An improved set-up for the characterization of multi-component sensors for force and moment is presented. It aims at calibrating such sensors under continuous sinusoidal excitation. Special focus is put on the design of load masses and adapting elements to activate uniaxial force and moment components where possible. To identify the motion and acceleration of the load mass with 6 degrees of freedom, a photogrammetric measurement system is implemented in the existing set-up. Using the set-up described, different experiments are performed to analyse a commercial multi-component sensor and perform a parameter identification for its force components.

The calibration and characterization of sensors for force and moment are
typically performed in a static manner

The aforementioned calibration approaches are limited to uniaxial sensors or
calibrations of only one axis of multiaxial or multi-component sensors
(MCSs). Such MCSs have become more popular in the last few years, which has
resulted in the need for new calibration procedures

For the dynamic calibration of MCSs, very few approaches exist.

In the following sections, an improved set-up for the dynamic analysis of
MCSs is described. The force generation is based on the periodic acceleration
of a load mass connected to an MCS. The theory behind sinusoidal calibration
is explained using the example of uniaxial force calibration in
Sect.

The aim of the dynamic calibration of force and moment
sensors is the identification of the parameters needed to describe the
dynamic behaviour and the sensitivity coefficient as a function of the
excitation frequency

Dual-mass oscillator as a model for uniaxial force sensors.

From the given parameters, the dynamic sensitivity

As an alternative to identifying the model parameters

The measurement set-up for the periodic excitation of MCSs
used in this work is based on the shaker set-up located in the dynamic force
calibration laboratory at the PTB

The electrodynamic shaker is only capable of generating accelerations in one
direction. To activate force and moment components in all six directions,
different adapting elements are needed. The design of such elements is
described in Sect.

The scanning vibrometer measures accelerations in the direction of the
excitation of the shaker system. Accelerations perpendicular to the
excitation direction, as well as twisting motions of the load mass, cannot be
detected. To rectify this, a photogrammetric measurement system is used to
detect accelerations that are not visible to the scanning vibrometer. Details
of the photogrammetric system, synchronization and data analysis are
described in Sect.

In past works

Due to its design, this set-up was only capable of generating uniaxial force
components in the axial force direction

To overcome these disadvantages, a special load mass was designed in order to
move the centre of gravity of the load mass to the origin of the coordinate
system of the sensor under test

Load mass and adapting element for transverse force components.

For the generation of moment components, a beam cross is used. The cross is
designed in a way to move the centre of gravity of the moment generating mass
components to the

Beam cross for generation of moment components.

To evaluate whether signal outputs of inactive
force or moment components of the sensor are a result of signal crosstalk
from the active component or of bending or twisting moments, a complete
three-dimensional movement of the set-up needs to be identified. The scanning
vibrometer is capable of measuring the acceleration in the direction of the
shaker axis at different points of the set-up. From the acceleration
distribution, rocking modes around the

To extend the acceleration measurement provided by the laser vibrometer and
the accelerometers, a photogrammetric measurement set-up is installed. It
consists of two stereo camera systems: one camera system observing the load
mass on top of the sensor, the other one directed at the surface of the
shaker. All four cameras are triggered simultaneously using a signal
generator. One master camera is used to trigger two LED strobe lights to
illuminate the observed area. This trigger signal is recorded by the analogue
to digital converter (ADC) in the junction box of the data acquisition PC of
the scanning vibrometer for synchronization purposes. The trigger signal also
serves as the time stamp of the image acquisition. The six force and moment
signals of the sensor under test are amplified by a 6-channel bridge
amplifier

Block diagram of the photogrammetric set-up in the dynamic shaker system.

Photogrammetric set-up in the shaker environment.

Vibrometer and camera trigger signals.

Random greyscale patterns are attached to the surfaces observed by the
cameras. The images of these patterns are analysed using digital image
correlation (DIC) algorithms

The rotation and translation of the rigid body transformation are calculated
using the singular value decomposition (SVD) method presented by

Fitted sine parameters.

The cameras used for DIC are industrial CMOS cameras with frame rates of up
to 166 images per second for 8-bit greyscale images. At 12-bit colour depth,
the frame rate is reduced to a maximum of 82 images per second. This frame
rate is too low to satisfy the Nyquist–Shannon sampling theorem for
frequencies higher than 40

The known frequency distribution resulting from the vibrometer measurement is
used to define a fitting function for the displacement of the load mass

An experimental evaluation of the set-up described
is performed using the axial force components. The excitation frequency is
set to

The difference between the camera frequency and the excitation frequency
results in a beat of 0.502

For each displacement measurement, the rigid body transformation is
calculated as described in Sect.

Sine fitting is performed using a nonlinear least squares fitting function,

Frequency spectrum of the vibrometer signal. The

Displacement in

The main contributions can be seen at the excitation frequency

Sine fitting of only the excitation frequency

Axial and angular displacement in

To calculate the resulting force and moment components on the sensor, the
axial and angular acceleration of the load mass is needed. Both values can be
calculated from the fitting function of the displacement as the second
derivative with respect to time. For a sine function, the second derivative
can easily be calculated by multiplying the function by the negative square
of the angular frequency

For uniaxial force sensors, a dual-mass oscillator
as shown in Fig.

The dynamic model for a three-component force
sensor is based on a superposition of three orthogonally aligned
spring-damper systems connected to two masses

Two-dimensional representation of the dual-mass oscillator for multi-component force sensors.

A dynamic model for a three-component moment
sensor is more complex than the model for a three-component force sensor.
Basically, the model from Fig.

To determine the internal mass

Force readings and sine fitting for

The described experiment is repeated with the load mass shown in
Fig.

The spring constant is calculated according to
the method presented by

Amplitude and phase of the transfer function

Figure

Amplitude of the transfer function

Resonance frequencies, spring constants and damping coefficients for force excitation in

In

To calculate the dynamic sensitivity of the
sensor, sinusoidal excitations at different frequencies are used.
Measurements are performed at 12 different frequencies from 53.7 to
1020.3

The phase shift between the acceleration of the top mass

Figure

Sensitivity of

An extended set-up for the dynamic calibration of multi-component sensors for force and moment measurement has been described. It is based on the periodic acceleration of a sensor and an attached load mass on an electrodynamic shaker. In comparison to earlier works, the design of the load mass and adapting elements was focused on activating single force and moment components where possible. Force components can be activated using a load mass with its centre of gravity in the origin of the sensor coordinate system. For the moment components, a beam cross with a movable load mass was designed.

To identify the movement of the load mass with 6 degrees of freedom, a
photogrammetric set-up was installed in the existing laboratory set-up. Two
stereo camera systems observe the load mass and the shaker surface. From the
displacement of the observed surfaces and the time stamp of the camera
images, accelerations can be calculated. With the additional information of
the excitation frequency and reference measurements using a laser
interferometer, the displacement of the set-up can be calculated even with
camera frame rates lower than the excitation frequency. From an experimental
evaluation, a deviation of

The dynamic parameters of the sensor are identified based on a
three-dimensional mass-spring-damper system. The internal mass of the sensor
was calculated in a static manner from measurements at different rotation
angles around the

The dynamic sensitivity of the sensor was analysed for the

As a next step, the analysis performed will be extended to the moment
excitation. The previously described beam cross and moment excitation mass
are currently being manufactured. Further optimization of the horizontal
set-up to reduce pitching motions is suggested to reduce signal crosstalk.
Alternatively, a three-dimensional shaker set-up can be used in combination
with the adapting element for

Datasets used in Sects. 3.3 and 5.2 are available at the PTB's public repository

JN designed and installed the setup, performed measurements and evaluation and wrote the manuscript. RK and RT raised the research problem, supervised the research, and discussed and proofread the paper.

Author Rainer Tutsch is a member of the editorial board of the journal.

The authors gratefully acknowledge the funding of this work by the Deutsche Forschungsgemeinschaft (DFG) under grants Tu 135/24 and Ku 3367/1. The authors thank Thomas Bruns and Leonard Klaus of the PTB for assistance with the dynamic moment model, sine fitting and hardware. Edited by: Ulrich Schmid Reviewed by: two anonymous referees