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Some theoretical aspects of error separation techniques in surface metrology

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1976 J. Phys. E: Sci. Instrum. 9 531

(http://iopscience.iop.org/0022-3735/9/7/007)

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 the out-of-roundness of the component. The effective absolute
RESEARCH PAPERS accuracy of the instrument is thereby improved.
One simple way of determining these errors (Bryan eta1 1967,
Donaldson 1972) on, say, a rotating spindle instrument is as
follows. Let the spindle error be e and the signal s; then one
probe will detect a voltage Vl(0):
Vl(0)=s(@ + e(@. (1)
Some theoretical aspects Turning the part through half a revolution, moving the probe
on its carriage through the axis of rotation without moving the
of error separation spindle and finally changing the direction of sensitivity of the
probe (figure 1) gives a voltage V,(0) where
techniques in surface V,( 0) = s( 0) - e( 0). (2)
metro Iogy Simple arithmetic gives
4 0 ) = ( vl - v 2 ) / 2 , S( e) = ( v1+ v2)/2. (3)
D J Whitehouse This method although simple does have its problems.
Research Department, Rank Taylor Hobson, Rank Perhaps the most important of these is that the pickup may
Precision Industries, Leicester LE5 3RG, UK

Received 27 October 1975, in final form 11 March 1976 ,Probe posit'ion I

Abstract There is an increasing demand for more accurate

surface metrology instruments and for techniques which will
allow complicated or large parts to be measured i f t situ. This
report analyses existing and new multi-orientation and
multiprobe techniques and shows how they can be used to
remove systematic and variable errors.

1 Introduction Figure 1 Method of isolating systematic errors using two

There is a growing need to measure surfaces down to ultrafine positions and a bias reversible probe
limits and also to measure large unwieldy components in situ.
This paper describes the theory behind some new methods of
achieving both. need to be handled when it is passed through centre and
The word 'signal' is taken to mean the texture or form of a reversed. Also it may be necessary to stop the rotation during
component. Noise is any extraneous effect which tends to the evaluation. These considerations could be important in
distort the signal or inhibit its measurement. The worst case highly critical cases, for instance when the instrument uses
situation where the signal and noise cannot be separated by hydrodynamic bearings. Another factor which may have to be
simple filtering will be considered here. taken into account on the rotating spindle instrument is that
Two types of noise will be studied, systematic errors which the loading of the bearing by the probe could be unbalanced
repeat and variable errors which do not. Systematic errors are when the probe is reversed through centre. This is not the case
often found in the precision bearing or datum shaft of an in the rotating table instrument where the probe is independent
instrument whereas variable errors can be a result of vibration of the axis of rotation.
or changes in the environment. How to deal with the noise At first glance there would appear to be two alternative
depends on which of these two categories it falls in. Variable approaches to that of the Donaldson method. These will be
errors can be removed by long term time averaging if the briefly described here because their analysis throws light on
experiment is repeatable (and the statistics of the variations subsequent techniques.
stable in time) or by instantaneous removal methods if not. Take first the possibility of using two probes simultaneously,
Systematic errors on the other hand can be removed step by at 180" to each other (figure 2(a)). If probe 1 gives VI and
step. This report will describe the theory behind multi-orienta- probe 2 gives VZthen
tion methods for removing systematic errors and multiprobe
methods for removing variable errors.
+
VI( 0) = s( 0) e(0) (4)

Vz(0) = s( 0 - T ) - e( 0). (5)

2 Systematic errors
A good example of a metrological situation in which identifica- What effect this has can best be seen by using Fourier analysis
tion and removal of systematic errors can be of benefit is in the techniques. These methods will be used extensively in this
measurement of roundness. In the aerospace industry for report to discover harmonic deficiencies in the probe output
example, there is a real need to improve the absolute accuracy signal. The possibility is also mentioned of having an instru-
of measurement. Normally a natural restriction is the accuracy ment which incorporates a computer. This computer would
of the reference rotation (this is either a spindle supporting the analyse the signals from the probes by Fourier methods,
probe or a table supporting the component). However, the compensate all coefficientsknown to be deficient and synthesize
need for smaller absolute uncertainty can be replaced by one of the results to produce a corrected out-of-roundness signal for
smaller uncertainty in repeatability if the systematic errors can display on the chart of the instrument, or for assessment.
be determined. Once ascertained the systematic errors can be Thus if the signal is analysed at discrete harmonics
offset against the incoming transducer voltage to reveal only n = 0, 1, 2 . . . where n = 1 corresponds to one undulation per

531
D J Whitehouse
less harmonic information. In fact combining the two
methods by adding equations (7) and (10) results in the
Probe I Donaldson output. This is because W becomes independent of
Reference on specimen n. This is tantamount to saying that the Donaldson method
could be achieved by using two probes and two orientations,
only one probe being used for one orientation. This would
Reference<: obviously reduce some of the mechanical problems associated
instrument with the method.
Remember that in the two alternatives to Donaldson's
method the combination signal C does not contain spindle (or
(0) Probe 2 table) error. It does contain somewhat scrambled information
of the component out-of-roundness. For those occasions where
either may be used in preference to Donaldson's method it is
necessary to show how the useful component signal can be
extracted. To do this it is essential to understand the nature of
the harmonic losses. Consider the two-orientation method. Let
the orientations be separated by an angle a. The combination
signal Fourier transform becomes Fc(n), where
Fc(n) = Fs(n)(l- e-jnz). (1 1 )
This will have zeros at 3in = 2 7 ~ Nwhere N is an integer. The
range of frequencies that can be obtained without having zeros
Figure 2 Alternative approaches to the problem of isolating
will extend from the first harmonic to 2nla. Hence, measuring
systematic errors in roundness. (a) Double probe method;
the Fourier coefficients of the combination signal up to 2 ~ / a ,
( b ) double orientation method
modifying them according to the weighting function contained
in equation ( 1 1 ) and then synthesizing them should produce
the out-of-roundness of the component, subject only to a high
circumference of the part and the results of the analysis for frequency cut at 27~/a.An instrument with an in-built com-
VI, V2, e and s are FI,F2, Fe, F, respectively, then puter could do this. If there is any danger of higher frequency
components causing aliasing the combination signal can be
Fdn) = Fdn) + Fe(n)
(6) attenuated by a filter at high frequencies prior to analysis. Thus
Fz(n) = Fs(n)e-jnn - Fe(??). Fs(n), the true coefficient of the component out-of-roundness
+
From this Fc(n) the transform of C= VI V2, the combination signal, can be obtained from equation ( 1 1 ) using an equaliza-
which eliminates e, is found by simple arithmetic to be tion technique.

Fc(iz)= F,(n)(l + e-jnn). (7)

Fs(n)=Fc(n)/(l- e-jnx), (12)
The amplitude modification necessary, A(n), is given by
The term ( 1 +e-jnn) will be called the harmonic weighting
function W. +
A(n)= [(l- cos na)z (sin na)Z]l,@ ( 1 30)
Examination of equation ( 7 ) shows that it has a magnitude
and the phase by
of 2 cos n.is/2, a much simpler formula. It can be seen from
these formulae that only the even harmonics of the com- $(n) = tan-1 [sin nai(1 -cos nm)]. (136)
ponent can be seen from an analysis of C; the odd harmonics
These equations can be conveniently simplified to A(n)=
are zero. The well known geometric implication of this state-
2 sin na/2 and $(n)= ( ~ - n a ) / 2 .To get the correct values of
ment is that only diametral information about the component
F,(n) the amplitudes of Fc(n) are divided by (13a) and the
can be detected if the spindle errors are removed by using two
phases shifted in the opposite direction to the phase angles
probes diametrally opposed.
given in equation (13b).
The second possibility uses two orientations of the com-
The number of zeros in the weighting function can be
ponent relative to the spindle and one probe rather than two
reduced by increasing the number of orientations. Take three
probes. This is shown in figure 2(b) where the orientations are
for example, at angles - z and p to the original one. Then,
at 180". The two outputs from the probe will be VI and V2
where v1(0)=s(0)+e(0)
Vl(0) =s( 0) + e ( @
(8) + +
V2(8)= s(0 a ) e( 0) (14)
V2(0) = s( B - .is) + e( 8) ~ ~ (=0S( )0 - p) + e(0).
aith their associated transforms The combination signal to remove e(6) is C=2V1- VZ- V3
+
Fl(n)= Fs(72) F e ( f 1 )
(9)
which has a transform
+
F'(n) = Fs(n)e-Jnn Fe(n). Fc(n)=Fs(n)(2-ejnx-e-j"P). (1 5 )
A combination signal C= V I - V2 is required to eliminate Equation ( 1 5 ) will only have zeros when both na and n p are
e( 0) yielding together a multiple of 2 ~There
. is a further degree of freedom
F~(n)=F,(n)(l-e-~%r). (10) which can be used to extend the coverage of harmonics and
this is the use of different probe sensitivities for different
This time the weighting function W shows a new feature,
orientations. For instance, taking the first orientation as unit
namely only odd harmonics can be detected. Here the magni-
gain, the second as a and the third as b, the transfer function
tude is 2 sin n7r/2.
corresponding to equation (15) is
Hence, despite the fact that both of these alternatives to
Donaldson's method are mechanically simpler they give rather Fc(n)=F,(n)(l - a ejna-b e-jng) (16)

532
 07125 pm

Figure 3 Example of systematic error determination and

repeatability of a precision spindle. The spread of each point
over six months is at worst 050050 pm

out-of-roundness of components down to N mm may

now be measurable.
Exactly similar techniques can be used for measuring the
systematic errors of straightness of a slideway. The equivalent
to the Donaldson method is well known. The part is measured
normally, it is then turned through 180‘ and the transducer
0 1 0 0 1 0 e2
adjusted to keep contact with the same track on the part. This
0
1
0
0
0
1
1
0
0
0
0
0
0
1
0
1
0
1
s1
e3 - “1
v21 *
(18)
mechanical trick changes the phase of e(x) with respect to s(x)
as in roundness (Spragg 1972, Hoffrogge et a1 1972). Alterna-
tively, to get the equivalent of a rotational shift of a in round-
ness (equation (11)) the part has to be moved linearly parallel
to the datum axis (Thwaite 1973). Obviously there are differ-
ences between the roundness and the straightness cases because
the latter signal does not join up. This means that end informa-
tion is lost. A more realistic method would be to move the
probe to another position on its carriage rather than to shift the
component bodily. This has the effect of presenting a different
s(x) to the same e(x).figure 4. It does not matter that some end

Shift o f
probe
,Oriqii
Datum shaft ?-?
I 8 ,

Probe carriage
Test piece 7 L 3 k
Figure 4 Systematic error removal in straightness -multi-
orientation technique

information is lost. All this does is to preclude any absolute

distances of the part to the datum from being determined. With
the two-position method chafiges in the distance between the
datum and part can be obtained from which the out-of-
straightness can be found.

3 Variable errors
This follows a similar pattern to that of systematic errors
except that multiple probes are used rather than multiple
orientations. The reason for this is that repetition cannot be
guaranteed. Variable errors have to be removed instantaneously
as they occur so that they are not allowed to infiltrate the,
signal. This is particularly important in the in situ measure-
ment of parts. Usually because of size or inaccessibility a part
may not be able to be moved to be measured. This poses a
problem of how to measure it properly without the accurate
instrument datum, e.g. no precise axis of rotation. The part
may have to be turned relatively crudely. The variations
produced by this can be considered to be random movements
of the axis of rotation relative to the measuring head, together

533
D J Whitehouse
with random angular variations. The former appear as random alignment of the probes are considerably reduced. Mathe-
variations in the eccentricity of the part relative to the measur- matically this mechanical repositioning of the probe can be
ing head. It is these eccentricity effects which must be removed simulated by changing the harmonic weighting function from
in order to get a reasonable assessment. In what follows it will +
JV1= (ejn" + ejnni3 e-jn79, which is obtained from having
be assumed that time averaging is not viable. This means that + + +
to ensure that e cos 6 e cos (6 a) e cos (6 - p) = 0, to
it is not possible to wait for random effects to cancel out by Wz=(l -ejnn:3-e-jn"i3) which is obtained from a similar
integration; they may be nonstationary in the statistical sense. angular imposition. In weighting function terms the removal
Consider first the case of roundness. The probe configuration of random eccentricity errors corresponds to the formula
has to be such that any random movement of the part is not giving a zero value for the case where n = 1, i.e. Fc(1) =O. This
seen. This is equivalent to making sure that a movement of the is true for WI and W2. However, there is a considerable
part as a whole during the measurement cycle is not detected mechanical advantage to be gained by using the second
and that neither does it upset the signal. Suppose that there configuration. Summarizing, a very important point has
are two probes at angles of - a and p to a datum angle. If emerged from this analysis which is that multiprobe methods
the part, considered to be a simple circle, is moved an amount are subject to one more constraint than multi-orientation
e at an angle 6 the two probes will suffer displacements of methods. In the weighting function there has to be complete
+
e cos (6 a>and e cos (6 - p) assuming that e < 10-2Rwhere R suppression for n = 1 otherwise the variable errors would not
is the part radius, and providing that the movement is con- be removed. (This is not necessary in the multi-orientation
fined to a region, within 10-2R, about the centre of action of case; eccentricity errors corresponding to n = 1 can be removed
the probes. For an instantaneous movement not to be seen for each data set, the data aresimplycentred.) This constraint in
the multiprobe method is in effect a fixing of the probe angles.
ecos (S+a)tecos(6-,B)=O. (20) To minimize the harmonic suppression of this technique it is
This can only be true if ci=.i~+P, the diametral probe con- possible to use probes having different sensitivities (Ozono
figuration analysed in equations (4) and (5). 1974). The general weighting function W then becomes
To get more out-of-roundness information another probe is w=(l - a ejns-b e-jnb)
needed. Let this other one be the reference for angle and the (22)
two existing probes be at angles of --CL and p from it as in which has the same amplitude and phase characteristics as
figure 5. The multiprobe signals corresponding with equation equation (17) but a and /3 are constrained. A point to note is
(14) for the multi-orientation method become that because of these constraints imposed by the different
error removal methods the weighting function for the multi-
V1(8)=s(O)+e cos 6
probe method need not be the same as the same order multi-
Vz(8)= s(8 + a)+ e cos (6 a)+ (21) orientation.
v3(0)=~(8-p), e COS (6- p). One important consequence of the first-order constraint
Fc(l)=O is that the zeroth-order Fc(0) is not zero. Thus
To eliminate part movement some combination of the probe
output voltages must be chosen so as to make the terms in e 1 - a - b) # 0.
FC(0)= FS(Oj( (23)
The case a+ b= 1 is not allowed because this makes Q: and
./ Probe ' Probe 3
p = 0. Mechanically, equation (23) means that the multiprobe
method is necessarily sensitive to radius differences between
Probe parts. This apparent shortcoming can be made use of.
Measuring the average value of the combination signal over
one revolution gives Fc(0). This can be converted to Fs(0) by
dividing by (1 -a-b). This in turn is the difference in radius
between the part under test and the radius at which the probe
configuration gives zero output, i.e. the radius value for which
Fs(0)= 0. The range of this radius measurement depends only
on the range of the transducers and the accuracy depends on
Figure 5 Variable error removal in roundness
how accurately any one radius has been calibrated relative to a
master part. This new technique is inherently more accurate
and 6 zero. Notice that e represents an error movement at a than conventional methods because it is substantially indepen-
particular moment in time; it is not dependent on 8, unlike dent of random errors in the measuring system. In the multi-
e ( @ in equation (14). The zero condition can be met if the orientation case the radius terms cannot necessarily be main-
probes are equispaced around the circumference at 120" but tained from one run to the next in sequence because of drift
this has a serious disadvantage which is that they totally etc,so for every orientation the dataare preprocessed to remove
enclose the part. However use can be made of the fact that a the zero-order radius term (and the eccentricity term),
movement of a part always affects at least two probes in the How the probe angles are fixed in the general case can be
opposite sense. This is because of their spatial configuration worked out from equation (23) with n= 1. Thus
around the part. By moving one probe through 180' and
l=a2+b2+2abcos (a+,B) (24)
changing its direction of sensitivity by 180" the same result can
be accomplished; and the component is no longer totally a= cos-1 (1 - 62 + a2)/2a
(25)
enclosed ! p = cos-1 (1 - a2 + b')/26.
This new feature is of great importance in measuring the
roundness of large inaccessible objects to a high harmonic From these expressions some relationships can be determined.
content, in situ. The advantage of having a probe assembly Forol+p=rr/2,a2+b2=1 andfor a+,B<rr,a+b>l. Should
subtending a total angle of less than w cannot be over- there be any confusion, the vee block method (Witzke 1967) is
emphasized; it means that the probes do not have to be fitted not the same as the multiprobe method. It is essentially a low
to a split collar. Consequently the problems of register and order lobe measuring technique.

534
Aspects of error separation techtziques
There is an important practical limit to using Fourier Notice that equation (28)corresponds to the simple formula for
synthesis. This is caused by the random angular movement the third numerical differential where the measured ordinates
between the part and the probe system. On a typical instru- f i , f 2 etc have been replaced by probes. Thus,
ment this restricts the order of coefficients to less than one
hundred. Such movement not only smoothes out the higher h3f”’=:(f+2-2J1+2f_l-f-2). (29)
coefficients but more dramatically introduces phase variations.
Problems of removing or identifying random errors also exist In this case there is a gap of 2h between V2 and V3, h is the unit
in surface texture, straightness, flatness, etc. of spacing between probes. Making all the probes equidistant
In linear and other measurements there are equivalent gives a = - 3, b = 3, c = - 1 which reduces the overall distance
problems to removing random variations in the eccentricity of from 4h to 3h.
the part relative to the datum. What needs to be removed is Such a system has a harmonic weighting function W given
often not best expressed in terms of Fourier coefficients. For by
example in straightness there may be variations in the linear W=(e-j2flLl/h+a e-j2nZ2!h+b ej2nZs!h+c ejZnZ4:A).
distance between the part and the datum, there may be varia-
tions in the tilt between the two and there may even be relative (30)
curvature variations. Such a situation involving all these might W is important when texture is being measured; it has a
arise in measuring the surface texture on a moving flexible slightly different form to the roundness weighting function
sheet of material for example. because a general wavelength h is used rather than a wave-
number M. This is due to the continuous nature of the spectrum
4 Generalized probe configurations for variable errors of texture rather than the discrete roundness spectrum.
Because of the diversity of problems of this nature met within However, despite the fact that the number of values of h
metrology it is necessary to identify a more generalized needed to characterize texture is greater than the number
configuration. Take the variations mentioned above for of wavenumbers required to describe roundness, harmonic
instance: the distancey between the datum and the test object suppression is less serious.
can be expressed as a random variable in time. Thus Obviously the usefulness of this more generalized approach
y = d ( t ) + m ( t ) x + c(t)x2 (26) depends upon being able to formulate the random variables
into a suitable power series. Dependent on this formulation
where d, m, c are independent random variables representing the total number of probes to be used can be increased to meet
average separation, tilt and curvature respectively, x is distance. any foreseeable requirement.
To eliminate this number of random variables four probes One very simple example of this technique is in the measure-
are needed in the general case. The probes would have sensi- ment of thin film deposits. Instruments for measuring these
tivities of 1, a, b, c and they would be at distances 11, le, 13 and 14 generally have to have gains of 106 in order to evaluate steps of
from the centre of the measuring system, figure 6. Three a few nanometres. At such sensitivities random vibration
equations need to be satisfied between the probe and the workpiece is critical. Consideration
1+a+b+c=O (274 of the mechanics shows that this random vibration manifests
itself as variations in the average distance between the pickup
- 11- a12 + bh+ cl4= 0 (276) and workpiece. Therefore the effect can be removed by using
h2+ ah2+ bb2 + ch2= 0. (27c) two probes instead of one. These probes should be coupled
differentially, any variation in gap acts on both probes simul-
Equation (27a) takes account of average separation, (27b) the taneously. But a genuine step height on the surface affects
tilt and (27c) the quadratic terms. The tilt term has to have odd each probe sequentially during traverse. It is possible to
extract the necessary step height from the record without the
need to synthesize the profile from Fourier coefficients as in
roundness. Two independent probes are needed if micro-
texture is required.
There are other types of measurement which also do not
require a frequency synthesis; typical examples are in straight-
ness and flatness. In these the differential form of probes can
be used to measure curvature etc from which straightness or
Test piece flatness may be determined.
Figure 6 Numerical analysis configuration for removal of
general variable errors. A four-probe example
5 Limitations
5.1 Multi-orientation techniques
Because a time element is involved, environmental effects are
symmetry about the midpoint of the carriage, hence the very important. Standard room conditions are desirable with
negative signs in the equation. Obviously just the same order of temperature, air flow and vibration controlled. Positional
systematic error can be determined by having multishifts accuracy of the respective orientations is very important both
instead of multiprobes. Solving these equations reveals a very in the axial and angular coordinates. The degree of accuracy
useful fact that formulae for numerical differentiation can be depends on the spatial bandwidth of the specimen being
used in multiprobe technology. For example, consider the case measured. Chetwynd and Siddall (1976) have carried out a
where 11= 2h, 12 = h, 13= h and 14 = 2h. The probe combination detailed examination of the potential of the multi-orientation
signal to satisfy equations (27)with these constraints imposed is technique based on the theory outlined above. They find that
given by C where considerable improvements in roundness measurement are
now possible; none of the problems mentioned are insuperable.
c=V1-2V2+21/3- v
4 (28) They even suggest that improvements greater than ten to one
and U= -2, b=2, c= -1. in accuracy o17er conventional methods should be attainable.

535.
D J Whitehouse
5.2 Multiprobe methods It seems highly likely that once probe technology has been
This technique is inherently more susceptible to instrumental improved there will be more scope for such methods. From
rather than environmental errors. In particular the probes what has been found to date only three or four probes will be
have to follow the same path as closely as possible, collinearity really practical; any more will present severe problems of
of less than 20 pm is not an unrealistic demand. Another point instrumentation. Also there is likely to be only a limited benefit
which needs attention is that of making sure that during a in having more probes. This is because errors produced by
traverse all the probes are always fully engaged. No probe can vibration, dirt, etc can generally only give rise to either tilts or
be allowed to hang free otherwise the instantaneous compensa- shifts, for which three probes will suffice.
tion provided by the technique will be lost. Also within the Notwithstanding other issues the major factor influencing
instrument system the matching of transducer frequency the use of these techniqes has already taken place. This is the
responses, linearities, temperature coefficients etc is vitally viable use of minicomputers in metrology (Kinsey and
important. Fortunately in most systems attempting such Chetwynd 1973) and the availability of fast computer
measurements one or two per cent is usual. Another problem algorithms for frequency synthesis.
with multiprobe systems is that of calibration of the whole
configuration, especially dynamically. Acknowledgments
Errors in Fourier synthesis caused through angular varia- I am grateful to the Directors of Rank Taylor Hobson for
tions, numerical noise etc are important in both methods. permission to publish this paper. I wish to thank also my
Such problems are currently being evaluated. colleagues in The Research and Development departments for
many useful discussions.
6 Conclusions
It has been shown how a multiprobe system can be designed to References
eliminate any desired order of variable error, Also it has been Bryan J, Clouser R and Holland E 1967 American Machinist
shown how multi-orientation methods can be devised to 612 149-64
isolate and identify systematic errors. Both techniques have Chetwynd D G and Siddall G J 1976 J. Phys. E: Sci.
been analysed to reveal ‘blind spots’ caused by harmonic Instrum. 9 537-44
suppression. Suitable harmonic weighting functions have been
Donaldson R R 1972 CIRP Annals 1’01 21 No. 1 pp 125-6
derived to illustrate such shortcomings. Representative
practical examples have been given of both techniques. Iizuka K and Goto M 1974 Proc. Int. Conf. Prod. Engng,
The use of such techniques gives a number of metrological Tokyo (Tokyo: Japan Society of Precision Engineering)
advantages not immediately obvious. First is the increase in part 1 pp 451-6
the accuracy of measurement - in roundness gains of up to one Hoffrogge C: Mann R and Rademacher H J 1972
million are now attainable and step height measurement can be Messtechnik 9 263-6
made having RMS noise levels of less than 1 nm as in figure 7. Kinsey D and Chetwynd D G 1973 Acta IMEKO VI,
Dresden vol 2 (Budapest : Hungarian Academy of Sciences)
pp 601-2
Mechanical loop Ozono S 1974 Proc. Int. ConJ Prod. Engng, TokjBo (Tokyo:
o f instrument
Japan Society of Precision Engineering) part 1 pp 457-62
Spragg R C 1972 Proc. Joint Measurement Conf. (Colorado:
National Bureau of Standards) pp 137-46
Spragg R C and Whitehouse D J 1968 Proc. Inst. Mech.
Engrs 182 397-405
Random noise
source Thwaite E G 1973 Messtechnik 10 317-8

-
/ Support
Whitehouse D J 1974 Proc. Int. Conf. Prod. Engng, Tokyo
(Tokyo : Japan Society of Precision Engineering) part 2
pp 39-44
Witzke F W 1967 Proc. Inst. Mech. Engrs 182 430-7

++l
LJ-J-+?
+4 ----LAr
0 025 pm

Figure 7 Random noise removal in step height

measurement

This implies a signal to noise level of about 2 to 3 % even with

steps of a few nanometres. These noise levels are close to the
theoretical values expected for Johnson and flicker noise with
transducers and amplifiers currently in use (Whitehouse 1974).
Second is the possibility of relaxing datum accuracy (Chet-
wynd and Siddall 1976). These considerations have meant
that in situations where a part cannot be accurately rotated
for assessment the multiprobe method can be used. Measuring
slideways in situ is now also possible with all the added speed Journal of Physics E: Scientific Instruments 1976 Volume 9
and convenience associated with probe methods. Printed in Great Britain 0 1976

536