Lecture 5
Formulating the System
Error Budget
5-1
Error Budget
• An error budget is formulated based on
connectivity rules that define the behavior of a
machine’s components and their interfaces, and
combinational rules that describe how errors of
different types are to be combined.
• First step: to develop a kinematic model of the
proposed system in the form of a series of
homogeneous transformation matrices (HTM).
• Second step: to analyze systematically each type of
error that can occur in the system and use the
HTM model to help determine the effect of the
errors on the toolpoint position accuracy with
respect to the workpiece.
5-2
Error & Tolerance Budgets
Errors in parts and the assembly are controlled
through the use of Error Budgets and Tolerance
Budgets
• Error budgets attempt to predict how a machine
will perform when it is assembled and running
¾ Each module is represented as a rigid body and has a
coordinate system assigned to it.
¾ Error budgets account for errors, geometric, thermal…,
in each module’s degrees of freedom (3 position and 3
orientation errors)
• Tolerance budgets attempt to predict what the
final assembled shape of the machine will be given
geometric errors in the parts
¾ Will the parts even have enough tolerance to make sure
5-3
they will all even fit
Error Assessment and Budgeting
• Given all the different types of errors that can affect all
different components:
¾ Keeping track of all the errors is such a daunting task:
& Most engineers don't bother and use "experience" to guide the design.
& It is left up to manufacturing and service to work the bugs out.
¾ This seems to be a major source of reliability and performance
problems.
•
The solution to a successful project for precision machine
design is a good budget:
¾ A project requires a good financial budget to make it feasible.
¾ A project requires a good time budget to make it feasible.
¾ A project requires a good error budget to make it feasible.
• In order to make a good error budget for the system, a
good mathematical model is needed.
5-4
Homogenous Transformation
Matrices
• Allows the designer to consider all errors
for one component of the machine at a time,
and then link them all together by specific
matrix.
• Based on rigid body model of a linear series
(open chain) of coordinate frames.
• Takes into account linear and angular
offsets between coordinate frames.
• Transforms XYZ coordinates of one frame
into XYZ coordinates of another frame.
5-5
Homogenous Transformation
Matrices
Reference coordinate frames
5-6
Homogenous Transformation
Matrices
• Coordinate frames are placed at bearings,
joints, and areas where other parameters
are lumped.
• Closed chains (e.g. a five point bearing
mount) need to be modeled with the
generation of constraint equations.
5-7
Sensitive Directions
• It is important to note which are the sensitive
directions of the machine:
A radial error in the
workpiece of magnitude
is 2·ε/r, which
is much smaller than ε.
Z
• Sensitive directions can be controllable:
¾ Z and r directions in a lathe.
5-8
Structure of a Homogenous
Transformation Matrix
Orientation of Xn with
respect to adjacent
coordinate frame
⎡ Oix
⎢O
jx
R
⎢
Tn =
⎢O kx
⎢
⎣ 0
Perspective transformation
Orientation of Yn with
respect to adjacent
coordinate frame
Oiy
Oiz
O jy
O ky
O jz
O kz
0
0
Orientation of Zn with
respect to adjacent
coordinate frame
Px ⎤
Py ⎥
⎥
Pz ⎥
⎥
Ps ⎦
Translation of Xn,
Yn, Zn with
respect to adjacent
coordinate frame
Scale Factor : usually set to unity to help
avoid confusion
5-9
Structure of a Homogenous
Transformation Matrix
• The first three columns are the direction cosines (unit
vectors i, j, k).
¾ They represent the orientation of the Xn, Yn, and Zn axes with
respect to an adjacent coordinate frame.
• Their scale factors are unity.
• The last column is the position of the rigid body's
coordinate system's origin with respect to the reference
frame. (Translation)
• The pre-superscript represents the reference frame in
which you want the result to be represented.
• The post-subscript represents the reference frame from
which you are transferring.
• The “O’s” are rotations (direction cosines” and the “P’s”
are translations.
5-10
Structure of a homogeneous
transformation matrix
• The equivalent coordinates of a point in a
reference frame n, in a reference frame R are:
Equivalent coordinates of a
point in reference frame R
⎡X R ⎤
⎡X n ⎤
⎢Y ⎥
⎢Y ⎥
⎢ R ⎥ = R Tn ⎢ n ⎥
⎢ ZR ⎥
⎢ Zn ⎥
⎢ ⎥
⎢ ⎥
⎣ 1 ⎦
⎣1⎦
Original coordinates of a point in
reference frame n
HTM that describes the
transformation of reference frame n
into reference frame R
5-11
Example: Translation in X
• The coordinate system X1Y1Z1 is shifted in
X by an amount a:
5-12
HTM of Translation in X, Y, Z
XYZ
TX1Y1Z1
⎡1
⎢0
=⎢
⎢0
⎢
⎣0
0 0 x⎤
1 0 0 ⎥⎥
,
0 1 0⎥
⎥
0 0 1⎦
XYZ
TX1Y1Z1
⎡1
⎢0
=⎢
⎢0
⎢
⎣0
a
XYZ
TX1Y1Z1
⎡1
⎢
⎢
⎢0
⎢
=⎢
⎢0
⎢
⎢
⎢0
⎣
0 0 0⎤
⎥
⎥
1 0 y⎥
⎥
⎥
0 1 0⎥
⎥
⎥
0 0 1 ⎥⎦
0 0 0⎤
1 0 0⎥
⎥
0 1 z⎥
⎥
0 0 1⎦
5-13
Example: Rotation about X-Axis
• The X1Y1Z1 coordinate system is rotated by
an amount θx about the X axis:
0
0
⎡1
⎢0 cosθ - sin θ
x
x
XYZ
TX1Y1Z1 = ⎢
⎢0 sin θx cosθx
⎢
0
0
⎣0
0⎤
0⎥
⎥
0⎥
⎥
1⎦
5-14
Example: Rotation about Y-Axis
• The X1Y1Z1 coordinate system is rotated by
an amount θy about the Y axis:
⎡ cosθy
⎢ 0
XYZ
TX1Y1Z1 = ⎢
⎢- sin θy
⎢
⎣ 0
0 sin θy 0⎤
1
0
0⎥
⎥
0 cosθy 0⎥
⎥
0
0
1⎦
5-15
Example: Rotation about Z-Axis
• The X1Y1Z1 coordinate system is rotated by
an amount θz about the Z axis:
⎡cosθz - sin θz
⎢sin θ cosθ
z
z
XYZ
TX1Y1Z1 = ⎢
0
⎢ 0
⎢
0
⎣ 0
0 0⎤
0 0⎥
⎥
1 0⎥
⎥
0 1⎦
5-16
Sequential Systems
• Transformation from the Nth axis to the
reference system will be the sequential
product of all the HTMs:
R
N
TN = ∏
m=1
m−1
Tm = T T T L
0
1 2
1 2
3
• The order of rotation is very important.
5-17
HTM between the Tool and
Workpiece
5-18
HTM between the Tool and
Workpiece
• The relative error HTM Erel between the
tool and workpiece in the tool coordinate
frame is:
Q Twork= TtoolErel
R
R
[
]
∴ Erel = Ttool
R
−1 R
Twork
• Erel is the transformation that must be done
to the toolpoint in order to be at the proper
position on the workpiece.
5-19
Error Correction Vector
• The error correction vector RPcorrection with
respect to the reference coordinate frame
can be obtained from:
R
R
⎡Px ⎤
⎢P ⎥
=
y
⎢ ⎥
⎢⎣Pz ⎥⎦correction
R
⎡Px ⎤
⎢P ⎥ −
⎢ y⎥
⎢⎣Pz ⎥⎦work
⎡Px ⎤
⎢P ⎥
⎢ y⎥
⎢⎣Pz ⎥⎦tool
5-20
Error Correction Vector
• Because of Abbe offsets and angular orientation errors of
the axes:
¾ RPcorrection will not necessarily be equal to the position vector P
component of Erel.
•
RP
correction
gives the motions the X, Y, and Z axes must
make to compensate for toolpoint location errors.
Y axis: Can be used to compensate
for straightness errors in the X axis.
X axis: Can be used to compensate for
straightness errors in the Y axis.
5-21
Rigid Body Models of Machine
Components
5-22
HTM of Linear Motion Carriage
• The HTM for a linear motion carriage with
small errors is:
Pure translations
R
Tnerr
⎡ 1
⎢ε
=⎢ z
⎢− ε y
⎢
⎣ 0
− εz
εy
1
εx
− εx
1
0
0
Translation errors
a + δx ⎤
b + δy ⎥
⎥
c + δz ⎥
⎥
1 ⎦
5-23
HTM of Linear Motion Carriage
• If a system is over constrained (non
deterministic), then in order to model
system errors, one has to:
¾Make an intelligent estimate.
¾Rely on experience or direct measurement.
¾Perform a sometimes-complicated analysis of the
system (model contact points as springs and solve
the constraint problem).
5-24
Estimating Position Errors from
Modular Bearing Catalogs
5-25
HTM of Modular Bearing System
• Consider the elements of the HTM we are
searching for:
⎡Xcs ⎤ ⎡ 1 − εz εy a + δx ⎤⎡XΔcs ⎤
⎥⎢ Y ⎥
⎢Y ⎥ ⎢ ε
−
+
1
ε
b
δ
x
y ⎥⎢ Δcs ⎥
⎢ cs ⎥ = ⎢ z
1 c + δz ⎥⎢ ZΔcs ⎥
⎢ Zcs ⎥ ⎢− εy εx
⎥
⎥⎢
⎢ ⎥ ⎢
0
0
1 ⎦⎣ 1 ⎦
⎣1⎦ ⎣ 0
5-26
HTM of Modular Bearing System
• The translational errors are based on the
average of the straightness errors
experienced by the bearing blocks:
δ x =δservo
δy =
δ y1 + δ y2 + δ y3 + δ y4
4
δ z1 + δ z2 + δ z3 + δ z4
δz =
4
5-27
HTM of Modular Bearing System
• Bearing block straightness is a function of bed
accuracy and running parallelism of the bearing
block to the bearing rail:
5-28
HTM of Modular Bearing System
• The angular errors are based on the differences in the
average straightness errors experienced by pairs of
bearing blocks acting across the carriage:
(δ
+ δy3) (δy1 + δy4 )
−
2
2
εx =
W
(δz3 + δz4 ) − (δz1 + δz2 )
2
2
εy =
L
(δy1 + δy2) − (δy3 + δy4)
2
2
εz =
L
y2
• Note we assumed for the
straightness we assumed all the
errors are acting in the same
direction.
• Here we assume one set of
errors acts up, and the other
acts down.
• This is very conservative
(makes up for other effects we
might miss).
5-29
HTM of Modular Bearing System
• Note we assumed for the straightness
assumed all the errors are acting in
same direction.
• Here we assume one set of errors acts
and the other acts down.
• This is very conservative ( makes up
other effects we might miss).
we
the
up,
for
5-30
Example – Linear Error Plot
• Straightness error of a kinematically (statically
determined) supported linear axis (five rolling
element bearings on a vee and flat).
5-31
Fourier Transformation of Linear
Error Plot
• Fourier transform of the
carriage's straightness error:
kinematic
Error Amplitude
as a function of
Wavelength
The wavelength of rolling elements is 2πD
5-32
Fourier Transform
• The Fourier transform is a generalization of
the complex Fourier series in the limit as
L→∞. Replace the discrete An with the
continuous F(k)·dk while letting n/L→k.
• Then change the sum to an integral, and the
equations become
∞
f ( x) = ∫ F (k )e
−∞
∞
2πikx
dk
F (k ) = ∫ f ( x)e − 2πikx dx
−∞
5-33
Fourier Transform
• Here,
• is called the forward (-i) Fourier transform, and
• is called the inverse (+i) Fourier transform.
5-34
Fourier Transformation of Linear
Error Plot
• The Fourier transform, when plotted as error
amplitude as a function of wavelength, is an
invaluable diagnostic tool.
¾ It can help identify the dominant sources of error, so
design attention can be properly allocated.
• In a kinematic system, once the source of error is
identified, it is more easily corrected.
¾ This is due to the fact that it is easier to see where the
error came from.
• Remember, for a rolling element:
¾ The center moves πD, while the element that rolls upon
it moves 2πD!
5-35
HTM of Rotary Axis
Radial displacement
measurements are
make parallel to the X
axis.
5-36
HTM of Rotary Axis
R Tnerr
cosε y cosθ z
− cosε ysinθ z
⎡
⎢
⎢ sinε sinε cosθ + cosε cosθ −
x
y
z
x
z
⎢
sinε x sinε ysinθ z
⎢ cosε x sinθ z
=⎢
⎢− cosε x sinε y cosθ z + sinε x cosθ z +
cosε x sinε ysinθ z
⎢sinε x sinθ z
⎢
⎢
0
0
⎣
sinε y
− sinε x cosε y
cosε x cosε y
0
δx ⎤
⎥
⎥
δy ⎥
⎥
⎥
δz ⎥
⎥
⎥
1 ⎥⎦
• This general result may also be used for the
case of a linear motion carriage if εz is
substituted for θz.
5-37
HTM of Rotary Axis
• Most often, second-order terms such as εx
εy are negligible and small-angle
approximations. (i.e., cosε=1, sinε=ε)
R Tnerr
cosθ z
⎡
⎢
⎢
⎢
sinθ z
⎢
=⎢
⎢− ε y cosθ z + ε x sinθ z
⎢
⎢
⎢
0
⎣
− sinθ z
εy
cosθ z
− εx
ε x cosθ z + ε ysinθ z
1
0
0
δx ⎤
⎥
⎥
δy ⎥
⎥
⎥
δz ⎥
⎥
⎥
1 ⎥⎦ 5-38
Components of the Total Error Motion
PC Center: Polar Chart
Center
MRS: Minimum Radial
Separation
5-39
Components of the Total Error
Motion
• The average error motion is indicative of
the form error that will be imparted to the
part when held in a spindle.
• The asynchronous error motion is indicative
of the surface finish that will be obtained.
5-40
Asynchronous Error Motion (AEM)
• Also known as non-repeatable runout or non-repetitive
runout (NRR), Asynchronous Error Motion (AEM)
describes the motion of a rotating shaft which is not the
same rotation after rotation. Whatever the motion is called,
its existence in precision spindles (such as disk drive
motors or CNC machine tools) is generally a detriment to
the application.
• All axes of rotation (i.e., rotating shafts) have repeatable
and non-repeatable components of displacement. The
repeatable component is referred to as Total Indicated
Runout (TIR) and is simply referred to as runout. Another
derived value similar to TIR is Synchronous Error Motion
(SEM), which is commonly referred to as average runout.
5-41
Asynchronous Error Motion (AEM)
• Non-repeating motion can be attributed in some
cases to “random statistical processes.” However,
a better explanation of the mechanism for the nonrepeatability is the slight variations of the
components of the spindle bearing.
• Ball bearing spindles exhibit the largest amount of
AEM which is directly attributable to the
deviations from “perfection” of the balls, races,
and ball cage. Fluid and air bearings exhibit
substantially less AEM than ball bearings.
5-42
FFT Analysis
• The FFT is a vital tool for identifying the source of errors
in so the designer can seek to minimize them:
‧ The spindle speed was 1680 rpm (28 Hz).
‧ The bearing inner diameter was 75 mm.
‧ The outer diameter was 105 mm.
‧ The number of balls was 20.
‧ The ball diameter was 10 mm.
‧ The contact angle was 15°.
5-43
Combinational Ruler for Three
Common Types
• Random - under apparently equal conditions at a given
position, errors that do not always have the same value,
and can only be expressed statistically.
• Systematic - which always have the same value and sign at
a given position and under given circumstances.
¾ Generally can be correlated with position along an axis and can be
corrected.
&If the relative accompanying random error is small enough.
• Hysteresis - a systematic error which in this instance is
separated out for convenience.
¾ Usually repeatable, sign depends on the direction of approach, and
magnitude partly dependent on the travel.
¾ May be compensated for if the direction of approach is known and an
adequate pre-travel is made.
5-44
Glossary for Error Motion
• B89.3.4M - 1985 “Axes of Rotation-Methods for
Specifying & Testing”
• Sensitive and nonsensitive directions
¾ The sensitive direction is perpendicular to the ideal
generated workpiece surface through the instantaneous
point of machining or gaging.
• The fixed sensitive direction is where the
workpiece is rotated by the spindle and the point
of machining or gaging is fixed (e.g., in a lathe).
• The rotating sensitive direction is where the
workpiece is fixed and the point of machining or
gaging rotates with the spindle (e.g., in a jig borer).
5-45
Glossary for Error Motion
• Radial motion - error motion in a direction
normal to the Z reference axis and at a specified
axial location.
¾ The radial motion will be a function of the position along
the Z axis, and the rotation angle. The term radial runout
includes errors due to radial motion, workpiece out-ofroundness, and workpiece centering errors; thus runout
is not equivalent to radial motion.
• Runout - the total displacement measured by an
instrument sensing against a moving surface or
moved with respect to a fixed surface.
¾ The term total indicator reading (TTR) is equivalent to
runout.
5-46
Glossary for Error Motion
• Axial motion - error motion colinear with
the Z reference axis.
• Face motion - error motion parallel to the Z
reference axis at a specified radial location.
¾Face motion includes sine errors caused by tilt
motions.
¾The term face runout includes errors in the
workpiece in a manner similar to radial runout;
thus face runout is not equivalent to face motion.
5-47
Glossary for Error Motion
• Tilt motion - error motion in an angular direction relative
to the Z reference axis.
¾ Tilt motion creates sine errors on the spindle which is why the radial
error motion is a function of Z position and face motion is a function
of radius.
¾ Tilt motion about the Y axis is in the sensitive direction because it
causes an error in X direction (along the assumed measurement axis).
¾ Note that "coning," "wobble," and "swash" are sometimes used to
describe tilt motion, but they are nonpreferred terms.
• Squareness - a plane surface is square to an axis of rotation
if coincident polar profile centers are obtained for an axial
and a face motion polar plot or for two face motion polar
plots at different radii." Squareness is equivalent to
orthogonality.
5-48
Glossary for Error Motion
• Error motion polar plot - a polar plot of
error motion made in synchronization with
the rotation of the spindle.
¾Error motion polar plots are often decomposed
into plots of various error components.
¾Note that it is also very important to consider the
frequency spectrum of the errors.
• Total error motion polar plot - the complete
error motion polar plot as recorded.
5-49
Glossary for Error Motion
• Average error motion polar plot - the mean
contour of the total error motion polar plot
averaged over the number of revolutions.
¾The average error motion has components that
include fundamental and residual error motion
components.
¾Note that asynchronous error motion components
do not always average out to zero, so the average
error motion polar plot may still contain
asynchronous components,
5-50
Glossary for Error Motion
• Fundamental error motion polar plot – the best-fit
reference circle fitted to the average error motion polar
plot. The fundamental error motion polar plot of an
eccentric workpiece would actually be a limacon (the polar
plot of a pure sinusoid). As the eccentricity decreases, the
limacon approaches being a circle.
• Residual error motion polar plot - the deviation of the
average error motion polar plot from the fundamental
error motion polar plot.
¾ For radial error motion measurements, this represents the sum of the
error motion and the workpiece (e.g., ball) out-of-roundness.
¾ The workpiece out-of-roundness can be removed using a reversal
technique.
5-51
Glossary for Error Motion
• Asynchronous error motion polar plot - the deviations of
the total error motion polar plot from the average error
motion polar plot.
¾ Asynchronous in this context means that the deviations are not
repetitive from revolution to revolution. Asynchronous error motions
are not necessarily random (in the statistical sense).
• Inner error motion polar plot - the contour of the inner
boundary of the total error motion polar plot.
• Outer error motion polar plot - the contour of the outer
boundary of the total error motion polar plot.
5-52
Glossary for Error Motion
• Polar chart (PC) center - the center of the polar
chart.
• Polar profile center - a center derived from the
polar profile.
• Minimum radial separation (MRS) center - the
center which minimizes the radial difference
required to contain the error motion polar plot
between two concentric circles.
• Least squares center - the center of a circle which
minimizes the sum of the squares of a sufficient
number of equally spaced radial deviations
measured from it to the error motion polar plot.
5-53
Glossary for Error Motion
• Total error motion value - the scaled difference in radii of
two concentric circles from a specified error motion center
just sufficient to contain the total error motion polar plot.
• Average error motion value - the scaled difference in radii
of two concentric circles from a specified error motion
center just sufficient to contain the average error motion
polar plot.
¾ The average error motion value is a measure of the best roundness
that can be obtained for a part machined while being held in the
spindle (or the roundness of a hole the spindle is used to bore).
• Fundamental error motion value –twice the scaled distance
between the PC center and a specified polar profile center
of the average error motion polar plot.
¾ This value represents the once-per-revolution sinusoidal component
of an error motion polar plot. Thus when a perfect workpiece is
perfectly centered, the fundamental radial error motion value will be
5-54
zero.
Glossary for Error Motion
• Residual error motion value - the average error motion
value measured from a specified polar profile center. This
represents the difference between the average and
fundamental error motions.
• Asynchronous error motion value - the maximum scaled
width of the total error motion polar plot, measured along
a radial line through the PC center.
• Inner error motion value - the scaled difference in radii of
two concentric circles from a specified error motion center
just sufficient to contain the inner error motion polar plot.
• Outer error motion value - the scaled difference in radii of
two concentric circles from a specified error motion center
just sufficient to contain the outer error motion polar plot.
5-55
Glossary for Error Motion
• Axis average line - a line passing through
two axially separated radial motion polar
plot centers. Note that the default is the
MRS center.
• Preferred centers: Unless otherwise
specified, the following motions are assumed
to be measured with respect to:
5-56