Research on MEMS-IMU signal denoising technology

The emergence and rapid development of MEMS (Micro Electro Mechanical System) inertial sensors are promoting the miniaturization of inertial navigation systems. The strategic and tactical value and significance of MEMS inertial technology and micro navigation systems in national defense and military have become prominent.

MIMU is a key unit in inertial systems, and its accuracy determines the performance of the system to a large extent. However, the output signal is usually noisy and non-stationary, and includes influences such as the experimental environment. Therefore, the output signals of the gyroscope and accelerometer obtained from the MIMU unit should be denoised to improve the output accuracy of the MIMU. This article focuses on MEMS-IMU signal denoising technology and uses three different methods: median average method, IIR digital filtering method and wavelet denoising technology to remove noise in the MIMU output signal. Through comparative research on the processing results, the applicability and superiority of wavelet transform in signal denoising processing are confirmed, and the output accuracy of MIMU is improved. This improves the performance of the inertial navigation system.

1.Traditional filtering and denoising method

1.1 Median average filtering denoising

The median average filtering algorithm is one of the simplest signal processing methods. The specific algorithm steps are as follows:

(1) Sort the signals and take the middle value as the median.

(2) Subtract each data point in the signal from the median to obtain the difference.

(3) Average the differences to obtain the average value.

(4) Add the median and average values to obtain the filtering result.

The formula of the median average filtering algorithm is as follows:

The data below is the data of 1000s under static conditions of MEMS-IMU, and the actual sampling frequency is 50Hz. This data is smoothed at 10 sampling points in the experiment. The processing results are shown in the figure below. It can be seen from the figure that the processed signal is relatively smooth.

                  Comparison of gyroscope and accelerometer signals before and after median averaging processing

1.2 IIR digital filtering and denoising

Digital filters are divided into IIR (infinite impulse response filter) and FIR (finite impulse response filter). The IIR filter has zero points and can achieve better filtering effects with smaller orders and requires less calculation. Smaller, so IIR filter design is used. Since the frequency band of micromechanical devices is narrow and the Butterworth low-pass filter has the largest flatness characteristics in the low frequency band, the Butterworth low-pass filter is considered.

(1) Spectrum analysis of MIMU output signal

The data below are the output signals of the gyroscope and accelerometer acquired by the navigation computer and subjected to fast Fourier transform.

                                                            Gyroscope and accelerometer signal spectrum plots

Observing their spectrum diagrams, as shown in the figure above, it can be seen that the high-frequency interference noise of the gyroscope is very large, almost equal to the amplitude of the useful low-frequency signal;

(2) Digital filter design

Designing an IIR filter generally involves designing an analog filter first, and then converting the analog filter into a digital filter. The square amplitude-frequency characteristic function of the simulated Butterworth filter is:

The transfer function and frequency response of an analog system are related by s=j Ω, so the transfer function Hn (s) of its normalized low-pass filter can be obtained as:

The design steps for an analog filter are:

(1) Given the passband Ω1 and stopband frequency Ω2, the corresponding attenuation coefficients ap and as;

(2) Find the order N of the filter according to the following formula;

Find the cutoff frequency Ωc from the following formula;

Obtain the filter order N and cutoff frequency Ωc according to the above method, so that the transfer function Hn (s) of the analog filter can be obtained, and the analog filter is converted into a digital filter through bilinear mapping. Taking into account the overall requirements, the system requirements Design a digital low-pass filter with technical specifications: fp=10Hz, fs=30Hz, ap=4dB, as=20dB, and a sampling frequency of Fs=50Hz. Substituting into the design formula, the filter order N=5 can be obtained. Use this filter to filter the original signal. The filtered signal spectrum and signal diagram are shown in Figure 3.3. From the figure, it can be seen that the high-frequency noise in the signal is significantly reduced.

                           Comparison of gyroscope and accelerometer signals before and after IIR digital filtering

2.Wavelet denoising technology

2.1 Principle of wavelet denoising

Wavelet transform theory is a branch of applied mathematics developed in the late 1980s. It is a time-frequency analysis method of signals, which has the characteristics of multi-resolution analysis, and has the ability to characterize the local characteristics of the signal in both time and frequency domains. It is a window with a fixed size but a shape that can be changed. The time window and It is a time-frequency localized analysis method in which the frequency domain window can be changed, so it is known as a signal analysis microscope.

Inspired by the tower algorithm for image decomposition and reconstruction, Mallet proposed a fast algorithm for wavelet decomposition and reconstruction based on the multi-resolution theory, called the Mallet algorithm. This algorithm’s status in wavelet transform is equivalent to FFT in Fourier transform. position in.

(1) Signal decomposition process of Mallet algorithm

The Mallat algorithm, also known as the tower algorithm, uses wavelet filters H0, H1 and G0, G1 to decompose and reconstruct the signal. The decomposition algorithm is as follows:

Among them, t is the discrete time sequence number, f(t) is the original signal; j is the number of layers, H0 and H1 are the wavelet decomposition filters in the time domain, and Aj is the approximate part of the signal f(t) in the jth layer. Wavelet coefficient, Dj is the wavelet coefficient of the detailed part of signal f(t) in the jth layer. Assume that the discrete signal f(t) is A0, and the signal f(t) is in the approximate part of the 2jth scale, that is, the wavelet coefficient of the low-frequency part is the wavelet coefficient Aj-1 of the approximate part of the 2j-1th scale and the decomposition filter H1 Convolution, and then sampling the convolution result at intervals, the detailed part of the signal f(t) on the 2j scale, that is, the wavelet coefficient Dj of the high-frequency part is the approximate part of the wavelet coefficient Aj-1 through the 2j-1 scale Convolved with the decomposition filter H0, the convolution result is obtained by sampling at every other point. By decomposition, at each scale 2j, the signal f(t) is decomposed into wavelet coefficients Aj of the approximation part (low-frequency subband) and wavelet coefficients Dj of the detail part (high-frequency subband). The wavelet decomposition algorithm is shown in the figure below.

                                                 Schematic diagram of wavelet decomposition algorithm

(2) Signal reconstruction composition of Mallet algorithm

The reconstruction algorithm is as follows:

In the above formula, G0 and G1 are wavelet reconstruction filters in the time domain. The wavelet coefficient Aj of the low-frequency part is convolved with the reconstruction filter G0 through the wavelet coefficient Aj+1 of the approximation part of the 2j+1 scale after zero interpolation at every other point, and the wavelet coefficient of the detail part of the 2j+1 scale is interpolated with zero at every other point. The reconstruction filter G1 is convolved and then summed. This process is repeated until the 20th scale to obtain the reconstructed signal. The wavelet reconstruction algorithm is shown in the figure below.

                                                         Schematic diagram of wavelet reconstruction algorithm

(3) Small waveform analysis and denoising of signals

The process of using wavelet analysis algorithm to denoise the signal is as follows: first perform wavelet decomposition on the signal, and then process the coefficients after wavelet decomposition in the form of threshold domain values; then perform wavelet reconstruction on the signal, so that elimination can be achieved noise purpose. Generally speaking, as the transformation scale increases, the resolution becomes higher and higher, and the filtering effect becomes better and better. However, in practical applications, factors such as the amount of calculation and rounding error of calculation must be taken into consideration, so the required wavelet transform scale is determined based on the actual accuracy requirements and various influencing factors. In the paper, based on the comparison of the effects of different wavelet bases and different decomposition layers, the 5-level Debauches2 wavelet was selected for the signal.

Summarize

Three different methods: median averaging method, IIR digital filtering method and wavelet transform all have certain effects on removing noise in MIMU output signals. Among them, the median average method is the simplest and has the shortest filtering time, so it is a more commonly used method. However, when the MIMU outputs a high dynamic signal, this method cannot be used, and the IIR digital filtering method or the wavelet denoising method must be used for denoising. The time of digital filtering is relatively short, while the wavelet denoising method requires decomposition and reconstruction of the signal, which takes a longer time, but the accuracy of wavelet denoising is higher than that of digital filtering. Therefore, when we use high-performance hardware, we can implement wavelet transform with fewer decomposition levels for denoising, thereby improving the inertial navigation accuracy to a certain extent. As a company specializing in inertial navigation products, ERICCO focuses on the development and research of high-performance MEMS IMUs and conducts professional signal denoising for navigation-level IMUs and tactical-level IMUs. For example, ER-MIMU-01 and ER-MIMU-05 have less signal noise interference and have many application fields, such as oil exploration, bridges, high-rise buildings, towers, dam monitoring, geotechnical monitoring, mining, etc. , can reduce interference through signal noise denoising. ​

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How to Improve the Impact Resistance of Quartz Accelerometer

 

The quartz accelerometer’s pendulum plate is formed by quartz material after laser cutting, acid etching and other special processing, and the thermal expansion coefficient is very small, which is 1/10 to 1/20 of ordinary glass, but the quartz glass is a brittle material, and the thickness of the flexible beam is 0.03mm, which is easy to break. In actual work, the acceleration sensor may often be in a harsh environment such as vibration, shock, and temperature upheaval, which has a great test for its accuracy and stability. In addition, the uneven thickness of the flexible beam edge will also reduce the reliability of the pendulum. Therefore, to study the impact resistance of quartz accelerometer is to study the impact resistance of pendulum components.

Measures to improve the impact resistance of quartz accelerometer

Material improvement

The technical requirements for materials used in flexible beams are:

1) The tensile strength is large, and the elastic modulus and stiffness are small.

2) Small elastic aftereffect. Elastic aftereffect refers to the phenomenon that when the object is subjected to a fixed load, its shape variable increases slowly with time, and when the load disappears, it cannot immediately return to the original state. The elastic aftereffect of the flexible beam directly affects the output hysteresis of the accelerometer.

3) High fatigue strength. Quartz flexible accelerometers are sensitive to both positive and negative accelerometers, and are often in an irregular reciprocating motion when working, especially in systems with pulse output, the vibration frequency of flexible beams is very high.

4) Easy to process. The thickness of the quartz flexible beam is only 0.03mm, which is made by special processing such as laser cutting and acid etching, and the processing is difficult and the efficiency is low. At present, the “strength elastic ratio” is commonly used to measure the quality of materials. It refers to the ratio of the strength limit to the elastic modulus, and within the allowable range, increasing the ratio can improve the stability of the accelerometer.

According to the relevant literature, the ion implantation method can improve the performance of flexible beams. When high-energy ions are injected into the surface of the material with an ion implantation machine, its mechanical and physical properties can be greatly improved, such as Au+, N +, etc. into the joint of the flexible beam, which can improve its fatigue life and corrosion resistance. In addition, with the popularity of carbon fiber, the emergence of short carbon fiber quartz composite materials, its fracture resistance is 4 times that of pure quartz glass.

Structural improvement

Through transient dynamic analysis, it can be seen that the quartz flexible accelerometer has the weakest impact resistance in the direction of output axis, because in the direction of pendulum axis, the flexible beam is mainly subjected to tensile or compressive stress and the value is small, while in the direction of input axis, there is a magnetic yoke limit, which can only swing 0.019mm maximum. Therefore, it is necessary to focus on improving the impact resistance of quartz pendulum in the direction of output axis.

1) In the structural design, try to make the center of the torquer, the center of gravity of the pendulum component and the limit support three points coincide in the direction of the input axis. However, due to the complexity of the pendulum assembly, it is almost impossible for the three points to coincide completely, and the impact force of the flexible beam is proportional to the distance of non-coincidence.

2) Increase the width of the flexible beam, the thickness is the most important factor affecting the stiffness, sensitivity and natural frequency of the flexible beam, and the smaller the allowed range, the better the performance, it is not appropriate to change its value. However, the increase of the width of the flexible beam has little effect on the above properties, and can greatly reduce the shear stress of the flexible beam in the direction of the output axis.

3) Increase the transition fillet of the fixed end of the flexible beam. Under the action of acceleration load, the stress at the junction of the swing plate beam and the outer ring is the largest. Rounded corners reduce stress concentration and increase service life.

4) Set a limit support in the direction of the output axis of the accelerometer, professionals in the relevant fields have designed an impact type accelerometer, the main features are: add a limit convex table on the inner side of the pendulum ring, or set a limit pin on the torque yoke to reduce the maximum deformation of the pendulum plate in the direction of the output axis.

summary

In this paper, based on the relevant literature research data, the impact resistance of acceleration sensors can be improved by means of material and structure improvement.

Quartz accelerometers produced by Ericco, such as ER-QA-03A, has good impact resistance and it’s bias repeatability is10-50μg and scale factor repeatability is 15-50 ppm. In the future inertial system application field, its performance will have a better breakthrough. 

REFENCE LINK:https://www.ericcointernational.com/application/how-to-improve-the-impact-resistance-of-quartz-accelerometer.html

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MEMS-IMU error analysis

 

MEMS-IMU has many advantages over traditional IMUs, such as small size, low cost, low power consumption and high integration. MEMS IMU is widely used in the military field, such as tactical guided weapons. In recent years, the application scope of MEMS-IMU in the civilian field has been continuously expanded, such as drones, underwater equipment, car navigation, etc. At present, the performance indicators of MEMS inertial sensors at home and abroad cannot meet the requirements of inertial navigation. The performance indicators of the most advanced MEMS gyroscopes can only reach the tactical level, and they are easily interfered in the actual working environment, reducing the output accuracy. MEMS accelerometer performance is also inferior to traditional quartz accelerometers. This chapter will comprehensively analyze various errors and compensation solutions of MEMS-IMU.

The following table shows the main indicators for judging gyroscope performance. Different grades have different accuracy levels.

The errors of MEMS-IMU mainly include errors of MEMS sensors and MEMS-IMU integration errors. The errors of MEMS IMU will be analyzed in detail from these two aspects below.

1.MEMS inertial sensor error analysis

1.1 Zero offset

The bias errors of MEMS accelerometers and MEMS gyroscopes mainly include bias, bias stability and bias repeatability. Zero-bias repeatability and zero-bias can be eliminated through initial alignment. Zero-bias stability refers to the degree of drift of its output angular velocity or acceleration over time under a certain input. This is related to the structure, design and external environment of the MEMS sensor.

1.2 Scale factor error

Since MEMS gyroscopes and MEMS accelerometers convert signals between signals through a scale factor, the error in the scale factor will directly affect the output error of the MEMS accelerometer and MEMS gyroscope. The error of the scale factor is divided into temperature drift and nonlinear error. It is difficult to measure the relationship between the scale factor and temperature through experimental testing for MEMS accelerometers, because centrifuges generally do not have temperature control devices, while MEMS gyroscopes can be temperature controlled. The relationship between the scale factor and temperature was measured using the turntable.

1.3 Non-sensitive axis coupling error

The non-sensitive axis coupling error refers to the error output caused by the non-orthogonality of the sensor structure itself when there is input on the non-sensitive axis. The non-sensitive axis mutual coupling error can be expressed by Equation

                                                                    Formula1：Insensitive axis mutual coupling error1-1

Among them, VX, VY, and VZ represent the output voltages of the x, y, and z-axis sensors, Input is the external input, and K is the mutual coupling error coefficient. It can be seen that the expression of the coupling error of the MEMS accelerometer and the MEMS gyro is the same as the installation error expression of the IMU can be processed together.

1.4 Acceleration sensitivity

The acceleration sensitivity of the MEMS gyroscope refers to the output of the MEMS gyroscope’s sensitive acceleration, which is an error term with a large impact. Because most MEMS gyroscopes are based on mechanical vibration, they may be affected by acceleration, especially in working environments with large accelerations. For example, when the acceleration of the carrier is 20g and the duration is 10s, when the acceleration sensitivity is 0.05 (° /s)/g, the angle error produced by this is approximately 10°. Such a large angle error has a great impact on the MEMS-IMU attitude solution, so acceleration sensitivity is an error term that cannot be ignored.

1.5 Random noise

The random noise of the structure and the random noise of the circuit are the main components of the random noise of the MEMS inertial sensor. The random noise of the structure is mainly mechanical thermal noise. The random noise of the circuit includes the thermal noise of the circuit, 1/f noise, shot noise and g-r. Noise, etc., the biggest impact on the performance of MEMS sensors is mechanical thermal noise and circuit thermal noise, which are the main research objects.

Brown’s force is the source of mechanical thermal noise. Its principle is that gas molecules or liquid molecules produce random collisions with mechanical particles. This effect directly affects the sensitivity and resolution of the MEMS sensor and increases the random noise during measurement. Because the structure of the MEMS sensor is on the micron or even nanoscale, the impact of molecular motion cannot be ignored.

For capacitive MEMS inertial sensors, the equivalent Brown noise acceleration is

                                                                    Formula2：Equivalent Brown noise acceleration1-2

                                                                                                Formula3： 1-3 

 

Circuit thermal noise refers to the irregular thermal movement of carriers in a conductor when the temperature is above zero. Due to this irregular thermal movement, the current in the circuit deviates from the average fluctuation, resulting in voltage fluctuations. The power spectrum distribution of this thermal noise is

                                                                 Formula4： The power spectrum of thermal noise1-4

Among them, R is the resistance of the conductor. From the above formula, we can know that the power spectrum of random noise is constant in the entire frequency band. However, the noise can be suppressed through low-pass filtering to prevent it from spreading in the form of integrals in navigation and positioning.

2.MEMS-IMU integrated error analysis

Sensor mounting non-orthogonality errors and lever-arm effect errors are the main components of MEMS-IMU integration errors.

2.1 Sensor installation error

The sensor installation error of MEMS-IMU is mainly due to the non-orthogonality of the MEMS-IMU shell, the sensor installation error and the non-orthogonality of the sensor itself. As shown below.

 where xByBzB is the reference orthogonal coordinate system, xyz is the coordinate system of the gyroscope group or accelerometer group, θij (i, j=x, y, z) represents the installation error angle, where i represents the measurement axis, j represents the measurement axis around j The installation error angle caused by shaft rotation is positive in counterclockwise direction. The transformation from the reference coordinate system to the axis coordinate system is as follows.

 

The form is consistent with the formula (1-1) and does not need to be distinguished. The installation error of the MEMS gyroscope can be evaluated using the turntable test method. Given different rotational speeds, the installation error angle parameters can be obtained by measuring the output at different rotational speeds. The static tumbling test method can be used to evaluate the installation error of the MEMS accelerometer, and the installation error angle parameters of the MEMS accelerometer can be solved by measuring the output at multiple positions.

2.2 Lever arm effect error

Since the sensor of the combined MEMS-IMU is installed separately, when the carrier rotates around a certain rotation axis, the sensor will be subject to additional centrifugal acceleration and tangential acceleration, resulting in output errors of the MEMS accelerometer and MEMS gyroscope. The error is related to the rotation angular rate. Directly proportional.

 

3 Calibration

According to the above calibration method, the single-axis turntable can be used to complete the angular rate calibration experiment and position calibration of the MEMS gyroscope and MEMS accelerometer, and solve the various error coefficients in the error model. The experimental platform is shown in the figure, and the performance indicators of the experimental platform as follows.

4 Calibration method verification

Use the obtained error coefficients to compensate the MEMS gyroscope and MEMS accelerometer, then install the MEMS-IMU on the turntable with the Z-axis facing upward and fix it, and control the temperature of the turntable to rise from -40° to 80°, and then from 80° ° drops to -40°, collect the output data of MEMS-IMU respectively and save them. Use MATLAB to draw the saved data into a graph, and the results are shown in the figure below.

   

                                                   Comparison of data before and after MEMS IMU compensation

As can be seen from the data before and after X-axis compensation in the figure above, the maximum output error of the gyro before compensation reached 0.025°/s, and after compensation it was reduced to 0.02°/s, and it can also be seen from the figure that the error is increasing. The data at 0.01°/s-0.025°/s is significantly reduced, and the errors in the Y and Z axes are also reduced. This shows that the calibration method in this article is feasible.

Summarize

Analyze various error sources of MEMS-IMU, including device errors and integration errors. Based on the main error sources of MEMS gyroscopes and MEMS accelerometers, corresponding error models were established, a calibration experimental plan was designed, and the calibration experimental plan was experimentally verified, confirming that the given calibration method is feasible and can improve MEMS-IMU measurement accuracy. Regarding the accuracy of MEMS IMU, I have to say that the MEMS IMU independently developed by ERICCO has high accuracy, small size, light weight and low power consumption. For example, the gyroscopes and accelerometers in ER-MIMU01 and ER-MIMU-02 are also more accurate. Strict measures have also been taken for the error calibration of the IMU.

If you want to learn about or purchase an IMU, please contact our relevant personnel.

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Research on Temperature Characteristics of Tilt Sensor

 

1. Influence of temperature on tilt sensor

Tilt sensor is widely used in various angle measurement, such as high precision laser instrument level, ship navigation attitude measurement, geological equipment tilt monitoring, satellite communication vehicle attitude detection. However, in the harsh working environment, the inclination sensor is easily affected by temperature, and there are zero point and sensitivity temperature drift.
Because the temperature of the working environment of the tilt sensor changes greatly, and the heat output caused by the temperature change will bring large measurement errors. At the same time, temperature changes also affect the size of zero and sensitivity values, and then affect the static characteristics of the sensor, so measures must be taken to reduce or eliminate the impact of temperature changes, that is, temperature compensation must be carried out.

2. Tilt sensor temperature compensation method
Inclination sensor temperature compensation methods are generally divided into hardware compensation method and software compensation method. Hardware compensation method is mainly achieved by changing device structure, material, working environment and technology to improve the reliability of measurement results. But in practical application, the working environment is bad, the structure of hardware compensation method is complicated and it is difficult to achieve the ideal effect. The idea of software compensation is to separate and compensate the error through experiment. Because the temperature error of tilt sensor is a nonlinear error, it is difficult to achieve a high compensation accuracy. RBF neural network has strong curve fitting ability. Compared with BP neural network, we use RBF neural network for temperature compensation of inclinometer sensor, which greatly reduces the influence of temperature on tilt sensor and achieves good compensation effect.

3. Temperature compensation based on radial basis function RBF neural network
3.1 RBF neural network compensation principle
RBF neural network is a kind of forward feedback-free network with excellent performance. It can approximate continuous function with arbitrary precision and has been widely used in pattern recognition, function approximation and so on. RBF neural network is composed of input layer, hidden layer and output layer. The hidden layer nodes are composed of Gaussian radial basis functions, as shown in equation (1) :

i=1，2，…，h （1）Where: ci is the center of the i-th basis function and has the same dimension as x; σi is the extension constant or width of the i-th basis function, and the smaller σi is, the smaller the width of the radial basis function and the more selective the basis function is. Neurons in the output layer adopt a linear activation function, and the output of the KTH neuron is shown in equation (2) :

Where: yk is the output of the KTH neuron in the output layer; W2ik is the connection weight of the I-th neuron in the hidden layer and the K-th neuron in the output layer.
In the parameter design of RBF neural network learning algorithm, it is generally necessary to design three parameters: data center of each basis function, expansion constant and weight of output node. We use K clustering algorithm to determine the basis function of the data center. Recursive least square method is used for the weight between hidden layer and output layer to ensure faster convergence speed.

3.2 RBF neural network temperature compensation results
In the simulation experiment, the output value T1 of the inclination sensor and the temperature of the temperature box are taken as the input of the model, and the rotation Angle of the tilt sensor is taken as the output of the model. The 100 sets of data in the following table are samples of the RBF neural network two-input-output model.

The input layer of the neural network is composed of two neurons, and the output layer is composed of one neuron. The number of hidden layer neurons is automatically added by the software through checking the output error until the error requirement or the maximum number of hidden layer neurons is reached. Through several experiments, the radial basis function distribution density SPREAD=0.5 and the training target error was set as EGOAL=1e-6. After 25 training sessions, the target training accuracy was achieved. Compensation results are shown in the following table.

The output RBF neural network modeling graph is shown in the following figure

Comparing Table 1 and Table 2, it can be seen that the temperature drift and full scale error of each axis are greatly reduced after modeling by RBF neural network. As can be seen from the RBF network fitting effect diagram in FIG. 3, the simulation graphics transition smoothly at each point, and the model compensation effect is relatively ideal. In order to compare the compensation effect of the RBF network, we also set up the BP neural network compensation model which is often used for temperature compensation. The input layer of BP neural network is composed of two neurons and one implicit layer. After many experiments, it is determined that the optimal number of hidden layer is 5, and the output layer is composed of 1 neuron. The goal error of the training is EGOAL=1e^(-4), and the learning efficiency is LP.lr=0.2. In the experiment, 100 groups of samples in Table 1 were used to train the established BP network. After about 50 iterations, the training was completed. The training pairs of RBF and BP network were shown in the following figure.

The target difference of BP neural network training is 1e^(-4), and that of RBF neural network training is 1e^(-6). However, the training time of RBF neural network is only about 1/2 of that of BP network, which shows that RBF convergence speed is relatively ideal. At the same time, the zero point temperature drift and sensitivity temperature drift before and after compensation are shown in following table. BP network and RBF network reduce the zero point and sensitivity temperature drift by 1 and 2 orders of magnitude respectively. The full scale error diagram of the inclination sensor under the action of -20~70℃ is shown in the figure below.

The maximum full scale error before compensation is about 2.34%, after BP neural network modeling compensation is about 0.85%, and after RBF modeling compensation is reduced to 0.158%. The simulation results show that the output error of the tilt sensor decreases after temperature compensation, and the output error decreases from 2.8° before compensation to 0.23° after conversion to angle degree. The modeling effect of RBF neural network is much better than that of BP network model, and it is very close to the actual expected value. Therefore, for the inclination sensor in this paper, RBF neural network modeling compensation can obtain better compensation effect.

Concluding discussion
We use RBF neural network compared with BP network to realize the research of temperature compensation of inclination sensor. The experimental results show that the RBF neural network modeling compensation achieves good compensation results in this system: for zero temperature drift, the X-axis decreases from 1.61×10^(-4) before compensation to 7.41×10^(-6), and the Y-axis decreases from 1.94×10^(-4) before compensation to 5.56×10^(-4). For sensitivity temperature drift, the X-axis decreased from 2.05×10^(-4) before compensation to 5.56×10^(-6), and the Y-axis decreased from 1.84×10^(-4) before compensation to 4.63×10^(-6). The RBF neural network is used to compensate the temperature of the inclinometer sensor, which reduces the influence of temperature on the sensor, overcomes the disadvantage that BP network is easy to fall into local optimal, improves the stability and measurement accuracy of the system, and lays a foundation for the application of the inclination sensor in various industries.

For example, Ericco’s ER-TS-12200-Modbus and ER-TS-32600-Modbus, both of which are dual-axis monitoring, we can completely carry out temperature compensation through RBF neural network modeling, and after compensation, the zero temperature drift and sensitivity temperature drift value of X axis and Y axis will be greatly reduced. In this way, we can reduce or even eliminate the adverse effects of temperature on the inclinometer sensor.