mapping geometric and thermal errors in turning

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

● “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
   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

● 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
- coupling error between feedback unit and ball

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

                     turret   X

            tool              roll

            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
● translational error in two directions
perpendicular to Z (moving axis)



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

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



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

      - R: radius of the best-fit circle
      - ri: sensor output at angle i
      - n: number of data
● 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 
         ( R21  R11 ) ( R12  R22 )
             z               z
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
● 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
● reversal technique cannot be used
because of machine ⇒ surface should be
                0   Cross-slide
calibratedex         straightness


● 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
       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
       z ( x)  (m1 ( x)  m2 ( x)) / 2

● orthogonality can be calculated from
   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
- 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
  6.4 calibration measurement
● 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"

● 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
        carriage (Z) linear
        displacement error
● initial backlash of 200min ⇒ different
backlash due to nonlinear lead screw

● 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
          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

● 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
● best fit line slope (squareness between
cross-slide and spindle axis)
        0  345  7.34T  0.0512T      2
      spindle radial and tilt
          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
● 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
● 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
    ● HTM

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

               Y ( z )  X ( z )     1       c   Z ( z) 
                                                             
              0             0          0           1         
 6.6 real-time implementation
  of the error compensation
● 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
 6.6 real-time implementation
  of the error compensation
● To calculate error, 3 independent
variables (position, direction, temperature)

● error calculated is sent to machine
            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

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

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