- Email: [email protected]
Laser measurement of form and dimensions of transparent tubular elements
Measurement ELSEVIER
Measurement 13 (1994) 13-22
Laser measurement of form and dimensions of transparent tubular elements Richard Jablonski*, Marek Dzwiarek Center of Metrology and Measuring Systems, Warsaw University of Technology, Chodkiewicza Str. 8, PL-02 525 Warsaw, Poland
Abstract
A laser scanning-reflection method enables to measure the total geometry of transparent tubular objects. By using an optimum approximation function, introducing an iteration procedure and optimizing the number of sampling points on the circumference an accuracy of 3 lam for the outer and 5 lain for the inner surface is obtained. The results are given by two correlated roundness graphs and mean values of both radii. The measurement is noncontact, does not need any reference standard and can be used for in-process control.
Key words: Geometrical measurement; Transparent workpieces; Scanning-reflection method; Approximation
1. Introduction
Modem technology very often requires fast, accurate and noncontact measurement of the dimensions and form of various products--also in the fabrication process. Especially high demands are made on optical and electro-optical elements and components, such as rings, lenses, slabs, flat and shaped plates, rods, tubes, etc. The number of instruments measuring both dimensions and form is very limited. Except for coordinate measuring machines (CMM) and some very special systems, the measurement of form is performed on stylus instruments (see ISO6318) and that of dimensions on gauges of various types. Recently developed noncontact optical probes, e.g. [1,2], can replace the stylus pick-ups but still the measurement of long tubes or in-process control is impossible. Among the methods specially developed for the *Corresponding author. 0263-2241/94/$7.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0263-2241 (93)E0019-O
measurement of transparent tubes only three have been published. The best known, the reflection method [3], uses a collimated beam from a He-Ne laser, which is incident on an object perpendicular to its axes and laterally offset, so that reflections from the outer and inner surfaces are obtained. These are collected by the imaging lens, converged to a focus at the image plane and detected by the detector array (Fig. la). The image of the reflections appears as two lines separated by a distance which is a function of the changing wall thickness (obtained accuracy is 5 Bm). The immersion method [4], consists in submerging an object in a fluorescent liquid, illuminating it with a laser beam and analyzing the contrast distribution. The accuracy obtained is about 10 pro, but the proposed procedure precludes wider application of this method, considering the necessity of washing and drying. In the shadow method [5] the tube is illuminated through a rectangular aperture and on the detector plane the shadows of inner and outer surfaces are measured. The accuracy obtained is
14
R. JablonskL M. Dzwiarek / Measurement 13 (1994) 13-22
ILens _.__-__~_~.1goe Dete
CuSe2
~ane
ArrayT splitter Alignment ] cube
7~ector
etector tray
Detector Fig. 1. Development of reflection methods: (a) basic lay-out; (b) with V-block;(c) interfacemethod; (d) combined method. 0.5%. None of the above described methods can measure the full geometry of a tube.
2. Development of scanning and reflection methods The reflection method has some modifications: the laser beam incident on the object can be collimated or focussed, the reflections from the inner and outer surfaces are collected by the imaging lens and converged to a focus at the image plane or are falling on the linear photodiode array directly. When the measurement of absolute diameters is required the position of the tube axis with respect to the incident beam must be defined (y). To fulfill this condition a V-block can be introduced as a reference basis and then the outer surface of a tube also becomes a basis (Fig. lb). It was proved that the transformation function is then represented by a set of implicit equations of 5th order, significantly increasing the computing time, which is followed by worsening of the accu-
racy (30 gm). In addition, the measurement is not purely noncontact since the tube is slightly pressed against the V-block [6]. When the diameter of the collimated laser beam is large and no imaging lens is used, the reflection method becomes an interference method (Fig. l c) and an interference pattern is observed in the detector plane. Big errors of cylindricity and coaxiality and also irregularities and impurities of the material cause significant deformations of the pattern, making its interpretation impossible. The results obtained [7] showed that this method can be used only for high-quality products and only for relative or quantitative measurements. When an accurate, contactless measurement is required, it is very convenient to apply a combination of the reflection method for dimensional measurement and the shadow technique for determination of the axis position (Fig. l d). The shadow technique is well known in fiber manufacturing processes, mostly for measuring the fibre diameter and position. It was proven that this technique can be used also for measuring larger diameters and displacements (up to a few hundred millimeters) of nontransparent or transparent elements [8]. All methods and instruments described above allow to measure only the dimension and wall thickness in one or two diametrically located points. The idea of combining two methods was further developed and finally resulted in the laser scanningreflection technique [9], providing automatic measurement of external and internal roundness, curvatures and wall thickness. The principle of the scanning-reflection method is given in Fig. 2a. An object to be measured, defined by outer and inner radii RD and Rd and refractive index n, rotates step by step relative to a stationary measuring set-up (or vice versa). A focused laser beam is scanned at a constant velocity perpendicular to the object. There are three dominant rays ( 1, 2, 3 ), which represent the two reflected waves from the outer (ray 1) and inner (ray 2) surfaces and the wave tangent to the outer surface (ray 3). These are collected by the detector lenses Lm and LD2 and converged to a focus in the planes of detectors D~ and D 2. During scanning (see Fig. 2b), first a signal appears at the output of detector D 2 (ray 3) and then detector D1 registers rays 2 and 3. By
K Jablonski, M. Dzwiarek / Measurement 13 (1994) 13-22
(a)
15
far. In the polar coordinates system, under the assumption that the axis of the object is perpendicular to the system and goes through its centre and that nominal values of outer and inner radii are Rno and Rdo, this cross-section can be described by the following set of equations:
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
L
Xd RD(a ) = RDO , Rd(a ) = Rdo ,
where a is the amplitude, RD(a) the radius vector of the outer surface, and Rd(a ) the radius vector of the inner surface. The complete measurement of the cross-section may have been done during one positioning through a single reading of XD and Xd. In the case of real surfaces the procedure is far more complex since their profile is the resultant of a theoretical profile and of roundness errors. Therefore, the measurement must bc performed for the entire cross-section and thus rotation of the object or the measuring set-up is necessary. Real surfaces can be most conveniently described by the Fourier series:
(b) [D2 I,~-I I ~d IF - ~ c ° ~ ',
o,
'
(2)
-~
5
RD(a) = RDo + ~ Mj sin(ja + ~bj), Fig. 2. (a) The principle of the scanning-reflection method. (b) The signals from the photodetectors.
j=1
5
(3)
Rd(a) = P~0 + ~ Mj sin(ja + ~bj). measuring the times AZD and ATd and assuming constant scanning velocity v,, the outer and inner radii of curvature of the measured object can be easily determined from: RD = xD( 1 -- sin A/2)- 1, Rd = RI~ sin G[n sin(G + A/2 - B)]- 1,
(1)
where B = arcsin ( 1 - Xd/RD), G = arcsin (sin B/n), X D -----Vs~TD,
Xd ~
VsAT d.
It is important to notice that the detector D 1 is designed to receive only rays parallel to its axis.
3. Recording of the profile of real surfaces Cross-sections of objects of ideal tubular form have been considered in the analyses conducted so
j=l
The accuracy of this representation depends on the number of harmonics j. It is obvious that i f j increases, the calculation procedure is prolonged. During the research, samples of tubes made by means of various technologies, from different types of material and by diverse manufactures were analyzed up to j = 20. The results obtained were so different that it was decided to establish the optimum number of harmonics for various types of products exclusively via experiments by measuring the series of samples. The following procedure was suggested: (1) carrying out the measurements; (2) analyzing the results by harmonics; (3) defining the decisive amplitudes; (4) comparing the results with the required measurement accuracy and attainable accuracy of the method (i.e. presently ca 2 ~m). To prove the conformity of the procedure, a series of drawn quartz tubes were tested. Their outer dimensions were 18-22mm and inner
16
R.
JablonskL M. Dzwiarek / Measurement 13 (1994) 13 22
12-19 m m (measurements were carried out using Talyrond TR-250, R T H ) . The following results were obtained: M1 < 100 ~tm,
M2 < 50 ~rl,
M 3 <
k
I
R
j
~o,
30 ~tm, (4)
M4 < 10 tam,
M5 < 5 ~tm.
The optimum number of harmonics was j = 5. Further on in the paper these results will constitute reference data serving to estimate the suitability of the particular approximation procedures. The measurement method requires a stepwise rotation of the object around an axis situated close to its geometrical axis by a constant angle da = 2rtm, where m is the number of measurement cycles. In each position xo and xd are measured. The procedure is repeated m times until the object assumes its initial position. For each cycle a set of equations is formulated which describes the dependences between the locations of reflected laser spots and the radius vectors of the measurement points. In order to determine the value of the radius located between the measurement points, the profile is approximated by segments. Approximation functions should be selected in such a way so as to ensure the continuity of approximation at the edges of the intervals and to secure the assumed measurement accuracy. The real paths of rays reflected from the outer and inner surfaces are shown in Fig. 3. Based on the geometrical dependences for the ith position of the object two sets of equations can be formulated: (1) For the outer surface (see Fig. 3a): RD1 cOSfD1 --
RD2 sin(A/2
+fD2)
=
-
I
~RD
RD 1
~a
- - - -
)~t
/ -
-
Fig. 3. Geometrical analysis of real paths of rays: (at reflected from the outer surface of the object; (b) reflected from the inner surface of the object.
RD1 cosfD
Xoi,
1 --
Xdi
= R 3
sin(Be+f3),
R2 R1 sin G1 - sin(Gx + be)'
A/2 +fDe = re/2 -- (a 2 - - a~ - f o l ), tgfm =
/
al
'
(5)
G1 + b2 + J2 = G2 + b3 -J'2, rt/2 + f r o - bl + B1 - f l = A,
- -1 o _~RD _ a2' t g f D e = -- RDe Oa
where a is the amplitude, Ro(a) the outer radius and f(a) the angle between the normal to the surface and the radius vector. (2) For the inner surface (see Fig. 3b): R2
sin G2
R3
sin(G2 + b D '
B2 + f 3 + bl + b2 + b3 = rt/2 + f r o , sin(G1 + f l ) = (sin B1)/n, sin(G2 - J ~ ) = (sin B2)/n, where
1
~RD bl+al
tgj~ = -- R~" ~a
'
(6)
R. Jablonski, M. Dzwiarek / Measurement 13 (1994) 13-22
17
R tgf2=
R2
tgf3=
R3
~a bl+b2+,l
1 ~RD I
~a b~+bz+b~+~"
After each full rotation of the tube a set of m sets of equations is obtained. Their solution provides distances of points on the surface of the measured object from the point of rotation. On the basis of that the mean-square circles as well as the values of both surfaces' form errors can be determined. The proper choice of the number of measurement cycles m has a similar influence on the measuring accuracy and the time of handling the data as the establishment of the number of harmonics. To optimize m, the processing of data within the range m = 15-200 was performed, using the simplest way of approximation. It became evident that an increase of m is accompanied by an exponential increase of the time for computation and only an insignificant rise in the measurement accuracy. Eventually, it was decided that for forms of objects described by parameters (4) it suffices to assume m = 36 for mean-accurate measurement and m = 72 for accurate measurement.
Fig. 4. Examples of radius variation approximation: (a) by fragments of a spiral of Archimedes; (b) by fragments of a cubic spiral ensuring the continuity of the derivative on the boundaries of approximation intervals.
of the approximation ranges (in some cases additional conditions may be formulated). The presented results indicate that for meanaccurate measurements the simplest approximation by means of the spiral of Archimedes may be applied. If exact determination of the radius of the mean-square circle is essential, then the approximation by means of the cubic spiral should be used; and for the determination of form errors only the approximation by means of the quadratic spiral is sufficient.
5. Solution of the set of equations 4. Approximation of the profile From the various possible types of approximation four most characteristic ones were taken into consideration, namely: the spiral of Archimedes, exponential spiral, quadratic spiral and cubic spiral (see Appendix). Figure 4 shows the results of two exemplary procedures of approximation for considerable variations of the radius of curvature and small quantization density to emphasize the differences between the approximations. Each method of approximating the radius variations between the ith and (i + 1)th measuring point must fulfill the conditions: RaP(0) = R i ,
RaP(da) = Ri+ 1 •
(7)
These are the conditions of continuity on the edges
The measured surface is given by a set of m equations. These are implicit equations, strongly nonlinear and only an approximate solution of them is possible. From the well-known methods of solving this type of equations the iterative method was chosen. The application of this procedure to the discussed problems proves efficient for all objects satisfying conditions (4). First, the outer radius is determined on the basis of Eqs. (5). For this purpose the initial point is assumed: 72
Y xDk R~ =
k=l 72[ 1 -- sin(A/2)]
(8)
and substituted into Eqs. (5). In this way the values of right sides of these equations are found, ~ , where i is the number of the measured point
18
R. Jablonski, M. Dzwiarek / Measurement 13 (1994) 13-22
and k is the number of the iteration. Corrections Pi to the value Xo are as follows: Pi = Xo - x~.
(9)
After having taken these corrections into consideration the next approximation of the searched values R/k, R/k = R/k-1 -[-Pi,
(10)
is being iterated. The process is being repeated until satisfactory results are obtained. The accuracy of the results may be satisfied on the basis of the value of the determined corrections. For each k it may be written: P~ < P . . . .
( 11 )
where Pmax is the assumed accuracy. As soon as this condition is satisfied, the iterative procedure can be terminated, and the solution will be as follows: (12)
R,=R~.
The outer radius having been determined, the inner radius is being found in an analogous way (from Eqs. (6)). In practice it turned out that the simple application of Eq. (10) not always ensures the convergence of the iterative process. An attempt was thus undertaken aiming at its modification. To this end a theoretical surface of a constant radius R~ = 10 mm was assumed and from Eq. ( 1 ) for A = rt/2 the value of xo~ was calculated: XDj = Ri[1 -- sin (14/2)] = 2929 pm.
(13)
Then, a simulated deformation of the form: R2o = 10000 lam + 1 lam = 10001 ~tm
(14)
was introduced and substituted into Eqs. (5) and finally its influence on the position of spots xD was tested. The obtained course of changes is shown in Fig. 5. It can be seen that the alteration of the radius RI) causes t h e biggest change in the positions of spots XD(i_9) and XD(i_lO). The changes are considerably greater than the introduced deformation and equal respectively: PlO=-10~tm,
pll=llp.m
(p9=l~m).
(15)
I I~10)--10
( c,.
Fig. 5. Influence of radius R D alterations on the position of
spot xt~.
Compensation of this deformation requires modification of Eq. (10) to the form: R~ +1 = R~ + (pl- lo/20) - (p~ 9/20).
(16)
An analogous dependence is obtained for the inner surface. It should be noticed that Eq. (16) was derived for A = rt/2; at another angle of observation the displacement between the introduced radius deformation and the change in the spot position will be different. For the inner surface this displacement depends also to a small extent upon the difference between radii RD and Rd.
6. Errors A full error analysis comprises the inaccuracies of measuring Xd (exd) and x o (exo), inaccuracy of setting the angle A, inaccuracy of angular sampling da, errors introduced by the approximation method and data processing (described above) and instability of the index of refraction. The first two errors mainly depend on the rotational stability of the drive unit and also on aberrations of the scanning lense, laser spot size, diffraction effects occurring at the beginning of intersection, perpendicularity error (perpendicularity between scan plane and object axis), laser beam intensity fluctuations, position of the object within the measuring area and error of the timing measurment circuit (these problems were discussed in Ref. [9]). In the experimental set-up a scanning unit was driven by three-phase synchronous motors. The stability of rotation was better than 0.05%, and the errors obtained were: exd = exo = 0.9 lam. The inaccuracy of setting A is about 30 arcsec and it was assured by means of a polygon mirror and
R. Jablonski, M. Dzwiarek / Measurement13 (1994) 13-22 autocollimator. The refractive index of a tube was given with an accuracy of 0.1%. The obtained accuracy is 3 Inn for the outer and 5 lain for the inner surface.
19
Appendix--Four types of approximations
A.I. Spiral of Archimedes The easiest way of approximation is by the spiral of Archimedes described by the formula:
7. Conclusion
a
RaY(a) = Ri + (Rt+l - Ri) -~a' The modification of Eq. (10) can be performed only under a significant change of parameters of the measured objects. During experiments such necessity did not occur. An additional source of error is the eccentricity which results from the displacement of the axis of rotation of an object measured relative to the centre of the mean-square circle. The smaller the eccentricity, the more the angle between the radius vector and the normal to the surface departs from n/2, and therefore the calculational error is greater. The method can be used satisfactorily for fast noncontact measurement of transparent objects. It is possible to extend the number of applications, for example to the measurement of the outer radius of nontransparent objects, or to interrupted surfaces.
References [1] M.C. Leu and R.M. Pherwani, Vision system for threedimensional position measurement based on stereo disparity, Opt. Laser Technol. 21(5) (1989) 319-324. [2] A.J.C. Brown, Surface testing with a second generation optical profilometer, IMEKO TC Set. 28 (1990) 213-223. [3] J. Sanchez, R.W. Heebner, D.H. Smithgall and L.S. Watkins, On-line measurement of glass thickness deposited in the MCVD fiber preform process, IOOC '81, San Francisco, CA, April 28, 1981, paper TUG2. [4] H.M. Presby, Geometric measurement of preform rods and starting tubes, AppL Opt. 19 (1982) 3528-3530. [5] W.A. Pilipovitz, A.K. Esman, W.K. Kuleshov and W.P. Dubrovski, Noncontact optical method of measurement of inner diameter of transparent tubes (in Russian), Izmer. Tekh. N. 6 (1990) 13-14. [6] R. Jablonski, A laser reflection method for transparent tube measurement, Proc. 1st Int. Symp. on MTIIP, Wuhan, China, 1989, pp. 109-116. [7] R. Jablonski, Report 17/Ci/87, OTO, Poland, 1987. [8] R. Jablonski, Scanning laser measurementof fibre manufacturing process, VDI Ber. 761 (1989) 93-99. [9] R. Jablonski, A scanning-reflectionmeasurement of curvature and thickness, IMEKO TC Set. 28 (1990) 129-134.
(A.1)
where 0 < a < da, Ri is the value of the radius determined by the ith point, da = 2n/m is the rotation angle between successive measuring points, a n d / P P ( a ) is the approximate value of the radius observed at an angle (i - 1)da + a. Taking Eqs. (3), (5) and (A.8) into account, it m a y be written:
IR[(i-1)da+a] -Rav(a)l~
~
j2Mj • (A.2)
As can be seen, the accuracy of approximation depends above all upon da, that is the number of measurement cycles, and the number and value of harmonics occurring in equations R = R(a) defining the surface of the measured object. For example, for the reference parameters of Eq. (4) and m = 72 we get the foUowing: AR = IR[(i - l ) d a + a] - RaP(a)l < 0.8 ~ra. (A.3) The obtained result proves that the error of radius approximation is far lower than the assumed measurement error and practically does not influence the accuracy of calculations. On the other hand, the error of Af, the angle between the normal to the surface and the radius vector, becomes essential. Making use of the relation defining the angle between the radius vector and the normal to the surface in polar coordinates as well as utilizing Eqs. (3), (7) and (A.1), we arrive at:
\[ Z j M j c o s [ j ( i - 1 ) d a + j a + q ~ j ] ) f(a) = - atn / J= 1
R
'
(A.4)
R. JablonskL M. Dzwiarek / Measurement 13 (1994) 13-22
20
iZ Ri+ I -- Ri" ~
(A.5)
f~P(a) = - atn ~d--~. ~%-~(a)),
;da E jZMj A f = Jf~P(a) - f ( a ) l ~<
j= i 2Ro
(A.6)
The achieved value may exert a considerable impact on the accuracy of calculations, especially when measuring objects for whose description higher harmonics are needed and objects with small mean radius R o. Under assumed da and M values and for Ro = 10 mm, we have: Aft< 0.0036 rad.
(A.7)
The change of angle f on the boundaries of successive ranges of approximation poses an additional problem. It entangles the solution of Eqs. (5) and (6) in the case when the scanned ray falls in the vicinity of the boundary between the ith and the (i+ 1)th area. It is particularly essential at measuring the inner radius. The quantity of this change is given by the following relation:
Errors arising from the approximation by means of the spiral of Archimedes are illustrated in Fig. A.1. Their maximum values are: A ~ = 2.309 ~tm, ADi = -- 1.269 ~tm, Ad~ = 5.508 tam, Ad~ = --2.514 ~tm. (Indexes e and i designate outer and inner measured surface, respectively.) They consist of two components: ARc, the accuracy of roundness errors measurement, and ARm, the accuracy of mean-square radius measurement and may be roughly determined from the relations:
(d2 - d, )/2 ~
(A. 11 )
The mentioned quantities for the object satisfying conditions (4), derived from Eqs. (5), (6), (A.11) and compared with theoretical values are: For the outer surface: - 1.9 }am < ZlSeD < 1.9 ~tm,
Z~RmD =
0.51 pm; (A.12)
(a)
['Ri+2-- Ri+l"~
f~P+l(0) = - atn ! . . . .
1,
\ da'Ri+a J
(A.8)
f Ri + a -- Ri"~
fTP+ i (da) = - atn ! . . . .
I,
25 ],
\ da" Ri+ 1J
As = If~+Pa(0) -f~P(da) l ~<
R0
1.
(A.9)
It is evident that the change of angle on the boundary of approximation ranges depends as well mainly on the number of harmonics, and only in the second place on the radius Ro. For the parameters of Eq. (4) we achieve: As < 0.007 rad.
(A.10)
From (A.10) it follows that the alteration of angle on the boundary of ranges may be two times higher than the error of this angle. It is associated with the change of angle sign inside the approximation range.
Fig. A.1. Results ofcalculatingthe theoreticalelement approximated by the spiral of Archimedes: (a) outer surface; (b) inner surface.
R Jablonski, M. Dzwiarek
continuity on the boundaries of successive ranges of approximation:
and for the inner surface: -4.1 p m < A R ~ < 4 . 1 pan,
21
Measurement13 (1994) 13-22
Aarad= 1.5 gm. (A.13)
~R~ p (a) Oa
0R~P+ 1 (a) a=da --
-~a
(A.17) a=O"
As it can be noticed, the results for the outer surface are contained in the permissible limit of error, whereas in the case of the inner surface the limit is considerably exceeded (though these are only errors of approximation, while other factors are not taken into account).
This condition, however, is admissible only for an odd number of measurement cycles m. (When the number m is even, the condition requires an additional limitation:
A.2. Exponential spiral
i=1
The approximation by the exponential spiral is provided by the following equation:
which is not always fulfilled.) A more beneficial condition is obtained by the extension of the approximation to the (i + l)th range and then:
( - 1 ) i . Ri = 0,
RaP(a) = R i exp
log(Ri+ 1/Ri) •
(A.14)
It is the approximation preserving a constant value of the angle f throughout the whole range of approximation: f~P(a) = - atn [l°g (R~---~/R')].
(A.15)
The obtained results are close to the ones achieved by the approximation by the spiral of Archimedes, yet the calculational procedures are more complicated, and the time of handling the data is significantly prolonged. Therefore, the approximation by means of the exponential spiral was considered inadvisable. A.3. Quadratic spiral
Far more advantageous is the approximation by the quadratic spiral. It is described by the following equation:
aP 2, RaP(2da) = Ri+
al =
ao = Ri,
4Ri + 1 - 3Ri - Ri + 2 2 ' Ri+2 -
a2 =
(A.18)
(A.19)
2Ri+l + Ri 2
If an analysis similar to the one used for the spiral of Archimedes is conducted, then the following is achieved: AR~<0.15~m,
Af~<0.001rad,
As~<0.001rad. (A.20~
The obtained accuracy of both the radius and the angle approximation are considerably higher, and the change of derivative at the edges of intervals is significantly lower than for the spiral of Archimedes approximation. The determined errors of form and mean square radius are: For the outer surface: - 1 lam
A ~ o = 1.3 lain, (A.21)
a
RaP(a)=ao+aldaa +a2 daa "
(A.16)
This type of approximation provides a supplementary degree of freedom which, apart from conditions (7), allows to add, e.g., a condition of angle
and for the inner surface: - 1 . 9 p m < ZJRer < 1.9 prn,
AR~ = 1.9 pro. (A.22)
These results show that the approximation by
22
1~ Jablonski, M. Dzwiarek / Measurement 13 (1994) 13-22
means o f the quadratic spiral with the extension of ranges may be utilized for the accurate measurement o f outer and inner surfaces.
intervals may be assumed. As a result we get:
ao=Ri,
al=l(Ri+l-Ri_l),
a2 = I ( R i + z -[- 4Ri+ 1 - 5Ri + 2 R i - 1 ),
a3 = ½(RI+2 - 3Ri+l + 3Ri - Ri- 1).
A.4. Cubic spiral Still more advantageous is the approximation by means o f the cubic spiral. It is described by the following equation:
a (ay R ( a ) = a o + a l d a +az ~
(A.24)
+a3 ~
•
(A.23)
This equation gives two additional degrees of freedom. Thus, e.g. apart from the conditions of continuity (7), two supplementary conditions for the continuity of the derivative at the edges of
It is the most accurate way of approximation. It is then easy to calculate that AR ~<0.03 ~tm, Aft< 0.0006 rad, As = 0.
(A.25)
The results for the conditions (4) are the following: For the outer surface: - 0 . 9 ~tm < Z1ReD < 0.9 I~m, ARm o = 0 . 6 5 ~tm; (A.26) and the inner one: -
1.6 lam < ziR~r< 1.6 ~tm, Z~Rmr = 0 ~tm.
(A.27)
Measurement 13 (1994) 13-22
Laser measurement of form and dimensions of transparent tubular elements Richard Jablonski*, Marek Dzwiarek Center of Metrology and Measuring Systems, Warsaw University of Technology, Chodkiewicza Str. 8, PL-02 525 Warsaw, Poland
Abstract
A laser scanning-reflection method enables to measure the total geometry of transparent tubular objects. By using an optimum approximation function, introducing an iteration procedure and optimizing the number of sampling points on the circumference an accuracy of 3 lam for the outer and 5 lain for the inner surface is obtained. The results are given by two correlated roundness graphs and mean values of both radii. The measurement is noncontact, does not need any reference standard and can be used for in-process control.
Key words: Geometrical measurement; Transparent workpieces; Scanning-reflection method; Approximation
1. Introduction
Modem technology very often requires fast, accurate and noncontact measurement of the dimensions and form of various products--also in the fabrication process. Especially high demands are made on optical and electro-optical elements and components, such as rings, lenses, slabs, flat and shaped plates, rods, tubes, etc. The number of instruments measuring both dimensions and form is very limited. Except for coordinate measuring machines (CMM) and some very special systems, the measurement of form is performed on stylus instruments (see ISO6318) and that of dimensions on gauges of various types. Recently developed noncontact optical probes, e.g. [1,2], can replace the stylus pick-ups but still the measurement of long tubes or in-process control is impossible. Among the methods specially developed for the *Corresponding author. 0263-2241/94/$7.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0263-2241 (93)E0019-O
measurement of transparent tubes only three have been published. The best known, the reflection method [3], uses a collimated beam from a He-Ne laser, which is incident on an object perpendicular to its axes and laterally offset, so that reflections from the outer and inner surfaces are obtained. These are collected by the imaging lens, converged to a focus at the image plane and detected by the detector array (Fig. la). The image of the reflections appears as two lines separated by a distance which is a function of the changing wall thickness (obtained accuracy is 5 Bm). The immersion method [4], consists in submerging an object in a fluorescent liquid, illuminating it with a laser beam and analyzing the contrast distribution. The accuracy obtained is about 10 pro, but the proposed procedure precludes wider application of this method, considering the necessity of washing and drying. In the shadow method [5] the tube is illuminated through a rectangular aperture and on the detector plane the shadows of inner and outer surfaces are measured. The accuracy obtained is
14
R. JablonskL M. Dzwiarek / Measurement 13 (1994) 13-22
ILens _.__-__~_~.1goe Dete
CuSe2
~ane
ArrayT splitter Alignment ] cube
7~ector
etector tray
Detector Fig. 1. Development of reflection methods: (a) basic lay-out; (b) with V-block;(c) interfacemethod; (d) combined method. 0.5%. None of the above described methods can measure the full geometry of a tube.
2. Development of scanning and reflection methods The reflection method has some modifications: the laser beam incident on the object can be collimated or focussed, the reflections from the inner and outer surfaces are collected by the imaging lens and converged to a focus at the image plane or are falling on the linear photodiode array directly. When the measurement of absolute diameters is required the position of the tube axis with respect to the incident beam must be defined (y). To fulfill this condition a V-block can be introduced as a reference basis and then the outer surface of a tube also becomes a basis (Fig. lb). It was proved that the transformation function is then represented by a set of implicit equations of 5th order, significantly increasing the computing time, which is followed by worsening of the accu-
racy (30 gm). In addition, the measurement is not purely noncontact since the tube is slightly pressed against the V-block [6]. When the diameter of the collimated laser beam is large and no imaging lens is used, the reflection method becomes an interference method (Fig. l c) and an interference pattern is observed in the detector plane. Big errors of cylindricity and coaxiality and also irregularities and impurities of the material cause significant deformations of the pattern, making its interpretation impossible. The results obtained [7] showed that this method can be used only for high-quality products and only for relative or quantitative measurements. When an accurate, contactless measurement is required, it is very convenient to apply a combination of the reflection method for dimensional measurement and the shadow technique for determination of the axis position (Fig. l d). The shadow technique is well known in fiber manufacturing processes, mostly for measuring the fibre diameter and position. It was proven that this technique can be used also for measuring larger diameters and displacements (up to a few hundred millimeters) of nontransparent or transparent elements [8]. All methods and instruments described above allow to measure only the dimension and wall thickness in one or two diametrically located points. The idea of combining two methods was further developed and finally resulted in the laser scanningreflection technique [9], providing automatic measurement of external and internal roundness, curvatures and wall thickness. The principle of the scanning-reflection method is given in Fig. 2a. An object to be measured, defined by outer and inner radii RD and Rd and refractive index n, rotates step by step relative to a stationary measuring set-up (or vice versa). A focused laser beam is scanned at a constant velocity perpendicular to the object. There are three dominant rays ( 1, 2, 3 ), which represent the two reflected waves from the outer (ray 1) and inner (ray 2) surfaces and the wave tangent to the outer surface (ray 3). These are collected by the detector lenses Lm and LD2 and converged to a focus in the planes of detectors D~ and D 2. During scanning (see Fig. 2b), first a signal appears at the output of detector D 2 (ray 3) and then detector D1 registers rays 2 and 3. By
K Jablonski, M. Dzwiarek / Measurement 13 (1994) 13-22
(a)
15
far. In the polar coordinates system, under the assumption that the axis of the object is perpendicular to the system and goes through its centre and that nominal values of outer and inner radii are Rno and Rdo, this cross-section can be described by the following set of equations:
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
L
Xd RD(a ) = RDO , Rd(a ) = Rdo ,
where a is the amplitude, RD(a) the radius vector of the outer surface, and Rd(a ) the radius vector of the inner surface. The complete measurement of the cross-section may have been done during one positioning through a single reading of XD and Xd. In the case of real surfaces the procedure is far more complex since their profile is the resultant of a theoretical profile and of roundness errors. Therefore, the measurement must bc performed for the entire cross-section and thus rotation of the object or the measuring set-up is necessary. Real surfaces can be most conveniently described by the Fourier series:
(b) [D2 I,~-I I ~d IF - ~ c ° ~ ',
o,
'
(2)
-~
5
RD(a) = RDo + ~ Mj sin(ja + ~bj), Fig. 2. (a) The principle of the scanning-reflection method. (b) The signals from the photodetectors.
j=1
5
(3)
Rd(a) = P~0 + ~ Mj sin(ja + ~bj). measuring the times AZD and ATd and assuming constant scanning velocity v,, the outer and inner radii of curvature of the measured object can be easily determined from: RD = xD( 1 -- sin A/2)- 1, Rd = RI~ sin G[n sin(G + A/2 - B)]- 1,
(1)
where B = arcsin ( 1 - Xd/RD), G = arcsin (sin B/n), X D -----Vs~TD,
Xd ~
VsAT d.
It is important to notice that the detector D 1 is designed to receive only rays parallel to its axis.
3. Recording of the profile of real surfaces Cross-sections of objects of ideal tubular form have been considered in the analyses conducted so
j=l
The accuracy of this representation depends on the number of harmonics j. It is obvious that i f j increases, the calculation procedure is prolonged. During the research, samples of tubes made by means of various technologies, from different types of material and by diverse manufactures were analyzed up to j = 20. The results obtained were so different that it was decided to establish the optimum number of harmonics for various types of products exclusively via experiments by measuring the series of samples. The following procedure was suggested: (1) carrying out the measurements; (2) analyzing the results by harmonics; (3) defining the decisive amplitudes; (4) comparing the results with the required measurement accuracy and attainable accuracy of the method (i.e. presently ca 2 ~m). To prove the conformity of the procedure, a series of drawn quartz tubes were tested. Their outer dimensions were 18-22mm and inner
16
R.
JablonskL M. Dzwiarek / Measurement 13 (1994) 13 22
12-19 m m (measurements were carried out using Talyrond TR-250, R T H ) . The following results were obtained: M1 < 100 ~tm,
M2 < 50 ~rl,
M 3 <
k
I
R
j
~o,
30 ~tm, (4)
M4 < 10 tam,
M5 < 5 ~tm.
The optimum number of harmonics was j = 5. Further on in the paper these results will constitute reference data serving to estimate the suitability of the particular approximation procedures. The measurement method requires a stepwise rotation of the object around an axis situated close to its geometrical axis by a constant angle da = 2rtm, where m is the number of measurement cycles. In each position xo and xd are measured. The procedure is repeated m times until the object assumes its initial position. For each cycle a set of equations is formulated which describes the dependences between the locations of reflected laser spots and the radius vectors of the measurement points. In order to determine the value of the radius located between the measurement points, the profile is approximated by segments. Approximation functions should be selected in such a way so as to ensure the continuity of approximation at the edges of the intervals and to secure the assumed measurement accuracy. The real paths of rays reflected from the outer and inner surfaces are shown in Fig. 3. Based on the geometrical dependences for the ith position of the object two sets of equations can be formulated: (1) For the outer surface (see Fig. 3a): RD1 cOSfD1 --
RD2 sin(A/2
+fD2)
=
-
I
~RD
RD 1
~a
- - - -
)~t
/ -
-
Fig. 3. Geometrical analysis of real paths of rays: (at reflected from the outer surface of the object; (b) reflected from the inner surface of the object.
RD1 cosfD
Xoi,
1 --
Xdi
= R 3
sin(Be+f3),
R2 R1 sin G1 - sin(Gx + be)'
A/2 +fDe = re/2 -- (a 2 - - a~ - f o l ), tgfm =
/
al
'
(5)
G1 + b2 + J2 = G2 + b3 -J'2, rt/2 + f r o - bl + B1 - f l = A,
- -1 o _~RD _ a2' t g f D e = -- RDe Oa
where a is the amplitude, Ro(a) the outer radius and f(a) the angle between the normal to the surface and the radius vector. (2) For the inner surface (see Fig. 3b): R2
sin G2
R3
sin(G2 + b D '
B2 + f 3 + bl + b2 + b3 = rt/2 + f r o , sin(G1 + f l ) = (sin B1)/n, sin(G2 - J ~ ) = (sin B2)/n, where
1
~RD bl+al
tgj~ = -- R~" ~a
'
(6)
R. Jablonski, M. Dzwiarek / Measurement 13 (1994) 13-22
17
R tgf2=
R2
tgf3=
R3
~a bl+b2+,l
1 ~RD I
~a b~+bz+b~+~"
After each full rotation of the tube a set of m sets of equations is obtained. Their solution provides distances of points on the surface of the measured object from the point of rotation. On the basis of that the mean-square circles as well as the values of both surfaces' form errors can be determined. The proper choice of the number of measurement cycles m has a similar influence on the measuring accuracy and the time of handling the data as the establishment of the number of harmonics. To optimize m, the processing of data within the range m = 15-200 was performed, using the simplest way of approximation. It became evident that an increase of m is accompanied by an exponential increase of the time for computation and only an insignificant rise in the measurement accuracy. Eventually, it was decided that for forms of objects described by parameters (4) it suffices to assume m = 36 for mean-accurate measurement and m = 72 for accurate measurement.
Fig. 4. Examples of radius variation approximation: (a) by fragments of a spiral of Archimedes; (b) by fragments of a cubic spiral ensuring the continuity of the derivative on the boundaries of approximation intervals.
of the approximation ranges (in some cases additional conditions may be formulated). The presented results indicate that for meanaccurate measurements the simplest approximation by means of the spiral of Archimedes may be applied. If exact determination of the radius of the mean-square circle is essential, then the approximation by means of the cubic spiral should be used; and for the determination of form errors only the approximation by means of the quadratic spiral is sufficient.
5. Solution of the set of equations 4. Approximation of the profile From the various possible types of approximation four most characteristic ones were taken into consideration, namely: the spiral of Archimedes, exponential spiral, quadratic spiral and cubic spiral (see Appendix). Figure 4 shows the results of two exemplary procedures of approximation for considerable variations of the radius of curvature and small quantization density to emphasize the differences between the approximations. Each method of approximating the radius variations between the ith and (i + 1)th measuring point must fulfill the conditions: RaP(0) = R i ,
RaP(da) = Ri+ 1 •
(7)
These are the conditions of continuity on the edges
The measured surface is given by a set of m equations. These are implicit equations, strongly nonlinear and only an approximate solution of them is possible. From the well-known methods of solving this type of equations the iterative method was chosen. The application of this procedure to the discussed problems proves efficient for all objects satisfying conditions (4). First, the outer radius is determined on the basis of Eqs. (5). For this purpose the initial point is assumed: 72
Y xDk R~ =
k=l 72[ 1 -- sin(A/2)]
(8)
and substituted into Eqs. (5). In this way the values of right sides of these equations are found, ~ , where i is the number of the measured point
18
R. Jablonski, M. Dzwiarek / Measurement 13 (1994) 13-22
and k is the number of the iteration. Corrections Pi to the value Xo are as follows: Pi = Xo - x~.
(9)
After having taken these corrections into consideration the next approximation of the searched values R/k, R/k = R/k-1 -[-Pi,
(10)
is being iterated. The process is being repeated until satisfactory results are obtained. The accuracy of the results may be satisfied on the basis of the value of the determined corrections. For each k it may be written: P~ < P . . . .
( 11 )
where Pmax is the assumed accuracy. As soon as this condition is satisfied, the iterative procedure can be terminated, and the solution will be as follows: (12)
R,=R~.
The outer radius having been determined, the inner radius is being found in an analogous way (from Eqs. (6)). In practice it turned out that the simple application of Eq. (10) not always ensures the convergence of the iterative process. An attempt was thus undertaken aiming at its modification. To this end a theoretical surface of a constant radius R~ = 10 mm was assumed and from Eq. ( 1 ) for A = rt/2 the value of xo~ was calculated: XDj = Ri[1 -- sin (14/2)] = 2929 pm.
(13)
Then, a simulated deformation of the form: R2o = 10000 lam + 1 lam = 10001 ~tm
(14)
was introduced and substituted into Eqs. (5) and finally its influence on the position of spots xD was tested. The obtained course of changes is shown in Fig. 5. It can be seen that the alteration of the radius RI) causes t h e biggest change in the positions of spots XD(i_9) and XD(i_lO). The changes are considerably greater than the introduced deformation and equal respectively: PlO=-10~tm,
pll=llp.m
(p9=l~m).
(15)
I I~10)--10
( c,.
Fig. 5. Influence of radius R D alterations on the position of
spot xt~.
Compensation of this deformation requires modification of Eq. (10) to the form: R~ +1 = R~ + (pl- lo/20) - (p~ 9/20).
(16)
An analogous dependence is obtained for the inner surface. It should be noticed that Eq. (16) was derived for A = rt/2; at another angle of observation the displacement between the introduced radius deformation and the change in the spot position will be different. For the inner surface this displacement depends also to a small extent upon the difference between radii RD and Rd.
6. Errors A full error analysis comprises the inaccuracies of measuring Xd (exd) and x o (exo), inaccuracy of setting the angle A, inaccuracy of angular sampling da, errors introduced by the approximation method and data processing (described above) and instability of the index of refraction. The first two errors mainly depend on the rotational stability of the drive unit and also on aberrations of the scanning lense, laser spot size, diffraction effects occurring at the beginning of intersection, perpendicularity error (perpendicularity between scan plane and object axis), laser beam intensity fluctuations, position of the object within the measuring area and error of the timing measurment circuit (these problems were discussed in Ref. [9]). In the experimental set-up a scanning unit was driven by three-phase synchronous motors. The stability of rotation was better than 0.05%, and the errors obtained were: exd = exo = 0.9 lam. The inaccuracy of setting A is about 30 arcsec and it was assured by means of a polygon mirror and
R. Jablonski, M. Dzwiarek / Measurement13 (1994) 13-22 autocollimator. The refractive index of a tube was given with an accuracy of 0.1%. The obtained accuracy is 3 Inn for the outer and 5 lain for the inner surface.
19
Appendix--Four types of approximations
A.I. Spiral of Archimedes The easiest way of approximation is by the spiral of Archimedes described by the formula:
7. Conclusion
a
RaY(a) = Ri + (Rt+l - Ri) -~a' The modification of Eq. (10) can be performed only under a significant change of parameters of the measured objects. During experiments such necessity did not occur. An additional source of error is the eccentricity which results from the displacement of the axis of rotation of an object measured relative to the centre of the mean-square circle. The smaller the eccentricity, the more the angle between the radius vector and the normal to the surface departs from n/2, and therefore the calculational error is greater. The method can be used satisfactorily for fast noncontact measurement of transparent objects. It is possible to extend the number of applications, for example to the measurement of the outer radius of nontransparent objects, or to interrupted surfaces.
References [1] M.C. Leu and R.M. Pherwani, Vision system for threedimensional position measurement based on stereo disparity, Opt. Laser Technol. 21(5) (1989) 319-324. [2] A.J.C. Brown, Surface testing with a second generation optical profilometer, IMEKO TC Set. 28 (1990) 213-223. [3] J. Sanchez, R.W. Heebner, D.H. Smithgall and L.S. Watkins, On-line measurement of glass thickness deposited in the MCVD fiber preform process, IOOC '81, San Francisco, CA, April 28, 1981, paper TUG2. [4] H.M. Presby, Geometric measurement of preform rods and starting tubes, AppL Opt. 19 (1982) 3528-3530. [5] W.A. Pilipovitz, A.K. Esman, W.K. Kuleshov and W.P. Dubrovski, Noncontact optical method of measurement of inner diameter of transparent tubes (in Russian), Izmer. Tekh. N. 6 (1990) 13-14. [6] R. Jablonski, A laser reflection method for transparent tube measurement, Proc. 1st Int. Symp. on MTIIP, Wuhan, China, 1989, pp. 109-116. [7] R. Jablonski, Report 17/Ci/87, OTO, Poland, 1987. [8] R. Jablonski, Scanning laser measurementof fibre manufacturing process, VDI Ber. 761 (1989) 93-99. [9] R. Jablonski, A scanning-reflectionmeasurement of curvature and thickness, IMEKO TC Set. 28 (1990) 129-134.
(A.1)
where 0 < a < da, Ri is the value of the radius determined by the ith point, da = 2n/m is the rotation angle between successive measuring points, a n d / P P ( a ) is the approximate value of the radius observed at an angle (i - 1)da + a. Taking Eqs. (3), (5) and (A.8) into account, it m a y be written:
IR[(i-1)da+a] -Rav(a)l~
~
j2Mj • (A.2)
As can be seen, the accuracy of approximation depends above all upon da, that is the number of measurement cycles, and the number and value of harmonics occurring in equations R = R(a) defining the surface of the measured object. For example, for the reference parameters of Eq. (4) and m = 72 we get the foUowing: AR = IR[(i - l ) d a + a] - RaP(a)l < 0.8 ~ra. (A.3) The obtained result proves that the error of radius approximation is far lower than the assumed measurement error and practically does not influence the accuracy of calculations. On the other hand, the error of Af, the angle between the normal to the surface and the radius vector, becomes essential. Making use of the relation defining the angle between the radius vector and the normal to the surface in polar coordinates as well as utilizing Eqs. (3), (7) and (A.1), we arrive at:
\[ Z j M j c o s [ j ( i - 1 ) d a + j a + q ~ j ] ) f(a) = - atn / J= 1
R
'
(A.4)
R. JablonskL M. Dzwiarek / Measurement 13 (1994) 13-22
20
iZ Ri+ I -- Ri" ~
(A.5)
f~P(a) = - atn ~d--~. ~%-~(a)),
;da E jZMj A f = Jf~P(a) - f ( a ) l ~<
j= i 2Ro
(A.6)
The achieved value may exert a considerable impact on the accuracy of calculations, especially when measuring objects for whose description higher harmonics are needed and objects with small mean radius R o. Under assumed da and M values and for Ro = 10 mm, we have: Aft< 0.0036 rad.
(A.7)
The change of angle f on the boundaries of successive ranges of approximation poses an additional problem. It entangles the solution of Eqs. (5) and (6) in the case when the scanned ray falls in the vicinity of the boundary between the ith and the (i+ 1)th area. It is particularly essential at measuring the inner radius. The quantity of this change is given by the following relation:
Errors arising from the approximation by means of the spiral of Archimedes are illustrated in Fig. A.1. Their maximum values are: A ~ = 2.309 ~tm, ADi = -- 1.269 ~tm, Ad~ = 5.508 tam, Ad~ = --2.514 ~tm. (Indexes e and i designate outer and inner measured surface, respectively.) They consist of two components: ARc, the accuracy of roundness errors measurement, and ARm, the accuracy of mean-square radius measurement and may be roughly determined from the relations:
(d2 - d, )/2 ~
(A. 11 )
The mentioned quantities for the object satisfying conditions (4), derived from Eqs. (5), (6), (A.11) and compared with theoretical values are: For the outer surface: - 1.9 }am < ZlSeD < 1.9 ~tm,
Z~RmD =
0.51 pm; (A.12)
(a)
['Ri+2-- Ri+l"~
f~P+l(0) = - atn ! . . . .
1,
\ da'Ri+a J
(A.8)
f Ri + a -- Ri"~
fTP+ i (da) = - atn ! . . . .
I,
25 ],
\ da" Ri+ 1J
As = If~+Pa(0) -f~P(da) l ~<
R0
1.
(A.9)
It is evident that the change of angle on the boundary of approximation ranges depends as well mainly on the number of harmonics, and only in the second place on the radius Ro. For the parameters of Eq. (4) we achieve: As < 0.007 rad.
(A.10)
From (A.10) it follows that the alteration of angle on the boundary of ranges may be two times higher than the error of this angle. It is associated with the change of angle sign inside the approximation range.
Fig. A.1. Results ofcalculatingthe theoreticalelement approximated by the spiral of Archimedes: (a) outer surface; (b) inner surface.
R Jablonski, M. Dzwiarek
continuity on the boundaries of successive ranges of approximation:
and for the inner surface: -4.1 p m < A R ~ < 4 . 1 pan,
21
Measurement13 (1994) 13-22
Aarad= 1.5 gm. (A.13)
~R~ p (a) Oa
0R~P+ 1 (a) a=da --
-~a
(A.17) a=O"
As it can be noticed, the results for the outer surface are contained in the permissible limit of error, whereas in the case of the inner surface the limit is considerably exceeded (though these are only errors of approximation, while other factors are not taken into account).
This condition, however, is admissible only for an odd number of measurement cycles m. (When the number m is even, the condition requires an additional limitation:
A.2. Exponential spiral
i=1
The approximation by the exponential spiral is provided by the following equation:
which is not always fulfilled.) A more beneficial condition is obtained by the extension of the approximation to the (i + l)th range and then:
( - 1 ) i . Ri = 0,
RaP(a) = R i exp
log(Ri+ 1/Ri) •
(A.14)
It is the approximation preserving a constant value of the angle f throughout the whole range of approximation: f~P(a) = - atn [l°g (R~---~/R')].
(A.15)
The obtained results are close to the ones achieved by the approximation by the spiral of Archimedes, yet the calculational procedures are more complicated, and the time of handling the data is significantly prolonged. Therefore, the approximation by means of the exponential spiral was considered inadvisable. A.3. Quadratic spiral
Far more advantageous is the approximation by the quadratic spiral. It is described by the following equation:
aP 2, RaP(2da) = Ri+
al =
ao = Ri,
4Ri + 1 - 3Ri - Ri + 2 2 ' Ri+2 -
a2 =
(A.18)
(A.19)
2Ri+l + Ri 2
If an analysis similar to the one used for the spiral of Archimedes is conducted, then the following is achieved: AR~<0.15~m,
Af~<0.001rad,
As~<0.001rad. (A.20~
The obtained accuracy of both the radius and the angle approximation are considerably higher, and the change of derivative at the edges of intervals is significantly lower than for the spiral of Archimedes approximation. The determined errors of form and mean square radius are: For the outer surface: - 1 lam
A ~ o = 1.3 lain, (A.21)
a
RaP(a)=ao+aldaa +a2 daa "
(A.16)
This type of approximation provides a supplementary degree of freedom which, apart from conditions (7), allows to add, e.g., a condition of angle
and for the inner surface: - 1 . 9 p m < ZJRer < 1.9 prn,
AR~ = 1.9 pro. (A.22)
These results show that the approximation by
22
1~ Jablonski, M. Dzwiarek / Measurement 13 (1994) 13-22
means o f the quadratic spiral with the extension of ranges may be utilized for the accurate measurement o f outer and inner surfaces.
intervals may be assumed. As a result we get:
ao=Ri,
al=l(Ri+l-Ri_l),
a2 = I ( R i + z -[- 4Ri+ 1 - 5Ri + 2 R i - 1 ),
a3 = ½(RI+2 - 3Ri+l + 3Ri - Ri- 1).
A.4. Cubic spiral Still more advantageous is the approximation by means o f the cubic spiral. It is described by the following equation:
a (ay R ( a ) = a o + a l d a +az ~
(A.24)
+a3 ~
•
(A.23)
This equation gives two additional degrees of freedom. Thus, e.g. apart from the conditions of continuity (7), two supplementary conditions for the continuity of the derivative at the edges of
It is the most accurate way of approximation. It is then easy to calculate that AR ~<0.03 ~tm, Aft< 0.0006 rad, As = 0.
(A.25)
The results for the conditions (4) are the following: For the outer surface: - 0 . 9 ~tm < Z1ReD < 0.9 I~m, ARm o = 0 . 6 5 ~tm; (A.26) and the inner one: -
1.6 lam < ziR~r< 1.6 ~tm, Z~Rmr = 0 ~tm.
(A.27)