MEMS IMU Error Compensation Methods

1. Introduction to MEMS IMU and Error Significance

In the field of mobile robot positioning and inertial navigation, MEMS IMU is the most widely used inertial information measurement sensor. The gyroscope in MEMS IMU is responsible for sensing the rotational changes of the vehicle, while the accelerometer measures the force information of the vehicle. However, the raw measurement data of low-cost and medium-cost MEMS IMU is affected by a large amount of zero bias instability errors and random noise. Over time, uncorrected sensor errors will accumulate into motion state errors. Generally speaking, the gyroscope bias causes attitude, velocity, and position errors to increase with time in a linear, quadratic, and cubic manner, respectively. In addition, the bias of the accelerometer will cause velocity and position errors to increase respectively at the first power and second power of time ¹. Therefore, the error of inertial sensors is the primary factor affecting the accuracy of inertial sensing, and controlling the error of inertial sensors is of crucial importance.

2. Calibration Methods for Deterministic Errors

For deterministic errors, commonly used error calibration methods include: discrete calibration methods and system-level calibration methods.

2.1 Discrete Calibration Methods

Most discrete calibration methods model the measurement output of MEMS IMU as a linear polynomial equation of systematic error. In the laboratory, reference information is obtained by using external high-precision alignment equipment, such as a three-axis turntable. Then, by comparing the bias between the measurement output of MEMS IMU and the reference information, the coefficients of the polynomial are calculated. Thereby calibrating the output of the MEMS IMU ^{2 3 4 5}. When only significant errors are considered, almost all system-level deterministic errors can be compensated, and some random errors can also be estimated and compensated online through navigation algorithms. However, laboratory calibration is costly and overly idealistic, failing to reflect the actual application environment.

2.2 System-Level Calibration Methods

Unlike the discrete correction method, the system-level calibration method corrects the IMU error by estimating the inertial navigation result error based on the Kalman filter. In 2017, Qureshi et al. ⁶ proposed a field calibration method for MEMS IMU. The greatest advantage of this method is that it does not rely on external calibration equipment. By using the local gravity signal as a stable external reference data, a simpler and faster calibration process can be completed simply by placing the sensor in an approximate direction and performing a simple rotation. System-level calibration methods typically use zero velocity as a reference for estimating error parameters. However, in the case of base swaying, zero speed is ineffective, and traditional methods will reduce calibration accuracy. For the oscillating base environment, Dai et al. ⁷ proposed a filtering method for three-axis RINS in inertial systems, achieving full parameter calibration on the oscillating base based on the velocity error in the inertial system.

2.3 Challenges and Advances in Practical Calibration

In most cases, The deterministic error calibration of MEMS IMU is carried out in an ideal environment free from interference and disturbance. However, such an ideal environment and conditions are not always applicable in real-world scenarios. In addition, the calibration of MEMS IMU usually requires an expensive and bulky calibration system, which is unaffordable for the average user. Therefore, Peng et al. ⁸ developed a low-cost, lightweight and portable IMU calibration embedded platform, and proposed a calibration algorithm considering external perturbations, using an iterative weighted Levenberg-Marquardt algorithm to handle perturbations. Although system-level calibration methods reduce the reliance on external devices and are suitable for online calibration, it is difficult to denoise and compensate for nonlinear and non-Gaussian random errors.

3. Techniques for Identifying and Modeling Random Errors

To identify and model random errors, researchers have proposed many representative MEMS IMU signal denoising techniques, which can mainly be divided into traditional signal processing-based methods and data-driven methods in recent years.

4. Traditional Signal Processing-Based Denoising Methods

Auto Regressive Moving Average Method (ARMA), Allan Variance method (AV), Kalman Filter (KF) and the Wavelet Transformation (WT) are the most representative signal processing methods.

4.1 The ARMA Method

The ARMA method is mainly used to analyze and study a set of random data arranged in sequence, and to establish mathematical models of different orders based on different error sequences. However, this method cannot identify the components of random errors one by one, and it is difficult to distinguish the sources of random errors.

4.2 The Allan Variance Method

The Allan variance method can identify various random error components and subdivide them into five parts: quantization noise, angular random walk, zero-bias instability, velocity random walk, and rate ramp. Therefore, the greatest advantage of Allan variance is that it can draw a double logarithmic curve to correlate and analyze the time-domain and frequency-domain characteristics of the data, thereby enabling intuitive and quantitative observation of random errors. When the data volume is large enough, the double logarithmic curve drawn will be more accurate ⁹.

4.3 The Kalman Filter Method

Kalman Filter is an effective linear quadratic estimator that can estimate the angular velocity or specific force information of the output of MEMS IMU through a series of observations with noise ¹⁰. When Kalman Filter conducts iterative estimation, it is necessary to conduct mathematical modeling of the MEMS IMU in advance, construct the corresponding system equations and observation equations, and manually set the covariance matrices of the system noise and observation noise to represent the confidence of the predicted values and observed values. However, the random error of MEMS IMU is a nonlinear and non-Gaussian signal, which will drift slowly over time and temperature, making it difficult to establish an accurate mathematical model. Therefore, the Kalman filter performs average in noise reduction for MEMS IMU.

4.4 The Wavelet Transform Method

The wavelet transform method is a commonly used signal processing method, which can be used to reduce the error of the high-frequency part of the random noise in MEMS IMU. Through wavelet transform, the original signal can be decomposed into wavelet coefficients of different frequencies. Then, according to actual needs, some of the wavelet coefficients can be retained or discarded, and ultimately a smoother signal can be reconstructed. However, the effectiveness of the wavelet transform method depends on the type and parameters of the selected wavelet function, as well as the threshold selection of the wavelet coefficients. For the low-frequency part error in the random noise of MEMS IMU, since the error in the low-frequency part is usually widely distributed in the wavelet coefficients, it is difficult to eliminate it through simple threshold selection or coefficient discarding, resulting in the poor effect of the wavelet transform method in removing the low-frequency error ¹¹.

5. Machine Learning Approaches for Enhanced Noise Reduction

To enhance the noise reduction capability of traditional signal processing-based methods, many researchers have adopted machine learning approaches to improve the traditional noise reduction algorithms, such as Support Vector Machine (SVM) and Neural Network, and have achieved better noise reduction results than those based on traditional signal processing methods.

5.1 Applications of SVM and Neural Networks

In 2012, Zhang et al. ¹² used SVM to model and compensate the angular rate error of MEMS gyroscope MG31-300, verifying that the SVM model has high accuracy and good ability in MEMS IMU noise reduction. Due to the optimal radial basis function neural network’s excellent nonlinear, adaptive, and self-learning characteristics, Leung et al.
¹³ successfully applied it in 2001 to predict the noisy time series signals of MEMS IMU. Gonzalez et al. ¹⁴ proposed a time-delay multilinear regression technique for reducing white noise in MEMS IMU in 2019, and conducted a strict feasibility analysis and proof for it.

5.2 Limitations of Static Machine Learning Models

Although these improvement methods have achieved certain results, their noise reduction effects mainly depend on the ability to identify complex patterns by learning the advanced features of the data, and most of the methods are based on static statistical models. This static model only models the current and previous moments of inertial information and does not consider the more historical moments of MEMS IMU dynamic information. However, MEMS IMU data is a time series data, and the errors in historical moments will inevitably affect the current measurement values.

6. Deep Learning in Inertial Navigation and Noise Reduction

In recent years, deep learning technology has demonstrated outstanding performance in computer vision ¹⁵ and natural language processing ¹⁶ due to its powerful feature representation and nonlinear modeling capabilities. Some researchers have introduced deep learning into inertial odometry, such as OriNet ¹⁷, IONet ¹⁸, and TLIO ¹⁹. These methods have achieved superior positioning performance compared to traditional methods. However, research on using deep learning to reduce random noise in MEMS IMU has just begun. Jiang et al. ²⁰ in 2018 applied recurrent neural networks (RNN) to the raw signals of MEMS IMU for noise reduction and used the Allan variance tool to calculate the main error factors, namely quantization noise, angular random walk, and zero bias instability. However, due to the gradient vanishing and gradient explosion problems ²¹ of RNN, it performs poorly in long sequences.

6.1 Advancements with LSTM and Hybrid Models

In 2012, long short-term memory (LSTM) neural networks ²², ²³ were proposed to enhance the learning ability of neural networks in handling long sequence problems. LSTM uses its unique internal input, forget, and output gate structures to selectively learn time series features based on current and historical data. In 2018, Jiang et al. ²⁴ processed MEMS IMU output signals as time series and used LSTM to reduce noise in MEMS gyroscope output data. Experimental results showed that this denoising scheme effectively improved the measurement accuracy of the gyroscope in MEMS IMU. To further explore the role of LSTM in signal denoising of MEMS IMU, Han et al. ²⁵ in 2021 evaluated the denoising effect of hybrid deep recurrent neural networks in both static and dynamic environments, including LSTM neural networks and gated recurrent units (GRU) neural networks. In 2021, Li et al. ²⁶ and Zhu et al. ²⁷ combined LSTM with Kalman filters to real-time estimate and compensate for the random drift of MEMS IMU gyroscope.

6.2 The Role of Convolutional Neural Networks

RNN can learn time correlations from the useful signals of the original data, but it is difficult to extract local features from noise components. In 2020, Brossard et al.²⁸ compensated for the errors of MEMS IMU through a carefully designed convolutional neural network (CNN) structure and further improved the accuracy of the estimated attitude angles of MEMS IMU.

7. Current Research Status and Limitations

The current research status of the error compensation methods for MEMS IMU indicates that the noise reduction method based on deep learning is superior to the traditional signal processing-based noise reduction methods in most cases. However, most of the work based on data-driven principles to suppress the random errors of MEMS IMU is based on a single RNN model, without simultaneously considering the local features of the MEMS IMU measurement signals and the long-term dependencies in the time series. Moreover, the computational efficiency of RNN is low, and the importance of the time series features of MEMS IMU at different times has not been explored.

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