WHITEPAPER – This Is How PI Does Measuring - Part 3-- Axel Grabowski, Peter Diez, Lukas Rau --This Is How PI Does Measuring - Part 3
Resolution / Noise / Minimum Incremental Motion

Physik Instrumente (PI) GmbH & Co. KG, Auf der Roemerstrasse 1, 76228 Karlsruhe, Germany
Phone +49 721 4846-0, Fax +49 721 4846-1019, Email info@pi.ws, www.pi.ws

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WHITEPAPER – This Is How PI Does Measuring - Part 3-- Axel Grabowski, Peter Diez, Lukas Rau --reflector mirrors which are used by the interferometer for its
measurements.

1 Motivation
2.2
Micro- and nanopositioning systems from PI are characterized
by their ability to execute the smallest positioning steps as well
as stand still with as little change in position as possible. This
document presents three variables which PI uses as areference to make these system properties both visible andcomparable:◼
◼
◼

Resolution
Noise
Minimum incremental motion

For an internal measurement, no external measuring
equipment is used. Instead, sensor systems that are present
within the system itself are also used for position control. Data
can be obtained from these sensor systems by recording theposition data via the controller’s data recorder in digitalsystems, for example, and reading it out from there. In analog
controller systems, the signals are tapped at the monitor
output and evaluated.

2.3

2 Performing the measurements
Before presenting the individual variables, we will briefly
describe the measuring equipment used. For further
information on measuring equipment, please refer to thewhite paper “This Is How PI Does Measuring - Part I”.

2.1

Measurements using externalmeasuring equipment

The external acquisition of measurement data is typically doneusing interferometers to determine the linear deflection or –for rotatory systems – the angular deflection.

Internal measurements

Measuring position and otherinfluencing factors

The measurements described here are typically performed ata position within the positioning range of the axis.
Measurements at other positions may result in deviating
measured values.
Furthermore, the values may differ depending on thecontroller used and the chosen settings for the control andmeasuring equipment. For example, it makes a difference
whether the position control for the system has been
optimized for low noise or for high dynamics. Ideally, theexternal measuring equipment should not influence themeasurement results, but in practice this is not feasible in all
situations and systems.
The measurements described here are typically performed ona time scale of a few seconds.

3 Resolution
3.1
Figure 1 Setup for measurements using external measuring
equipment: positioning axis with mount for a reflector mirror,
interferometer setup to measure the position
One such measurement setup is shown in Figure 1,whereby
the linear deflection of the positioning axis is determined using
an interferometer. The positioning axis is equipped with

Definition and special features ofpositioning system

In general, the resolution is defined as the smallest perceptible
difference, for example to characterize measuring equipment.
Translating this widely used term into the world of micro- andnanopositioning requires further delineation.

Physik Instrumente (PI) GmbH & Co. KG, Auf der Roemerstrasse 1, 76228 Karlsruhe, Germany
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WHITEPAPER – This Is How PI Does Measuring - Part 3-- Axel Grabowski, Peter Diez, Lukas Rau --With regard to the resolution of positioning systems, it matterswhether the data in the system is processed in digital or analog
form.
Furthermore, a positioning system consists of different
components that must be considered separately. Parts of thepositioning system to which a resolution can be assigned:◼
◼
◼
◼

Electronics to control the actuator technology
Drive element to generate the movement
Sensor and internal measuring system to read out theposition data
Overall system, which is typically stabilized via a servo
loop

Finally, it must be distinguished whether the underlying data isrecorded internally in the system or externally via an externalmeasuring device such as an interferometer; see chapter 2.

3.2

Digital or analog?

If a digital data stream is available, the resolution corresponds
to the smallest possible digital unit. This may be, for example,
one bit from a digital sensor system or from the data stream at
the digital interface of a controller, but also the smallest
possible unit of control with a digital-to-analog converter.
If a digital data stream is not available (e.g., in completelyanalog systems), the resolution always corresponds to a
standard deviation of a measured value. To allow this, ofcourse, the resolution of the measuring equipment must be
sufficiently high.

3.3

Components of a positioning system,
design resolution

In some systems, the drive element itself has no limitation withregard to triggering. This applies to piezo actuators, forexample. In other systems, however, the drive element has aspecific resolution, such as a stepper motor that has a definedresolution per step. In all cases, the resolution of the overallsystem is ultimately determined by the interaction between
the control electronics and the drive element. In a closed servoloop, the resolution of the position sensor also contributes tothe position resolution. Therefore, the resolution of the overallsystem may differ depending on whether the system isoperated in servo-loop or open-loop mode.

Based on these considerations, a theoretical resolution limit
can be calculated for certain systems, the so-called “designresolution”. For a linear axis with a stepper motor, for example,
the design resolution would be the resolution of one step
converted to a linear movement. This calculated value
indicates a theoretical lower limit which, however, is usuallydifficult to achieve in the overall system due to the interactionof the large number of individual components.

3.4

Internal or external measurement

In the case of an external measurement, for example themeasurement of a complete system in front of aninterferometer (see Figure 1), the resolution corresponds to a
standard deviation of the measured values determined withthe external measuring system. The resolution of the externalmeasuring system is typically higher than the resolution of thesystem to be measured and can be regarded as quasi
continuous.

4 Noise
4.1Definition

At PI, the noise is defined as the standard deviation
determined from the measured values of a positioning systemby means of statistical evaluation. If it is not possible tocalculate such a standard deviation in a meaningful way, 0 is
specified as the value. This may be the case, for example, if theincremental sensor of a system provides a noise-free signal, or
if a stepper motor is only controlled in single steps, thus
making the drivetrain noise-free.

4.2

Noise vs. resolution

There are a number of cases where noise and resolution canbe equated. These include, for example, all fully analog
positioning systems, since no discretization is available in thiscase.

4.3

Noise measurement

In the standard case, the noise measurement is performed bydetermining the noise power density in the relevant frequency

Physik Instrumente (PI) GmbH & Co. KG, Auf der Roemerstrasse 1, 76228 Karlsruhe, Germany
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WHITEPAPER – This Is How PI Does Measuring - Part 3-- Axel Grabowski, Peter Diez, Lukas Rau --range. In the simplified case, only a cumulative single value isdetermined over a specific frequency range.
In terms of the overall system, PI distinguishes between
internal and external noise. The internal noise is measured
using a signal generated by the system itself (e.g., analog
output of an analog controller, internal data recorder in adigital system), while the external noise is measured using an
external measuring device (e.g., an interferometer) on one axisof the positioning system. These relationships are shown in
Figure 2.

Figure 2 Illustration of the difference between the internal and external noise. The internal noise is measured via the feedback sensor
installed inside the positioning system; the external noise is measured using external measuring equipment, e.g., an interferometerPhysik Instrumente (PI) GmbH & Co. KG, Auf der Roemerstrasse 1, 76228 Karlsruhe, Germany
Phone +49 721 4846-0, Fax +49 721 4846-1019, Email info@pi.ws, www.pi.ws

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WHITEPAPER – This Is How PI Does Measuring - Part 3-- Axel Grabowski, Peter Diez, Lukas Rau --4.4Internal noiseFor purely analog systems, the internal noise is measured witha digital multimeter in AC-coupled mode at the monitor
output.
An example of internal noise measurement with a digital
controller in the complete system case is shown in Figure 3.
Here, position data was recorded using the controller's internal
data recorder. The resolution of the complete system is small
compared to its noise, so the position noise is normally
distributed and a standard deviation can be determined(Figure 3, bottom right). The data was recorded at themaximum sampling rate for a defined period (typically 1second).
Figure 3 illustrates the difference between the internal
resolution and the internal noise when the data was recordedwith a digital system. According to the definition in chapter 3,
the resolution corresponds to the smallest incremental
movement of the system that can be detected in the digital
data. Accordingly, it is evident that a meaningful histogram asshown in Figure 3 (bottom right) can only be created if theinternal resolution is correspondingly small. If the internal
resolution is larger than the signal noise, then the noise cantake the value 0. The value of the noise is therefore limited bythe value of the resolution.

Figure 3 Example of an internal position measurement of apositioning system. The obtained measurement data isnormally distributed (bottom right). The value of one standard
deviation of this position data corresponds to the value of thenoise (bottom right, orange). One digital increment
corresponds to the resolution (top).

Physik Instrumente (PI) GmbH & Co. KG, Auf der Roemerstrasse 1, 76228 Karlsruhe, Germany
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WHITEPAPER – This Is How PI Does Measuring - Part 3-- Axel Grabowski, Peter Diez, Lukas Rau --4.5

External noise

A measurement of the external noise is generally performed
with external measuring equipment, e.g., an interferometer.
For this purpose, the system is held at a specific position while
simultaneously recording position data over a defined period(typically one second). To compensate for environmentdependent drifts during the measurement time, the data may
be filtered with a high-pass filter (default: 1 Hz) or smoothedwith a polynomial up to a maximum of the 5th order. If thenoise is to be measured over longer periods, this high-passfilter must be adjusted accordingly. The standard deviation,
which corresponds to the external noise of the overall system,
is then determined from this position data.

discussed in detail here). If backlash steps have been inserted
during a measurement, this can be seen from the log. Figure 4
shows an example measurement profile with 3 backlash steps
and 6 measurement steps per direction of motion.

The external noise defined here is also called “standstill jitter”.

5 Minimum incremental motion
5.1

Definition

PI defines the minimum incremental motion as the size of thesmallest reliably detectable steps that the positioning system
is capable of executing. To determine the smallest step size forthe system, PI measures a predefined number of steps inforward and backward motion. For this purpose, measurement
profiles with staircase functions for different step sizes are
created and commanded. The obtained measurement data isthen evaluated using predefined criteria.

5.2

Figure 4 Typical measurement profile for determination of theminimum incremental motion
As standard, only the measured position information obtained
during the second half of the individual steps is taken into
account in the evaluation (marked in green in Figure 4). This is
to ensure that the step-and-settle of the positioning axis inrelation to the target position is not included in the measured
value. An attempt is also made to separate factors that are
already determined by other measurements, such as noise.
To determine the minimum incremental motion, the mean
incremental motion and its standard deviation are first
determined using the obtained measurement data. This isshown as an example in Figure 5 with 10 measuring steps andwithout backlash steps.

Measurement methods and evaluation
criteria

The procedure for measuring the incremental motion and thecriteria for the reliable detectability of a step are explained
below.
Each staircase function is defined with a fixed step width andnumber of forward and backward steps. The profile ends again
at the defined starting position after the corresponding steps
have been completed. Ten steps are selected by default.
Additional steps can be inserted which take into account any
backlash and ensure that this backlash does not influence thesubsequent measurement results (the backlash can be
determined via a separate measurement, which will not be

Physik Instrumente (PI) GmbH & Co. KG, Auf der Roemerstrasse 1, 76228 Karlsruhe, Germany
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WHITEPAPER – This Is How PI Does Measuring - Part 3-- Axel Grabowski, Peter Diez, Lukas Rau --A commanded step size is then accepted as the minimum
incremental motion if all of the following criteria are met after
a measurement run:1.

|𝛿| < 20 %2.

4 ∙ SSTD < S and3.

4 ∙ N𝑆𝑇𝐷,𝑖 < 𝑆, for each single step iAuthors

Figure 5 Evaluation of a measurement to determine theminimum incremental motion. The target course is shown inblack, the measured data in blue, and green indicates the rangeof the measured data that is used for the evaluation. The
average size per step is shown in orange as an example. Redshows that the measured size of the first step differs slightly
from the commanded step size.

Dr. Axel Grabowski, Head of Sensor Technology Development

First, the incremental motion for each individual step isdetermined for both the forward and reverse directions, with
only the area marked in green being included in thecalculation. In Figure 5, the measured incremental motion for
the first step is shown in red as an example. With n measured
steps, this results in a data set of 2n step sizes, since the steps
in the forward and reverse directions are included equally inthe evaluation. The mean value of the measured step sizes SMand the standard deviation of the measured step sizes SSTD arecalculated from these 2n step sizes.
The mean deviation in the incremental motion is thendetermined from the mean value of these step sizes:Peter Diez, Senior Industrial Engineer in the Measuring andManufacturing Technology Group𝛿 = (𝑆𝑀 − 𝑆)/𝑆Furthermore, the noise of the signal is determined. For thispurpose, the raw measured values can be filtered with a lowpass filter (4th-order Butterworth filter). On the basis of thisfiltered data, the noise is determined separately for eachindividual step for the area shown in green by calculating thestandard deviation. This results in the values NSTD,i for eachsingle step i of the 2n steps.
Finally, it must be verified whether the measurement meetsthe requirements for the minimum incremental motion.

Lukas Rau, Product Marketing Manager for Piezo Drives andSystems

Physik Instrumente (PI) GmbH & Co. KG, Auf der Roemerstrasse 1, 76228 Karlsruhe, Germany
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WHITEPAPER – This Is How PI Does Measuring - Part 3-- Axel Grabowski, Peter Diez, Lukas Rau --PI at a glance
For many years, PI (Physik Instrumente), founded in 1970, hasbeen a market and technology leader for high-precision
positioning technology and piezo applications in thesemiconductor industry, life sciences; photonics, and inindustrial automation. In close cooperation with customers
from all over the world and for 50 years now, PI's specialists(approx. 1,300) have been pushing, again and again, theboundaries of what is technically possible and developing
customized solutions from scratch. Technologies from PI
achieve reproducible accuracies in the millionth of a millimeter
range. More than 350 granted and registered patents
underline the company's claim to innovation.
PI develops, manufactures, and qualifies all core technologies
in-house, thereby constantly setting new standards for
precision positioning: Piezoceramic patch transducers andactuators, electromagnetic drives, and sensors working in thenanometer range. As the majority owner of ACS Motion
Control, PI is also a leading global manufacturer of modular
motion control systems for multi-axis drive systems anddevelops customized complete systems for industrial
applications with the highest precision and dynamics.
With six manufacturing sites and 15 sales and service offices inEurope, North America, and Asia, PI is represented whereverhigh-tech solutions are developed and manufactured.

Physik Instrumente (PI) GmbH & Co. KG, Auf der Roemerstrasse 1, 76228 Karlsruhe, Germany
Phone +49 721 4846-0, Fax +49 721 4846-1019, Email info@pi.ws, www.pi.ws

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