Influence of sensor accuracy on the simulation results

In order to test the influence of sensor accuracy on the accuracy of simulation results, an artificial error was added to the input data of the remote surveillance model. The error was generated as a pseudo-random real number evenly distributed in the predefined range. It was added to the sensor data received from the client-side simulation program

and stored in the database (see Figure 6.6) and represented the noise introduced by the sensor inaccuracy. The same conditions of the experiment were used as described in the Chapter 6.3.2.

Figure 6.11 shows the examples of the position sensor data from the lift cylinder with two levels of an artificially generated error: 1% and 10%. In order to account for the measurement errors, some kind of smoothing should be applied to the input data. Two kinds of smoothing were tested: simple moving average (SMA) and Savitzky-Golay filter.

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Figure 6.11: Examples of the sensor data with artificially added measurement errors

SMA is commonly used to smooth out short-term fluctuations in the input signal by creating a series of averages of different subsets of the full data set. A series of real numbers that represent a signal received during some period of time is scanned with the use of a "window" that selects a subset of data. The window moves (runs) along the data giving the name for the method. The first element of the moving average is obtained by taking the average of the initial N signal values where N is the size of the window. Then the window "shifts forward" modifying the subset by excluding the first number of it and including the next value in the series. The size of the window defines the ability of smoothing the signal fluctuations. The more is the window size the more fluctuations can be excluded from the signal. Figure 6.12 shows the dependency of the resulting normalized root mean square error of the calculation of the position and force of the lift and jib cylinders using remote surveillance technique on the window size used in SMA.

Figure 6.12: Normalized root mean square error of calculated positions and forces as a function of window size used in SMA for smoothing input data

The input data were generated using the pattern of the control signal shown in Figure 6.7. The data from the position and pressure sensors were disturbed by the artificially generated noise with the value of normalized root mean square error of 1%. The window size equal to 13 data points provides the minimum error of position and force calculation.

Savitzky-Golay filter (Savitzky and Golay, 1964) is a widely used digital filter that performs smoothing by fitting successive sub-sets of adjacent data points with a low-degree polynomial by the method of linear least squares. This method uses subsets of data of the length m (window size) to calculate m convolution coefficients that define a polynomial function fitting the subset. SMA can be considered as a special case of Savitzky-Golay filter when all convolution coefficients are equal to 1/m. Figure 6.13 shows the dependency of the normalized root mean square error of calculation of the position and the force in each cylinder on the window size when the smoothing of data is performed by Savitzky-Golay filter. The window size equal to 27 data points provides the minimum error of position and force calculation.

Figure 6.13: Normalized root mean square error of calculated positions and forces as a function of window size used in Savitzky-Golay filter

Three sets of experiments were performed: with the error added to the cylinder position data, the error added to the pressure data and the error added to the data obtained from all position and pressure sensors.

Figure 6.14 shows the dependency of the normalized root mean square error of calculation of the position and the force in each cylinder on the level of normalized root mean square error added to the position sensor data. The level of added error was being changed from 0 to 10% of the range of the measured parameter with the step of 0.1%.

As the most of position and pressure sensors provide the accuracy which is less than 10% (the values less than 1% are common), this range covers the majority of sensors available on the market.

Input data containing the noise from the artificially introduced errors were smoothed by Savitzky-Golay filter with the optimum window size equal to 27 data points. The solid lines on the plots represent an approximation of the experimental data with the third degree polynomials created using an implementation of the nonlinear least-squares Marquardt-Levenberg algorithm provided by the gnuplot software.

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Figure 6.14: Dependency of the normalized root mean square error of calculation of the position and the force in each cylinder on the accuracy of the position sensors

Figure 6.15 shows the dependency of the normalized root mean square error of calculation of the position and the force in each cylinder on the level of normalized root mean square error added to the pressure sensor data. As the position of each cylinder is calculated by interpolation of the position sensor data, an artificial error added to the pressure data does not influence the accuracy of position calculation.

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Figure 6.15: Dependency of the normalized root mean square error of calculation of the position and the force in each cylinder on the accuracy of the pressure sensors

The total error of calculation of the position and the force in each cylinder as a function of the maximum error introduced by the position and pressure sensors is presented in Figure 6.16. The experimental data show that the total error of calculation of positions and forces linearly increases with the decrease of sensor accuracy for the values of sensor accuracy that are most common for the commercially available sensors (less than 5%). In the experiment the total error of calculation did not exceed the error value introduced by the position and pressure sensors more than twice, if the sensor data were smoothed by Savitzky-Golay filter.

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Figure 6.16: The total error of calculation of the position and the force in each cylinder as a function of a maximum error of input data from position and pressure sensors

In order to compare the influence of two popular smoothing methods, the simple moving average and Savitzky-Golay filter, on the accuracy of calculations, the experiments were performed using the optimum window sizes for both methods. Figure 6.17 shows the dependency of the normalized root mean square error of the cylinder position calculation on the accuracy of the position and pressure sensors for the cases of using two different smoothing methods. The window size for SMA was 13 data points and for Savitzky-Golay filter it was 27 data points.

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Figure 6.17: Dependency of the normalized root mean square error of the cylinder position calculation on the accuracy of the position and pressure sensors for two different smoothing methods

Figure 6.18 shows the dependency of the normalized root mean square error of the force calculation for each cylinder on the accuracy of the position and pressure sensors for the cases of using two different smoothing methods.

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Figure 6.18: Dependency of the normalized root mean square error of the force calculation in each cylinder on the accuracy of the position and pressure sensors for two different smoothing methods

The use of Savitzky-Golay filter provided better accuracy of the force calculation than the SMA. For the values of the normalized root mean square error of input data less than 5% the use of Savitzky-Golay filter provided better accuracy for the position and the force calculation.

The forces calculated by the digital twin constitute the input for the calculation of the load history which can be used for the fatigue life estimation of the machine.

6.4 Test of Fatigue Life Estimation of Hydraulically Actuated Mobile