Technical Information

Rotary Encoders on Servo Motors for Linear Axes:
Influence on the Surface Quality of Milled Workpieces
In many areas of the machine tool industry, particularly in mold making, workpieces with tight tolerances and flawless surfaces are thegoal. To achieve this aim in the shortest possible time is an extraordinary challenge. Mold making demands large cutting volumes during
roughing, followed by high surface quality in the subsequent finishing process (Figure 1). Only if the quality of the milled surfaces is highis it possible to avoid subsequent cost due to manual polishing for example.
In addition to the machine mechanics, the control and the tools used, the applied axis measuring technology is decisive for a high-value
milling result. Particularly in a waveIength range between 0.5 mm and 5 mm, periodically repeating surface defects on the workpiecesurface are visible to the human eye and therefore bothersome—especially in mold making. Interpolation errors of the axis encoders,among other causes, can be responsible for these errors. Besides high-end linear encoders, rotary encoders with especially smallinterpolation error in the linear feed axes are crucial in order to attain the required high workpiece surface quality.

December 2015

Figure 1: Mold making workpiece during millingCauses of visible surface defects
Interpolation error and its effects

Surface defects stand out
The human eye is very sensitive to structuralchanges or even the smallest defects on asurface. A single pixel error on the monitor,
for example, with a display comprising up
to five million pixels, is immediately visible.
The human eye is even more sensitive toperiodic surface faults.
Particularly in mold making, periodic surface
defects are disturbing and require expensive
rework. Surface defects are visible on partsurfaces as shadows or fluctuations incontrast. These can be attributed to various
causes, e.g.:• vibrations in the machine that disturb themachining process (see [1]),• interpolation error of the axis encoders
used (see [2])Signal deviations cause interpolation
error
The axis encoders generally built into
today’s machine tools establish the position
value of a feed axis principally from twocomponents:• an absolute part, which after switch-on
ascertains the feed-axis position withlimited accuracy,• an incremental part, which provides twosine waves (A and B) with a 90° phaseshift
To attain the required resolution, the periodic
analog signals A and B are subdivided further
with interpolation methods. The interpolation
processes operate without error as long as
the two sinusoidal output signals are ideal
and are electrically phase-shifted to eachother by exactly 90°. Deviations generateerrors that repeat themselves with eachperiod of the scanning signals (signal period).
This is therefore referred to as interpolation
error (Figure 2).

Position error

Accuracy of the interpolation

Interpolation
error

Signal levelSignal period
360° elec.

PositionFigure 2: Position errors within one signal period (interpolation error)

2

There are many factors critical for the sizeof the interpolation error, such as:• the size of the signal period,• the homogeneity and period definition of
the graduation,• the quality of scanning filter structures,• the characteristics of the sensors, and• the stability and dynamics of further
processing of the analog signals
Effects of interpolation error
The mold making industry is demanding
workpieces with increasingly complex
geometries. In five-axis machining, allcombinations of axis movements arecommon. If an inclined or curved
machining surface is manufactured throughthe interpolation of multiple NC axes, theinterpolation error can be seen directly on
the workpiece.
This becomes particularly apparent when
an inclined surface with a small angle ismachined. The interpolation error of anencoder (e.g. in Z direction) can be madevisible by projection on the inclined
workpiece surface (Figure 3). Because of
the inclination, an n-fold stretching of thesignal period appears in the tool path.
While the axis in Z direction moves by only
one signal period, the X axis moves n times
more. A wave appears on the inclined
workpiece surface with a wavelength that
corresponds to the n-fold signal period of
the Z-axis encoder. A wavelength of 0.5 mm
to 5 mm is especially easy for the human
eye to detect.

Ideal path
Milled path

Znom+– ZactPositioncontroller

Velocity controller

+–ddt

SP =
A =n =λ =λ = n · SP

φact

Signal period
Amplitude
1/inclination = b/h
WavelengthhAb

Figure 3: Illustration of interpolation error on a workpiece incline

Position and velocity measurement infeed axes
Figure 4 shows the basic design of a
servo-controlled feed axis with servo
motor, recirculating ball screw, spindle axis
structure and axis measuring technology.
The position of the spindle axis is measured
with a linear encoder and returned to theposition control loop of the axis as an
actual position value. A rotary encoder as asecond measured value encoder is locatedon the servo motor of the feed axis formeasuring the motor shaft angle. The
signal from this rotary encoder is used todetermine the actual speed.
Source 2 shows results from machining
tests in which the interpolation errors oflinear encoders in magnitudes of ±200 nm
become visible as surface defects on theworkpiece. The HEIDENHAIN LC 100 linear
encoder used for the comparison, with aninterpolation error of significantly less than±100 nm, caused no visible surface waves.

Figure 4: Schematic representation of a feed axis

The machine tool used in the tests outlined
below uses HEIDENHAIN type LC 483
linear encoders in all linear axes. These also
have an interpolation error of distinctly lessthan ±100 nm, indicating that no visible
surface waves may be expected. This
principle is the basis for the following
observation and description of the influenceof rotary encoder interpolation error on thesurface defects of milled workpiece planes.
The interpolation error of a rotary encodercorresponds to a short range angle error
within a signal period. To achieve thedesired levels of accuracy for workpiecegeometries and surfaces, machine tools
with high geometric accuracy are normallyused. With a mechanically stiff andbacklash-free feed axis, the interpolation
error of the rotary encoder is converted via
the kinematics of the ball screw into alinear error. This means that from a specificrotary encoder interpolation error size andunder further conditions as specified
below, surface defects on the workpiece
are to be expected.

An increasing travel speed of the encoder
head over the scale with a periodicallyrecurring error will lead to an increase infrequency. Therefore, the interpolation error
frequency depends on the feed rate.
Because of the differing cutoff frequencies(bandwidth) of the position control loop andspeed control loop, the interpolation errorof the encoder becomes noticeable tovarying degrees on the tool center point
depending on the feed rate. With low feed
rates (low interpolation error frequency),
the interpolation error of the rotary encoder
is corrected by the position controller. If thefrequency of the rotary encoder interpolation
error exceeds the cutoff frequency of theposition control loop (higher feed rate), nomore adjustment takes place. Increased
occurrences of this interpolation error at
the tool center point can then be expected.

3Factors influencing the interpolation error
Feed rate and bandwidth of the positionand speed controller

Figure 5: Measuring test setup with KGM 181 grid encoder

Test setup with grid encoder
In order to be able to separate the effectsof the machining process and the movementof the tool center point, a HEIDENHAIN
KGM 181 grid encoder is used to determine
the contouring deviations prior to machining.
The grid encoder permits non-contacting
determination of the contouring deviations
between the tool center point and machinetable in the plane of the two moving feedaxes. Figure 5 shows the test setup. The
tests were conducted with a high-end dieand mold machine.
The measuring plane is formed via theX axis and Z axis of the machine. The servo
motor of the Z axis is coupled directly tothe recirculating ball screw spindle, so thatno further transmission lines in the drive
train exist apart from the ball screw. The pitchof the ball screw is P = 20 mm.

4The Z axis of the machine is relevant forthe observations. The rotary encoder of theservo motor is varied on the Z axis for thetests. Two high-end, mounting-compatiblerotary encoders are used, each with 2048
graduations on the circular scale. Rotary
encoder 1 has an interpolation error 3 times
higher than that of rotary encoder 2.
As described above, the interpolation errorof the rotary encoder are converted via thecirculating ball screw drive into linear error.
The signal period of the linear movement iscalculated via the ball screw pitch and theline count of the encoder:
SPRElin = P = 20 mm  9.8 µm (Equation 1)
LC 2048
For the workpieces to be machined, atraverse path of b = 60 mm and h = 0.4 mm
is defined for the grid measurement. These
parameters result from the equation
specified in Figure 3 on page 3 for calculating
the wavelength under consideration of
SPRElin and the sensitivity of the humaneye for periodic form errors with λ = 0.5 mm
to 5 mm.

Thus, with excessive interpolation error, asignal period of 9.8 µm causes a contouring
deviation of λ ≈ 1.5 mm.
Measuring results confirm interrelations
Figure 6 shows the measuring results using
rotary encoder 1 and two different contour
speeds. The lower speed (500 mm/min) is
generally for machining steel workpieces
with small cutter diameters (e.g. Ø 6 mm),and the higher speed of 4000 mm/min formachining aluminum.
The motion error traces show that at afeed rate of 4000 mm/min, wave-shaped
contouring deviations occur in the Z direction.
By zooming the view, a wavelength of about
1.5 mm becomes visible. This corresponds
to the expected wavelength based on theinterpolation error of the rotary encoder on
the servo motor of the Z axis.

Figure 6: Measuring
results for rotary
encoder 1 with a 3-foldhigher interpolation
error

0.3

500 mm/min

1

0.5Error in Z [µm]01.5 mm0.2

4000 mm/min

0.1
0-0.5

-0.1-101020304050

60-0.250

5152535455

505152535455

X [mm] 

Figure 7: Measuring
results with rotary
encoder 20.3500 mm/min

10.2

4000 mm/min

Error in Z [µm]

0.50.100-0.5-0.1-10102030405060-0.2

X [mm] As described above, the interpolation error
frequency depends on the feed rate.
Whether the interpolation error of the rotary
encoder occurs as a contouring deviationon the tool center point or is adjusted
continues to depend on the bandwidth ofthe position and speed controller. Equation 2(calculation of Z axis speed) and Equation 3
are used to determine the frequency of theinterpolation error in the rotary encoder
signal and the frequency with which it isreturned to the speed control loop of theZ axis:¹ ¹2vB=21+ bhmm= 26.7minvz =fSP =()2( )[ ]mm
40002 min60
1+0.4(Equation 2)vz
26.7 mm/min=
SPRElin
9.8 µm 45 HzAt a feed rate of 4000 mm/min with theexisting inclined plane, the interpolation
error is manifested with a frequency ofapprox. 45 Hz in the rotary encoder signal.
The feedback and differentiation of therotary encoder signal (Figure 4, page 3)
does not influence the frequency of theencoder interpolation error and therefore
also exists in the actual speed signal.
The speed controller with a bandwidth ofgreater than 80 Hz follows the oscillation of
the rotary encoder interpolation error. The
feed movement implemented on the Z axis
structure now has oscillation corresponding
to the frequency of the interpolation error.
In the present case, the rotary encoderinterpolation error frequency of 45 Hz
exceeds the cutoff frequency of the position
control loop (fl_PC < 40 Hz). If the amplitudeof the interpolation error is now high enough,a corresponding contouring deviation on
the tool center point is noticed (cf. themeasuring results with rotary encoder 1).

Figure 7 shows the measuring results forrotary encoder 2 mounted on the servo
motor of the Z axis with otherwise
identical conditions. Due to the 3-fold lower
interpolation error with rotary encoder 2,
superimposed waves no longer occur inthe curves of the low or high feed rates.
The interpolation error of the rotary encoder
on the servo motor of the Z axis is now sosmall that no detectable periodic contouring
deviations are caused between the tool
center point and the machine table. The
following milled workpieces show that thisalso applies for the relative movements
between the tool and workpiece, i.e. themachined surface.

(Equation 3)

5

Low interpolation error avoids contouring deviations
Surface quality starts with the rotary encoder

Workpiece machining
To show the influence of the interpolation
error on the machined surface quality, thepreviously specified inclined plane withb = 60 mm and h = 0.4 mm is machined
on a workpiece with use of rotary encoder1 or rotary encoder 2. The workpiece
undergoes multipass milling in the Y direction
with a downward cut to produce this incline.
The line spacing in the Y direction andfinishing allowance each consist of 0.1 mm.
Figure 8 shows the machining situation in
the machine tool with the set technology
parameters.
Figure 9 shows the milled surfaces. The
surface with use of rotary encoder 1 isshown in the top image. The wave forms
with λ ≈ 1.5 mm caused by the interpolation
error of this encoder create an objectionableshadowing effect despite the low amplitudeof only around 0.1 µm. This means that thecontouring deviations determined previouslywith the grid encoder measurements leadto an obviously inferior surface quality on
the workpiece.

Technological parameters
Tool diameter in mm

6

Number of teeth

2

Spindle speed in rpm

18 000Feed rate in mm/min

4000

Cutting material

PCD

Workpiece material

Aluminum

Figure 8: Machining situation and technological parameters

6

The change to rotary encoder 2 on theservo motor of the Z axis with a 3-foldlower interpolation error—with otherwise
identical conditions—results in a significant
improvement of surface quality (Figure 9,below). Thanks to the lower interpolation
error of the rotary encoder, the previously
existing regular surface waves are nolonger visible. The HEIDENHAIN ERN 1387
rotary encoder has an even lower
interpolation error than rotary encoder 2.
The unit is used among other applications
for the speed control of electrical drives.
Conclusion
Axis encoders with low interpolation error
are needed to produce high quality sur­faces. Because the interpolation errors
are manifested as contouring deviations
between the tool and workpiece withdifferent feed rates, this not only concerns
the linear encoders of the feed axes but
also the encoders on the servo motors. To
avoid the contouring deviations outlined
above, high quality axis encoders with low
interpolation error are needed throughout
in the feed axes of a machine tool.

Figure 9: M
 illed surfaces: Rotary encoder 1 (above) with 3-fold greaterinterpolation error than rotary encoder 2 (below)

7

Rotary encoders with very small interpolation error

ERN 1387
Incremental rotary encoders with integral
bearing and mounted stator couplingIncremental
ERN 13871)

Interface 1 VPPLine count/System accuracy

2048/±20“

Reference markOne

Output frequency
Cutoff frequency –3 dB– 210 kHz

Commutation signals

 1 VPP1)

WidthZ1 track2)

Voltage supply

DC 5 V ±0.25 VShaft

Tapered shaft  9.25 mm; taper 1:10

Mechanically permissible
speed n 15 000 rpm

Starting torque 0.01 Nm (at 20 °C)

Moment of inertia of rotor

2.6 · 10–6 kgm2

Natural frequency of thestator coupling 1800 Hz

Permissible axial motion ofmeasured shaft±0.5 mm

Vibration 55 Hz to 2000 HzShock 6 ms

 300 m/s2 3) (EN 60 068-2-6)
 2000 m/s2 (EN 60 068-2-27)

Max. operating temperature 120 °C
Protection EN 60 5291)

IP40 when mounted

Signal amplitude:
0.8 VPP to 1.2 VPP
Asymmetry:
0.05
Amplitude ratio:
0.9 to 1.1
Phase angle:
90° ±5° elec.
Signal-to-noise ratio E, F: 100 mV2)
One sine and one cosine signal per revolution; see the brochure Interfaces ofHEIDENHAIN Encoders
3)
As per standard for room temperature;applies for operating temperature Up to 100 °C:  300 m/s2Up to 120 °C:  150 m/s2           


1157576-20 · 4 · 12/2015 · F&W · Printed in Germany

Restricted tolerances

More information:[1] Technical Information: DynamicPrecision[2] Technical Information: PerfectSurfaces with HEIDENHAIN
Encoders