Evaluation and correction of systematic effects in a simultaneous 3-axis vibration calibration system


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 388 

ACTA IMEKO 
ISSN: 2221-870X 
December 2020, Volume 9, Number 5, 388 - 393 

 
 

EVALUATION AND CORRECTION OF SYSTEMATIC EFFECTS  

IN A SIMULTANEOUS 3-AXIS VIBRATION CALIBRATION SYSTEM 
 

A. Prato1, F. Mazzoleni1, A. Schiavi1 

 
1 INRiM – National Institute of Metrological Research, Torino, Italy, a.schiavi@inrim.it  

 

Abstract: 

This paper presents a calibration method, recently 

realized at INRIM, suitable for the calibration of 3-

axis accelerometers in frequency domain. The 

procedure, allows to simultaneously evaluate the 

main and transverse sensitivities on three axes by 

means of a single-axis vibration excitation of inclined 

planes. Nevertheless, the excitation system is 

subjected to spurious motions mainly due to the 

vibrational modes of the inclined planes and to the 

horizontal motions of the shaker. In order to provide 

the proper sensitivities to the 3-axis sensors, the 

evaluation of systematic effects is experimentally 

carried out and the related correction is proposed. 

Keywords: calibration, 3-axis accelerometer, 

systematic effects 

1. INTRODUCTION 

The 3-axis accelerometers, especially low-cost 

unconventional-shaped transducers, such as MEMS 

sensors, are largely used in a wide range of advanced 

industrial, environmental, energy and medical 

applications, and in particular within extensive 

sensor and multi-sensor networks [e.g., 1 – 10]. For 

example, in the context of Industry 4.0, a huge 

number of sensors is needed for an effective 

implementation of smart factories, learning machine 

and intelligent manufacturing systems, as well as for 

traditional application such as early failure detection 

and predictive maintenance; low-power devices and 

battery-operated systems are practical and useful in 

IoT applications, such as for smart cities, for accurate 

navigation/positioning systems and in environmental 

monitoring and survey; moreover, accurate 

measurements are of paramount importance in 

medical applications, in remote surgery and remote 

diagnoses. The possibility to have many accurate, 

low-power consuming and low-cost sensors present 

undoubted advantages, in terms of costs reduction 

and energy saving, in the control processes, 

monitoring or measurements and being flexible in 

providing enhanced data collection, automation and 

operation. By way of example, the calibration of 

digital MEMS sensors, with the associated 

uncertainty budget, allows to ensure traceability and 

measurement accuracy of nodes in sensor networks, 

as well as in other innovative implementations. 

Moreover, the evolving improvement of the technical 

performance and the reliability of MEMS sensors are 

emerging quality attributes of interest for 

manufacturers, costumers and end-users.  

However, in the particular case of digital MEMS 

accelerometers, the sensitivity is generally provided 

by manufacturer without traceable calibration 

methods and is obtained in static conditions, whereas 

dynamic response, as a function of frequency, is often 

barely known or completely disregarded. Given this 

condition, a simultaneous 3-axis vibration calibration 

system with a single axis excitation to be exploitable 

by manufacturers is proposed. This paper deals with 

the evaluation of systematic effects of this system. 

2. DESCRIPTION OF THE WORK  

Traceable calibration methods for digital sensors, 

and smart sensors, in metrological terms [11 - 13], 

including sensitivity parameter and an appropriate 

uncertainty evaluation, are necessary in order to 

consider low-cost and low-power consuming 

accelerometers as actual measurement devices in 

frequency domain [14]. 

The calibration system, developed at INRiM, 

allows to simultaneous evaluate the main and the 

transverse sensitivity, in frequency domain, of 3-axis 

accelerometers, by means of a single-axis vibration 

excitation, by using inclined planes rigidly fixed to 

the vibrating table [14, 15]. The calibration procedure 

is based on the comparison to a reference transducer 

(in analogy to ISO 16063-21[16]). A preliminary 

version of the system was previously investigated for 

the characterization of analog MEMS accelerometers 

performance in operative conditions [17 – 23]. 

Measurements are performed at a nearly constant 

amplitude of 10 m∙s-2, from 5 Hz up to 3 kHz. The 

mechanical calibration system, composed of the 

shaker and the inclined planes (with tilt angles of 15°, 

35°, 55° and 75°), is characterized in order to take 

into account systematic effects. Similar procedure is 

also applied to the shaker at 0° and 90°. 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 389 

Measurements of systematic effects is carried out by 

means of a Laser-Doppler vibrometer. The detailed 

uncertainty budged is evaluated according to GUM 

[24]. 

3. CALIBRATION SET-UP  

The calibration set-up here proposed, consisting 

of a single-axis vibrating table on which aluminum 

inclined planes are screwed, allows to generate a 

projection of the reference acceleration along three 

axes simultaneously. A single vertical sinusoidal 

acceleration at nearly-constant amplitude acts as 

reference acceleration aref along the vertical z’-axis of 

the system. In this way, accelerations of proportional 

amplitudes, are simultaneously generated on the 

inclined surface plane, along the three axes.  

In figure 1, the geometrical principle of the 

proposed method on which the 3-axis accelerometer 

is fixed during calibration, is schematically depicted.  

 
Figure 1: Inclined plane – scheme  

From simple trigonometrical laws, the reference 

accelerations detected by the sensor in calibration, 

along its three sensitive axes, are expected to be:  

ax,theor =  |aref sin(𝛼) cos(𝜔)|  (1) 

ay,theor =  |aref sin(𝛼) sin(𝜔)|  (2) 

az,theor = |aref  cos(𝛼)|  (3) 

 

where,  is the tilt angle, ω is the angle of rotation, 
aref is the Root Mean Square (RMS) reference 

acceleration along the vertical z’-axis of the system, 

and ax,theor, ay,theor, az,theor are the RMS reference 

accelerations spread along x-, y- and z-axis of the 

MEMS accelerometer in calibration.  

In the experimental set-up, the inclined plane is 

screwed on the vertical vibrating table (PCB 

Precision Air Bearing Calibration Shaker), and the 3-

axis accelerometer is fixed to the inclined plane and 

located along the vertical axis of excitation. The 

experimental configuration is shown in figure 2. 

The acceleration along vertical z’-axis aref is 

measured by a single axis reference transducer (PCB 

model 080A199/482A23), calibrated according to 

ISO 16063-11:1999 [25], against INRiM primary 

standard, located within the stroke of the shaker and 

is acquired by an acquisition board NI 4431 

(sampling rate of 50 kHz) integrated in the PC and 

processed through LabVIEW software to provide 

the RMS reference value in m s-2. 

The digital MEMS output is acquired by an 

external microcontroller at a maximum sampling rate 

of 6.660 kHz and saved as binary files. 

 

 
Figure 2: The calibration set-up: the MEMS fixed to the 

inclined plane on the vibrating table.   

4. EVALUATION OF SYSTEMATIC 
EFFECTS AND CORRECTION 

As described in the previous Section, the reference 

accelerations along MEMS accelerometer 

sensitivities axes are given by equations (1)-(3), by 

using trigonometrical laws. However, in dynamic 

conditions, systematic effects caused by spurious 

oscillating components along the three axes of the 

reference system (x’-, y’- and z’) at MEMS position 

need to be taken into account. These spurious 

components affect the actual reference accelerations 

aref splitted on the three sensitivity axes of the 

MEMS. In Figure 3, a schematic representation of the 

occurring phenomenon is shown. 

 

 

Figure 3: Representation of the spurious oscillating 

components combination along the three axes of the 

MEMS during the calibration.   

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Such components are mainly due to the vibrational 

modes of the inclined aluminum planes and due to 

small but not negligible horizontal motions of the 

shaker. Each spuroius component along reference 

system x’-, y’- and z’-axis, has to be decomposed 

along the 3-axis accelerometer x-, y- and z-axis and 

summed to the reference accelerations ax,theor, ay,theor, 

az,theor according to wave interference laws. The 

actual decomposition of the spurious components, 

acting along the three axes of the reference system, is 

schematically depicted in Figure 4. 

 

 

Figure 4: Representation of the spurious oscillating 

component decomposition, along the three axis of the 

reference system, at a given frequency. 

 

By way of example, considering the general case 

of four overlapping waves, 𝐸1 = 𝐸1,0𝑒
𝑖(2𝜋𝑓𝑡) , 𝐸2 =

𝐸2,0𝑒
𝑖(2𝜋𝑓𝑡+𝜑2) , 𝐸3 = 𝐸3,0𝑒

𝑖(2𝜋𝑓𝑡+𝜑3) , 𝐸4 =

𝐸4,0𝑒
𝑖(2𝜋𝑓𝑡+𝜑4) , oscillating at a same frequency f, 

with different amplitudes and phase differences  with 

respect to the reference signal E1, along a particular 

direction, their interference can be opportunely 

defined according to equation (4), where E2, E3 and 

E4 are the amplitude- and phase-dependant spurious 

components along x-, y- and z-axis of the MEMS.   

 

𝐸𝑡𝑜𝑡 = |𝐸1,0 + 𝐸2,0𝑒
𝑖𝜑2 + 𝐸3,0𝑒

𝑖𝜑3 + 𝐸4,0𝑒
𝑖𝜑4 |   (4) 

 

In this way, it is possible to correct reference 

theoretical accelerations along MEMS axes in 

equations (1)-(3), into ax, ay, az of equations (8)-(10), 

where ax’,syst, ay’,syst, az’,syst and φx’,syst, φy’,syst, φz’,syst are, 

respectively, the amplitudes and the phase 

differences as shown in equations (5)-(7), with 

respect to the reference signal aref, of the spurious 

components along x’-, y’- and z’-axis. As it will be 

shown in Section 5, the amplitude of spurious 

components vary as a function of frequency between 

0.1% and 10% of the reference acceleration aref. 

Experimental evaluation of systematic effects due to 

spurious components is carried out by means of a 

laser-Doppler velocimeter (Polytec OFV 505).  

Amplitude and phase measurements along the x’-, 

y’- and z’-axis of the reference system are evaluated 

for each inclined plane and for all frequencies, at 

reference vertical amplitude of 10 m s-2. Laser signal, 

during measurements of spurious components 

amplitude, is acquired by a NI 4431 board (sampling 

rate of 50 kHz) integrated into the PC, while 

measurement of phase differences between reference 

acceleration and spurious components are measured 

by means of a dynamic signal analyzer (KEYSIGHT 

35670A).  

Since the digital MEMS accelerometer is too small 

to be used as a reflective surface, the beam spot of the 

laser directly hits a small aluminum triangular-based 

parallelepiped located at MEMS position and fixed to 

the different inclined planes, as shown in figure 5. 

The volume of the triangular-based parallelepiped is 

around 0.5 cm3, which is 0.6% of the total volume of 

the inclined plane, i.e., negligible with respect to the 

total mass. 

The values of the measured acceleration 

amplitudes ax’, ay’ and az’ along x’-, y’- and z’-axis are 

related to the actual systematic effects acting on the 

axes of the reference system, and are expressed, as a 

function of frequency and experimental phase-shift, 

by the following equations: 

  

𝑎𝑥′ = |𝑎𝑥′,𝑡ℎ𝑒𝑜𝑟 + 𝑎𝑥′,𝑠𝑦𝑠𝑡 ∙ 𝑒
𝑖(2𝜋𝑓𝑡+𝜑𝑥′,𝑠𝑦𝑠𝑡)| (4) 

 

𝑎𝑦′ = |𝑎𝑦′,𝑡ℎ𝑒𝑜𝑟 + 𝑎𝑦′,𝑠𝑦𝑠𝑡 ∙ 𝑒
𝑖(2𝜋𝑓𝑡+𝜑𝑦′,𝑠𝑦𝑠𝑡)|  (5) 

 

𝑎𝑧′ = |𝑎𝑧 ′,𝑡ℎ𝑒𝑜𝑟 + 𝑎𝑧 ′,𝑠𝑦𝑠𝑡 ∙ 𝑒
𝑖(2𝜋𝑓𝑡+𝜑𝑧′,𝑠𝑦𝑠𝑡)|  (6) 

 

where ax’,theor and ay’,theor are 0 (i.e., no acceleration is 

generated along the horizontal system plane x’-y’) 

and the vertical component az’,theor  aref.  

In Figure 5 the experimental method used to 

quantify the spurious components from accelerations 

ax’, ay’ and az’, and the related phase-shift φx’, φy’ and 

φz’, with respect to the reference acceleration aref, 

acting along the vertical axis z’, is shown.   

 

 

ax = √
(ax,theor + |a𝑥′,𝑠𝑦𝑠𝑡  cos(𝛼) cos(𝜔)| cos 𝜑𝑥′,𝑠𝑦𝑠𝑡 + |a𝑦′,𝑠𝑦𝑠𝑡 sin(𝜔)| cos 𝜑𝑦′,𝑠𝑦𝑠𝑡 + |a𝑧′,𝑠𝑦𝑠𝑡  sin(𝛼) cos(𝜔)| cos 𝜑𝑧′,𝑠𝑦𝑠𝑡 )

2
+

+(|a𝑥′,𝑠𝑦𝑠𝑡  cos(𝛼) cos(𝜔)| sin 𝜑𝑥′,𝑠𝑦𝑠𝑡 + |a𝑦′,𝑠𝑦𝑠𝑡 sin(𝜔)| sin 𝜑𝑦′,𝑠𝑦𝑠𝑡 + |a𝑧′,𝑠𝑦𝑠𝑡  sin(𝛼) cos(𝜔)| sin 𝜑𝑧′,𝑠𝑦𝑠𝑡 )
2

 (8) 

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ay =  √
(ay,theor + |a𝑥′,𝑠𝑦𝑠𝑡  cos(𝛼) sin(𝜔)| cos 𝜑𝑥′,𝑠𝑦𝑠𝑡 + |a𝑦′,𝑠𝑦𝑠𝑡 cos(𝜔)| cos 𝜑𝑦′,𝑠𝑦𝑠𝑡 + |a𝑧′,𝑠𝑦𝑠𝑡  sin(𝛼) sin(𝜔)| cos 𝜑𝑧′,𝑠𝑦𝑠𝑡 )

2

+(|a𝑥′,𝑠𝑦𝑠𝑡  cos(𝛼) sin(𝜔)| sin 𝜑𝑥′,𝑠𝑦𝑠𝑡 + |a𝑦′,𝑠𝑦𝑠𝑡 cos(𝜔)| sin 𝜑𝑦,𝑠𝑦𝑠𝑡′ + |a𝑧′,𝑠𝑦𝑠𝑡  sin(𝛼) sin(𝜔)| sin 𝜑𝑧′,𝑠𝑦𝑠𝑡 )
2

 (9) 

 

az =  √
(az,theor + |a𝑥′,𝑠𝑦𝑠𝑡  sin(𝛼)| cos 𝜑𝑥′,𝑠𝑦𝑠𝑡 + |a𝑧′,𝑠𝑦𝑠𝑡  cos(𝛼)| cos 𝜑𝑧′,𝑠𝑦𝑠𝑡 )

2
+

+(|a𝑥′,𝑠𝑦𝑠𝑡  sin(𝛼)| sin 𝜑𝑥′,𝑠𝑦𝑠𝑡 + |a𝑧′,𝑠𝑦𝑠𝑡  cos(𝛼)| sin 𝜑𝑧′,𝑠𝑦𝑠𝑡 )
2

 

(10) 

 

 

 

 
 

Figure 5: The laser beam hitting the aluminum triangular-

based parallelepiped located at the MEMS position.  

5. EXPERIMENTAL RESULTS 

In the graphs of Figures 6 – 9, the values of the 

amplitudes of ax’, ay’ and az’, are normalized with 

respect to the reference acceleration aref amplitude. 

Mesurement are performed from 5 Hz and 3 kHz. 

 

 
Figure 6: Normalized accelerations along x’-, y’- and z’-

axis at 15° of tilt angle.  

 

 
Figure 7: Normalized accelerations along x’-, y’- and z’-

axis at 35° of tilt angle.  

 

 
Figure 8: Normalized accelerations along x’-, y’- and z’-

axis at 55° of tilt angle.  

 

 
Figure 9: Normalized accelerations along x’-, y’- and z’-

axis at 75° of tilt angle.  

The experimental values of the spurious 

components amplitude of accelerations ax’, ay’ and az’, 

along the axes of the system, are combined in order 

to calculate the values of systematic effects due to the 

acceleration amplitudes ax’,syst, ay’,syst and az’,syst, along 

the MEMS sensitivity axes.  

In the graphs of the Figures 10 – 13, the 

experimental values of phase-shift φx’, φy’ and φz’, 

with respect to the reference acceleration aref, acting 

along the vertical axis z’, are shown. In this case the 

values of phase-shift allow to evaluate the phase 

differences, with respect to the reference signal aref, 

in terms of φx’,syst, φy’,syst and φz’,syst. Since φz’,syst is close 

to 0° in every configuration, it is not shown in the 

graphs. 

 

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Figure 10: Phase-shifts on the x’-y’ horizontal plane, with 

respect to the vertical axis z’, at 15° of tilt angle.  

 

 

Figure 11: Phase-shifts on the x’-y’ horizontal plane, with 

respect to the vertical axis z’, at 35° of tilt angle.  

 

 

Figure 12: Phase-shifts on the x’-y’ horizontal plane, with 

respect to the vertical axis z’, at 55° of tilt angle.  

 

 

Figure 13: Phase-shifts on the x’-y’ horizontal plane, with 

respect to the vertical axis z’, at 75° of tilt angle.  

Once quantified the acceleration amplitudes ax’, 

ay’ and az’, and the values of the phase-shift φx’, φy’ 

and φz’, the reference theoretical accelerations along 

MEMS axes in equations (1)-(3), can be opportunely 

expressed into ax, ay, az of equations (8)-(10), by 

taking into account the systematic effects due to the 

vibrational modes of the inclined planes and to 

horizontal motions of the shaker. In particular, it can 

be observed an increasing of acceleration amplitudes, 

as a function of frequency, along the horizontal axes 

x’, y’ and also along vertical axis z’, this last mainly 

due to resonant modes. Moreover, the lateral motions 

of the shaker, occurring around 80 Hz, are 

presumably the cause of the large phase-shifts 

observed at low frequencies, as depicted in Figure 10 

– 13; these motions are independent from the 

resonant modes of the inclined planes, but they are 

affecting in any case, the whole behavior of the 

inclined plane, in terms of amplitude and phase 

differences. On the other hand, the analysis of the 

systematic effects, is an aggregation of both shaker 

spurious motions and resonant modes.  

Standard uncertainties associated to the amplitude 

of the spurious components, u(ax’,syst), u(ay’,syst), 

u(az’,syst), are considered as type B uncertainty 

contributions with an average error of 0.0025 m s-2, 

from three repeated measurements, and a uniform 

rectangular distribution. Standard uncertainties 

associated to the phase difference due to the spurious 

components, u(φx’,syst), u(φy’,syst), u(φz’,syst), are 

considered as type A uncertainty contributions with a 

maximum standard deviation of 2° from five repeated 

measurements, as shown in [14]. 

This correction allows to univocally define the 

actual projection of the reference acceleration aref on 

the three axes, thus the “standard” calibration can be 

achieved, by comparison to the reference transducer 

within the stroke of the shaker and it can be finally 

related to the primary Standard, as declared.  

6. SUMMARY 

In this paper is presented a technical insight on the 

calibration system, recently realized at INRIM, 

suitable for the calibration of 3-axis accelerometers 

in frequency domain. The procedure, allows to 

simultaneously evaluate the main and transverse 

sensitivities on three axes by means of a single-axis 

vibration excitation of inclined planes. The 

mechanical calibration system, composed of the 

shaker and the inclined planes, is characterized in 

order to take into account systematic effects occuring 

during dynamical excitation. The evaluation of 

systematic effects, due to the vibrational modes of the 

inclined aluminum planes and due to small but not 

negligible horizontal motions of the shaker, are 

carried out from 5 Hz up to 3 kHz at an amplitude of 

10 m s-2. The amplitudes of the acceleration spurious 

components ax’, ay’ and az’, and the values of the 

phase-shift φx’, φy’ and φz’, with respect to the 

reference acceleration aref, acting along the vertical 

axis z’, are accurately measured by means of a Laser-

Doppler vibrometer. This correction allows to 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 393 

univocally define the actual projection of the 

reference acceleration aref on the three axes, thus the 

“standard” calibration can be achieved and related to 

the primary Standard. 

7. NOTE 

This work has to be considered as an addendum to 

the paper: “Prato, A., Mazzoleni, F., & Schiavi, A. 

(2020). Traceability of digital 3-axis MEMS 

accelerometer: simultaneous determination of main 

and transverse sensitivities in the frequency 

domain. Metrologia, 57(3), 035013” [14], which 

shows the detailed extension of the determination of 

the systematic effects of the calibration system.  

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[23] A. Prato, A. Schiavi, F. Mazzoleni, A. Touré, G. Genta, M. 

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measurements: a case study on calibration of digital 3-axis 

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[24] JCGM 100 2008 Evaluation of Measurement Data — 

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(GUM), Joint Committee for Guides in Metrology, Sèvres, 

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[25] ISO 16063-11 1999 Methods for the calibration of 

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