Estimation of Uncertainty for the Torque Transducer in MNm Range—Classical Approach and Fuzzy Sets

Abstract

The article provides an analysis of the metrological properties of a 5 MN·m torque transducer. The relative electrical signal (given in electrical units, mV/V) as a function of torque measured in both the clockwise and anticlockwise directions was monitored. To fit the data, the weighted method of least squares with both a straight-line and a cubic spline curve was used. The results of the analysis indicated that the straight-line fitting method produced smaller values of expanded uncertainty than the cubic spline fitting method. Additionally, the study confirmed the assumptions that the Monte Carlo method for propagating uncorrelated distributions was more accurate than the uncertainty propagation method, regardless of the type of curve fitting used. From the estimated uncertainty coverage corridor at selected measurement points, confidence intervals–expanded uncertainties were determined. Additionally, the fuzzy sets approach to the evaluation of uncertainty was presented, and the approximate value of the expanded uncertainty was calculated.

1. Introduction

Wind energy has emerged as a key participant in the renewable energy sector in recent years [1,2,3], with wind turbines accounting for an increasing proportion of global electricity production [4,5]. As these turbines continue to grow in size and capacity, it becomes increasingly vital to precisely monitor and comprehend the forces in action within them. One critical area of assessment is the torque encountered by the rotor blades, which can affect the turbine’s overall efficiency and dependability. It is vital not only for the safety of the turbine and the personnel engaged in designing, manufacturing of the turbine and the blades, and their maintenance, but also for the surrounding environment and the wind farms where the turbines are installed [6,7,8]. It also affects the whole sector, with the end users being the most important factor in energy safety. This is particularly pronounced now in the current volatile markets, with changes in the prices of various fuels, geopolitical tensions, and wars reminiscent of the oil crisis of the 1970s. The latter was also a driving force for research and technological development [9,10].

Since a long-term measurement of torque in wind turbines is considered nonpractical or not economically feasible [11], the torque in wind turbines is mainly measured in test benches for wind turbine drive trains [12]. However, most of the measurements, especially in strategic sectors, should be performed in a traceable and reliable manner. This can be provided by National Metrological Institutes (NMIs) and accredited calibration laboratories. They offer highest possible accuracy of measurements and the lowest uncertainty, at the same time ensuring traceability to SI units [13,14,15,16]. To provide the highest quality calibration services, NMIs continually develop new apparatuses, experimental stands, and machines to meet the growing demand [16,17]. However, advancing these machines and procedures to accommodate beyond current ranges of measurable values requires significant financial investment from both the providers and recipients. Consequently, in cases where the value of the measurand is extremely high or low, extrapolation may be necessary based on the available measuring equipment [18,19].

The extrapolation method is widely used in experiments and measurements as an effective tool to forecast the behaviour of different devices and systems beyond the actual range of work [18,20,21]. This method is crucial in calibrating and estimating uncertainty for large values of torque [22]. It is critical for wind industry laboratories and NMIs, which continue to face a need for precise torque measurements with larger values as the sector expands rapidly [23] and experiences values of torque above the current limit [22]. Providing a suitable solution for every customer or action is not always feasible, and developing massive calibration equipment for every torque measurement can be difficult and expensive.

The world’s largest torque calibration equipment, owned by one of the NMIs, Physikalisch-Technische Bundesanstalt (PTB), can measure values up to 1.1 MN·m, making extrapolation the preferred method for values beyond this range, as developing new equipment would be costly and time-consuming. Therefore, the proper method of extrapolation for large values could lead to satisfying results, and can be treated as a reasonable approach. While extrapolation is not a unique method in measurements, it has produced satisfactory results for force transducers above their calibration range [19,24]. Therefore, it can be applied to torque transducers as well.

This paper presents the metrological characteristics of the large torque transducer situated at PTB. The results of these measurements are extrapolated to gain a better understanding of the torque transducer behaviour, particularly for wind turbines operating under different conditions.

The aim of the research is to present comprehensive information about a developed transfer standard for mechanical power measurement. This standard encompasses a traceable torque measurement capability, facilitating the assessment of torque up to 5 MN·m. Additionally, the study explores how the direction of rotational speed impacts torque measurement, particularly under alternating rotation conditions (clockwise and anticlockwise). Furthermore, the research endeavours to outline the process of evaluating measurement uncertainty through distinct methodologies, namely, the Monte Carlo method and the law of propagation of uncertainty. For the 5 MN·m transducer, the values of the electrical signal at the measuring points for the set torques values were measured. Subsequently, both linear and nonlinear characteristics, along with the uncertainty coverage corridor, were adjusted through regression. An alternative approach to uncertainty evaluation using the fuzzy set method is also presented. This approach, though unconventional, proves to be a valuable and complementary addition to the repertoire of well-established methods.

2. Materials and Methods

A 5 MN·m torque transducer owned by PTB has been established as a torque transfer standard to trace torque measurement under rotation in wind turbine drive train test benches to national standards. Unfortunately, it is not currently possible to calibrate the transducer in its nominal range. In order to still be able to characterise the torque transducer, it was calibrated up to 1.1 MN·m and additional tests were carried out to analyse the contributors to the overall uncertainty.

2.1. PTB’s 1.1 MN·m Reference Torque Standard Machine

Since 2004, it has been possible to calibrate torque measurement up to 1.1 MN·m at the German NMI PTB in Braunschweig, Germany. To date, it remains the largest torque calibration machine in the world. The 1.1 MN·m torque standard machine (Figure 1) has a vertical measuring axis and is comprised of a drive unit at the bottom, an additional reference torque transducer underneath the transducer being calibrated, and a measuring arm including force transducers at the top. At the bottom, two servo-electric spindles apply the torque by driving a double-sided lever arm, which is arranged in a free-floating matter, in the same direction and in parallel. The measuring arm at the top is also a double-sided arm that splits the torque into two equal forces, which are measured. A multi-component strain-controlled hinge is present between the measuring arm and each force sensor, and strain gauges on these hinges measure the parasitic bending moments and transverse forces, which are reduced to a minimum amount. The machine serves as a national standard, with an expanded measurement uncertainty of 0.08% ($k=2$) in the measuring range between 100 kN·m and 1100 kN·m [25].

2.2. The 5 MN·m Torque Transducer

The 5 MN·m torque transducer is a specially designed and manufactured torque transducer for measuring torques up to 5 MN·m (nominal torque sensitivity of about 1.75 mV/V at 6 MN·m). The torque is measured in the form of strains, using strain gauges that are attached to the body that is being deformed. Here, the body is a hollow shaft with flanges for easier assembly (Figure 2). The transducer was developed for torque measurement under rotation in wind turbines and in a special system of test benches for wind turbine drive trains. In addition, it was equipped with a telemetry system for data transmission, which was necessary for this task [23].

3. Calibration Results and Discussion

Due to the lack of calibration devices with a calibration range greater than 1.1 MN·m, the 5 MN·m torque transducer can previously only be calibrated up to 1.1 MN·m. The calibration was carried out according to the DIN 51309 standard [26] both in clockwise and anticlockwise directions. To better characterise the transducer metrologically, partial range calibrations up to 600 kN·m and 800 kN·m were also carried out. When a linear regression curve for increasing torque (Case I-B in DIN 51309) is calculated, the sensitivity of the 5 MN·m torque transducer shows a relative expanded measurement uncertainty $U(k=2)$ that ranges from 0.08% (uncertainty of the calibration device) at 1.1 MN·m of the normalised signal [mV/V] to approx. 0.15% in the lower measurement range [27].

3.1. Evaluation of Uncertainty according to the GUM Guide

The measurement uncertainty equation for every single measurement point takes into account the correction errors of signal Y as a function of the value of torque M = X and is composed of eight components for TCM and five components related to the measurement procedure and measurement setup. The corresponding equation of the measurement of the signal y for the the error budget has the following form:

$$\delta y=\delta y_{\mathrm{TCM}}+\delta y_r+\delta y_b+\delta y_{b^{\prime }}+\delta y_0+\delta y_{\mathrm{fa}}$$

(1)

where $\delta y$ is the total error of the signal during the measurement at the given point of torque x, $\delta y_{\mathrm{TCM}}$ is the sum of the device calibration error components with uncertainty $u_{\mathrm{TCM}}$, and

$$\delta y_{\mathrm{TCM}}=\sum _{i=1}^8\delta y_i$$

(2)

where

Symbol	Description
$\delta y_1$	The error of y of the calibration results of the reference transducers, when cubic fitting functions are used. The random variable is described by a normal/Gaussian distribution with standard uncertainty $u\left(\delta y_1\right)=u_1$
$\delta y_2$	The error of y due to the short-term creep of the reference transducers. The random variable is described by a rectangular distribution and with standard uncertainty $u\left(\delta y_2\right)=u_2$
$\delta y_3$	The error of y associated with the long-term drift of the reference transducers. The exact value of the long-term drift will be calculated using results of the next calibrations performed on the same torque standard machines at PTB. The random variable is described by a rectangular distribution and standard uncertainty denoted by $u\left(\delta y_3\right)=u_3$
$\delta y_4$	The error of y due to the misalignment of the device under calibration. The random variable is described by a rectangular distribution and standard uncertainty is denoted by $u\left(\delta y_4\right)=u_4$
$\delta y_5$	The error of y associated with the resolution and stability of the indicating device (amplifier). The random variable is described by a normal/Gaussian distribution and standard uncertainty is denoted by $u\left(\delta y_5\right)=u_5$
$\delta y_6$	The error of y associated with using reference transducers in partial ranges; the implementation of the scaling transformation of polynomial interpolation Equations (3rd degree). The random variable is described by a normal/Gaussian distribution and standard uncertainty is denoted by $u\left(\delta y_6\right)=u_6$
$\delta y_7$	The error of y due to the stability of the torque transmission on the shafts; related with the performance of the DC motor and the gearbox. The random variable is described by a normal/Gaussian distribution and standard uncertainty is denoted by $u\left(\delta y_7\right)=u_7$
$\delta y_8$	The error of y due to influence of the variation in the temperature on the reference transducers. The random variable is described by a rectangular distribution and also type B method uncertainties and standard uncertainty is denoted by $u\left(\delta y_8\right)=u_8$
$\delta y_r$	The error of resolution r and $u\left(\delta y_r\right)=u\left(r\right)=\frac{r}{2\sqrt{3}}$—the standard uncertainty of the resolution r of the display unit described by a rectangular distribution
$\delta y_b$	The error of the reproducibility b and $u\left(\delta y_b\right)=u\left(b\right)=\frac{b}{2\sqrt{3}}$—the standard uncertainty of the reproducibility b described by a rectangular distribution
$\delta y_{b^{\prime }}$	The error of the repeatability $b^{\prime }$ and $u\left(\delta y_{b^{\prime }}\right)=u\left(b^{\prime }\right)=\frac{b^{\prime }}{2\sqrt{3}}$—the standard uncertainty of the repeatability $b^{\prime }$ described by a rectangular distribution
$\delta y_0$	The error of zero point deviation $f_0$ and $u\left(\delta y_0\right)=u\left(f_0\right)=\frac{f_0}{2\sqrt{3}}$—the standard uncertainty of the zero point deviation $f_0$ described by a rectangular distribution
$\delta y_{fa}$	The error of the interpolation deviation $f_a$ and $u\left(\delta y_{fa}\right)=u\left(f_a\right)=\frac{f_a}{2\sqrt{6}}$—the standard uncertainty of the interpolation deviation $f_a$ described by a triangular distribution

We assumed that the four error components TCM $\delta y_1,\delta y_5,\delta y_6,\delta y_7$ are probabilistic due to the fact that these errors can be observed as a dispersion of measured values. The component of interpolation deviation $\delta y_{f_a}$ has a triangular distribution because these components can be regarded as the difference between two systematic components characterised by a rectangular a priori distribution. The other components are systematic errors with an a priori systematic distribution.

As a result, the final equation for the combined uncertainty of the signal $u_y$ measurement, resulting from the application of the uncertainty propagation law, is as follows:

$$u_y=\sqrt{u_{\mathrm{TCM}}^2+2·u^2\left(r\right)+u^2\left(b\right)+u^2\left(b^{\prime }\right)+u^2\left(f_0\right)+u^2\left(f_a\right)}$$

(3)

where

$$u_{\mathrm{TCM}}=\sqrt{\sum _{i=1}^8{u}_i^2}$$

(4)

The expanded relative uncertainties determined by this method in the whole range from −1100 kN·m to 1100 kN·m are at a level of 0.08–0.1% (

Table 1. Measurement single points ($x_i,y_i$), for anticlockwise torque [27].

${\mathit{x}}_{\mathit{i}}$ [kN·m]	${\mathit{y}}_{\mathit{i}}$ [mV/V]	${\mathit{U}}_{\mathit{r}}\left({\mathit{y}}_{\mathit{i}}\right)={\mathit{U}}_{\mathit{yi}}/{\mathit{y}}_{\mathit{i}}=2{\mathit{u}}_{\mathit{yi}}/{\mathit{y}}_{\mathit{i}}$ [%]
−1100	−0.285722	0.080
−1000	−0.259750	0.080
−800	−0.207792	0.080
−600	−0.155834	0.081
−400	−0.103900	0.080
−300	−0.077912	0.085
−200	−0.051949	0.080
−100	−0.025986	0.119

and

Table 2. Measurement single points ($x_i,y_i$), for clockwise torque [27].

${\mathit{x}}_{\mathit{i}}$ [kN·m]	${\mathit{y}}_{\mathit{i}}$ [mV/V]	${\mathit{U}}_{\mathit{r}}\left({\mathit{y}}_{\mathit{i}}\right)={\mathit{U}}_{\mathit{yi}}/{\mathit{y}}_{\mathit{i}}=2{\mathit{u}}_{\mathit{yi}}/{\mathit{y}}_{\mathit{i}}$ [%]
100	0.025956	0.166
200	0.051919	0.144
300	0.077903	0.097
400	0.103879	0.089
600	0.155825	0.086
800	0.207793	0.08
1000	0.259765	0.081
1100	0.285745	0.081

).

Coefficient 2 in Equation (3) arises from the fact that the calibration result is determined by the difference between the reading under load and the zero reading. Therefore, the influence of the resolution on the reading should be taken into account twice. The relative value of the combined uncertainty of the signal $u_r$ is determined by normalizing the standard combined uncertainty $u_y$ to $y\left(x\right)$, which is given by $u_r=\frac{u_y}{y\left(x\right)}$. The expanded uncertainty value is obtained by multiplying it by a coverage factor of $k=2$, resulting in $U_r=k·u_r$.

Due to the relative expanded measurement uncertainty of the calibration device ($U_{rTCM}$ = 0.08%, $k=2$), the transducer was classified as class 0.5.

3.2. Alternative Method for Estimation Uncertainty at the Single Point of Measurement—Fuzzy Sets Approach to Evaluation of Uncertainty

The guide [28] distinguishes two methods of evaluating uncertainty: method A describing random processes manifested by scattering of data, and method B describing the metrologist’s decision-making processes. The combination of both methods requires assumptions that the GUM guide does not discuss. The papers [29,30] present the transformation of probability to fuzzy, which allows combining methods A and B within fuzzy set theory.

Fuzzy sets were proposed by Zadeh [31] as mathematics for the description of decision-making based on expert knowledge. Zadeh proposed a linguistic interpretation for the application to approximate reasoning, but for measurement sciences, the conception of a fuzzy variable [32] is more adequate [29,30]. The basic difference between a random variable and a fuzzy variable is that the measure of possibility (equivalent to probability in fuzzy set theory) is a maximal measure, not an additive one as in probability theory. Uncertainty propagation in probability theory is a consequence of the arithmetic of random variables. Equation (1) is an equation for the standard deviation of the sum of independent random variables. Within the theory of fuzzy sets, the total error should be written as the sum of the fuzzy variables (1), and the expanded uncertainty is equal to the radius of $\alpha$-cut of the fuzzy set representing the result of the measurement [33]. According to the papers [29,30,34,35], the expanded uncertainty is equal to the algebraic sum of two terms:

• The expanded uncertainty of random components calculated as a radius of a confidence interval of sum of random errors;
• The maximal bound error of systematic components of errors.

The first component is calculated in accordance to the guide [28], i.e., using the probability-to-possibility transformation proposed in the papers [29,30,34,35], the expanded uncertainty is obtained as in method A according to the guide [28]. This transformation consists in transforming the cumulative distribution function into a fuzzy set in such a way that the $\alpha$-cut of the fuzzy set is equal to the confidence interval of estimator of the measurand (excepted value). The second component is equal to the maximal bound error due to the assumption that this error is modelled by a rectangular distribution (distribution fuzzy set). In the fuzzy set, the $\alpha$-cut of the sum of two fuzzy sets is equal to the sum of the $\alpha$-cuts of these components if the one component is a rectangular fuzzy set. Therefore, we can group the error components into two groups: those for which the data dispersion is described by the empirical distribution (type A) and those for which only the maximum bound error is known.

The errors $\delta y_1$, $\delta y_5$, and $\delta y_7$ are random errors with a normal distribution, the remaining errors have a rectangular distribution and we treat them as systematic. In the framework of fuzzy set philosophy [29], the expanded uncertainty is equal:

$$\begin{array}{c}U\left(y\right)=\sqrt{U^2{\left(\delta y_1\right)}^2+U^2{\left(\delta y_5\right)}^2+U^2\left(\delta y_7\right)}+\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\Delta y_{m2}+\Delta y_{m3}+\Delta y_{m4}+\Delta y_{m6}+\Delta y_{m8}+\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\Delta y_{mr}+\Delta y_{mb}+\Delta y_{m{b}^{\prime }}+\Delta y_{m{f}_0}+\Delta y_{m{f}_a}\end{array}$$

(5)

where $\Delta y_{m*}$ denotes the maximal bound error of the corresponding components in Equation (1). Assuming that the errors from [36] are similar in proportion, an approximate value of the expanded uncertainty can be calculated. After inserting the data, we obtain the expanded fuzzy uncertainty $0.12\%$ (calculated for the range up to 1.1 MN·m).

Equation (5) can be obtained from Equation (1) assuming that all components of errors are independent random variables. Such an assumption requires additional research; if we assume a strong correlation of components with a rectangular distribution, we will obtain a value similar to the fuzzy set method.

3.3. Fitting by Linear Regression of Transducer Standard

The classification criteria for anticlockwise torque are significantly below the limits for class 0.05, except for the linear regression deviation. Equations (6) and (7) show the linear regression equation for clockwise and anticlockwise torque, respectively, while Equation (8) is for the combination of clockwise and anticlockwise torque.

$$S_{ai}=0.0002597·M_i,$$

(6)

$$S_{ai}=0.00025968·M_i,$$

(7)

$$S_{ai}=0.00025969·M_i,$$

(8)

where $S_{ai}$ is the output signal in mV/V and $M_i$ is the torque load in kN·m.

The linearity deviation of the measured load curves relative to the sensitivity at maximum torque load, as shown in Figure 3, Figure 4 and Figure 5, is a very characteristic representation for torque calibration. From this representation, the hysteresis, zero point deviation, repeatability, and reproducibility can be derived in addition to the linearity deviation. For the 5 MN·m torque transducer, the hysteresis is consistently in the range of 2 · ${10}^{-5}$ for clockwise torque in all partial range calibrations. According to the literature, a scaling of the maximum torque would be expected. It is noteworthy that the course of the linearity deviation remains the same for each subrange and is not scaled, as expected from the literature.

4. Calculations

4.1. Background

General considerations for the direct linear regression of y on x are presented, where the y-dependent coordinate represents the signal in mV/V and the x-independent variable represents the torque in kN·m. The linear or nonlinear functions are described by the polynomial expansion of a Taylor series $f(x,\mathit{\theta })={\theta }_0+{\theta }_{1}x+{\theta }_2{x}^2+\dots +{\theta }_m{x}^m$, where the measurements of the variable $y_i$ are carried out with the associated uncertainties $u\left(y_i\right)$ and $i=1,\dots ,n$, while the coordinate $x_i$ is determined with an uncertainty close to zero. These considerations lead to the use of the weighted least squares (WLS) method, which minimises the criterion function (9) [37]:

$${\displaystyle \varphi \left(\mathit{\theta }\right)=\sum _{i=1}^n\frac{{(y_i-f(x,\mathit{\theta }))}^2}{u^2\left(y_i\right)}⟶min}$$

(9)

where the parameters of the function are defined by an $m+1$ dimensional vector $\mathit{\theta }={[{\theta }_0,{\theta }_1,{\theta }_2,\dots ,{\theta }_m]}^T$. It should be noted that Equation (9) does not take into account the correlations between the coordinates, as it is assumed that there are no correlations between coordinates during the measurements.

The condition for minimising Equation (9) implies that the gradient of the criterion function is zero, meaning that the first derivatives with respect to all parameters are equal to zero, i.e., ${\nabla }_{\mathit{\theta }}\varphi \left(\mathit{\theta }\right)=$ 0. This leads to obtaining $m+1$ linear equations with respect to the parameters of the vector $\mathit{\theta }$:

$$\sum _{i=1}^n\frac{y_i-{\theta }_m{x}_i^m-\dots -{\theta }_1{x}_i-{\theta }_0}{u^2\left(y_i\right)}{x}_i^j=0$$

(10)

and $j=0,\dots ,m$.

The above linear equations, which depend on the parameters ${\theta }_i$, can be solved using the determinant method. This is because they can be expressed in the following form:

$$\left[\begin{array}{cccc}{S}_x^{2m}& \dots & S_x^{m+1}& S_x^m\\ \dots & \dots & \dots & \dots \\ S_x^m& \dots & S_x^1& S_x^0\end{array}\right]\left[\begin{array}{c}\begin{array}{c}{\theta }_m\\ \begin{array}{c}\\ ...\end{array}\end{array}\\ \begin{array}{c}\\ {\theta }_1\end{array}\\ \begin{array}{c}\\ {\theta }_0\end{array}\end{array}\right]=\left[\begin{array}{c}\begin{array}{c}{S}_{xy}^m\end{array}\\ \begin{array}{c}\dots \\ S_{xy}^0\end{array}\end{array}\right]$$

(11)

where

${\displaystyle S_x^j=\sum _{i=1}^n\frac{x_i^j}{u^2\left(y_i\right)},S_{xy}^l=\sum _{i=1}^n\frac{y_i{x}_i^l}{u^2\left(y_i\right)}}$

$j=0,\dots ,2m,l=0,\dots ,m$

4.2. Fitting

In the following section, a straight line with $m=1$ and a cubic spline with $m=3$ were fitted to the coordinates of the measurement points using the WLS method.

4.2.1. Linear Function Adjustment

For the linear function $f\left(x\right)={\theta }_{1}x+{\theta }_0(m=1)$, the following system of two equations written in matrix form is obtained:

$$\left[\begin{array}{cc}{S}_{xx}& S_x\\ S_x& S\end{array}\right]\left[\begin{array}{c}{\theta }_1\\ {\theta }_0\end{array}\right]=\left[\begin{array}{c}{S}_{xy}\\ S_y\end{array}\right]$$

(12)

where auxiliary parameters $S=S_x^0,S_x=S_x^1,S_y=S_{xy}^0,S_{xx}=S_x^2,S_{xy}=S_{xy}^1$ have been defined as:

$${\displaystyle S=\sum _{i=1}^n\frac{1}{u^2\left(y_i\right)},S_x=\sum _{i=1}^n\frac{x_i}{u^2\left(y_i\right)},S_y=\sum _{i=1}^n\frac{y_i}{u^2\left(y_i\right)},S_{xx}=\sum _{i=1}^n\frac{x_i^2}{u^2\left(y_i\right)},S_{xy}=\sum _{i=1}^n\frac{x_i{y}_i}{u^2\left(y_i\right)},}$$

(13)

and expressions for the slope ${\theta }_1$ and intercept ${\theta }_0$ have the form:

$${\theta }_1=\frac{S{S}_{xy}-S_x{S}_y}{\Delta }\mathrm{and}{\theta }_0=\frac{S_y{S}_{xx}-S_x{S}_{xy}}{\Delta }$$

(14)

where $\Delta =S{S}_{xx}-{\left(S_x\right)}^2$.

If it is desired to express the uncertainties of the parameters of a simple line, then the dependencies should be linearised ${\theta }_1$ and ${\theta }_0$ as a function of coordinates $x_i$ and $y_i$. Using the law of propagation of uncertainty (LPU) for uncorrelated variables, the squares of sensitivity coefficients (the first partial derivatives after the coordinates) can be used to determine the variance and covariance elements for the parameters ${\theta }_1$ and ${\theta }_0$.

In such a way, the variances $u_{{\theta }_1}^2$ and $u_{{\theta }_0}^2$ and the covariance with a correlation coefficient ${\rho }_{{\theta }_1{\theta }_0}$ are:

$$u_{{\theta }_1}^2=\frac{S}{\Delta },u_{{\theta }_0}^2=\frac{S_{xx}}{\Delta },{\rho }_{{\theta }_1{\theta }_0}{u}_{{\theta }_1}{u}_{{\theta }_0}=-\frac{S_x}{\Delta }$$

(15)

respectively.

The coverage interval of the expanded uncertainty $U_y$ around the straight line $y=ax+b$ can be determined by using the uncertainty propagation method, as follows:

$$U_y=t_{1-\frac{\alpha }{2},n-2}\sqrt{{x^{2}u}_{{\theta }_1}^2+2\left|x\right|u_{{\theta }_1}{u}_{{\theta }_0}{\rho }_{{\theta }_1{\theta }_0}+u_{{\theta }_0}^2}$$

(16)

where $t_{(1-\alpha /2,n-2)}$ denotes the inverse distribution for the Student’s t-distribution with n − 2 degrees of freedom and a probability of $1-\alpha /2$ (where $\alpha =0.05$). To estimate the coverage interval of uncertainty more precisely, the Monte Carlo method (MCM) can be used. This involves generating, for example, ${10}^6$ to ${10}^7$ samples for the y-coordinates normal distributions $N(y_i,u^2\left(y_i\right))$, and then deriving the parameter distributions for a and b using Equation (15). Finally, the distributions for the coordinate $y=ax+b$ can be obtained to cover the interval with a probability of 0.95 at fixed coordinate values of y.

4.2.2. Matching with Cubic Spline—A Polynomial of Third Degree

Regarding the possibility of the nonlinear behaviour of the characteristic of the calibrated transducer, it is necessary to investigate whether higher-order curve fitting will give better results than linear alignment. Therefore, the use of polynomials with the smallest degree–cubic splines is the most sensible option because they have saddle points.

In this case ($m=3$), the interpolating function is described by four parameters $A,B,C$, and D:

$$y=A{x}^3+B{x}^2+Cx+D$$

(17)

The criterion function (9) can be reduced to the following form:

$$\varphi \left(A,B,C,D\right)=\sum _{i=1}^n\frac{{(y_i-A{x}_i^3-B{x}_i^2-C{x}_i-D)}^2}{u^2\left(y_i\right)}\to min$$

(18)

The following function (18) reaches the minimum when the system of four linear equations is satisfied, where the first derivatives with respect to parameters $A,B,C$, and D are equal to zero:

$$\sum _{i=1}^n\frac{y_i-A{x}_i^3-B{x}_i^2-C{x}_i-D}{u^2\left(y_i\right)}{x}_i^j=0$$

(19)

where j = 0, 1, 2, 3.

Solving the system of four linear equations obtained for the parameters $A,B,C$, and D results in a linear system of four equations with four unknowns:

$$\left[\begin{array}{cccc}{S}_x^6& S_x^5& S_x^4& S_x^3\\ S_x^5& S_x^4& S_x^3& S_x^2\\ S_x^4& S_x^3& S_x^3& S_x^1\\ S_x^3& S_x^2& S_x^1& S_x^0\end{array}\right]\left[\begin{array}{c}A\\ B\\ C\\ D\end{array}\right]=\left[\begin{array}{c}{S}_{xy}^3\\ S_{xy}^2\\ S_{xy}^1\\ S_{xy}^0\end{array}\right]$$

(20)

A more simplified version of cubic spline fitting can be achieved with only three parameters, without the intercept D. This is achieved by solving a reduced system of only three equations (see Equation (20)). The adjustment function passes through the point (0, 0) and takes the form of $y=A{x}^3+B{x}^2+Cx$. However, it is important to note that the limitation associated with the removal of the intercept means that a zero signal for the zero of torque is also observed in the measurements.

The expanded uncertainty interval $U_y$ around the curve described by (17) can be determined by using the uncertainty propagation method, as follows:

$$U_y=t_{1-\alpha /2,n-4}{u}_y$$

(21)

where by $u_y$, the composed standard uncertainty is denoted. The number of degrees of freedom in this case of four parameters of spline is $n-4$. Assuming that the standard uncertainties $u_A,u_B,u_C$, and $u_D$ are known for four distributions of the cubic spline curve coefficients A, B, C, and D, and the correlation coefficients between them, ${\rho }_{AB}, {\rho }_{AC}, {\rho }_{AD}, {\rho }_{BC}, {\rho }_{BD}$, and ${\rho }_{CD}$, the square of the composed uncertainty can be expressed as follows:

$$\begin{array}{c}{u}_y^2=u_A^2{x}^6+2{\rho }_{AB}{u}_A{u}_B{\left|x\right|}^5+(u_B^2+2{\rho }_{AC}{u}_A{u}_C)x^4+2({\rho }_{AD}{u}_A{u}_D+\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\rho }_{BC}{u}_B{u}_C{\left)\right|x|}^3+(u_C^2+2{\rho }_{BD}{u}_B{u}_D)x^2+2{\rho }_{CD}{u}_C{u}_D\left|x\right|+u_D^2\end{array}$$

(22)

The determination of the correlation coefficients or standard uncertainties can be performed analytically after solving a system of Equation (20), or using the Monte Carlo method to generate parameter distributions of $A,B,C$, and D. Then, using standard library functions, for example, in the R environment, the basic parameters of distributions for the parameters $A,B,C$, and D are determined: expected values–function mean(), standard deviations–function sd(), and correlation coefficients–function cor().

To determine the uncertainty interval more accurately, the MCM can be used by generating ${10}^6$ to ${10}^7$ samples for the y-coordinates’ normal distributions $N(y_i,u^2\left(y_i\right))$. From Equation (20), the parameter distributions for $A,B,C$, and D can be derived to finally obtain distributions for the y-coordinates from the (17), while covering the interval with a probability of 0.95 at fixed coordinate values y.

4.3. Results of Fitting to a Straight Line and 3-Degree Polynomial

As a result of the torque calibration measurements, the transducer signals listed in Table 1 and Table 2 were obtained [27].

On the basis of Section 4.2.1, straight line fitting is carried out. The obtained characteristics are shown in Figure 6, where measurement points are marked in the form of squares, points are measured in the range from −1100 kN·m to 1100 kN·m, and two fitted linear characteristics above the calibration range are plotted using the weighted method of least squares (WLS) (dashed blue line for anticlockwise torque and dashed red line for clockwise torque). The equations of both matched straight lines are also given.

In order to show the surroundings of the measuring point and the line adjustments from Figure 6, the last calibration measurement point for clockwise torque is included in the magnified area in Figure 7 below. For the selected calibration point, i.e., torque = 1100 kN·m, the area from Figure 6 has been enlarged, where the WLS line is adjusted, marked with a red dash-dot line. You can also see two pairs of curved lines determining a narrower coverage region in the case of using the MCM method (red dotted lines and a wider one in the form of blue dashed lines for estimating the corridor using the law of propagation of uncertainty). The following surrounding corridor coverage for the selected measured point looks analogous for all other points, including points measured in the anticlockwise range.

Figure 6. Straight-line adjustment of the relationship of the tared transducer signal as a function of torque from the WLS method. Data from Table 1, Table 2 and

Table 3. Expanded absolute uncertainties (U) for the transducer signals for clockwise and anticlockwise torque measurement using the LPU method and MCM linear adjustment.

Torque [kN·m]	Expanded Uncertainty ${\mathit{U}}_{\mathit{y}}$ [mV/V] Monte Carlo Method MCM	Expanded Uncertainty ${\mathit{U}}_{\mathit{y}}$ [mV/V] Law of Propagation of Uncertainty LPU
−1100	0.000105	0.000131
−1000	0.000093	0.000117
−800	0.000071	0.000089
−600	0.000050	0.000062
−400	0.000030	0.000038
−300	0.000023	0.000029
−200	0.000021	0.000027
−100	0.000025	0.000031
100	0.000035	0.000044
200	0.000030	0.000037
300	0.000030	0.000037
400	0.000034	0.000043
600	0.000052	0.000065
800	0.000075	0.000093
1000	0.000099	0.000123
1100	0.000111	0.000139

.

Figure 6. Straight-line adjustment of the relationship of the tared transducer signal as a function of torque from the WLS method. Data from Table 1, Table 2 and Table 3.

In addition, an extrapolation of the linear characteristics of the tared transducer signal was performed as a function of torque in the range from −100 MN·m up to 100 MN·m. The results are illustrated in Figure 8. Figure 8 includes linear extrapolations of straight line fitting in the range from −1100 kN·m to 1100 kN·m of the torque to the range from −100 MN·m to 100 MN·m of the torque in both the anticlockwise and clockwise directions. In addition, there is a table containing selected characteristic points with extended uncertainties of the signal in steps of 0.5 MN·m in the range from −5 MN·m to 5 MN·m of the torque.

In order to check the nonlinearity, the adjustment to the measured coordinates was achieved with cubic splines, i.e., polynomials of the third degree, which, due to their characteristics, can adapt to various types of nonlinearity. Then, a cubic spline curve fit is performed. The expanded uncertainties of the signal $U_y$ are obtained by transforming it from Equations (21) and (22). The results are shown in Table 4, Table 5 and Table 6. Figure 9 and Table 4, Table 5 and Table 6 shows the full fitting characteristics of the cubic spline. Figure 9, on the other hand, shows two matching cubic characteristics in the calibrated range with measurement points, allowing for possible nonlinearities in the characteristics of the tested transducer. In terms of calibration, formulas describing these curves with cubic splines in both measured directions are given. The penultimate calibration point for the clockwise direction is also indicated, whose enlarged surroundings are shown in Figure 10. It is easy to see that the coverage corridor is broader for both the MCM method and the method of law of propagation of uncertainty (LPU) compared to the coverage corridor for the linear fit (Figure 7). As before, for linear fitting, the MCM method determines a narrower corridor than the LPU method. The following surrounding corridor coverage for the selected measured point looks analogous for all other points, including points measured in the anticlockwise range.

In addition, a cubic spline curve was fitted to the three parameters for the data in Table 1, and the resulting equation for the curve is $y=2.19·{10}^{-13}·x^3+2.91·{10}^{-10}·x^2+0.00025982·x$.

For anticlockwise signal measurements, a cubic spline curve was fitted to the three parameters for the data in Table 2, and the resulting equation for the curve is $y=-1.86·{10}^{-13}{x}^3+3.99·{10}^{-10}{x}^2+0.00025955·x$.

In Figure 11, the absolute expanded uncertainties, and in Figure 12, the relative expanded uncertainties determined by the LPU method and MCM are presented.

Figure 11 shows the characteristics of half of the corridor of the expanded absolute uncertainty of the signal for linear and cubic spline fitting in both directions for both the MCM and LPU methods. These characteristics indicate a nonlinear increase in the expanded uncertainty of the signal with an increase in the measured torque setpoint. Due to the anticlockwise and clockwise measurement directions, the characteristics of the measured transducer are not symmetrical, and there is a nonzero uncertainty in the signal for a zero torque. The smallest extended uncertainties occur for MCM estimation for linear fitting, slightly higher for LPU estimation, and for spline matching in the measured range, a significant increase in signal uncertainty is observed, in particular for the LPU method.

Table 4. Adjustment for clockwise signal measurements, adjustment factors $A=-2.32·{10}^{-14}$, $B=1.21·{10}^{-10}$, $C=0.00025968$, $D=-1.21·{10}^{-5}, y=-2.32·{10}^{-14}{x}^3+1.21·{10}^{-10}{x}^2+0.00025968·x-1.21·{10}^{-5}$. The expanded absolute uncertainties $U_y$ and relative uncertainties ${\delta }_y$ for 0.95 probability of signal—coverage factor $k=t(0.95,8-4)=2.78$ for 0.95 probability of signal $U_y$.

Torque [kN·m]	Expanded Uncertainty of Signal Monte Carlo Method MCM		Expanded Uncertainty of Signal—The Law of Propagation of Uncertainty LPU
	Absolute  ${\mathit{U}}_{\mathit{y}}$  [mV/V]	Relative  ${\mathit{\delta }}_{\mathit{y}}$  [%]	Absolute  ${\mathit{U}}_{\mathit{y}}$  [mV/V]	Relative  ${\mathit{\delta }}_{\mathit{y}}$  [%]
100	0.000041	0.159	0.000058	0.225
200	0.000040	0.077	0.000056	0.109
300	0.000050	0.064	0.000071	0.091
400	0.000053	0.051	0.000076	0.073
600	0.000084	0.054	0.000119	0.076
800	0.000113	0.055	0.000161	0.077
1000	0.000121	0.047	0.000171	0.066
1100	0.000193	0.068	0.000274	0.096

Table 4. Adjustment for clockwise signal measurements, adjustment factors $A=-2.32·{10}^{-14}$, $B=1.21·{10}^{-10}$, $C=0.00025968$, $D=-1.21·{10}^{-5}, y=-2.32·{10}^{-14}{x}^3+1.21·{10}^{-10}{x}^2+0.00025968·x-1.21·{10}^{-5}$. The expanded absolute uncertainties $U_y$ and relative uncertainties ${\delta }_y$ for 0.95 probability of signal—coverage factor $k=t(0.95,8-4)=2.78$ for 0.95 probability of signal $U_y$.

Torque [kN·m]	Expanded Uncertainty of Signal Monte Carlo Method MCM		Expanded Uncertainty of Signal—The Law of Propagation of Uncertainty LPU
	Absolute  ${\mathit{U}}_{\mathit{y}}$  [mV/V]	Relative  ${\mathit{\delta }}_{\mathit{y}}$  [%]	Absolute  ${\mathit{U}}_{\mathit{y}}$  [mV/V]	Relative  ${\mathit{\delta }}_{\mathit{y}}$  [%]
100	0.000041	0.159	0.000058	0.225
200	0.000040	0.077	0.000056	0.109
300	0.000050	0.064	0.000071	0.091
400	0.000053	0.051	0.000076	0.073
600	0.000084	0.054	0.000119	0.076
800	0.000113	0.055	0.000161	0.077
1000	0.000121	0.047	0.000171	0.066
1100	0.000193	0.068	0.000274	0.096

Table 5. Adjustment for anticlockwise signal measurements, adjustment factors $A=-1.37·{10}^{-13}$, $B=-3.01·{10}^{-10}$, $C=0.00025956$, and $D=-2.72·{10}^{-5}$, $y=-1.37·{10}^{-13}{x}^3-3.01·{10}^{-10}{x}^2+0.00025956·x-2.72·{10}^{-5}$. Expanded absolute uncertainties ($U_y$) and relative uncertainties ${\delta }_y$—coverage factor $k=t(0.95,8-4)=2.78$ for $0.95$ probability of the signal $U_y$.

Torque [kN·m]	Expanded Uncertainty of Signal Monte Carlo Method MCM		Expanded Uncertainty the Signal—The LAW of Propagation of Uncertainty LPU
	Absolute  ${\mathit{U}}_{\mathit{y}}$  [mV/V]	Relative  ${\mathit{\delta }}_{\mathit{y}}$  [%]	Absolute  ${\mathit{U}}_{\mathit{y}}$  [mV/V]	Relative  ${\mathit{\delta }}_{\mathit{y}}$  [%]
−1100	0.000188	0.066	0.000267	0.093
−1000	0.000188	0.046	0.000169	0.065
−800	0.000106	0.051	0.000151	0.073
−600	0.000080	0.051	0.000134	0.073
−400	0.000045	0.043	0.000063	0.061
−300	0.000038	0.049	0.000054	0.070
−200	0.000029	0.056	0.000041	0.079
−100	0.000030	0.114	0.000042	0.161

Table 5. Adjustment for anticlockwise signal measurements, adjustment factors $A=-1.37·{10}^{-13}$, $B=-3.01·{10}^{-10}$, $C=0.00025956$, and $D=-2.72·{10}^{-5}$, $y=-1.37·{10}^{-13}{x}^3-3.01·{10}^{-10}{x}^2+0.00025956·x-2.72·{10}^{-5}$. Expanded absolute uncertainties ($U_y$) and relative uncertainties ${\delta }_y$—coverage factor $k=t(0.95,8-4)=2.78$ for $0.95$ probability of the signal $U_y$.

Torque [kN·m]	Expanded Uncertainty of Signal Monte Carlo Method MCM		Expanded Uncertainty the Signal—The LAW of Propagation of Uncertainty LPU
	Absolute  ${\mathit{U}}_{\mathit{y}}$  [mV/V]	Relative  ${\mathit{\delta }}_{\mathit{y}}$  [%]	Absolute  ${\mathit{U}}_{\mathit{y}}$  [mV/V]	Relative  ${\mathit{\delta }}_{\mathit{y}}$  [%]
−1100	0.000188	0.066	0.000267	0.093
−1000	0.000188	0.046	0.000169	0.065
−800	0.000106	0.051	0.000151	0.073
−600	0.000080	0.051	0.000134	0.073
−400	0.000045	0.043	0.000063	0.061
−300	0.000038	0.049	0.000054	0.070
−200	0.000029	0.056	0.000041	0.079
−100	0.000030	0.114	0.000042	0.161

Table 6. Table of parameters of 3-degree curves $A, B, C, D$—uncertainties and correlations coefficients obtained by the MCM.

	Anticlockwise	Clockwise
$A=$	$-1.37·{10}^{-13}$	$-2.29·{10}^{-14}$
$B=$	$-3.01·{10}^{-10}$	$1.21·{10}^{-10}$
$C=$	0.00025956	0.00025968
$D=$	$-2.72·{10}^{-5}$	$-1.39·{10}^{-5}$
$u_A=$	$7.46·{10}^{-13}$	$5.59·{10}^{-13}$
$u_B=$	$1.14·{10}^{-9}$	$1.36·{10}^{-9}$
$u_C=$	$4.58·{10}^{-7}$	$5.67·{10}^{-7}$
$u_D=$	$4.54·{10}^{-5}$	$5.79·{10}^{-5}$
${\rho }_{AB}=$	−0.987	−0.988
${\rho }_{AC}=$	0.918	0.926
${\rho }_{AD}=$	−0.796	−0.791
${\rho }_{BC}=$	−0.967	−0.971
${\rho }_{BD}=$	0.864	0.856
${\rho }_{DC}=$	−0.947	−0.937

Table 6. Table of parameters of 3-degree curves $A, B, C, D$—uncertainties and correlations coefficients obtained by the MCM.

	Anticlockwise	Clockwise
$A=$	$-1.37·{10}^{-13}$	$-2.29·{10}^{-14}$
$B=$	$-3.01·{10}^{-10}$	$1.21·{10}^{-10}$
$C=$	0.00025956	0.00025968
$D=$	$-2.72·{10}^{-5}$	$-1.39·{10}^{-5}$
$u_A=$	$7.46·{10}^{-13}$	$5.59·{10}^{-13}$
$u_B=$	$1.14·{10}^{-9}$	$1.36·{10}^{-9}$
$u_C=$	$4.58·{10}^{-7}$	$5.67·{10}^{-7}$
$u_D=$	$4.54·{10}^{-5}$	$5.79·{10}^{-5}$
${\rho }_{AB}=$	−0.987	−0.988
${\rho }_{AC}=$	0.918	0.926
${\rho }_{AD}=$	−0.796	−0.791
${\rho }_{BC}=$	−0.967	−0.971
${\rho }_{BD}=$	0.864	0.856
${\rho }_{DC}=$	−0.947	−0.937

Figure 11. Plot of absolute expanded uncertainties using LPU and MCM for the straight line fit and cubic spline fit—signal values near zero are not included in the graph because the concept of relative error loses its meaning. Data from Table 3, Table 4 and Table 5.

Figure 11. Plot of absolute expanded uncertainties using LPU and MCM for the straight line fit and cubic spline fit—signal values near zero are not included in the graph because the concept of relative error loses its meaning. Data from Table 3, Table 4 and Table 5.

Figure 12. Plot of relative expanded uncertainties using LPU and MCM for the straight line fit and cubic spline fit—signal values near zero are not included in the graph because the concept of relative error loses its meaning. Data from Table 4, Table 5 and

Table 7. Expanded relative uncertainties (${\delta }_y$ = $U_y$$100/y$) for transducer signals for clockwise and anticlockwise torque measurement using the LPU method and MCM linear adjustment.

Torque [kN·m]	Expanded Uncertainty ${\mathit{\delta }}_{\mathit{y}}$ [%] Monte Carlo Method MCM	Expanded Uncertainty ${\mathit{\delta }}_{\mathit{y}}$ [%] Law of Propagation of Uncertainty LPU
−1100	0.0367	0.0459
−1000	0.0360	0.0450
−800	0.0342	0.0428
−600	0.0317	0.0397
−400	0.0291	0.0364
−300	0.0300	0.0376
−200	0.0409	0.0511
−100	0.0960	0.1201
100	0.0576	0.0720
200	0.0379	0.0473
300	0.0329	0.0412
400	0.0336	0.0419
600	0.0379	0.0451
800	0.0380	0.0475
1000	0.0388	0.0485
1100	0.0576	0.0720

.

Figure 12. Plot of relative expanded uncertainties using LPU and MCM for the straight line fit and cubic spline fit—signal values near zero are not included in the graph because the concept of relative error loses its meaning. Data from Table 4, Table 5 and Table 7.

Figure 12 contains expanded relative uncertainties, which, for both matches and both methods below −300 kN·m and above 300 kN·m of the calibrated range up to −100 kN·m and above 100 kN·m, are from 0.03% to a maximum of 0.08%. Near torque values close to zero, the relative uncertainty of measuring a virtually close-to-zero signal loses its meaning. It also turned out that extrapolated expanded uncertainties when fitting using the cubic spline method increase significantly relative to the extrapolated expanded uncertainties for a linear fit.

Table 8. The relative expanded measurement uncertainty for the 5 MN·m torque transducer designed to measure the clockwise and anticlockwise directions [38].

Performance Work and Method Used	Anticlockwise/Clockwise Direction	Range	Expanded Relative/Uncertainties $\mathit{\delta }$ [%]
Current work LPU linear [38]	Anticlockwise	from $-5\mathrm{MN}·\mathrm{m}$ up to $-500\mathrm{kN}·\mathrm{m}$	0.05 ÷ 0.06
	Clockwise	from $500\mathrm{kN}·\mathrm{m}$ up to $5\mathrm{MN}·\mathrm{m}$	0.04 ÷ 0.06
Current work MCM linear [38]	Anticlockwise	from $-5\mathrm{MN}·\mathrm{m}$ up to $-500\mathrm{kN}·\mathrm{m}$	0.03 ÷ 0.05
	Clockwise	from $500\mathrm{N}·\mathrm{m}$ up to $5\mathrm{MN}·\mathrm{m}$	0.03 ÷ 0.05

presents the relative expanded measurement uncertainty for the 5 MN·m torque transducer designed to measure clockwise and anticlockwise torque calculated by different methods. Finally, taking into account only the linear nature of the transducer, Table 8 gives the estimated values of expanded relative uncertainty in both the calibrated and extrapolated ranges, to an absolute torque value of |MK| = 5 MN·m. These values are below 0.06% for the LPU method and 0.05% for the MCM method in both directions of measurement.

5. Summary

The work provides an analysis of the metrological properties of a 5 MN·m torque transducer. The relative electrical signal (given in electrical units, mV/V) as a function of torque measured in both the clockwise and anticlockwise directions was monitored.

To fit the data, the weighted method of least squares with both a straight-line and a cubic spline curve was used. The work presents the results of estimating both the relative and absolute uncertainty using two methods: uncertainty propagation and Monte Carlo simulation. The results of the analysis indicated that the straight-line fitting method produced smaller values for the expanded uncertainty than the cubic spline fitting method. Moreover, the study confirmed the assumptions that the MCM for propagating uncorrelated distributions was more accurate than the LPU method, regardless of the type of curve fitting used. In the torque range above 200 kN·m, the expanded relative uncertainty of the measured signals was at the level of (0.03 ÷ 0.04)% for the straight-line fitting method and about 0.05% for the cubic spline fitting method. In the calibrated range below 1 MN·m, with linear fitting in both directions, the absolute expanded uncertainty of the normalised signal was below 0.0001 mV/V.

In addition, matching to the measured coordinates was performed with cubic splines, i.e., polynomials of the third degree, which, due to their characteristics, can adjust to various types of nonlinearities. In interpolating problems, cubic spline interpolation is often preferred to other polynomials with a higher degree. But this interpolation, in general, cannot be used for analytic continuation—the extrapolation of characteristics using cubic spline in a range not measured by the instrument, e.g., the torque transducer. Additionally the expanded uncertainties estimated from the Monte Carlo method for all cases were much higher for cubic spline than for linear interpolation.

Finally, we explored how to evaluate uncertainty using fuzzy sets, and an approximation of the expanded uncertainty’s value was computed.

Overall, the study provides a detailed analysis of the metrological properties of the torque transducer and presents a comparison of different methods for estimating the uncertainty above a calibratable range. The findings can be useful for researchers and engineers working with torque transducers and those in the wind turbine sector expanding the development of more precise measurement techniques.