; mapping geometric and thermal errors in turning reciprocating screw
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# mapping geometric and thermal errors in turning reciprocating screw

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```									         Chap 6
mapping geometric and
thermal errors in a turning
center
6.1 introduction
● Ph.D. thesis of a researcher in NIST(미국표준
연구소) ('85 Purdue)
- First in error mapping (a function of position
and temperature)
- Error compensation algorithm based on S/W
⇒ compensating geometric and thermal errors

● “A general methodology for machine tool
accuracy enhancement” by Dr. Alkan Donmez of
the NIST
6.1 introduction
● machine tool:
2 axis turning center ("Superslant“ lathe)
6.1 introduction
● slanted bed  to remove chips easily

● 8 tools mounted on a turret
- upper turret: outside surface cutting
- lower turret: end surface cutting
6.1 introduction
6.1 introduction
● spindle (connected to bed) - revolute joint
● carriage (connected to bed) - sliding joint
● cross-slide (connected to carriage) - sliding
joint
● turret (connected to cross-slide) - revolute
joint
● cutting tool (connected to turret) - fixed
● workpiece (connected to spindle) - fixed
6.3 machine tool metrology
● error at toolpoint is represented by
combination of errors in machine elements

● all errors measured or predicted (for all
positions and temperature) ⇒ error
compensation at the tool tip is possible

● various errors as functions of position and
temperature  construct error map by
measuring errors at position and temperature
6.3 machine tool metrology
● reversal error (ex, hysteresis) can be obtained
(ex, backlash)

 For each axis,
- in one direction, table is moved and error is
measured
- in the other direction, same procedure
6.3 machine tool metrology
● when measurement interval is selected,
measurement position is set as a (even number)
multiple of lead ⇒ periodic error can be
separated

● to remove the effect of temperature in periodic
error determination  for small interval (1-2
times of lead) separate experiment  assume
the same periodic error in entire range
Preliminary temperature test
● temperature rise during operation
(temperature change at several parts)
Preliminary temperature test
● error measurement is needed in order to find
out the effect of temperature on errors

● warm-up: reciprocating the slide

● temperature measurement position (near heat
sources): bearing housing, slide way, motor, bed,
fixture, environment
4 groups of errors
● linear displacement error ⇒ length change

● angular error ⇒ angle change

● straightness, parallelism, squareness

● spindle thermal drift error
linear displacement error
● linear error along moving axis of machine element 
cause: geometric incorrectness

● ball screw
- Lead error: distance variation per 1 rotation
- misalignment (between rotating shaft and center axis):
error occurs perpendicular to moving axis
- Geometric inhomogeneous: machining error in ball
screw
- coupling error between feedback unit and ball screw

● laser interferometer measurement is desirable
angular errors
● cause: geometric incorrectness in slide,
misalignment during assembling machine
structural elements
yaw               Y

Z
turret   X

tool              roll

pitc h
angular errors
● roll and pitch errors are in nonsensitive
direction (depending on machine tools)

● yaw error is important (sensitive direction,
making the tool move in radial direction)

● laser interferometer (or autocollimator) is
normally used
straightness
● translational error in two directions
perpendicular to Z (moving axis)
Y

Z

X

● non-contact capacitance sensor, and
precision test arbor are used
(laser interferometer is hard to mount because of
allowable space and size of optical device)
straightness
● test arbor is attached to spindle
-Sensor attached to carriage moves with
carriage, and measure the gap between sensor
and arbor

● measurement contains straightness error, test
arbor non-straightness, and misalignment
- reversal technique is used to remove arbor
profile error
- misalignment can be removed by deleting best-
fit line slope
parallelism
● parallelism between Z motion and center axis
of the spindle

● procedure
- Two measurements at two positions along test
arbor  difference/distance
parallelism
● to measure parallelism without other errors
affecting measurement
 two sensors 0 ° and 180° (top and bottom of
artifact) are used
probe 1

spindle

artifac t

probe 2
parallelism
● with small spindle rotation (in order for spindle
error motion small)
- Measuring distance = 12"
-512 points per 1 rotation ⇒ best-fit circle is
determined
ri
R
n

- R: radius of the best-fit circle
- ri: sensor output at angle i
- n: number of data
parallelism
● To remove straightness error of the shaft, two
sensors (0° and 180 °) are used  best-fit circle
is determined from two outputs

● parallelism error
( R21  R11 )  ( R22  R12 )
p 
2z
( R21  R11 ) ( R12  R22 )

      z               z
2
parallelism
R11: least square radius at position 1 and sensor 1
R21: least square radius at position 2 and sensor 1
R12: least square radius at position 1 and sensor 2
R22: least square radius at position 2 and sensor 2
z: distance from sensors 1 and 2
orthogonality
● To measure straightness of X motion wrt the Z
direction, and orthogonality between x axis and
spindle center line, one more test arbor is used.
(7" diameter, lapped flat surface)
● reversal technique cannot be used because
of machine ⇒ surface should be calibrated
0   C ross- slide
ex          straightness

X

Z
orthogonality
● In order to remove arbor misalignment and
squareness error,

- As cross-slide moves against test arbor surface,
measurement is done at the interval
- Repeat measurement with spindle rotated 180 °
 Orthogonality and Z straightness of X motion
can be determined
orthogonality
m1 ( x)   z ( x)  e( x)
m2 ( x)   z ( x)  e( x)
m1(x): 1st measurement
m2(x): 2nd measurement (spindle at 180 °)
dz: Z straightness of X motion
e(x): arbor squareness and misalignment error

 z ( x)  (m1 ( x)  m2 ( x)) / 2
● orthogonality can be calculated from best-fit
line slope averaged from m1 (x) and m2 (x)
6.3.4 spindle thermal drift
● thermal drift: distance between two bodies
changes due to temperature change (internal or
external heat sources)

● 3 components of spindle thermal drift
- axial thermal drift: spindle deformation in the Z
direction
- radial thermal drift: perpendicular to Z and
deformation in sensitive direction
- tilt thermal drift: spindle’s tilting motion in X-Z
plane
6.3.4 spindle thermal drift
● capacitance sensor and precision test arbor
are sued

● investigate position and orientation of spindle
wrt time

● positions of temperature measurement:
sensitive to temperature change
6.3.4 spindle thermal drift
● operate spindle for 8 hours at constant speed
- temperature, spindle thermal drifts in radial and
axial direction at every 10 minutes
- stop after 8 hours, and same measurement is
repeated during cooling

● to remove roundness error of test arbor, and
fundamental spindle error motion, measure at
the same spindle angle

● tilt is determined radial displacement
difference/distance between two sensors
6.4 calibration measurement
results
● given condition of Superslant
- lead of the screw = 0.2"
- travel range = 13" (carriage, Z axis),
3.4" (cross-slide, X axis)
- measuring interval = 1" (carriage), 0.2" (cross-
slide)

● interval = multiple of lead  periodic error is
separated

● periodic error is obtained for 0.4"(2 times of
lead) with 0.002“ of interval (200 points)
6.4 calibration measurement
results
carriage (Z) linear
displacement error
● initial backlash of 200min ⇒ different backlash

● backlash compensation⇒ calibrations
different for forward and reverse directions

● home position change (drift) ⇒ home position
is unstable ⇒ limit switch is needed

● geometric error due to state ⇒ as warm-up,
error curve slope changes
carriage (Z) linear
displacement error
● best positions for temp measurement (sensitive
position) : - bearing housing at ends of ball screw
- ballnut assembly
carriage (Z) linear
displacement error
● 2 parameter nonlinear least square
regression analysis for position and temperature
⇒ fails

● error behavior wrt temperature at each
position (Z axis: 1", X axis: 0.2") is analyzed ⇒
temperature is only parameter

 z ( z )  a0  a1T  a2T 2  a3T 3  a4T 4
carriage (Z) linear
displacement error
carriage (Z) linear
displacement error
● 12 sets of 5 coefficients for each position is
necessary to map linear displacement
performance of the carriage

● interpolation scheme is used to determine
error between two positions

● prediction of periodic displacement error is
required to determine accurate Z position
⇒ determined from 0.02" interval experiment
carriage (Z) linear
displacement error
● to determine periodic error (without temperature
effect) ⇒ measurement is conducted for small
range (0.4") (assuming homogeneous periodic
error in all range)
carriage (Z) linear
displacement error
● net error motion data and fitted curve per one
revolution
carriage (Z) linear
displacement error
● forward

 'z ( z )  3.19  0.164 cos(31.4 z )  3.54sin(31.4 z )
301z  1694 z   2

z: incremental nominal position

● reverse
 'z ( z )  15.1  6.45cos(31.4 z )  4.42sin(31.4 z )
1209 z  5953 z    2
carriage (Z) linear
displacement error
● sinusoidal interpolation procedure based on
superposition can be applied to find the periodic
error at any point

● when combined with thermal error, the total
linear displacement error for the carriage is
obtained
⇒ same procedure applied to cross-slide
carriage yaw error
● 1 stage
yaw error of carriage cross-slide assembly was
measured when the machine was at its home
position (machine gradually warmed up from
cold state)

● 2 stage
yaw error can be determined as a function of
distance (as cross-slide and carriage are away
from home position)
carriage yaw error
carriage yaw error
carriage yaw error
● depending on direction  analyzed
separately

● effect of temperature on yaw erroris constant
over carriage motion  one parameter(z)
regression
forward:  y ( z )  15.1  2.64 z  0.109 z  0.00397 z
2                3

backward:  y ( z )  16.3  ...z 3

● applied to cross-slidesimilarly
X straightness of the Z motion
X straightness of the Z motion
● sample raw data from two probes ⇒ X
direction straightness of carriage as a function of
Z ⇒ linear regression analysis to calculate best
fit line

● temperature effect is not significant

● irregular curve  least square curve fitting
does not give satisfactory correlation ⇒ look-up
table
X straightness of the Z motion
X straightness of the Z motion
● to determine parallelism error between
carriage motion and spindle shaft average line,
best fit circle is calculated (two probes at 0 and
180)
 temperature effect is small
⇒ average -14 mrad ⇒ constant value is used to
error compensation
orthogonality
● best fit line slope (squareness between cross-
slide and spindle axis)

 0  345  7.34T  0.0512T         2
thermal drift
● data was obtained using two probes mounted
8” apart along the test arbor
- Difference between the radial displacements
measured by two probes divided by the distance
 tilt
- Using this value, the pure radial displacement
at the spindle nose was also calculated
● noise occurs due to spindle error motion ⇒
amplified by tilt
● radial thermal drift is more complicated
(temperature and rotating speed influence)
6.6 real-time implementation of
the error compensation system
● error of each element depending on
temperature and position ⇒ HTM ⇒ error vector
at machining point ⇒ error compensation signal
to the controller⇒ accuracy improves

● error compensation algorithm ⇒ into micro
computer
6.6 real-time implementation of
the error compensation system
● HTM

 1           Z ( z )  Y ( z ) a   X ( z ) 
  ( z)        1                                
 X ( z ) b   Y ( z ) 
R
Tnerr     Z

  Y ( z )  X ( z )     1       c   Z ( z) 
                                                
 0             0          0           1         
6.6 real-time implementation of
the error compensation system
● position command signal is calculated
- Compare command with position feedback
- for speed control, velocity feedback is
monitored or speed feedback signal is
determined from position feedback signal

● real-time error compensation system: put
error compensation signal to position servo loop
6.6 real-time implementation of
the error compensation system
● To calculate error, 3 independent variables
(position, direction, temperature)

● error calculated is sent to machine controller
Cutting tests
● real time error compensation is constructed 
cutting tests is conducted at unsteady state
(error compensation effect is to be found)
- With or without error compensation system

● significant precision improvement in diameter
and length (up to 20 times)

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