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The Abbé principle revisited: An updated interpretation
The Abb6 Principle Revisited: An Updated Interpretation J. B. Bryan*
Slideway displacement and angular motion The Abb~ principle was first made public in a paper published by Professor Abb~ in the Journal for Instrumental Information (Vol X, 1890). In this paper Professor Abb~ discussed the benefits of using divided scales as length measuring systems. He also discussed the benefits of arranging the measuring scale to be in line with the distance to be measured. The concept of designing a machine tool or measuring machine with the displacement measuring system in line with the displacement to be measured is referred to as the Abb~ principle. It has also been called the first principle of machine tool design and dimensional metrology. A strict interpretation of the Abbe principle leaves us with no options other than to design things to be in line. I think that this was a minor oversight caused by Abb~'s concentration on single axis measuring problems at the time. I feel confident that Prof=ssor Abb~ would happily endorse a restatement of his principle that would cover those situations where it is not possible to design 'inline', but the end result of the alternative is the same. By the end result being the s.ame, I mean that the accuracy of the displacement measuring system is unaffected by slideway angular motion or changes thereof. I therefore propose that the generalized Abb~ Principle should be stated as: "The displacement measuring system should be in line with the functional p o i n t whose displacement is to be measured. + I f this is n o t possible, either the slideways that transfer the displacement must be free o f angular motion or angular motion data must be used to calculate the consequences o f the offset. "
The first sentence represents Abb~'s original recommendation. We
can call this option number one. The second sentence gives us two additional options. No angular motion?
The second option (no angular motion in the slideways) might superficially appear to be meaningless because we know there is no such thing as a slideway without angular motion. It is possible, however, to use two displacement measuring systems parallel to each other to read the displacement of one slide. The difference in reading between the two systems is a measure of angular motion. This signal can be used to drive some kind of mechanism tp introduce a correcting angular motion which will then make the difference between the two systems zero. If the spacing between the two systems is greater than the maximum Abb~ offset, I propose that we arbitrarily agree that such a system complies with the Abb~ principle since the error in displacement accuracy due to slideway angular motion will be less than the difference of the two displacement measuring systems. I feel confident that Professor Abb~ would also endorse this further compromise to his original principle on the grounds that there is no other practical solution for situations involving varying Abb~ offset. This concept (the second option) was used by Professor Harrison in the early 1950's at Massachusetts Institute of Technology. He used two interferometers to measure the displacement of a diffraction grating ruling machine slideway. The difference signal between the two interferometers was used to rotate a platform mounted on top of the slide which supported the workpiece and the moving mirrors. The platform was then free of angular motion.
A similar scheme was developed by Dr. Erwin Loewen at Bausch and Lomb in 1959. He used two white light interferometers to measure displacement and two leadscrews to drive the slide. The difference signal between the two interferometers was used to drive one screw faster or slower than the other to eliminate angular motion. Using angular motion data
The third option of the generalized Abb~ Principle calls for using angular motion data to calculate and correct for the consequences of Abb~ offset. Fixed Abb~ offsets are easier to deal with than offsets which vary during a machining or inspection procedure. An example of varying Abb~ offset is a classical jig boring machine that is used to bore holes that are located at a varying height above the table of the machine. Modern CNC control systems can be programmed to accept angular motion data, and then using their own continuous knowledge of the varying Abb~ offset, multiply the two values and add or subtract the result to the displacement measuring system. To the best of our knowledge this capability has not yet been demonstrated, but Lawrence Livermore Laboratory now has a contract with a control builder to provide this feature. The system will aequire angular motion data in the form of the differences between two parallel laser interferometers that are spaced at a distance greater than the maximum Abb~ offset. It is possible to develop angular motion data for either options two or three by techniques other than the differences between two linear displacement devices. A sensitive autocollimator could, for example, be built into a machine for this purpose. A question then arises as to how accurate the autocollimator has to be before the system can be regarded as complying with the Abb~ principle. I propose that we answer this question by estimating the angular accuracy that would be achieved if two linear
Livermore, California 94550, USA
+A more rigorous but longer phrasing of the first sentence should read "The path of the effective point of a displacement measuring system should be colinear with the path of the functional point whose displacement is to be measured. ""
PR EC IS ION ENG IN E E R I N G
0 1 4 1 - 6 3 5 9 / 7 9 / 0 3 0 1 2 5 - 0 4 $02.00 @ 1979 IPC Business Press
~Lawrence Livermore Laboratory,
University of California, PO Box 808,
129
-
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Angular motion //
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'~'- Straightness : ~ / i l i l l l l l l l l /
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+,
displacement measurement systems, similar to the ones being used on the machine in question, were spaced at a distance equal to the maximum Abbe offset. The generalized Abbe principle has been used to advantage in recent years at the Lawrence Livermore Laboratory. It is a great practical convenience to be able to speak about whether or not a design complies with the Abbe principle and if it does, which of the three options has been selected. I hope that it will be accepted by the precision engineering community and find its way into our common vocabulary.
Slideway straightness and angular motion Fig 1 Stationary straightedge," no angular motion
11
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"slide straightness error is the non linear movement that an indicator sees when it is either stationary and reading against a perfect straightedge supported on a moving slide or moved by the slide along a perfect straightedge which is stationary. ""
Fig 2 Moving straightedge," no angular motion
Angular motion Straightness-~-j" ,"
~'l
/ f" ,J
Ir--o-qr-m r-o-q r--m J ~,p Auto collimator
The Abbe principle for location of displacement measuring systems has been known for 89 years. It is remarkable that so few engineers, physicists, and mechanics are aware of its existence or its importance. It is a simple idea, but far from being obvious. The massive nature of machine tool slides misleads us into believing that one point on the mass moves in the same manner as other points. The same thing applies to slideway straightness. If angular motion is present in a slide, the straightness of different points is different for the same reason that displacement of different points is different. A definition for slide straightness is needed at this point in the discussion, and is proposed as:
P
The purpose of the term 'non. linear movernel~t' is to recognize the fact that the straightedge does not need tJ be perfectly parallel to the slid, travel. If the slide is perfectly straight but no[ parallel to the straightedge, the resulting trace wilt be a straight line, Figs 1 to 6 illustrate a few of the motions that can occur in slideways. The motions are exaggerated for clarity, but I have observed each condition in real machine tools in small but significant magnitudes.
Fig 3 Stationary straigh tedge," oscillating angular motion 130
PRECISION
ENGINEERING
Fig 1 illustrates a slide moving along a bed which has a sinusoidal shape. The feet of the slide have a spacing equal to an even number of wavelengths. The movement of the slide is free of angular motion, but is not straight. Since there is no angular motion all points move with the same amount of straightness error, and indicator positions 1, 2 and 3 show the same values. Fig 2 is the same as Fig 1 except that the straightedge is moving. Here again, the straightness error is unaffected by indicator position. In Fig 3 the feet of the slide have been changed to a spacing equal to a half wavelength longer. The slide now has oscillating angular motion as the autocollimator strip chart indicates. This condition leads to the rather amazing result that the straightness error depends entirely on the location of the indicator. A t position No. 2 there is no error (or second order). Position No. 3 has a different magnitude and phase than position No. 1. Fig 4 is the same as Fig 3 except that the straightedge is moving. The straightness error is now a combination of the profiles in Fig 3. When the indicator tip is opposite the centre of the slide there is no error (or second order), when it is at the end of the travel it shows a maximum. The exact shape of the profile is not represented accurately in the illustration. For the purpose of this paper it is sufficient to show that for the same slide motion there can be different straightness error profiles depending on the set-up and location of the indicator. Fig 5 shows a slide having uniform angular motion. The bedway is circular with a radius of R. The autocollimator trace is an inclined straightline. The different indicator positions all show the same straightness error magnitude S, but the inclination of a straightline passed through the end points is different. The general direction of the slide travel changes with the position of the indicator. This can create some terrible confusion when a slide is being adjusted square or parallel w i t h another slide or possibly a spindle. Fig 5 also shows two approximations for the value of S as a function of either the travel of the slide T a n d the angular motion 0, or the travel of the slide T a n d the radius of curvature R, These approximations are accurate to a few percent as long as the radius R is at least 10
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Angular • motion
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Fig 4 Moving straightedge," oscillating angular motion -
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S -~ 061 T8~- 8R S,ffin 8, Arc seconds T = Slide travel, in R = Radius,in
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Fig 5 Stationary straightedge; uniform angular motion T~
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8, Arc seconds T = Slide travel, in R = Radius,in
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Fig 6 Moving straightedge," uniform angular motion
PRECISION ENGINEERING
131
times as large as the travel 7-. These approximations are not relevant to the main subject of this paper, but I have found them to be enormously useful in acquiring a feel for the magnitudes involved when dealing w i t h uniform angular motion. Fig 6 is the same as Fig 5 except that the straightedge is moving. Changing the indicator position does not affect the straightness error or the general direction of travel. It should now be clear that an ambiguity exists regarding straightness measurement in the presence of angular motion that is analogous to the problem of displacement measurement in the presence of angular motion. If straightness error values and directions of travel of slides are dependent on the location of the indicator tip, what is the correct location? The answer can only be found in the function of the machine and the functional point of importance. In a machine tool the functional point is the tool: in a measuring machine it is the sensor, (stylus tip, microscope focus point, etc). tn many machines the functional point changes either during the procedure or during different set-ups. What is the correct location in those cases? By substituting 'straightness' for 'displacement' the wording of the generalized Abb~ principle appears to provide the answer:
"'The straightness measuring system s h o u l d be in l i n e w i t h the f u n c t i o n a l p o i n t whose straightness is to be measured. * I f this is n o t
132
possible, e i t h e r the slideways t h a t transfer the straightness m u s t be free o f angular m o t i o n o r angular m o t i o n data m u s t be used to calculate the consequences o f the offset. ""
Applications This principle will probably have greater immediate application to machine tool and measuring machine metrology than it will to design. However, in the design of special machines optimised for accuracy, this principle should dictate the design. In the case of the ultraprecision machines such as the 84-inch Diamond Turning Machine (Precision Engineering, Vol 1, No. 1 pp 13-17) and the Error-Corrected Y-Z Coordinate Measuring Machine (discussed on page 125 of this issue), this principle did dictate the design. I am proposing that this principle be named the Bryan Principle rather than treating it as a corollary to the Abb~ Principle on the grounds that Professor Abb~ was not particularly interested in the question of slide straightness, and that displacement and straightness should be ,well separated subjects to avoid confusion. I am absolutely convinced that fundamental principles and methods such as
*A more rigorous but longer phrasing of the first sentence should read "'The effective point of a straightness measuring system should be along a line which is perpendicular to the direction of slideway travel and passes through the functional point whose straightness is to be measured. ""
this should have well defined names to facilitate precise and rapid communication. The name should be linked with the method as soon as possible. There is always the possibility that the method or principle will be discredited in the future. Similar sounding, but valid, principles should not be discredited by association.
Acknowledgement I would like to recognize the contributions of my colleague, Donald L. Carter, to the development of this paper. Mr. Carter worked out the concept of the illustrations and provided invaluable assistance in editing the text. I would also like to thank Dr. Erwin Loewen of Bausch and Lomb for his advice and encouragement. The work was performed under the auspices of the US Department of Energy by the Lawrence Livermore Laboratory under contract number W-7405-ENG-48.
Notice This report was prepared as an account of work sponsored by the United States Government. Neither the United States nor the United States Department of Energy, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness or usefulness of any information, apparatus, product or process disclosed, or represents that its use would not infringe privatelyowned rights. Reference to a company or product names does not imply approval or recommendation of the product by the University of California or the U.S. Department of Energy to the exclusion of others that may be suitable.
PRECISION
ENGINEERING
Slideway displacement and angular motion The Abb~ principle was first made public in a paper published by Professor Abb~ in the Journal for Instrumental Information (Vol X, 1890). In this paper Professor Abb~ discussed the benefits of using divided scales as length measuring systems. He also discussed the benefits of arranging the measuring scale to be in line with the distance to be measured. The concept of designing a machine tool or measuring machine with the displacement measuring system in line with the displacement to be measured is referred to as the Abb~ principle. It has also been called the first principle of machine tool design and dimensional metrology. A strict interpretation of the Abbe principle leaves us with no options other than to design things to be in line. I think that this was a minor oversight caused by Abb~'s concentration on single axis measuring problems at the time. I feel confident that Prof=ssor Abb~ would happily endorse a restatement of his principle that would cover those situations where it is not possible to design 'inline', but the end result of the alternative is the same. By the end result being the s.ame, I mean that the accuracy of the displacement measuring system is unaffected by slideway angular motion or changes thereof. I therefore propose that the generalized Abb~ Principle should be stated as: "The displacement measuring system should be in line with the functional p o i n t whose displacement is to be measured. + I f this is n o t possible, either the slideways that transfer the displacement must be free o f angular motion or angular motion data must be used to calculate the consequences o f the offset. "
The first sentence represents Abb~'s original recommendation. We
can call this option number one. The second sentence gives us two additional options. No angular motion?
The second option (no angular motion in the slideways) might superficially appear to be meaningless because we know there is no such thing as a slideway without angular motion. It is possible, however, to use two displacement measuring systems parallel to each other to read the displacement of one slide. The difference in reading between the two systems is a measure of angular motion. This signal can be used to drive some kind of mechanism tp introduce a correcting angular motion which will then make the difference between the two systems zero. If the spacing between the two systems is greater than the maximum Abb~ offset, I propose that we arbitrarily agree that such a system complies with the Abb~ principle since the error in displacement accuracy due to slideway angular motion will be less than the difference of the two displacement measuring systems. I feel confident that Professor Abb~ would also endorse this further compromise to his original principle on the grounds that there is no other practical solution for situations involving varying Abb~ offset. This concept (the second option) was used by Professor Harrison in the early 1950's at Massachusetts Institute of Technology. He used two interferometers to measure the displacement of a diffraction grating ruling machine slideway. The difference signal between the two interferometers was used to rotate a platform mounted on top of the slide which supported the workpiece and the moving mirrors. The platform was then free of angular motion.
A similar scheme was developed by Dr. Erwin Loewen at Bausch and Lomb in 1959. He used two white light interferometers to measure displacement and two leadscrews to drive the slide. The difference signal between the two interferometers was used to drive one screw faster or slower than the other to eliminate angular motion. Using angular motion data
The third option of the generalized Abb~ Principle calls for using angular motion data to calculate and correct for the consequences of Abb~ offset. Fixed Abb~ offsets are easier to deal with than offsets which vary during a machining or inspection procedure. An example of varying Abb~ offset is a classical jig boring machine that is used to bore holes that are located at a varying height above the table of the machine. Modern CNC control systems can be programmed to accept angular motion data, and then using their own continuous knowledge of the varying Abb~ offset, multiply the two values and add or subtract the result to the displacement measuring system. To the best of our knowledge this capability has not yet been demonstrated, but Lawrence Livermore Laboratory now has a contract with a control builder to provide this feature. The system will aequire angular motion data in the form of the differences between two parallel laser interferometers that are spaced at a distance greater than the maximum Abb~ offset. It is possible to develop angular motion data for either options two or three by techniques other than the differences between two linear displacement devices. A sensitive autocollimator could, for example, be built into a machine for this purpose. A question then arises as to how accurate the autocollimator has to be before the system can be regarded as complying with the Abb~ principle. I propose that we answer this question by estimating the angular accuracy that would be achieved if two linear
Livermore, California 94550, USA
+A more rigorous but longer phrasing of the first sentence should read "The path of the effective point of a displacement measuring system should be colinear with the path of the functional point whose displacement is to be measured. ""
PR EC IS ION ENG IN E E R I N G
0 1 4 1 - 6 3 5 9 / 7 9 / 0 3 0 1 2 5 - 0 4 $02.00 @ 1979 IPC Business Press
~Lawrence Livermore Laboratory,
University of California, PO Box 808,
129
-
A
i
J J 8
Angular motion //
I/
I/I
.52J
J
'~'- Straightness : ~ / i l i l l l l l l l l /
Illil
0,~ ,, ~-~'(~,, ,, Auto collimator
+,
displacement measurement systems, similar to the ones being used on the machine in question, were spaced at a distance equal to the maximum Abbe offset. The generalized Abbe principle has been used to advantage in recent years at the Lawrence Livermore Laboratory. It is a great practical convenience to be able to speak about whether or not a design complies with the Abbe principle and if it does, which of the three options has been selected. I hope that it will be accepted by the precision engineering community and find its way into our common vocabulary.
Slideway straightness and angular motion Fig 1 Stationary straightedge," no angular motion
11
r
~[-
+
+
J
Angular motion
B J /
/
,
i
/
Straightness~
~
[i -i Cs--j
"\\
E-sq I
Auto collimator
"slide straightness error is the non linear movement that an indicator sees when it is either stationary and reading against a perfect straightedge supported on a moving slide or moved by the slide along a perfect straightedge which is stationary. ""
Fig 2 Moving straightedge," no angular motion
Angular motion Straightness-~-j" ,"
~'l
/ f" ,J
Ir--o-qr-m r-o-q r--m J ~,p Auto collimator
The Abbe principle for location of displacement measuring systems has been known for 89 years. It is remarkable that so few engineers, physicists, and mechanics are aware of its existence or its importance. It is a simple idea, but far from being obvious. The massive nature of machine tool slides misleads us into believing that one point on the mass moves in the same manner as other points. The same thing applies to slideway straightness. If angular motion is present in a slide, the straightness of different points is different for the same reason that displacement of different points is different. A definition for slide straightness is needed at this point in the discussion, and is proposed as:
P
The purpose of the term 'non. linear movernel~t' is to recognize the fact that the straightedge does not need tJ be perfectly parallel to the slid, travel. If the slide is perfectly straight but no[ parallel to the straightedge, the resulting trace wilt be a straight line, Figs 1 to 6 illustrate a few of the motions that can occur in slideways. The motions are exaggerated for clarity, but I have observed each condition in real machine tools in small but significant magnitudes.
Fig 3 Stationary straigh tedge," oscillating angular motion 130
PRECISION
ENGINEERING
Fig 1 illustrates a slide moving along a bed which has a sinusoidal shape. The feet of the slide have a spacing equal to an even number of wavelengths. The movement of the slide is free of angular motion, but is not straight. Since there is no angular motion all points move with the same amount of straightness error, and indicator positions 1, 2 and 3 show the same values. Fig 2 is the same as Fig 1 except that the straightedge is moving. Here again, the straightness error is unaffected by indicator position. In Fig 3 the feet of the slide have been changed to a spacing equal to a half wavelength longer. The slide now has oscillating angular motion as the autocollimator strip chart indicates. This condition leads to the rather amazing result that the straightness error depends entirely on the location of the indicator. A t position No. 2 there is no error (or second order). Position No. 3 has a different magnitude and phase than position No. 1. Fig 4 is the same as Fig 3 except that the straightedge is moving. The straightness error is now a combination of the profiles in Fig 3. When the indicator tip is opposite the centre of the slide there is no error (or second order), when it is at the end of the travel it shows a maximum. The exact shape of the profile is not represented accurately in the illustration. For the purpose of this paper it is sufficient to show that for the same slide motion there can be different straightness error profiles depending on the set-up and location of the indicator. Fig 5 shows a slide having uniform angular motion. The bedway is circular with a radius of R. The autocollimator trace is an inclined straightline. The different indicator positions all show the same straightness error magnitude S, but the inclination of a straightline passed through the end points is different. The general direction of the slide travel changes with the position of the indicator. This can create some terrible confusion when a slide is being adjusted square or parallel w i t h another slide or possibly a spindle. Fig 5 also shows two approximations for the value of S as a function of either the travel of the slide T a n d the angular motion 0, or the travel of the slide T a n d the radius of curvature R, These approximations are accurate to a few percent as long as the radius R is at least 10
I /
Angular • motion
\,
[I o Io
Straightness
I
Auto collimator
Fig 4 Moving straightedge," oscillating angular motion -
TZ
S -~ 061 T8~- 8R S,ffin 8, Arc seconds T = Slide travel, in R = Radius,in
4-
/i
A
;
/ I / I ;-,f,/.
I
//
8
Angular motion
B
Y \~ Straightness"~/
ro Auto collimator
Fig 5 Stationary straightedge; uniform angular motion T~
S~--O61 TOm 8R S,Fin
-
8, Arc seconds T = Slide travel, in R = Radius,in
+
\
I -~
Angular ~ [ ~ . motion
1/i1~
+
s
Straightness
Auto collimator R
Fig 6 Moving straightedge," uniform angular motion
PRECISION ENGINEERING
131
times as large as the travel 7-. These approximations are not relevant to the main subject of this paper, but I have found them to be enormously useful in acquiring a feel for the magnitudes involved when dealing w i t h uniform angular motion. Fig 6 is the same as Fig 5 except that the straightedge is moving. Changing the indicator position does not affect the straightness error or the general direction of travel. It should now be clear that an ambiguity exists regarding straightness measurement in the presence of angular motion that is analogous to the problem of displacement measurement in the presence of angular motion. If straightness error values and directions of travel of slides are dependent on the location of the indicator tip, what is the correct location? The answer can only be found in the function of the machine and the functional point of importance. In a machine tool the functional point is the tool: in a measuring machine it is the sensor, (stylus tip, microscope focus point, etc). tn many machines the functional point changes either during the procedure or during different set-ups. What is the correct location in those cases? By substituting 'straightness' for 'displacement' the wording of the generalized Abb~ principle appears to provide the answer:
"'The straightness measuring system s h o u l d be in l i n e w i t h the f u n c t i o n a l p o i n t whose straightness is to be measured. * I f this is n o t
132
possible, e i t h e r the slideways t h a t transfer the straightness m u s t be free o f angular m o t i o n o r angular m o t i o n data m u s t be used to calculate the consequences o f the offset. ""
Applications This principle will probably have greater immediate application to machine tool and measuring machine metrology than it will to design. However, in the design of special machines optimised for accuracy, this principle should dictate the design. In the case of the ultraprecision machines such as the 84-inch Diamond Turning Machine (Precision Engineering, Vol 1, No. 1 pp 13-17) and the Error-Corrected Y-Z Coordinate Measuring Machine (discussed on page 125 of this issue), this principle did dictate the design. I am proposing that this principle be named the Bryan Principle rather than treating it as a corollary to the Abb~ Principle on the grounds that Professor Abb~ was not particularly interested in the question of slide straightness, and that displacement and straightness should be ,well separated subjects to avoid confusion. I am absolutely convinced that fundamental principles and methods such as
*A more rigorous but longer phrasing of the first sentence should read "'The effective point of a straightness measuring system should be along a line which is perpendicular to the direction of slideway travel and passes through the functional point whose straightness is to be measured. ""
this should have well defined names to facilitate precise and rapid communication. The name should be linked with the method as soon as possible. There is always the possibility that the method or principle will be discredited in the future. Similar sounding, but valid, principles should not be discredited by association.
Acknowledgement I would like to recognize the contributions of my colleague, Donald L. Carter, to the development of this paper. Mr. Carter worked out the concept of the illustrations and provided invaluable assistance in editing the text. I would also like to thank Dr. Erwin Loewen of Bausch and Lomb for his advice and encouragement. The work was performed under the auspices of the US Department of Energy by the Lawrence Livermore Laboratory under contract number W-7405-ENG-48.
Notice This report was prepared as an account of work sponsored by the United States Government. Neither the United States nor the United States Department of Energy, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness or usefulness of any information, apparatus, product or process disclosed, or represents that its use would not infringe privatelyowned rights. Reference to a company or product names does not imply approval or recommendation of the product by the University of California or the U.S. Department of Energy to the exclusion of others that may be suitable.
PRECISION
ENGINEERING