This paper presents a delay-free tracking differentiator based on variational mode decomposition (VMD) for extracting the useful signal from a noisy measurement of gyroscope. Sigmoid function-based tracking differentiator (STD) is a novel tracking differentiator with the advantages of noise-attenuation ability and dynamical performance. However, there is a contradiction in STD; i.e., selecting a larger acceleration factor may cause faster convergence but bad random noise reduction whereas selecting a smaller acceleration factor may lead to signal delay but effective random noise reduction. Here, multiscale transformation is introduced to overcome the contradiction of STD. VMD is selected to decompose the noisy signal into multiscale components, and the correlation coefficients between each component and original signal are calculated, then the component with biggest correlation coefficient is reserved and other components are filtered by the proposed adaptive STD algorithm based on the correlation coefficient of each component, and finally the denoising result is obtained after reconstruction. The prominent advantages of the proposed algorithm are as follows: (i) compared to traditional tracking differentiators, better noise suppression ability can be achieved with suppression of time delay; (ii) compared to other widely used denoising methods, a simpler structure but better denoising ability can be obtained.
In the early of 1990s, MEMS gyroscope was introduced by the Draper Laboratory. Due to the extensive research and advancements in fabrication technologies and readout electronics, MEMS gyroscope’s performance has been improved over the last twenty years [
In the application of MEMS gyroscope, the noise becomes the main bottleneck which degrades of the signal accuracy. Therefore, it is very important to study the denoising technique to improve the performance of MEMS gyroscope. Many literatures have been dedicating great effort to remove MEMS gyroscope noises. Multiscale transformation method is a widely used technique for gyroscope denoising. In the mentioned wavelet-based techniques [
In this paper, to develop a novel denoising algorithm for MEMS gyroscope, a Sigmoid function-based tracking differentiator (STD) based on multiscale decomposition is proposed with advantages of strong noise suppression ability and delay-free. In addition, through both simulations and experiments, the superiority and effectiveness of the proposed denoising algorithm in significantly reducing the noise are verified.
In order to provide the accurate estimation of derivative of virtual control, STD was first developed with the advantages of being simple structure, global fast convergence, and chattering-free in differential estimation. Here only a brief introduction of STD will be given. In [
The following system is considered:
The following novel tracking differentiator is considered:
It can be concluded that
Although STD has already shown advantages of dynamical performance and noise-attenuation ability, note that STD is still constrained in supplying a relatively smooth denoising results without signal delay. This statement motivates us to pursue for an improved algorithm model which can radically relax the contradictory and give a better solution in suppressing noises. Here, a simulation noisy signal
Denoising results of STD with different
In Figure
The results of STD applied on the three components.
After analysis, how to decompose the noisy signal into different component and filter the components with different but appropriate acceleration factors become the key problem that needs to be resolved. As a novel nonrecursive signal processing technique, VMD can adaptively decompose a real valued signal into discrete set of band-limited subsignals intrinsic mode functions (BLIMFs) owing specific sparsity properties [
There is another important issue that needs to be resolved, which is how to determine the acceleration factor (
The VMD is utilized to decompose the signal into BLIMFs, which are
VMD is a novel method of signal decomposition; multicomponent signals are decomposed into BLIMFs, so as to minimize the sum of the bandwidth estimation of each mode. VMD can be represented as a constrained variational problem which is given by
Calculate
Judge the biggest
After
Design
The last step is to get the final denoising result by reconstruction. Note that
Since the proposed denoising algorithm is the combination of STD and VMD, it is named as VMD-STD algorithm. The flowchart of VMD-STD algorithm is shown as Figure
Flowchart of VMD-STD algorithm.
In order to present the superiority of the proposed VMD-STD denoising algorithm, the simulated signal
Decomposition results of
The decomposition is the first step of the proposed VMD-STD algorithm, and the second step is to calculate the CC of each BLIMF. The calculation results are shown in Table
The correlation coefficients of BLIMFs in Figure
BLIMF1 | BLIMF2 | BLIMF3 | BLIMF4 | BLIMF5 | BLIMF6 | BLIMF7 | BLIMF8 | BLIMF9 | |
---|---|---|---|---|---|---|---|---|---|
Correlation coefficient (CC) | 0.75 | 0.32 | 0.21 | 0.20 | 0.19 | 0.20 | 0.21 | 0.20 | 0.18 |
| |||||||||
Acceleration factor (R) | 0.8 | 0.36 | 0.23 | 0.21 | 0.20 | 0.21 | 0.23 | 0.22 | 0.19 |
The next step is the application of STD on BLIMFs with determined
Denoising results of each BLIMFs by using STD with determined R.
The last step is reconstruction. All the filtered components in Figure
Reconstruction results and comparisons.
But beyond that, the prior work conducts a simulation process with the single VMD-STD algorithm. To verify our proposed algorithm feasible indeed, some other advanced denoising methods are applied for comparison, like adaptive robust Kalman filter (ARKF), detrended fluctuation analysis-VMD (DFA-VMD), and empirical mode decomposition–forward linear prediction (EMD-FLP). These methods can be also well applied for noise suppression with gyroscope output signal. In Table
Comparison of the VMD-STD with other methods with different SNRs.
SNR (dB) | -20 | -16 | -9 | -3 | 0 | 3 | 6 | 9 | 11 | 13 | 16 | 20 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
EMD-FLP | -14. 51 | -10. 23 | -3. 63 | 1. 76 | 3. 29 | 7. 85 | 9. 67 | 11. 96 | 14. 64 | 15. 79 | 18. 34 | 21. 15 |
DFA-VMD | -16. 55 | -12. 35 | -5. 59 | 0. 38 | 6. 54 | 6. 48 | 12. 77 | 12. 47 | 14. 25 | 16. 73 | 18. 90 | 22. 56 |
ARKF | -12. 71 | -9. 21 | -0. 41 | 4. 69 | 7. 12 | 10. 30 | 13. 49 | 15. 68 | 17. 73 | 19. 59 | 21. 30 | 24. 87 |
VMD-STD | -9. 70 | -3. 98 | 1. 77 | 6. 75 | 10. 14 | 13. 15 | 13. 87 | 17. 06 | 17. 96 | 19. 71 | 22. 57 | 25. 61 |
In order to assess the complexity of the proposed VMD-STD algorithm, the computation time and space complexity are analysed. Firstly, we assume that the time cost of each operator is the same; therefore only the performance and running hardware are concerned. All required operations, such as addition (ADD), subtraction (SUB), multiplication (MUL), definition (DEF), and division (DIV) are considered.
Evaluating time and space complexity for the VMD algorithm.
function | T | M |
---|---|---|
Initialize | O ( | [(3+K· |
Update | (6ADD+2MUL+2DIV) ·K· | |
Update | (2CMP+3MUL+2ADD) ·K· | 0 |
Dual ascent | (4ADD+1MUL) · | 0 |
Convergence | (4ADD+2MUL) · | 0 |
Complexity | O ( | O ( |
In our VMD algorithm, the initialization parameters are set as follows: alpha = 2000, tau = 0, tol = 1e-7, and N = 1000. In Table
In the STD algorithm, the latter relates specifically to the parameter initialization and system function output. From the detailed computation, the STD’s time and space complexity are both of linear order O (N) and linear order O (N).
As shown in Table
Time and space complexity for the STD based on the VMD algorithm.
Function | T | M |
---|---|---|
STD | O ( | [S+5N·S +11] float |
VMD | O ( | [(4+K·N+N) ·2S+K·N] float |
STD based on VMD | O ( | [(4+K |
Complexity | O ( | O ( |
To compare the execution time of different denoising algorithms, a simulation with signals of lengths ranging from
Comparing these current algorithms with VMD-STD, the same experimental conditions were applied in the tests. The actual execution times are listed in Table
Execution times for a simulated signal applying different noise suppression methods.
n | EMD-FLP | DFA-VMD | ARKF | VMD-STD |
---|---|---|---|---|
28 | 0.1514 | 0.1847 | 0.2094 | 0.1643 |
29 | 0.2632 | 0.3028 | 0.7036 | 0.2871 |
210 | 0.4876 | 0.6178 | 1.0962 | 0.5974 |
211 | 0.9247 | 1.3842 | 2.4621 | 1.3625 |
212 | 1.2148 | 1.9634 | 2.8693 | 1.8712 |
213 | 2.5642 | 3.5613 | 6.1679 | 3.2964 |
214 | 5.0469 | 7.0951 | 15.1413 | 6.9451 |
215 | 9.8346 | 16.2846 | 31.5973 | 12.8753 |
216 | 22.3765 | 32.8897 | 66.8432 | 26.3154 |
In this section, the output of gyroscope is employed for verifying the effectiveness of VMD-STD denoising algorithm. MEMS S-springs vibrating ring gyroscope (MSVRG) [
Schematic of the MSVRG. (a) Whole structure of MSVRG; (b) silicon capacitor electrodes; (c) the glass substrate with patterned electrode leads [
(a) Drive mode shapes and frequencies of ring resonator; (b) sensitive mode shapes and frequencies of ring resonator [
As Figure
Experimental equipment arrangement of MSVRG.
In this work, one set of data is collected from gyroscope with temperature changing from +10°C to -10°C to +10°C, where the temperature change rate is less than 1°C/min. From Figure
Comparison results of different denoising methods.
From Figure
To evaluate the denoising ability of the proposed algorithm, Allan variance analysis is introduced to quantitative comparison. By using Allan variance analysis, the noise coefficients of MEMS gyroscope can be identified and evaluated. Normally, the identified noise coefficients are Q (quantification noise), N (angle random walk), B (bias instability), K (rate random walk), and R (angular rate ramp), respectively, where, N and B are the most important parameters to characterize the noise performance of gyroscope. Simply speaking, N stands for the white noise and B means the 1/f noise or other noises induced by environment. From Figure
Comparison of random noise coefficients by Allan variance analysis.
Q ( | N (°/ h1/2) | B (°/h) | K (°/h3/2) | R (°/h2) | |
---|---|---|---|---|---|
Original | 103.94 | 6.84 | 166.60 | 126.65 | 189.24 |
STD | 54.87 | 0.98 | 58.94 | 96.89 | 188.90 |
DFA-VMD | 48.90 | 0.87 | 52.34 | 85.67 | 189.18 |
EMD-FLP | 8.54 | 0.15 | 9.28 | 79.20 | 187.47 |
ARKF | 13.11 | 0.47 | 28.57 | 71.72 | 187.33 |
VMD-STD | 3.65 | 0.10 | 6.55 | 60.88 | 187.12 |
Allan variance analysis of gyro’s output after compensation.
In order to demonstrate the effectiveness of the proposed VMD-STD denoising algorithm, dynamic (step rotation) and bandwidth tests are carried out. Figure
Dynamic test results.
Figure
Frequency band of sample signal.
Denosing result.
To minimize the random noise of MEMS gyroscope, a VMD-STD denoising algorithm is proposed. The main contributions of this paper include the following: firstly, a novel denoising method, named as VMD-STD algorithm, is studied for gyroscope signal processing; secondly the signal delay of STD is solved by combining VMD and STD together; thirdly a VMD-based denoising method is given which makes VMD-based denoising method expanded. The proposed VMD-STD algorithm is verified by the collected MEMS gyroscope data, and the experimental results show that the best denoising result is obtained by using VMD-STD compared to other advanced denoising algorithms, and the temperature drift of MEMS gyroscope can be extracted without signal delay. Additionally, it can be concluded by stationary simulated data and nonstationary experimental output that the proposed denoising algorithm is effective for both stationary and nonstationary signals.
The [Original.mat] data used to support the findings of this study are included within the supplementary information file.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Xi Zhang and Huiliang Cao contributed equally to this paper.
This work was supported in part by the National Natural Science Foundation of China (61603353, 51705477), the Pre-Research Field Foundation (6140518010201), the Scientific and Technology Innovation Programs of Higher Education Institutions in Shanxi (201802084), the Program for the Top Young Academic Leaders of Higher Learning Institutions of Shanxi, the Young Academic Leaders Foundation in North University of China, Science Foundation of North University of China (XJJ201822), the Fund for Shanxi “1331 Project” Key Subjects Construction, and the Shanxi Province Outstand Researcher (2016M180018).
The original data are collected from gyroscope with temperature changing from +10°C to -10°C to +10°C, where the temperature change rate is less than 1°C/min.