtain the ASMUKF filtering algorithm, scale mini-skewness single line sampling is used to replaced the traditional symmetrical sampling method and an adaptive scale factor is adapted into the Unscented Kalman Filter (UKF) to correct the real-time transformation sampling process; (2) the improved ASMUKF algorithm is combined with KF to form KF-ASMUKF hybrid filtering model; (3) the hybrid filtering model is divided into linear and nonlinear parts. The linear filtering part adopts the KF filtering model and the nonlinear filtering part adopts the ASMUKF model. Then, the calculation steps of the hybrid filtering algorithm is designed in this paper. The simulation and experimental results show that the hybrid filtering algorithm proposed has certain advantages over the traditional algorithm, and it can reduce the ship's SINS calculation amount and improve alignment accuracy under mooring conditions.

Modeling and Calibration for Dithering of MDRLG and Time-Delay of Accelerometer in SINS.

At present, the design and manufacturing technology of mechanically dithered ring laser gyroscope (MDRLG) have matured, the strapdown inertial navigation systems (SINS) with MDRLG have been widely used in military and business scope. When the MDRLG is working, high-frequency dithering is introduced, which will cause the size effect error of the accelerometer. The accelerometer signal has a time delay relative to the system, which will cause the accelerometer time delay error. In this article, in order to solve the above-mentioned problem: (1) we model the size effect error of the mechanically dithering of the MDRLG and perform an error analysis for the size effect error of the mechanically dithering of the MDRLG; (2) we model the time delay error of accelerometer and perform an error analysis for the time delay error of accelerometer; (3) we derive a continuous linear 43-D SINS error model considering the above-mentioned two error parameters and expand the temperature coefficients of accelerometers, inner lever arm error, outer lever arm error parameters to achieve high-precision calibration of SINS. We use the piecewise linear constant system (PWCS) method during the calibration process to prove that all calibration parameters are observable. Finally, the SINS with MDRLG is used in laboratory conditions to test the validity of the calibration method.

An Improved Calibration Method for the IMU Biases Utilizing KF-Based AdaGrad Algorithm.

In the field of high accuracy strapdown inertial navigation system (SINS), the inertial measurement unit (IMU) biases can severely affect the navigation accuracy. Traditionally we use Kalman filter (KF) to estimate those biases. However, KF is an unbiased estimation method based on the assumption of Gaussian white noise (GWN) while IMU sensors noise is irregular. Kalman filtering will no longer be accurate when the sensor's noise is irregular. In order to obtain the optimal solution of the IMU biases, this paper proposes a novel method for the calibration of IMU biases utilizing the KF-based AdaGrad algorithm to solve this problem. Three improvements were made as the following: (1) The adaptive subgradient method (AdaGrad) is proposed to overcome the difficulty of setting step size. (2) A KF-based AdaGrad numerical function is derived and (3) a KF-based AdaGrad calibration algorithm is proposed in this paper. Experimental results show that the method proposed in this paper can effectively improve the accuracy of IMU biases in both static tests and car-mounted field tests.

A Novel Calibration Method for Gyro-Accelerometer Asynchronous Time in Foot-Mounted Pedestrian Navigation System.

Pedestrian Navigation System (PNS) is one of the research focuses of indoor positioning in GNSS-denied environments based on the MEMS Inertial Measurement Unit (MIMU). However, in the foot-mounted pedestrian navigation system with MIMU or mobile phone as the main carrier, it is difficult to make the sampling time of gyros and accelerometers completely synchronous. The gyro-accelerometer asynchronous time affects the positioning of PNS. To solve this problem, a new error model of gyro-accelerometer asynchronous time is built. The effect of gyro-accelerometer asynchronous time on pedestrian navigation is analyzed. A filtering model is designed to calibrate the gyro-accelerometer asynchronous time, and a zero-velocity detection method based on the rate of attitude change is proposed. The indoor experiment shows that the gyro-accelerometer asynchronous time is estimated effectively, and the positioning accuracy of PNS is improved by the proposed method after compensating for the errors caused by gyro-accelerometer asynchronous time.

Latitude Determination and Error Analysis for Stationary SINS in Unknow-Position Condition.

The initial geographic latitude information is the key to the self-alignment of the strapdown inertial navigation system (SINS), but how to determine the latitude when the latitude cannot be obtained directly or in a short time? The latitude determination (LD) methods are introduced, including magnitude method, geometric method, and analytical methods 1 and 2, to solve this situation only by the output of the SINS itself. Simulation and experimental test results validate the efficiency of these LD methods. In order to improve the accuracy of the LD, the error of the LD method is derived through comparative analysis. Based on the relationship between LD error and inertial measurement unit (IMU) bias. Partial bias estimation method is introduced and executed during latitude determination. After compensating the estimated IMU bias, the accuracy of the LD will be further improved. Latitude errors are also affected by the latitude where SINS is located. Comprehensive simulation and experimental tests verify the effectiveness of the method. The IMU determined latitude can not only be used to achieve the self-alignment of the SINS, but also to correct the navigation latitude of the long-term SINS, thereby improving the autonomy and positioning accuracy of the navigation system.

A Combined Gravity Compensation Method for INS Using the Simplified Gravity Model and Gravity Database.

In recent decades, gravity compensation has become an important way to reduce the position error of an inertial navigation system (INS), especially for a high-precision INS, because of the extensive application of high precision inertial sensors (accelerometers and gyros). This paper first deducts the INS's solution error considering gravity disturbance and simulates the results. Meanwhile, this paper proposes a combined gravity compensation method using a simplified gravity model and gravity database. This new combined method consists of two steps all together. Step 1 subtracts the normal gravity using a simplified gravity model. Step 2 first obtains the gravity disturbance on the trajectory of the carrier with the help of ELM training based on the measured gravity data (provided by Institute of Geodesy and Geophysics; Chinese Academy of sciences), and then compensates it into the error equations of the INS, considering the gravity disturbance, to further improve the navigation accuracy. The effectiveness and feasibility of this new gravity compensation method for the INS are verified through vehicle tests in two different regions; one is in flat terrain with mild gravity variation and the other is in complex terrain with fierce gravity variation. During 2 h vehicle tests, the positioning accuracy of two tests can improve by 20% and 38% respectively, after the gravity is compensated by the proposed method.

Error Analysis of the K-Rb-21Ne Comagnetometer Space-Stable Inertial Navigation System.

According to the application characteristics of the K-Rb-21Ne comagnetometer, a space-stable navigation mechanization is designed and the requirements of the comagnetometer prototype are presented. By analysing the error propagation rule of the space-stable Inertial Navigation System (INS), the three biases, the scale factor of the z-axis, and the misalignment of the x- and y-axis non-orthogonal with the z-axis, are confirmed to be the main error source. A numerical simulation of the mathematical model for each single error verified the theoretical analysis result of the system's error propagation rule. Thus, numerical simulation based on the semi-physical data result proves the feasibility of the navigation scheme proposed in this paper.

Accurate Compensation of Attitude Angle Error in a Dual-Axis Rotation Inertial Navigation System.

In the dual-axis rotation inertial navigation system (INS), besides the gyro error, accelerometer error, rolling misalignment angle error, and the gimbal angle error, the shaft swing angle and the axis non-orthogonal angle also affect the attitude accuracy. Through the analysis of the structure, we can see that the shaft swing angle and axis non-orthogonal angle will produce coning errors which cause the fluctuation of the attitude. According to the analysis of the rotation vector, it can be seen that the coning error will generate additional drift velocity along the rotating shaft, which can reduce the navigation precision of the system. In this paper, based on the establishment of the modulation average frame, the vector projection is carried out, and then the attitude conversion matrix and the attitude error matrix mainly including the shaft swing angle and axis non-orthogonal are obtained. Because the attitude angles are given under the static condition, the shaft swing angle and the axis non-orthogonal angle are estimated by the static Kalman filter (KF). This kind of KF method has been widely recognized as the standard optimal estimation tool for estimating the parameters such as coning angles (α1 , α2), initial phase angles (ϕ1,ϕ2), and the non-perpendicular angle (η). In order to carry out the system level verification, a dual axis rotation INS is designed. Through simulation and experiments, the results show that the amplitudes of the attitude angles' variation are reduced by about 20%-30% when the shaft rotates. The attitude error equation is reasonably simplified and the calibration method is accurate enough. The attitude accuracy is further improved.

Systematic Calibration for Ultra-High Accuracy Inertial Measurement Units.

An inertial navigation system (INS) has been widely used in challenging GPS environments. With the rapid development of modern physics, an atomic gyroscope will come into use in the near future with a predicted accuracy of 5 × 10(-6)°/h or better. However, existing calibration methods and devices can not satisfy the accuracy requirements of future ultra-high accuracy inertial sensors. In this paper, an improved calibration model is established by introducing gyro g-sensitivity errors, accelerometer cross-coupling errors and lever arm errors. A systematic calibration method is proposed based on a 51-state Kalman filter and smoother. Simulation results show that the proposed calibration method can realize the estimation of all the parameters using a common dual-axis turntable. Laboratory and sailing tests prove that the position accuracy in a five-day inertial navigation can be improved about 8% by the proposed calibration method. The accuracy can be improved at least 20% when the position accuracy of the atomic gyro INS can reach a level of 0.1 nautical miles/5 d. Compared with the existing calibration methods, the proposed method, with more error sources and high order small error parameters calibrated for ultra-high accuracy inertial measurement units (IMUs) using common turntables, has a great application potential in future atomic gyro INSs.

An Online Gravity Modeling Method Applied for High Precision Free-INS.

For real-time solution of inertial navigation system (INS), the high-degree spherical harmonic gravity model (SHM) is not applicable because of its time and space complexity, in which traditional normal gravity model (NGM) has been the dominant technique for gravity compensation. In this paper, a two-dimensional second-order polynomial model is derived from SHM according to the approximate linear characteristic of regional disturbing potential. Firstly, deflections of vertical (DOVs) on dense grids are calculated with SHM in an external computer. And then, the polynomial coefficients are obtained using these DOVs. To achieve global navigation, the coefficients and applicable region of polynomial model are both updated synchronously in above computer. Compared with high-degree SHM, the polynomial model takes less storage and computational time at the expense of minor precision. Meanwhile, the model is more accurate than NGM. Finally, numerical test and INS experiment show that the proposed method outperforms traditional gravity models applied for high precision free-INS.

A Novel Gravity Compensation Method for High Precision Free-INS Based on "Extreme Learning Machine".

In recent years, with the emergency of high precision inertial sensors (accelerometers and gyros), gravity compensation has become a major source influencing the navigation accuracy in inertial navigation systems (INS), especially for high-precision INS. This paper presents preliminary results concerning the effect of gravity disturbance on INS. Meanwhile, this paper proposes a novel gravity compensation method for high-precision INS, which estimates the gravity disturbance on the track using the extreme learning machine (ELM) method based on measured gravity data on the geoid and processes the gravity disturbance to the height where INS has an upward continuation, then compensates the obtained gravity disturbance into the error equations of INS to restrain the INS error propagation. The estimation accuracy of the gravity disturbance data is verified by numerical tests. The root mean square error (RMSE) of the ELM estimation method can be improved by 23% and 44% compared with the bilinear interpolation method in plain and mountain areas, respectively. To further validate the proposed gravity compensation method, field experiments with an experimental vehicle were carried out in two regions. Test 1 was carried out in a plain area and Test 2 in a mountain area. The field experiment results also prove that the proposed gravity compensation method can significantly improve the positioning accuracy. During the 2-h field experiments, the positioning accuracy can be improved by 13% and 29% respectively, in Tests 1 and 2, when the navigation scheme is compensated by the proposed gravity compensation method.

AMA- and RWE- Based Adaptive Kalman Filter for Denoising Fiber Optic Gyroscope Drift Signal.

An improved double-factor adaptive Kalman filter called AMA-RWE-DFAKF is proposed to denoise fiber optic gyroscope (FOG) drift signal in both static and dynamic conditions. The first factor is Kalman gain updated by random weighting estimation (RWE) of the covariance matrix of innovation sequence at any time to ensure the lowest noise level of output, but the inertia of KF response increases in dynamic condition. To decrease the inertia, the second factor is the covariance matrix of predicted state vector adjusted by RWE only when discontinuities are detected by adaptive moving average (AMA).The AMA-RWE-DFAKF is applied for denoising FOG static and dynamic signals, its performance is compared with conventional KF (CKF), RWE-based adaptive KF with gain correction (RWE-AKFG), AMA- and RWE- based dual mode adaptive KF (AMA-RWE-DMAKF). Results of Allan variance on static signal and root mean square error (RMSE) on dynamic signal show that this proposed algorithm outperforms all the considered methods in denoising FOG signal.

An In-Run Automatic Demodulation Phase Error Compensation Method for MEMS Gyroscope in Full Temperature Range.

The demodulation phase error will cause the quadrature error to be coupled to the rate output, resulting in performance deterioration of the MEMS gyroscope. To solve this problem, an in-run automatic demodulation phase error compensation method is proposed in this paper. This method applies square wave angular rate input to the gyroscope and automatically identifies the value of the demodulation phase error through the designed automatic identification algorithm. To realize in-run automatic compensation, the demodulation phase error corresponding to the temperature point is measured every 10 °C in the full-temperature environment (-40~60 °C). The relationship between temperature and demodulation phase error is fitted by a third-order polynomial. The temperature is obtained by the temperature sensor and encapsulated in the ceramic packages of the MEMS gyroscope, and the in-run automatic compensation is realized based on the fitting curve. The temperature hysteresis effect on the zero-rate output (ZRO) of the gyroscope is eliminated after compensation. The bias instability (BI) of the three gyroscopes at room temperature (25 °C) is reduced by four to eight times to 0.1°/h, while that at full-temperature environment (-40~60 °C) is reduced by three to four times to 0.1°/h after in-run compensation.

Optimizing Process Parameters of Direct Ink Writing for Dimensional Accuracy of Printed Layers.

Direct ink writing (DIW) belongs to extrusion-based three-dimensional (3D) printing techniques. The success of DIW process depends on well-printable ink and optimized process parameters. After ink preparation, DIW process parameters considerably affect the parts' dimensional accuracy, and process parameters optimization for dimensional accuracy of printed layers is necessary for quality control of parts in DIW. In this study, DIW process parameters were identified and divided into two categories as the parameters for printing a line and the parameter from lines to a layer. Then, a two-step method was proposed for optimizing process parameters. Step 1 was to optimize process parameters for printing a line. In Step 1, continuity and uniformity of extruded filaments and printed rectangular objects were observed in screening experiments to determine printability windows for each process parameter. Then, interaction effect tests were conducted and degree of freedom for experiments was calculated followed by orthogonal array selection for the Taguchi design. Next, main experiments of line printing based on the Taguchi method were conducted. Signal-to-noise ratio calculations and analysis of variance were performed to find the optimal combination and evaluate the significance, respectively. Step 2 was to optimize the parameter from lines to a layer. In Step 2, the average width of the printed line under optimal condition was first measured. Then, single-factor tests of rectangular object printing were conducted to find the optimal parameter from lines to a layer. After these two steps, confirmation results were conducted to verify the reliability of the proposed method and the method robustness on other shapes and other materials; parameter adaptability in 3D parts printing from printed layers' analyses for the proposed method; and parameter adaptability in constructs fabricated as 100% infill or with porosities.

A robust in-motion attitude alignment method for odometer-aided strapdown inertial navigation system.

A significant issue of land vehicle navigation is in-motion attitude alignment of the odometer (OD)-aided strapdown inertial navigation system (SINS). The consecutive OD outliers can occur due to sudden wheel slipping and skidding while vehicle maneuvering. They seriously reduce the robustness and precision of attitude alignment. In this paper, we investigate a robust in-motion attitude alignment method for the OD-aided SINS. The method consists of in-motion coarse alignment and in-motion fine alignment. In the in-motion coarse alignment process, we developed Huber's M-estimation and integral formula based robust Kalman filter (HRKF/IF-CA), which can restrain the interference of consecutive OD outliers on reconstructed observation vectors. Thus, HRKF/IF-CA can contribute to better coarse attitude results. The next process is in-motion fine alignment. Under the popular repeated backtracking scheme, we investigate HRKF based fine alignment (HRKF-FA) with the SINS/OD summed measurement model. HRKF-FA can refine attitude alignment and restrain the interference of consecutive OD outliers simultaneously. Finally, the proposed method is evaluated by simulation and vehicle test. The attitude alignment results show that this method can achieve reasonable attitude results, and the interference of consecutive OD outliers caused by sudden wheel slipping and skidding can be greatly restrained.

Fuzzy adaptive strong tracking scaled unscented Kalman filter for initial alignment of large misalignment angles.

In the initial alignment process of strapdown inertial navigation system (SINS), large misalignment angles always bring nonlinear problem, which can usually be processed using the scaled unscented Kalman filter (SUKF). In this paper, the problem of large misalignment angles in SINS alignment is further investigated, and the strong tracking scaled unscented Kalman filter (STSUKF) is proposed with fixed parameters to improve convergence speed, while these parameters are artificially constructed and uncertain in real application. To further improve the alignment stability and reduce the parameters selection, this paper proposes a fuzzy adaptive strategy combined with STSUKF (FUZZY-STSUKF). As a result, initial alignment scheme of large misalignment angles based on FUZZY-STSUKF is designed and verified by simulations and turntable experiment. The results show that the scheme improves the accuracy and convergence speed of SINS initial alignment compared with those based on SUKF and STSUKF.