Radial Basis Functions Intended to Determine the Upper Bound of Absolute Dynamic Error at the Output of Voltage-Mode Accelerometers

In this paper, we propose using the radial basis functions (RBF) to determine the upper bound of absolute dynamic error (UAE) at the output of a voltage-mode accelerometer. Such functions can be obtained as a result of approximating the error values determined for the assumed-in-advance parameter variability associated with the mathematical model of an accelerometer. This approximation was carried out using the radial basis function neural network (RBF-NN) procedure for a given number of the radial neurons. The Monte Carlo (MC) method was also applied to determine the related error when considering the uncertainties associated with the parameters of an accelerometer mathematical model. The upper bound of absolute dynamic error can be a quality ratio for comparing the errors produced by different types of voltage-mode accelerometers that have the same operational frequency bandwidth. Determination of the RBFs was performed by applying the Python-related scientific packages, while the calculations related both to the UAE and the MC method were carried out using the MathCad program. Application of the RBFs represent a new approach for determining the UAE. These functions allow for the easy and quick determination of the value of such errors.


Introduction
Acceleration, which is measured by accelerometers [1][2][3], is a feature of which instantaneous values are constantly changing [4]. A similar situation occurs when measuring other dynamic quantities (e.g., pressure, temperature, etc.) [5][6][7][8]. For those quantities, no explicit comparative criterion [9] has been developed so far, as is the case with the accuracy class of the instruments intended for static measurements [10,11]. When measuring acceleration, as is the case with other dynamic quantities [12][13][14][15], it is convenient to use the two best-known quality factors: The integral-square error [9,16,17] and the absolute error [12]. The first factor, for the assumed time of testing, allows one to determine the value of signal energy at the output of the sensor, while the second factor measures the maximum value of magnitude of this signal. However, both of these factors allow the determination of the numerical values of dynamic error for any measurement signal [9,[12][13][14][15][17][18][19]. After the time corresponding to the time of the steady state of the sensor impulse response, the time characteristic of the integral-square error is linear [16,19], while the absolute error takes a constant value in time [12,18]. The time that it takes for the steady state of impulse response can be considered to be transient due to its very short duration.
Since the set of dynamic signals at the sensor input is infinite [9], in order to determine the upper bound of dynamic error [9,[12][13][14] for the considered quality factor, the critical case of the input signal The procedure for determining the RBF function as well as an application of the MC method to establish the error UAE RBF constitute the novelty of this paper. Figure 1 shows the block diagram of the procedure intended for determining the RBF and then the value of UAE RBF . The procedure for determining the RBF function as well as an application of the MC method to establish the error UAE RBF constitute the novelty of this paper. Figure 1 shows the block diagram of the procedure intended for determining the RBF and then the value of UAE RBF . The procedure above involves the algorithm for determining the UAE, which is presented in detail in Section 4. The input data for this algorithm are the parameters of the voltage-mode accelerometer and the standard, as well as the value of the signal limitation. The cut-off frequency of the standard was selected to be equal to the operational frequency bandwidth of the accelerometer. In the first step, denoted by 1 in Figure 1, the parameters of the voltage-mode accelerometer are generated with the assumed quantization steps and from the assumed ranges of their variability. The value of the UAE was determined for each set of a such parameters. In this way, the matrix of UAE values was obtained, which then serves as the input data for the procedure intended for determining the RBF. The procedure for determining the RBF based on the RBF-NN is discussed in detail in Section 5.

General Guidelines for the Proposed Procedure
In the second step, denoted by 2, we can easily obtain the values of the error UAE RBF for the voltage-mode accelerometer by substitution of any set of parameters from the ranges assumed earlier. The RBF, therefore, allows the determination of the values of the UAE RBF without the need to use the procedures described in Section 4. If the values of parameters are known (without the associated uncertainties), then it is not necessary to carry out the parametric identification of the accelerometer model. Thus, the procedures described in Sections 3 and 6 are also not necessary. However, it should always be kept in mind that the RBF is valid only for the ranges of variability of the accelerometer parameters for which it was determined earlier.
For the assumed-in-advance ranges covering only the parameters obtained as a result of parametric identification (neglecting the uncertainties), the procedure shown in Figure 1 should be executed to determine the RBF. The block diagram of the procedure intended for determining the UAE RBF(max) is shown in Figure 2. The procedure above involves the algorithm for determining the UAE, which is presented in detail in Section 4. The input data for this algorithm are the parameters of the voltage-mode accelerometer and the standard, as well as the value of the signal limitation. The cut-off frequency of the standard was selected to be equal to the operational frequency bandwidth of the accelerometer. In the first step, denoted by 1 in Figure 1, the parameters of the voltage-mode accelerometer are generated with the assumed quantization steps and from the assumed ranges of their variability. The value of the UAE was determined for each set of a such parameters. In this way, the matrix of UAE values was obtained, which then serves as the input data for the procedure intended for determining the RBF. The procedure for determining the RBF based on the RBF-NN is discussed in detail in Section 5.
In the second step, denoted by 2, we can easily obtain the values of the error UAE RBF for the voltage-mode accelerometer by substitution of any set of parameters from the ranges assumed earlier.
The RBF, therefore, allows the determination of the values of the UAE RBF without the need to use the procedures described in Section 4. If the values of parameters are known (without the associated uncertainties), then it is not necessary to carry out the parametric identification of the accelerometer model. Thus, the procedures described in Sections 3 and 6 are also not necessary. However, it should always be kept in mind that the RBF is valid only for the ranges of variability of the accelerometer parameters for which it was determined earlier.
For the assumed-in-advance ranges covering only the parameters obtained as a result of parametric identification (neglecting the uncertainties), the procedure shown in Figure 1 should be executed to determine the RBF. The block diagram of the procedure intended for determining the UAE RBF(max) is shown in Figure 2. The procedure above employs the previously determined RBF, as well as the parameters of the testing accelerometer and the associated uncertainties assumed-in-advance or obtained as a result of modeling carried out by the WLS method. This procedure is based on the MC method, which was employed to determine the values of accelerometer parameters from the ranges set by the associated uncertainties. As a result of implementation of this procedure, the UAE RBF(max) was obtained. The MC-based procedure is presented in detail in Section 6.

Mathematical Models of the Voltage-Mode Accelerometer and the Standard
The mathematical model of the voltage-mode accelerometer is most often represented by the transfer function: while , , and are the voltage sensitivity (V/ms ), dimensionless damping ratio, and nondamped natural frequency (Hz), respectively.
The observer canonical form of the state-space representation associated with Equation (1) is where is the 2 × 2 dimensional identity matrix, while , , and are Let the model of the standard with the cut-off frequency be given by the K-th order Butterworth filter: The cut-off frequency of the standard is equal to the accelerometer operational frequency bandwidth.
The observer canonical form of the state-space representation related to the transfer Function (5) is where The procedure above employs the previously determined RBF, as well as the parameters of the testing accelerometer and the associated uncertainties assumed-in-advance or obtained as a result of modeling carried out by the WLS method. This procedure is based on the MC method, which was employed to determine the values of accelerometer parameters from the ranges set by the associated uncertainties. As a result of implementation of this procedure, the UAE RBF(max) was obtained. The MC-based procedure is presented in detail in Section 6.

Mathematical Models of the Voltage-Mode Accelerometer and the Standard
The mathematical model of the voltage-mode accelerometer is most often represented by the transfer function: where while S V , β, and f 0 are the voltage sensitivity (V/ms −2 ), dimensionless damping ratio, and nondamped natural frequency (Hz), respectively. The observer canonical form of the state-space representation associated with Equation (1) is where I is the 2 × 2 dimensional identity matrix, while A V , B V , and C V are Let the model of the standard with the cut-off frequency f c be given by the K-th order Butterworth filter: The cut-off frequency f c of the standard is equal to the accelerometer operational frequency bandwidth.
The observer canonical form of the state-space representation related to the transfer Function (5) is where and the variables in the last row of the matrix

Algorithm for Determining the UAE
The upper bound of absolute dynamic error is determined by the following formula [12]: where a and T are the magnitude limitation of the input signal and the time of the accelerometer testing, respectively, while L −1 denotes the inverse Laplace transformation. The component R L,L in Equation (10) is the bottom-right element of the Romberg array and allows avoiding the numerical integration of the first component of UAE-Equation (10).
The Romberg array can be determined by where and The value of L is determined by the stop condition for the Romberg method. The signal producing the error UAE is where sgn denotes the signum operation [12].

Procedure for Determining the RBF Based on the RBF-NN
The RBF-NN was proposed as a formal tool for mathematical modeling of error space [28][29][30][31][32]. The classical network architecture is applied where its structure consists of three layers: An input layer, a hidden layer with a nonlinear two-dimensional RBF activation function, and a linear output layer. This type of the network is characterized by the overall response function: where P denotes the number of radial neurons, while a p , c p , and ϕ are the inner parameters and function, respectively. The Gaussian kernel as the nonlinear RBF is presented as where γ and ||· || 2 denote the inner parameter and squared Euclidian distance, respectively. Given the assumptions, the RBF network can be treated as a universal approximator [28][29][30][31][32]. This means that such a system with a sufficient number of neurons is able to approximate any continuous function on a closed and bounded dataset with arbitrary precision. Here, the aim is to map the multivariate function of two arguments as follows: Hence, the network includes the input layer of size two and a single output. The hyperparameters a p , c p , and γ must be determined in a way that optimizes the match between ϕ and the given data. For model simplification, the parameter γ is fixed as the same for every Gaussian kernel function existing in the hidden layer.
The equations describing the approximator model can be presented in the form of a matrix notation as below: where the p × q dimensional matrix and the vectors denoted as G, w, and b respectively, have the following structure where p and q denote the number of RBF neurons in the hidden layer and the number of input samples, respectively.
The elements of matrix G are the values of the RBFs evaluated at the points indicated by the input data according to the formula where x j ∈ R 2 are samples of the input data and c i ∈ R 2 are centers of RBF for individual neurons. In turn, the vector b consists of the values of the original function known in the finite number of points, such that and the values of the linear output weights are stored as the vector w. Finally, the following training scheme was used to obtain the network hyperparameters relevant to the correct approximation: 1.
The RBF centers were randomly sampled among the domain of the input dataset.

2.
The value of parameter γ was selected from the set range with a given step.

3.
For every value of parameter γ, the appropriate weights were calculated using a pseudoinverse solution. After the RBF centers c i are fixed, the weights that minimize the error at the output can be directly computed using a linear pseudoinverse method: where G + denotes the Moore-Penrose pseudoinverse of the matrix G [38,39]. 4.
The determination coefficient (R 2 ) and the mean squared error (MSE) were calculated. 5.
Steps 2-4 were repeated for all indicated γ ranges to find the hyperparameters which optimize the value of the coefficient R 2 .

MC-Based Procedure for Determining the UAE RBF(max)
Let the variables S V , f 0 , and β denote the parameters of the mathematical model of the voltage-mode accelerometer assumed in advance or determined based on the WLS method [33,34], while the variables u( S V ), u( f 0 ), and u( β) are the uncertainties associated with these parameters. Also, let where (u) and (l) denote the upper and the lower ranges of the parameter changes by the values of associated uncertainties.
If the RBF was determined on the basis of the accelerometer model parameters for the assumed-in-advance ranges S V− , S V + , f 0− , f 0+ , and β − , β + , and if the below conditions are met, then it is possible to use the MC method to determine such values of the accelerometer model parameters S V max , f 0 max , and β max from the ranges S V l , S V u , f 0 l , f 0 u , and β l , β u , respectively, for which the value of UAE RBF(max) is obtained. It is carried out on the basis of RBF( S V max , f 0 max , β max ).  Table 1 includes the values of the obtained based on the parameters from the ranges ∈ 〈0.100, 0.150〉 and ∈ 〈0.0100, 0.0150〉, as well as for the constant value of parameter equal to 1 kHz.

Results and Verification
The quantization steps for the parameters and were equal to 0.002 and 0.0002, respectively. Taking into account the assumptions above, we have = 0.100, = 0.150, = 0.0100, and = 0.0150 according to Equation (38). The values of the UAE were obtained by utilizing the algorithm presented in Section 4 for the input parameters: = and = 0.1 s. The 15th-order Butterworth filter with the cut-off frequency was determined by solving the equation describing the amplitude response obtained based on Equation (1). Uncertainties associated with the parameters S V j , f 0 j and β j are determined based on the formula and δ denotes the variable that should be substituted by this parameter of the accelerometer model for which the uncertainty is determined [11,34]. The uncertainty associated with the UAE RBF(max) is determined in an analogous way (Figure 3). Table 1 includes the values of the UAE obtained based on the parameters from the ranges S v ∈ 0.100, 0.150 and β ∈ 0.0100, 0.0150 , as well as for the constant value of parameter f 0 equal to 1 kHz.   0100 0.0102 0.0104 0.0106 0.0108 0.0110 0.0112 0.0114 0.0116 0.0118 0.0120 0.0122  The quantization steps for the parameters S v and β were equal to 0.002 and 0.0002, respectively. Taking into account the assumptions above, we have S V− = 0.100, S V + = 0.150, β − = 0.0100, and β + = 0.0150 according to Equation (38). The values of the UAE were obtained by utilizing the algorithm presented in Section 4 for the input parameters: a = S v and T = 0.1 s. The 15th-order Butterworth filter with the cut-off frequency f c was determined by solving the equation describing the amplitude response obtained based on Equation (1).

Results and Verification
On the basis of the values of the UAE tabulated in Table 1 and by applying the procedure presented in Section 5, the RBF(S V , β) was determined. The cases of 5, 10, and 15 radial neurons based on Equations (28)-(30) were checked.

2.
For 10 radial neurons: 3. For 15 radial neurons: Computational experiments were carried out for three given network structures with different hidden layer parameters and including sizes of 5, 10, and 15 neurons. Optimal hyperparameters and relevant criteria (statistical measures) regarding the quality of the model were calculated in each experiment. The obtained results are presented in Table 2, where Max error, MSE, MAE, MedAE, and R 2 denote the maximum error, mean squared error, mean absolute error, median absolute error, and determination coefficient, respectively. For comparison, in the case of polynomial approximation presented in [18], the MSE was equal to 1.66 and 0.31, while in the case of using this approximation in [19], the fitting coefficient was equal to 0.797. Thus, it can easily be concluded that the fitting indexes obtained by the RBF are significantly better than those obtained with applying the polynomial approximation. In the case of 15 neurons, the values of UAE RBF obtained by substitution of the parameters S V and β from Table 1 into the RBF 15 (S V , β) were, in most cases, the same as the values of the UAE tabulated in this table. Figure 4a-d shows the values of the UAE tabulated in Table 1 and an approximation of the UAE using the RBF according to Equations (28)- (30).  The abovementioned approximation approach was implemented using Python 3.6 [40] and computed on a hardware configuration that included an Intel Core i5 M430, 2.27 GHz, 8 MB RAM and the operating system Windows 7 to obtain the experimental results. The following Python-related scientific packages were used in the implementation: NumPy, SciPy, iPython, Scikit-learn, Pandas, and Matplotlib for visualization purposes [41][42][43][44][45].
The values of UAE RBF determined based on RBF 15 (S V , β) for the values of parameters S V and β selected from the ranges S V− , S V + and β − , β + are shown in Table 3. Let us assume that the uncertainties associated with the parameters S V and β included in Table 3 are: u( S V ) = 0.001 and u( β) = 0.0001, respectively. In this case, according to Equation (23), we have S V u = 0.150, S V l = 0.100, β u = 0.0150, and β l = 0.0100. The conditions given by Equation (24) Table 4, are shown in the following order: , and the number m of the corresponding MC trial. Based on the obtained results, it can be easily concluded that, in most cases, the values of parameters S V max and β max were contained in the ranges S V− , S V + and β − , β + . It confirms the advisability of using the MC method to determine them. This is the only correct way to accurately determine the parameters of a voltage-mode accelerometer, which produce the UAE RBF(max) . When assuming the variability of all three parameters of the accelerometer, the functions of three variables are obtained, similar to those presented by Equations (28). When the values of parameters with associated uncertainties are obtained based on the WLS method, then the values of the parameters S V max , f 0 max , and β max and the value of UAE RBF(max) are determined using the MC method in an analogous way to that above.

Conclusions
This paper presents the procedure for determining the RBF based on the numerical values of the UAE calculated for a voltage-mode accelerometer as an example. These UAEs were determined for the both the assumed-in-advance ranges of variability of the parameters of the accelerometer model and the quantization steps of these parameters. When the RBF is obtained in this way, we can easily and quickly calculate the UAE for any values of the parameters of an accelerometer model from the ranges above. The error obtained in this way is denoted by UAE RBF . The above facility results from the fact that it is not always necessary to use the algorithm dedicated to determinining the UAE and the related necessity of determining the cut-off frequency of the standard applied as a reference for calculating the error.
The paper also discussed in detail the use of the MC method to determine the UAE (such error is denoted by UAE RBF(max) ) when considering the uncertainties associated with the parameters of an accelerometer model. To ensure the correct realization of the procedures for modeling a voltage-mode accelerometer by applying the parametric identification, such uncertainties should always be determined. The MC method is based on the previously determined RBFs for such ranges of change in the parameters of the accelerometer model, which contain all parameters for the accelerometer considered in a particular case. It is also important to underline that the parameters of such accelerometer, in the case of their decrease or increase by the values of the uncertainties associated with them, do not go beyond the lower and upper limits of the parameters for which the RBF was earlier determined.
The solutions presented in this paper regarding the determination of the RBF using an RBF-NN for an assumed number of radial neurons and the application of the MC method for determining the UAE RBF(max) are the first solutions in the subject of measurement traceability. Based on the results obtained for the solutions above, in the case of 15 radial neurons, it can be seen that the statistical ratios regarding the uncertainty of approximation of the values of UAE using the RBF are much higher than those obtained using the polynomial approximation presented in [18] and [19]. The obtained values of these ratios also confirm that the number of neurons equal to 15 is optimal in terms of the uncertainty of an approximation of the UAE using the RBF.
The RBFs for the assumed range of variability of two accelerometer parameters were determined in this paper. This assumption was required to limit the number of calculation results intended for the presentation here. However, based on the displayed procedures, the RBF can be easily determined for the assumed ranges of variability of all three parameters of the accelerometer model.
Based on the obtained low uncertainty of the approximation for 15 neurons, it can be concluded that the RBFs determined for such a number of neurons can be successfully applied for the mutual comparison of the UAE RBF(max) obtained for different types of accelerometers. It should be kept in mind, however, that the compared accelerometers should have the same frequency bandwidth of operation and that the parameters associated with their models must be within the ranges of parameter changes for which the RBF was determined.