On the regression of sensitivity characteristics of torque transducers


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 194 

ACTA IMEKO 
ISSN: 2221-870X 
December 2020, Volume 9, Number 5, 194 - 199 

 
 

ON THE REGRESSION OF SENSITIVITY CHARACTERISTICS   

OF TORQUE TRANSDUCERS 
 

A. Brüge1 

 
1 Department 1.2, Physikalisch-Technische Bundesanstalt, Braunschweig, Germany, andreas.bruege@ptb.de  

 

Abstract: 

A new method to estimate the influence of 

regression on the measurement uncertainty for the 

calibration of torque transducers will be presented 

in the following. This method is based on Monte 

Carlo simulations and on sampling calibration 

results. In some cases, the uncertainty contributions 

of regression thus obtained are considerably higher 

than those yielded by estimates according to widely 

accepted methods. The differences between the 

methods will be discussed and suggestions for 

simplified estimates within the scope of the new 

method will be made. 

Keywords: Regression; torque; calibration; 

measurement uncertainty; Monte Carlo simulation 

1. INTRODUCTION 

Calibration results of torque transducers are 

usually documented in the form of discrete 

measurement points. These discrete points are then 

used to calculate a sensitivity characteristic curve by 

means of a cubic polynomial function, so that users 

can apply traceability also between those values. In 

widely accepted methods ([1], [2], [3]), the 

associated uncertainty contribution is estimated by 

means of a global characteristic value 𝑢𝑓a,Std , 

assuming a triangular distribution. The conditions 

justifying such an estimation have not yet been 

investigated. The introduction of such conditions 

was mainly motivated by the necessity of keeping 

the calculation of measurement uncertainties 

straightforward for users in calibration laboratories. 

Due to the fact that analytical approaches require 

considerable effort (see article by D. Röske in these 

proceedings), systematic investigations have, to 

date, only been carried out either based on linear 

regressions or by means of simulation methods ([4], 

[5], [6]). As a rule, simulations provide results that 

are specific to a particular application. The special 

conditions of torque calibration have not yet been 

sufficiently investigated for this reason. In this 

paper, the influence of regression on the 

measurement uncertainty will therefore be 

examined by means of the Monte Carlo method 

(MCM); the results will then be discussed based on 

real measurement data. 

2. MONTE CARLO SIMULATION 

The dependence of the output signals of strain 

gauge torque transducers on the load torque is 

essentially linear. Since the nonlinear contributions 

are smaller than the linear ones by several orders of 

magnitude, characteristic curves are mostly 

represented as linearity deviations over the torque 

and can be modelled very conveniently by means of 

a cubic polynomial (Figure 1, green line). The 

uncertainties associated with each of the data points 

are used in the MCM to shift the data points 

accordingly. In the regression, these uncertainties 

lead to different polynomial functions. In Figure 1, 

two examples of these polynomials are shown 

(yellow and orange lines) which are based on the 

impact of all uncertainty bars in positive and 

negative directions respectively. 

 
Figure 1: Linearity deviations of a simulated series of 

measured values (blue dots) over a torque range to which 

uncertainties (blue bars) are assigned. A cubic regression 

of the dots provides the green curve. Taking into account 

the uncertainties, other regression curves are also 

possible, two of which are shown as examples (yellow 

and orange). 

MCM allows the behaviour of the polynomials 

to be investigated in detail over a wide range of 

parameter values. Synthetic measurements as data 

points are defined by the parameters of linearity 

deviation L, distortion D, noise N, and the relative 

-1.5E-03

-1.0E-03

-5.0E-04

0.0E+00

5.0E-04

1.0E-03

1.5E-03

0 50 100

L
in

e
a
ri
ty

 
d
e
v
. 
 i
n
 m

V
/V

Torque load in %

Meas.

Model

Regress+

Regress-

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 195 

position of the inflection point Pi/NV (Position of 

inflection point / Nominal Value). The data points 

are generated with a cubic polynomial in such a way 

that their maximum deviation from the linear course 

is given by L. With a normal distribution with the 

standard deviation D, the points, which until now 

have been ideally located on the regression curve, 

are randomly shifted individually. The simulated 

measurement curve obtained in this way is now 

varied randomly within the MCM by shifting the 

measurement points individually from the distorted 

centre position generated with D, using a normal 

distribution with the standard deviation N. The 

position of the inflection point Pi  can easily be 
calculated from the second and third order 

parameters 𝑐2 and 𝑐3 of the cubic polynomial: 

 Pi = 𝑐2 (3 𝑐3)⁄  . (1) 

For the simulation, the model function is adapted 

to the conditions of real transducers in certain fields 

of application by means of typical parameter 

combinations, as shown in Figure 2. The parameters 

are obtained by sampling a large number of 

calibrations (n  3000). From the results obtained 

by means of MCM, it is possible to gain findings 

concerning the regression of characteristic curves of 

transducers. 

 
Figure 2: Model parameters in mV/V for the simulation 

of the regression of characteristic curves with a nominal 

signal of 2 mV/V in typical fields of application (see the 

text in Section 4). The values show the expected values 

(horizontal lines) and the standard deviations (bars) 

obtained for a sampling of calibrations. Values for the 

combined standard uncertainty of the calibrations at the 

10 % data point 𝑢cert,0.1 are also indicated for comparison 
purposes. 

An analysis of the correlation shows that the 

measurement uncertainty contributions of interest 

are mainly influenced by noise N. For the standard 

deviation of the regression deviation 𝑆(𝑓a) , this 
yields a sensitivity of approximately 4. The 

systematic contribution of the regression deviation 

𝑢𝑓a also depends on the distortion D. The linearity 

and the position of the inflection point do not play a 

significant role (Figure 3). In the investigated range 

of values, the influences of N and D are strictly 

linear. The results of simulations using input 

parameters with defined values are therefore easily 

scalable to other input values. 

 
Figure 3: Correlation of measurement uncertainty 

contributions as output quantities of the MCM simulation 

of the regression with the model parameters. 

The variations of the reference points entering 

into the simulations 𝑢St are measured in accordance 
with the noise N which is composed of nonlinearity 

contributions of the amplifier [7], of its resolution 

and of the signal disturbance in the measurement 

chain. Although nonlinearity leads to contributions 

with beta distribution, the simulations have shown 

that in realistic scenarios, the contributions of other 

components with a Gaussian distribution prevail, so 

that a Gaussian distribution may be assumed. It was 

possible to confirm this assumption empirically, 

based on 5 × 104 real data sets. 

The uncertainty of the regression function 𝑢R 
must be taken into account when assessing stepped 

calibration processes between the reference points. 

This uncertainty is obtained by determining the 

uncertainty of the polynomial coefficients that 

define the regression function. For cubic 

polynomials without an absolute term, simulation 

between the coefficients of different orders yields 

normalized covariances with absolute values near 1: 

𝑐𝑜𝑣norm(𝑋𝑖 , 𝑋𝑗 ) = [
 −0.9600    0.9028

−0.9600  −0.9870
   0.9028 −0.9870  

]. (2) 

When calculating 𝑢R , the covariances may 
therefore not be neglected. 

3. CONTRIBUTIONS TO THE 
MEASUREMENT UNCERTAINTY 

3.1. User's Scenarios 
In current standardization procedures, the 

influence of the regression enters into the 

measurement uncertainty in a generalized manner 

via a plausible estimation. This method does not 

take the fact into consideration as to whether the 

user would like to use the regression at all. 

This paper therefore suggests considering these 

parameters separately. When traceability is required 

at the reference points only (scenario a), the 

regression and its contribution to the measurement 

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

V
a

lu
e

 in
 m

V
/V

TT

TW

AT

IT

Estim. L

Estim. P

L D N u_cert,0.1

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 196 

uncertainty are dispensable. Only when values 

between the reference points are also required 

(scenario b) must this uncertainty contribution be 

taken into account. 

In addition, instead of the generalized estimation 

of the regression influence, this paper breaks down 

the contributions into the individual contributions 

that are made accessible by the MCM simulation: 

the systematic contribution 𝑢𝑓a , the random 

contribution 𝑢S𝑓a , and the contribution that is 

attributed to the model 𝑢R. Typical curves of these 
different contributions in the torque range compared 

to the estimation according to accepted procedures 

𝑢𝑓a,Std  have shown that 𝑢R  cannot be used as the 

sole contribution to the uncertainty due to 

regression (Figure 4). Instead, 𝑢S𝑓a  contains both 

this contribution and the interaction of the 

regression with the uncertainty of the reference 

points, so that these two contributions cannot both 

enter into the measurement uncertainty budget 

together. 

 
Figure 4: Curves of the measurement uncertainty 

contributions from the regression for maximum 

simulation parameters in accordance with the data 

sampling (N = 2 × 10-4 mV/V) compared to the 

contribution for 𝑓a in accordance with DIN 51309. 

The simulations have also shown that the 

estimates as used in accepted procedures are 

systematically below the estimations that are 

determined with the new procedure. 

3.2. Best Estimator 
As long as the standard uncertainty of the 

systematic fraction 𝑢𝑓a  is not larger than the 

uncertainty of the reference points 𝑢St , it can be 
assumed that the reference points, in principle, 

follow a cubic polynomial curve which therefore 

represents the “true characteristic curve”. The 

deviations of the reference points from this curve 

must then be understood as noise, and, due to its 

averaging action, the regression is the best estimator 

for the characteristic curve (estimator L). This 

corresponds the practice of using the average as the 

estimator of the expected value for quantities with a 

Gaussian distribution. For the ratio 

𝑒 = 𝑢𝑓a 𝑢St > 1⁄ , the assumption of a purely cubic 

characteristic curve is no longer justified, and the 

reference points represent the best available 

estimator for the sensitivity of the transducer 

(estimator P). For practical reasons, the individual 

values are combined into a geometric sum across all 

torque steps for this test. 

The combined total measurement uncertainty 𝑢c 
is calculated differently, depending on the user’s 

scenario (a, b) and on the best estimator for the 

sensitivity (L, P). All contributions to the 

measurement uncertainty that are not dealt with in 

this paper are summarized as 𝑢rest . Since the 
sampled calibrations were carried out earlier than 

2018, the contribution of the sawtooth effect usaw 

[7] must be added to their uncertainties 𝑢′c,Std
2
. 

The fact that for torque values between the 

reference points, the curve of 𝑢𝑓a and 𝑢S𝑓a can only 

be determined by interpolation must additionally be 

taken into account for scenario b. The interpolation 

deviations 𝑢Int,S𝑓a and 𝑢Int,𝑓a can be determined at 

the data points, and an upper estimation with a 

constant absolute value can be found across the 

entire torque range (see Section 5.2). From these 

deviations, the absolute value  𝑢Int is calculated by 
geometric addition. 

𝑢c,Std
2 = 𝑢′ c,Std

2
+ 𝑢saw

2 (3) 

𝑢rest
2 =  𝑢c,Std

2 − 𝑢𝑓a
2 − 𝑢r

2 − 𝑢A
2 (4) 

𝑢c,a,L
2 = 𝑢rest

2 −  𝑢saw
2 + 𝑢R

2 (5) 

𝑢c,b,L
2 = 𝑢rest

2 −  𝑢saw
2 + 𝑢R

2 + 𝑢Int
2 (6) 

𝑢c,a,P
2 =  𝑢rest

2 + 𝑢r
2 + 𝑢A

2 (7) 

𝑢c,b,P
2 = 𝑢rest

2  −  𝑢saw
2 + 𝑢𝑓a

2 + 𝑢S𝑓a
2

+  𝑢Int
2 

(8) 

            =  𝑢rest
2 −  𝑢saw

2 + (
𝑓a
2

)
2

+𝑆(𝑓a)
2    

+  𝑢Int
2   

(9) 

4. EMPIRICAL RESULTS 

Real calibration results were analysed 

correspondingly to the evaluation for the synthetic 

measurement data. First, the data were classified 

into categories corresponding to the field of 

application of the torque transducers and exhibited 

different sample sizes (TT: 1466 transfer 

transducers, TW: 100 transfer wrenches, AT: 

117 application transducers, IT: 208 industrial 

transducers). Although this classification is 

arbitrary and does not provide any clear 

delimitation, it is nevertheless significantly 

reflected in the results. In addition, apart from the 

0,0E+00

5,0E-05

1,0E-04

1,5E-04

2,0E-04

2,5E-04

3,0E-04

3,5E-04

0 20 40 60 80 100

u
 in

 m
V

/V

torque load in %

u_fa_DIN u_fa,syst u_S(fa) u_R

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 197 

application categories, the data were also arranged 

according to the best estimator (1550 L, 341 P). 

When plotting across the interpolation 

uncertainty, it turned out that the categories TT and 

TW could mainly be treated with the estimator L 

(Figure 5). However, there are also many cases 

where 𝑒 >  1. For the categories AT and IT, the 
distribution is not clear. The correct approach for 

the sensitivity estimator must therefore be 

determined individually for each transducer by 

determining 𝑒 . If 𝑒  is not known, then only the 
estimator P is justified. 

 

Figure 5: Ratio 𝑒 of the regression deviation 𝑢𝑓𝑎  to the 

uncertainty of the reference points 𝑢St  against the 
interpolation uncertainty  𝑢Int . The geometric sums 
across all reference points were considered. The data are 

categorized by field of application of the transducers. For 

each category, the distributions are indicated on either 

side. 

The combined standard measurement 

uncertainties determined in accordance with 

equations (5) to (8) by varying the data points by 

means of MCM and normalized to 2 mV/V are 

compared in the following with the corresponding 

standard measurement uncertainty obtained by 

means of the accepted procedures: 

𝑢𝑐,Std
2 =  𝑢rest

2 + 𝑢𝑓a
2 + 𝑢r

2 + 𝑢A
2 . (10) 

The assessment of the calibration data in 

scenario a for estimator L shows that the 

measurement uncertainty in accordance with the 

accepted procedures is higher than with the new 

procedure. Here, the new procedure thus leads to 

lower measurement uncertainties, in particular for 

transducers with measurement uncertainties of less 

than 1 × 10-5 mV/V (Figure 6, top). 

For scenario a and estimator P, the accepted 

procedure is compatible with the new procedure 

(Figure 6, centre). In this setting, underestimations 

remain smaller than a factor of 2. 

 

 

 
Figure 6: Compared to the combined standard 

measurement uncertainty normalized to 2 mV/V from the 

sampling of real calibration data for the estimator L and 

scenario a, the combined standard measurement 

uncertainty in accordance with accepted procedures is an 

upper estimate (top).  

For the estimator P and scenario a, the combined standard 

uncertainty is a suitable estimate (centre), and for the 

estimator P and scenario b, it is an underestimate 

(bottom). The suitable data points that are plotted here for 

control purposes for the estimator L are unremarkable. 

In the case of scenario b with estimator P, 

underestimations of up to a factor of 4 occur with 

the accepted procedures. These underestimations 

are encountered over the whole uncertainty scale 

and are therefore not correlated with the quality of 

the transducers. Thus, it is always recommendable 

to analyse the uncertainty contribution of the 

regression in more depth if 𝑒 >  1 (i.e. in the event 
of a large regression deviation and small uncertainty 

of the reference points) and if the calibration result 

between the reference points is needed. 

 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 198 

5. SIMPLIFIED ESTIMATION 

Calculating the measurement uncertainty 

contributions from the use of the regression with 

MCM requires increased computational effort, 

which may impede the use of the procedure in a 

calibration laboratory. It is therefore desirable to 

find practicable estimation procedures that work 

without MCM. 

5.1. Estimating 𝒖S𝒇a  and 𝒖𝐑  with Boundary 

Values 

To determine the standard deviations of the 

residuals 𝑆(𝑓a) and of the polynomial coefficients 
of the regression function 𝑆(𝑎i) , using MCM 
requires numerous variations of the reference points 

within the limits of their measurement uncertainty 

and the associated regressions to be calculated. 

Simplified estimations 𝑆 ′  of these standard 
deviations are obtained by calculating only selected 

variations of the reference points which take the 

boundary values of the regression functions into 

account. For this purpose, each of the reference 

points is varied with the maximum absolute values 

𝑁 of the associated uncertainties and the standard 
deviation of the residuals and of the polynomial 

coefficients, respectively, is calculated. To estimate 

the standard deviation of the residuals, the reference 

points are varied alternately with +𝑁 and −𝑁, the 
regression is performed for each of them, and the 

standard deviation of these two states is calculated: 

𝑆 ′(𝑓a) =  𝑆(𝑓a,+−N, 𝑓a,−+N)  (11) 

For the simplified estimation of 𝑆 ′(𝑎i) , the 
reference points are continuously varied with +𝑁 
and −𝑁, respectively, the regression is performed 
for each of them, and the standard deviations of the 

resulting polynomial coefficients are calculated. 

The standard uncertainty for the entire torque range 

can then be calculated based on these coefficients: 

𝑆 ′(𝑎i) =  𝑆(𝑎i,+𝑁 , 𝑎i,−𝑁 )  (12) 

Compared with the corresponding MCM 

calculations, the simplified calculations generally 

turn out to be upper estimates in real measurement 

data (Figure 7). Also in simulations, the simplified 

calculation leads to combined uncertainties which 

are slightly higher than combined measurement 

uncertainties when using the MCM (Figure 8). 

5.2. Constant Estimation 
As shown by the MCM investigations with 

synthetic signals, the random contributions 𝑆(𝑓a) 
and 𝑢R primarily depend on the uncertainty of the 
reference points 𝑁  (see Figure 3). It therefore 
suggests itself to estimate the combination of the 

two contributions 𝑢p,c  using this quantity. 

Compared to the full MCM calculation, the 

simulation provides a reliable upper estimation in a 

contribution 𝑢p,c ≈ 2𝑁  that is constant across the 

torque range. The constant estimation thus yields a 

performance similar to that of the estimation of the 

boundary values. However, its disadvantage is that 

it leads to a slight overestimation towards the ends 

of the torque range. 

 
Figure 7: Relationships between simplified calculations 

of 𝑢𝑅
′  and 𝑢𝑆𝑓a

′  and MCM calculations of 𝑢𝑅  and 𝑢S𝑓a , 

for real measurement data. Both ratios remain generally 

above 1, so the simplified calculations are upper 

estimates. 

 
Figure 8: Simulated curves of the combined 

measurement uncertainties for regression with MCM 

(𝑢p,c), with boundary value estimation (𝑢p,c'), and with 

the estimation 2𝑁  compared to 𝑢𝑓a  in accordance with 

DIN 51309. The simulation parameters correspond to the 

maxima of the values obtained by sampling real 

measurements. 

5.3. Interpolation Uncertainties of 𝒖𝒇a and 𝒖S𝒇a 

The contributions to the measurement 

uncertainty of the residuals 𝑢𝑓a and of the variance 

of the residuals 𝑢S𝑓a may only be calculated at the 

reference points. In between the reference points, 

they have to be interpolated by means of a cubic 

regression (see Section 3). The resulting uncertainty 

is yielded from the residuals of these regressions, 

for which a constant absolute value can be found as 

an upper estimation by means of its maxima in the 

torque range. 

In the sampling, 𝑢S𝑓a,Int  appears to be most of 

the time considerably smaller than 𝑢S𝑓a  (Figure 9, 

top) and does not represent a dominant contribution 

in the overall uncertainty budget. In contrast to this, 

the value of 𝑢𝑓a,Int is similar to that of 𝑢𝑓a, which is 

1,0E-06

1,0E-05

1,0E-04

1,0E-03

0 20 40 60 80 100

u
 in

 m
V

/V

torque load in %

u_fa_DIN u_p,c 2·N u_p,c'

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 199 

most of the time smaller than 1 × 10-5 for 

calibrations in accordance with estimator L and thus 

does not compromise the limits of the overall 

measurement uncertainty. Contributions for 

calibrations in accordance with estimator P range up 

to 1 × 10-4 and are therefore not negligible 

(Figure 9, bottom). 

 

 
Figure 9: Interpolation uncertainty for 𝑆(𝑓a)  as upper 
estimation in the entire torque range, calculated as the 

maximum of the residuals of a cubic fit to the curve of 

𝑆(𝑓a), plotted against the value 𝑆(𝑓a) at the first load step 
(top).  

Interpolation uncertainty for 𝑓a as the upper estimation in 
the entire measuring range, calculated as the maximum 

of the residuals of a cubic fit to the curve of 𝑢𝑓a, plotted 

against the value 𝑢𝑓a at the first load step (bottom). 

6. SUMMARY 

Estimating the contribution of the regression to 

the measurement uncertainty as described in 

accepted procedures is not always appropriate 

according to the results of MCM simulations and to 

the assessment of numerous real calibrations. If the 

systematic uncertainty of the residuals is larger than 

the standard uncertainty of the reference points and 

if the calibration results between the reference 

points are needed, then the measurement 

uncertainty obtained by means of accepted 

procedures may be underestimated by more than a 

factor of 4. 

A detailed measurement uncertainty analysis can 

take the requirements of the user of a calibration 

into account (whether the calibration values at the 

reference points only or those between the reference 

points are needed) in different cases. Whether the 

reference points or the regression must be 

considered the best estimator for the characteristic 

curve of the transducer can be calculated in specific 

approaches. 

The new procedure avoids underestimation, but 

also involves high analytical effort. Simplified 

procedures, either in the form of the estimation of 

the boundary values or of a constant upper 

estimation, can reproduce the results of the 

suggested procedure with reasonable effort. 

7. REFERENCES 

[1] DIN 51309 – Materials testing machines – 
Calibration of static torque measuring devices, 

2005. 

[2] “Statische Kalibrierung von anzeigenden 
Drehmomentschlüsseln” (in German): Guideline 

DKD-R 3-7, 2018.  
Fehler! Linkreferenz ungültig.     

[3] “Statische Kalibrierung von Kalibriereinrichtungen 
für Drehmomentschraubwerkzeuge” (in German): 

Guideline DKD-R 3-8, 2018.  
Fehler! Linkreferenz ungültig. 

[4] R. R. Cordero, P. Roth, “On two methods to 
evaluate the uncertainty of derivatives calculated 

from polynomials fitted to experimental data”, 

Metrologia, vol. 42, pp. 39-44, 2005. 

[5] A. Silva Ribeiro, J. Alves e Sousa, C. Oliveira 
Costa, M. Pimenta de Castro, “Uncertainty 

propagation in multi-stage measurements using 

linear regression analysis and Monte Carlo 

simulation” in Proc. of the XVIII IMEKO World 

Congress, Rio de Janeiro, Brazil, 17-22 September 

2006. 

[6] M. Schalles, M. Hohmann, “Einsatz von Monte-
Carlo-Methoden zur Bestimmung der Kennlinien-

unsicherheit” (in German), VDI-Berichte Nr. 2319, 

pp. 151-162, 2017. 

[7] M. F. Beug, H. Moser, A. Kölling, “Bridge 
Amplifier Linearity Investigation with a Cascaded 

Inductive Voltage Divider Setup”, in Proc. of the 

XXII IMEKO World Congress, Belfast, United 

Kingdom, 3-6 September 2018. 

 

 
 

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