EP0607240A1 - Measuring the accuracy of multi-axis machines - Google Patents

Measuring the accuracy of multi-axis machines

Info

Publication number
EP0607240A1
EP0607240A1 EP92921141A EP92921141A EP0607240A1 EP 0607240 A1 EP0607240 A1 EP 0607240A1 EP 92921141 A EP92921141 A EP 92921141A EP 92921141 A EP92921141 A EP 92921141A EP 0607240 A1 EP0607240 A1 EP 0607240A1
Authority
EP
European Patent Office
Prior art keywords
orientation
machine
light beam
measuring
difference
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Withdrawn
Application number
EP92921141A
Other languages
German (de)
French (fr)
Inventor
Peter Henrikus Johannes Schellekens
Johannes Augustinus Soons
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Renishaw Transducer Systems Ltd
Original Assignee
Renishaw Transducer Systems Ltd
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Priority claimed from GB919121686A external-priority patent/GB9121686D0/en
Priority claimed from GB919121687A external-priority patent/GB9121687D0/en
Priority claimed from GB919121685A external-priority patent/GB9121685D0/en
Application filed by Renishaw Transducer Systems Ltd filed Critical Renishaw Transducer Systems Ltd
Publication of EP0607240A1 publication Critical patent/EP0607240A1/en
Withdrawn legal-status Critical Current

Links

Classifications

    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
    • G01B21/00—Measuring arrangements or details thereof, where the measuring technique is not covered by the other groups of this subclass, unspecified or not relevant
    • G01B21/02—Measuring arrangements or details thereof, where the measuring technique is not covered by the other groups of this subclass, unspecified or not relevant for measuring length, width, or thickness
    • G01B21/04—Measuring arrangements or details thereof, where the measuring technique is not covered by the other groups of this subclass, unspecified or not relevant for measuring length, width, or thickness by measuring coordinates of points
    • G01B21/042—Calibration or calibration artifacts
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S17/00—Systems using the reflection or reradiation of electromagnetic waves other than radio waves, e.g. lidar systems
    • G01S17/66—Tracking systems using electromagnetic waves other than radio waves

Definitions

  • Such machines include, for example, coordinate measuring
  • Such machines generally have a position transducer for each axis of movement (e.g. a linear encoder scale for linear axes, or an angle encoder for rotational axes).
  • the outputs of these transducers are taken to a numerical control or computer, which can thus determine the position in space of an end effector of the machine, such as a tool or probe.
  • a position transducer for each axis of movement e.g. a linear encoder scale for linear axes, or an angle encoder for rotational axes.
  • the outputs of these transducers are taken to a numerical control or computer, which can thus determine the position in space of an end effector of the machine, such as a tool or probe.
  • Various methods and apparatus are already known for determining the accuracy of such machines, e.g. for calibration purposes.
  • the artefact may incorporate known or unknown constant lengths, which are measured by the machine at various positions and
  • parameters can be determined by direct measurement using a laser interferometer, but this has the disadvantage that a different interferometer set up is required for each parameter, so that the procedure is time consuming. Also, it is difficult or expensive to measure roll using an interferometer.
  • One aspect of the present invention provides an apparatus and method using a laser interferometer to measure the accuracy of a number of distances generated on a multi-axis machine, in various orientations.
  • a second aspect of the present invention provides a method of determining parametric errors of a multi-axis machine, in which the machine is caused to generate a number of distances in various orientations, and the accuracy of these distances is ascertained.
  • the invention then provides a method to determine the required parametric errors from these distances.
  • the accuracy measurements can be made by the method of the first aspect of the invention, or they may be obtained from measurements upon artefacts.
  • Fig 1 is a schematic diagram of one embodiment of the invention
  • Fig 2 is a schematic diagram of a coordinate measuring machine with a second embodiment of the invention.
  • Fig 3 is a schematic diagram corresponding to Fig 1 but showing a modification
  • Fig 4 is a vector diagram for explaining a method according to the second aspect of the invention.
  • Fig 5 is a schematic diagram of a two-dimensional coordinate measuring machine.
  • FIG 1 illustrates schematically the apparatus required for making accuracy measurements on a machine having only two axes of movement (or a machine having more than two axes, the remaining axes being held stationary).
  • a retroreflector 10 is attached to a portion 12 of the machine which can move within the two axes, for example the end of a ram intended for holding a tool or probe.
  • a rotary table 14 is separately mounted on a stationary part of the machine, and can be rotated about an axis as indicated by arrows 16, by means of a motor 18, or manually.
  • a mirror 20 is mounted for rotation with the rotary table 14. Light from a laser 24 is passed to the rotatable mirror 20.
  • An interferometer 22 includes a retroreflector 23 in a reference arm, and the interferometer is placed between the laser 24 and the mirror 20, so as to measure changes in the path length of the light.
  • apparatus is controlled from a computer (not shown in Fig 1).
  • the above apparatus is (software) aligned by measuring the position and direction of the reflected beam for several angles of the rotary table 14. Each position and direction is estimated from the scales or other
  • position transducers are taken at at least two different, manually tuned, positions where the laser beam is optimally returned to the interferometer 22, as determined by
  • a "best fit" straight line is calculated by a "least squares" method.
  • a simple CNC program which moves the retroreflector 10 across the laser beam 28 and measures the interferometric signal strength. By obtaining such information for a sufficient number of angles of the rotary table, the position and direction of the reflected beam can be inferred for any other angle of the rotary table. For this, the orientation of the table itself needs to be measurable to a certain minimum accuracy. If this is not possible, the laser beam should be aligned at more angles, or at all angles which are to be used.
  • the apparatus is used as follows. In a certain fixed position of the rotary table, the machines ram 12 is automatically moved along the reflected laser beam 28, the position and direction of which are now known. At certain positions, the machine's travel along the line of movement (as determined by the computer from the scales of the machine) is compared with the reading given by the interferometer 22. This gives readings of the accuracy of the machine when making a number of length measurements, e.g. as indicated at 26 between start and finish points 26A,26B. As many such accuracy readings as desired can be made during this movement.
  • the rotary table 14 is rotated by the motor 18, under computer control, so that the laser beam 28 takes up a new orientation.
  • the above process is repeated, moving the ram 12 along the new orientation of the laser beam and taking as many comparative readings as desired.
  • the process is repeated again for as many different angles of orientation of the laser beam as required.
  • the accuracy of a large number of attained distances of the ram 12 in different directions can be measured fully automatically. Since the position and orientation of the mirror is the same at both end points 26A,26B of the measured
  • Fig 2 shows in more detail an arrangement for measuring the accuracy of a coordinate measuring machine 30, having three axes x,y,z. A similar arrangement may be used for other machines having three or more axes.
  • Fig 2 includes various components which are similar to those of Fig 1, including a retroreflector 10 attached to the ram 12 of the machine, a laser 24 and an interferometer 22.
  • a mirror 20 is also provided, as before, except that it is now mounted on a rotary table 14, e.g. in a gimballed arrangement, in such a way that it can be driven about two different axes of rotation (which need not be
  • Fig 2 also shows other components of the practical arrangement, including a computer 32 which provides the numerical control of the three axes of movement of the machine 30; and a computer 34 from which the accuracy measurements are controlled and which is linked to the computer 32, e.g. by an RS232 link 36.
  • the computer 34 receives readings from the laser interferometer 22,24 via an interface 40, and the readings may be compensated for atmospheric conditions detected by a sensor unit 42 if desired.
  • the two axes of rotation of the mirror 20 are preferably motorised and controlled
  • the apparatus of Fig 2 is used in exactly the same manner as the apparatus of Fig 1, except of course that the laser beam 28 is directed into a number of orientations in three dimensions, using the two axes of rotation of the mirror
  • a machine having three or more axes could also be measured with a mirror 20 having only one axis of rotation, as in Fig 1, by moving the laser 24 and interferometer 22 to obtain the other axis of beam orientation.
  • the laser and the mirror 20 must remain fixed relative to each other during each set of measurements at a given
  • Fig 3 shows a modification of Fig 1, in which the common components have been given the same reference numerals.
  • the modification lies in the provision of a position detector 44, which receives the returning laser beam via a beam splitter 42, after the laser beam has been rereflected by the mirror 20.
  • the output signal of the detector 44 controls the position of the mirror 20 via a servo connection 46 to the motor 18. This causes the mirror 20 to track the movements of the retroreflector 10.
  • the retroreflector 10 can be moved from the start position 26A to the finish position 26B of the displacement by any convenient route. This is useful in the case of manual or numerically controlled machines which are
  • the position detector 44 should be capable of detecting deviations in two dimensions and controlling the two axes of the rotary table 14 accordingly.
  • the position detector may be connected to the computer 34 via an
  • the position detector 44 is also useful during the initial alignment of the rotary table 14, since it facilitates the detection of the optimal position of the retroreflector 10 during the alignment procedure.
  • the retroreflector 10 can be a corner-cube retroreflector, or a cat's-eye retroreflector may be used for its wide angle properties.
  • a plane mirror may be used (with a plane mirror interferometer system 22). The plane mirror should be adjustably mounted to the ram 12 so that it can be adjusted to be normal to the beam
  • the retroreflector 10 can be mounted on an arm extending laterally from the ram 12. This is particularly desirable if it is desired to measure z-axis roll (in the arrangement depicted in Fig 2).
  • the parametric errors of a multi-axis machine can be estimated from the length errors in a large number of such distance measurements.
  • the large number of distance measurements may be obtained using the apparatus and method described above, or other interferometer length measurement arrangements. Alternatively, they may be obtained from conventional measurements upon artefacts of the type discussed above.
  • the method as discussed below can be applied to machines with prismatic and/or revolute joints in an arbitrary configuration, including large machines, software compensated machines and machines with significant finite stiffness related errors.
  • the position errors of the end effector (e.g. a tool or probe) of a multi-axis machine are caused by separate errors introduced in the components of the machine (e.g. squareness, straightness and scale errors, and angular errors of roll, pitch and yaw). These are the so-called parametric errors.
  • a mathematical model is built of the machine's error structure, including a
  • squareness error can be considered as a steady gradient in the parametric equations describing straightness errors, or it can be considered as a static rotation error. It should be modelled as one of these alternatives, but not both. In the description below, it is modelled as a static rotation.
  • this least squares method is used to allocate the errors between the parameters, as opposed to simply allocating a given error to a given parameter.
  • this linear regression may be performed using off-the-shelf software, either in the computer 34 (Fig 2) or in another computer using the data recorded by the computer 34.
  • Each parametric error is described as a linear combination of known functions.
  • the known functions are defined on the position of the machine's axes and other relevant variables (e.g., the output of a certain temperature sensor).
  • pk(x) is a known function defined on the relevant independent variables contained in the vector x.
  • Bk is the unknown parameter describing the contributions of this function to the parametric error Ei,j , and has to be estimated.
  • variable knot positions carries the practical danger of overfitting the data, and makes testing of the hypothesis concerning areas of structural change virtually impossible. Unless prior information is available, we use a basic model which contains enough polynomial pieces with a fixed length and a specified maximum degree, to accommodate the most complex error expected.
  • the relative position error of the end-effector of the machine is related to the parametric errors using an error model [3].
  • error model [3].
  • a large variety of such models have been described, these models are linear in the parametric errors, since the difference between the nominal and the actual machine geometry usually does not significant change the active arm of angular errors and the direction in which the various errors acts.
  • the model can be summarized as:
  • Equation 2 the vector Ei contains the three angular and three
  • the 3 ⁇ 6 matrix uFi describes how these parametric errors affect the errors ⁇ P in the relative position of the end-effector. This matrix is completely defined by the nominal geometry of the machine, the length of the
  • Vector eu contains the
  • ⁇ we use the observed errors in a large number of distance measurements. These measurements can be realized by artifacts, or by using a distance measuring instrument with a high relative accuracy (e.g., laser interferometer), for instance as described above .
  • a distance measuring instrument with a high relative accuracy (e.g., laser interferometer), for instance as described above .
  • the difference between the measured (by the machine) and actual position and orientation of the reference distance has a negligible effect on the measurement error of this distance.
  • ⁇ i represents the measurement error due to non-repeatable and non-modelled machine errors.
  • Vector xi contains the variables which describe the status of the machine and its environment during the measurement (e.g., the position of the machines axes for each measured point).
  • the known function igk(Xi) describes the effect of parameter ⁇ k on the feature's measurement error.
  • pk(x) in relation 1. that describes the effect of the of parameter ⁇ k on the respective parametric error, are calculated for the status of the machine when measuring the end points of the considered distance. These values are then inserted in relation 2 to obtain the effect of ⁇ k on the displacement error of the end-effector for various measured points.
  • igk(xi) is computed as the dot product of the difference between both displacement errors with the direction vector along the measured distance.
  • the vector ⁇ has to be expanded to include these dimensions.
  • at least one measurement of known dimensions has to be made, in order to obtain absolute accuracy.
  • Equation (1) above is a simplification, because it ignores cross-talk between different axes, that is, the way in which the position in one axis affects the errors in another.
  • the vector x can be generalised to include the effects seen in the x axis when y and z are varied:
  • equation (1A) will have further terms
  • a two-dimensional coordinate measuring machine depicted schematically in Fig 5.
  • the machine has an x slide 50 connected to a y slide 52.
  • a probe P is connected to the y slide 52, offset by a distance a.
  • the position of the probe P is measured by the machine as having coordinates (X,Y).
  • the discussion below also refers to a direction z, normal to the plane of Fig 5.
  • the parametric errors of the machine are:
  • Equations E2 and E3 constitute a formal statement of the properties of the machine. Substituting E1 into E3 gives:
  • ⁇ i is a term relating to the difference between the real error and the modelled error
  • the actual length of the artefact is defined as:
  • Equation E10 can now be described as:

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  • Physics & Mathematics (AREA)
  • General Physics & Mathematics (AREA)
  • Electromagnetism (AREA)
  • Engineering & Computer Science (AREA)
  • Computer Networks & Wireless Communication (AREA)
  • Radar, Positioning & Navigation (AREA)
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  • Length Measuring Devices By Optical Means (AREA)

Abstract

On mesure la précision d'une machine à plusieurs axes telle qu'une machine de mesure à coordonnées à trois axes en obtenant un grand nombre de mesures de longueur d'un article placé en différentes positions et selon différentes orientations dans le volume de travail de la machine. La précision de ces mesures est vérifiée par l'intermédiaire d'un interféromètre laser (22, 24). Un miroir (20) est monté sur une table rotative (14) afin d'entrer en rotation autour de deux axes, de sorte que le faisceau laser produit (28) peut être dirigé sur un rétroréflecteur (10) monté à l'extrémité d'un coulisseau (12) de la machine. Afin d'effectuer des mesures de longueur, le rétroréflecteur est déplacé dans le sens du faisceau laser (28). Ceci est ensuite répété alors que le faisceau laser (28) est orienté en un certain nombre de différentes positions. On décrit également un procédé permettant de calculer les erreurs paramétriques de la machine à partir des erreurs détectées dans ces mesures de longueur.The accuracy of a multi-axis machine such as a three-axis coordinate measuring machine is measured by obtaining a large number of length measurements of an article placed at different positions and in different orientations within the working volume of the machine. The accuracy of these measurements is checked by means of a laser interferometer (22, 24). A mirror (20) is mounted on a rotary table (14) to rotate about two axes so that the produced laser beam (28) can be directed onto a retroreflector (10) mounted at the end of a slider (12) of the machine. In order to perform length measurements, the retroreflector is moved in the direction of the laser beam (28). This is then repeated as the laser beam (28) is aimed at a number of different positions. A method is also described for calculating the parametric errors of the machine from the errors detected in these length measurements.

Description

MEASURING THE ACCURACY OF MULTI-AXIS MACHINES
FIELD OF THE INVENTION This invention relates to apparatus and methods for
measuring the accuracy of multi-axis machines. Such machines include, for example, coordinate measuring
machines, machine tools and industrial robots. DESCRIPTION OF PRIOR ART
Such machines generally have a position transducer for each axis of movement (e.g. a linear encoder scale for linear axes, or an angle encoder for rotational axes). The outputs of these transducers are taken to a numerical control or computer, which can thus determine the position in space of an end effector of the machine, such as a tool or probe. Various methods and apparatus are already known for determining the accuracy of such machines, e.g. for calibration purposes.
One such technique involves using the machine to make measurements on a predetermined artefact. The artefact may incorporate known or unknown constant lengths, which are measured by the machine at various positions and
orientations within its working volume. Deviations in the length as measured then indicate the volumetric accuracy of the machine. Examples of such artefacts include slip gauges, step gauges, hole plates, ball plates and ball bars of constant length. See for example U.S. Patent No.
4,819,339 (Kunzmann), PCT Patent Application No. WO91/03706 (Leitz), U.S. Patent No. 4,777,818 (McMurtry). It is also known to use artefacts having a variable length, with a transducer for measuring this variable length. The length as indicated by this transducer is then compared with the length as measured by the machine. Examples are shown in U.S. Patent No. 4,435,905 (Bryan) and U.K. Patent
Application No. 2,210,978 (CD Measurements). Measurements taken with such artefacts have the
disadvantage that if a very large number of measurements is to be taken, in order to give a good picture of the accuracy of the machine over its entire working volume, then intermediate manual set up operations are required to reposition the artefact in different parts of the working volume. This is particularly a problem on large machines.
It is also known to describe the accuracy of a multi-axis machine in terms of parameters which relate to the various component errors which can cause inaccuracy. For example, on a machine having three linear axes of movement, it is common to describe the errors in terms of 21 parameters: 6 parameters for each axis (linear position error, roll, pitch and yaw errors, straightness errors with respect to each of the other two axes), plus 3 parameters relating to the squareness between the three axes. Determining these parameters has the advantage of giving information about the sources of machine inaccuracies. Most of these
parameters can be determined by direct measurement using a laser interferometer, but this has the disadvantage that a different interferometer set up is required for each parameter, so that the procedure is time consuming. Also, it is difficult or expensive to measure roll using an interferometer.
It is also known to determine these parametric errors indirectly, by deriving them from length measurements made on artefacts. See for example "A Uniform Concept For
Calibration, Acceptance Test, and Periodic Inspection of Coordinate Measuring Machines Using A Reference Object", H Kunzmann et al, Annals of the CIRP, Volume 39/1, 1990, pages 561-564; "Coordinate Measuring Machines and Machine Tools Self-Calibration and Error Correction", G Belforte et al, Annals of the CIRP, Volume 36/1, 1987, pages 359-364; and "Estimation of Coordinate Measuring Machine Error
Parameters", J. Chen et al, Proc. IEEE Conf. on Robotics and Automation, 1987, pages 196-201. A similar method using displacement measurements taken with a laser
interferometer is described in "A Displacement Method For Machine Geometry Calibration", G Zhang et al, Annals of the CIRP, Volume 37/1, 1988, pages 515-518.
SUMMARY OF THE INVENTION
One aspect of the present invention provides an apparatus and method using a laser interferometer to measure the accuracy of a number of distances generated on a multi-axis machine, in various orientations.
A second aspect of the present invention provides a method of determining parametric errors of a multi-axis machine, in which the machine is caused to generate a number of distances in various orientations, and the accuracy of these distances is ascertained. This aspect of the
invention then provides a method to determine the required parametric errors from these distances. The accuracy measurements can be made by the method of the first aspect of the invention, or they may be obtained from measurements upon artefacts.
BRIEF DESCRIPTION OF DRAWINGS
Preferred embodiments of the invention will now be
described with reference to the accompany drawings in which: Fig 1 is a schematic diagram of one embodiment of the invention;
Fig 2 is a schematic diagram of a coordinate measuring machine with a second embodiment of the invention;
Fig 3 is a schematic diagram corresponding to Fig 1 but showing a modification;
Fig 4 is a vector diagram for explaining a method according to the second aspect of the invention; and Fig 5 is a schematic diagram of a two-dimensional coordinate measuring machine.
DESCRIPTION OF THE PREFERRED EMBODIMENTS
As a simple example. Fig 1 illustrates schematically the apparatus required for making accuracy measurements on a machine having only two axes of movement (or a machine having more than two axes, the remaining axes being held stationary). A retroreflector 10 is attached to a portion 12 of the machine which can move within the two axes, for example the end of a ram intended for holding a tool or probe. A rotary table 14 is separately mounted on a stationary part of the machine, and can be rotated about an axis as indicated by arrows 16, by means of a motor 18, or manually. A mirror 20 is mounted for rotation with the rotary table 14. Light from a laser 24 is passed to the rotatable mirror 20. From the mirror 20, the laser beam 28 is directed to the retroreflector 10, and returns on the same path. An interferometer 22 includes a retroreflector 23 in a reference arm, and the interferometer is placed between the laser 24 and the mirror 20, so as to measure changes in the path length of the light. The whole
apparatus is controlled from a computer (not shown in Fig 1).
In use, the above apparatus is (software) aligned by measuring the position and direction of the reflected beam for several angles of the rotary table 14. Each position and direction is estimated from the scales or other
position transducers which measure the movements of the machines axes. Readings from these scales or other
position transducers are taken at at least two different, manually tuned, positions where the laser beam is optimally returned to the interferometer 22, as determined by
measuring the strength of the interferometric signal.
Preferably, three or four such readings are taken, and a "best fit" straight line is calculated by a "least squares" method. On a CNC machine it is also possible to determine the optimal positions by a simple CNC program which moves the retroreflector 10 across the laser beam 28 and measures the interferometric signal strength. By obtaining such information for a sufficient number of angles of the rotary table, the position and direction of the reflected beam can be inferred for any other angle of the rotary table. For this, the orientation of the table itself needs to be measurable to a certain minimum accuracy. If this is not possible, the laser beam should be aligned at more angles, or at all angles which are to be used.
As an alternative to aligning the apparatus only from the position and orientation of the deflected laser beam, it is possible to use also the position and orientation of the incoming laser beam (i.e. between the laser 24 and the mirror 20). This enables one to reduce the number of angles of the deflected beam which must be measured. Once aligned, the apparatus is used as follows. In a certain fixed position of the rotary table, the machines ram 12 is automatically moved along the reflected laser beam 28, the position and direction of which are now known. At certain positions, the machine's travel along the line of movement (as determined by the computer from the scales of the machine) is compared with the reading given by the interferometer 22. This gives readings of the accuracy of the machine when making a number of length measurements, e.g. as indicated at 26 between start and finish points 26A,26B. As many such accuracy readings as desired can be made during this movement.
Next, the rotary table 14 is rotated by the motor 18, under computer control, so that the laser beam 28 takes up a new orientation. The above process is repeated, moving the ram 12 along the new orientation of the laser beam and taking as many comparative readings as desired. The process is repeated again for as many different angles of orientation of the laser beam as required. Thus the accuracy of a large number of attained distances of the ram 12 in different directions can be measured fully automatically. Since the position and orientation of the mirror is the same at both end points 26A,26B of the measured
displacement 26 along the line of the laser beam, errors in the location of the mirror only affect the measurement accuracy by cosine errors. Thus the accuracy requirements of the rotary table, mirror surface and retroreflector are relatively low. Due to the alignment procedure, the mirror does not have to be accurately mounted on the rotary table, and the position and orientation of the incoming laser beam and of the rotary table need not be known for any of the measurements.
Fig 2 shows in more detail an arrangement for measuring the accuracy of a coordinate measuring machine 30, having three axes x,y,z. A similar arrangement may be used for other machines having three or more axes.
Fig 2 includes various components which are similar to those of Fig 1, including a retroreflector 10 attached to the ram 12 of the machine, a laser 24 and an interferometer 22. A mirror 20 is also provided, as before, except that it is now mounted on a rotary table 14, e.g. in a gimballed arrangement, in such a way that it can be driven about two different axes of rotation (which need not be
perpendicular). It can therefore be oriented to direct the laser beam 28 into any desired direction in three
dimensions within the three dimensional working volume of the machine, so that it can be directed at the
retroreflector 10 wherever it may be positioned within the three dimensional working volume. Fig 2 also shows other components of the practical arrangement, including a computer 32 which provides the numerical control of the three axes of movement of the machine 30; and a computer 34 from which the accuracy measurements are controlled and which is linked to the computer 32, e.g. by an RS232 link 36. The computer 34 receives readings from the laser interferometer 22,24 via an interface 40, and the readings may be compensated for atmospheric conditions detected by a sensor unit 42 if desired. The two axes of rotation of the mirror 20 are preferably motorised and controlled
automatically via a controller 38 from the computer 34.
However, it is possible to have only one axis motorised and the other adjustable manually, or both adjustable manually, if desired.
The apparatus of Fig 2 is used in exactly the same manner as the apparatus of Fig 1, except of course that the laser beam 28 is directed into a number of orientations in three dimensions, using the two axes of rotation of the mirror
20, so that comparative length measurements are taken along the laser beam 28 throughout the working volume of the machine 30. A machine having three or more axes could also be measured with a mirror 20 having only one axis of rotation, as in Fig 1, by moving the laser 24 and interferometer 22 to obtain the other axis of beam orientation. However, the laser and the mirror 20 must remain fixed relative to each other during each set of measurements at a given
orientation.
Fig 3 shows a modification of Fig 1, in which the common components have been given the same reference numerals.
The modification lies in the provision of a position detector 44, which receives the returning laser beam via a beam splitter 42, after the laser beam has been rereflected by the mirror 20. The output signal of the detector 44 controls the position of the mirror 20 via a servo connection 46 to the motor 18. This causes the mirror 20 to track the movements of the retroreflector 10. During the measurement of a displacement 26, it is no longer necessary for the retroreflector to be moved in a straight line along the laser beam 28, since the laser beam will track the retroreflector should it deviate from this line. The retroreflector 10 can be moved from the start position 26A to the finish position 26B of the displacement by any convenient route. This is useful in the case of manual or numerically controlled machines which are
incapable of moving along an arbitrary straight line. The start and finish, points 26A,26B are of course selected such that the mirror 20 has the same orientation for each of these positions. It will be appreciated that the ability of the rotary table 14 to return repeatably to the same position will affect the accuracy of the measurement, and so it is desirable that this repeatability should be high (although it is still not necessary to know accurately the absolute position and orientation of the mirror 20).
When the tracking arrangement of Fig 3 is applied to a three dimensional arrangement such as shown in Fig 2, then of course the position detector 44 should be capable of detecting deviations in two dimensions and controlling the two axes of the rotary table 14 accordingly. The position detector may be connected to the computer 34 via an
interface unit 48, so that the servoing of the axes of the rotary table 14 occurs via the computer 34 and the
controller 38.
The position detector 44 is also useful during the initial alignment of the rotary table 14, since it facilitates the detection of the optimal position of the retroreflector 10 during the alignment procedure.
In place of a single mirror 20 having two rotary axes as shown in Fig 2, it is equally possible to use two mirrors each having one axis about which it can be rotated under the control of the computer 34. The laser beam is incident upon each of the mirror in turn. The above description has referred to a mirror 20, or to two such mirrors. It should be appreciated, however, that this is merely a convenient means for directing the laser beam 28 into the various required orientations, and any equivalent means can be used to reflect or refract the laser beam into the various desired orientations, e.g. a rotatably mounted prism.
The retroreflector 10 can be a corner-cube retroreflector, or a cat's-eye retroreflector may be used for its wide angle properties. Alternatively, a plane mirror may be used (with a plane mirror interferometer system 22). The plane mirror should be adjustably mounted to the ram 12 so that it can be adjusted to be normal to the beam
orientation each time the beam orientation is changed.
The retroreflector 10 can be mounted on an arm extending laterally from the ram 12. This is particularly desirable if it is desired to measure z-axis roll (in the arrangement depicted in Fig 2).
The measurements taken of numerous different displacements 26 in various different orientations, using the apparatus of Figs 1 to 3 , may if desired simply be taken to indicate the volumetric accuracy of the machine, in the same way as is commonly done with measurements taken upon artefacts as discussed in the above description of the prior art.
Alternatively, there will now be described a method by which the parametric errors of a multi-axis machine can be estimated from the length errors in a large number of such distance measurements. The large number of distance measurements may be obtained using the apparatus and method described above, or other interferometer length measurement arrangements. Alternatively, they may be obtained from conventional measurements upon artefacts of the type discussed above. The method as discussed below can be applied to machines with prismatic and/or revolute joints in an arbitrary configuration, including large machines, software compensated machines and machines with significant finite stiffness related errors.
The position errors of the end effector (e.g. a tool or probe) of a multi-axis machine are caused by separate errors introduced in the components of the machine (e.g. squareness, straightness and scale errors, and angular errors of roll, pitch and yaw). These are the so-called parametric errors.
In the first stage of the method, a mathematical model is built of the machine's error structure, including a
suitable description of its parametric errors. According to the nature of the measured distances, an algebraic linear relation is next established between the respective deviations in their measurement and the unknown parameters describing the parametric errors. (N.B. It should be borne in mind that a relationship such as Δx = βx3 is linear for the present purposes, since the unknown parameter is β , and Δx varies linearly with β . ) It is also important that each machine error should appear in only one parametric
equation. For example, squareness error can be considered as a steady gradient in the parametric equations describing straightness errors, or it can be considered as a static rotation error. It should be modelled as one of these alternatives, but not both. In the description below, it is modelled as a static rotation.
Finally, these parameters are estimated by standard linear regression techniques, such that the residual sum of squares between the modelled and observed errors in the measurements is minimised. Since the model is
overdetermined, this least squares method is used to allocate the errors between the parameters, as opposed to simply allocating a given error to a given parameter. In the case of non-repeatable machines one has to look at not only the mean error allocated to the parameter but also at its variance. The linear regression may be performed using off-the-shelf software, either in the computer 34 (Fig 2) or in another computer using the data recorded by the computer 34.
Each parametric error is described as a linear combination of known functions. The known functions are defined on the position of the machine's axes and other relevant variables (e.g., the output of a certain temperature sensor).
In this equation, pk(x) is a known function defined on the relevant independent variables contained in the vector x. Bk is the unknown parameter describing the contributions of this function to the parametric error Ei,j , and has to be estimated.
Although most mathematical functions can be used to describe the general trend of an error, their application is limited to model the frequently disjointed or disassociated nature of the remainder. Application for high accuracy calibration and software compensation requires the use of a special family of functions to comply with this characteristic. We have obtained good results using piecewise polynomials. In the description of piecewise polynomials we make use of truncated polynomials or "+"-functions [2]. The "+"- function is defined as:
· u+ = u if u > 0
· u+ = 0 if u≤ 0
In general, with k knots t1, ..., tk and k+1 polynomial pieces each of degree n. the truncated power representation of a piecewise polynomial p(x) with no continuity restrictions can be written as:
Note that the presence of a term allows a discontinuity in the
j-th derivative of p(x). Thus different continuity restrictions can be
imposed at different knots simply deleting the appropriate terms. Normally it is sufficient to ensure that each model is continuous with respect to the function value and its first derivative.
In practice parametric errors are often modelled as a linear interpolation between their values at different points. Such model can be easily described by piecewise polynomials.
The model's potential to accommodate irregular errors is to an extensive degree determined by the number and positions of the knots. If the positions of these knots are considered variable, that is: parameters to be estimated, they enter into the regression problem in a nonlinear fashion., and all the problems arising in nonlinear regression are present The use of variable knot positions also carries the practical danger of overfitting the data, and makes testing of the hypothesis concerning areas of structural change virtually impossible. Unless prior information is available, we use a basic model which contains enough polynomial pieces with a fixed length and a specified maximum degree, to accommodate the most complex error expected.
The relative position error of the end-effector of the machine is related to the parametric errors using an error model [3]. A large variety of such models have been described, these models are linear in the parametric errors, since the difference between the nominal and the actual machine geometry usually does not significant change the active arm of angular errors and the direction in which the various errors acts. For a multi-axis machine having n axes the model can be summarized as:
In equation 2. the vector Ei contains the three angular and three
displacement errors introduced in axis i and its surrounding components. The 3 × 6 matrix uFi describes how these parametric errors affect the errors ΔP in the relative position of the end-effector. This matrix is completely defined by the nominal geometry of the machine, the length of the
end-effector and the position of the axes. Vector eu contains the
displacement errors introduced by the end-effector (e.g.. the probe system).
For the estimation of the unknown parameters β we use the observed errors in a large number of distance measurements. These measurements can be realized by artifacts, or by using a distance measuring instrument with a high relative accuracy (e.g., laser interferometer), for instance as described above .
In order to relate these observed deviations to errors in the realized
(relative) positions of the end-effector, the following approximation is made:
The difference between the measured (by the machine) and actual position and orientation of the reference distance has a negligible effect on the measurement error of this distance.
Consider a distance measurement between the actual points and (see Figure 4), with actual distance L. These points are measured as and respectively. Using the above mentioned approximation, the observed error ΔL in the measured distance Lmcu can be modeled as a known linear combination of the errors and in the measured positions and
In relation 5, the observed error ΔL is approximated as the projection along the estimated feature of errors in the relative positions of the
end-effector, which results in a linear relation. Also the error model of the machine (equation 2) and the description for the various parametric errors (equation 1) are linear in respectively the parametric errors Ei,j and the unknown parameters β that describe these errors. An appropriate combination of these models yields a relation in which the measurement error ΔLi of an individual distance measurement is expressed as (known) linear combination of the unknown parameters β:
In equation 6, γi represents the measurement error due to non-repeatable and non-modelled machine errors. Vector xi contains the variables which describe the status of the machine and its environment during the measurement (e.g., the position of the machines axes for each measured point). The known function igk(Xi) describes the effect of parameter βk on the feature's measurement error. Basically the values of the related known function pk(x) in relation 1. that describes the effect of the of parameter βk on the respective parametric error, are calculated for the status of the machine when measuring the end points of the considered distance. These values are then inserted in relation 2 to obtain the effect of βk on the displacement error of the end-effector for various measured points. igk(xi) is computed as the dot product of the difference between both displacement errors with the direction vector along the measured distance.
The proposed measurement procedure, based on m distance measurements throughout the machine's workspace, can presented mathematically as:
In matrix notation:
ΔL = Xβ + γ (9)
The vector γ of 'random' errors is assumed to be independent and normally distributed with mean 0 and variance σ2. Since the above model is linear in the unknown parameters β, they can be estimated using linear regression analysis [4]:
In the case of using artifacts of unknown constant dimensions, the vector β has to be expanded to include these dimensions. Obviously, at least one measurement of known dimensions has to be made, in order to obtain absolute accuracy.
Equation (1) above is a simplification, because it ignores cross-talk between different axes, that is, the way in which the position in one axis affects the errors in another. To include cross-talk in equation (1), the vector x can be generalised to include the effects seen in the x axis when y and z are varied:
Consequently, equation (1A) will have further terms
describing the effect of variations in y and z axes upon p(x).
EXAMPLE
To illustrate the above method, consider a two-dimensional coordinate measuring machine depicted schematically in Fig 5. The machine has an x slide 50 connected to a y slide 52. A probe P is connected to the y slide 52, offset by a distance a. The position of the probe P is measured by the machine as having coordinates (X,Y). The discussion below also refers to a direction z, normal to the plane of Fig 5.
The parametric errors of the machine are:
- scale : xTx, yTy (position errors of scales) - angular : xRx, xRy, xRz (roll, pitch and yaw
yRx, yRy , yRz errors caused by small
rotations of slides)
- straightness: xTy, xTz, (out of straightness errors
yTx, yTz of slides) This notation is consistent with VDI 2617. xRx, xRy, yRx, yRy, xTz and yTz only produce cosine errors, therefore they are neglected, in this example.
As an example, consider the following over-simplified error model (used to obtain a first approximation of major error sources): xTx = β1X (E1) yTy = β2Y
xRz = β3 + β4X
yRz = x Ty = y Tx = 0 It will be noted that the yaw xRz includes a constant component β3 to describe the squareness error between x and y axes.
The errors ΔP experienced at the probe P can be described by:
Equations E2 and E3 constitute a formal statement of the properties of the machine. Substituting E1 into E3 gives:
Now consider the measured errors ΔLi in a number of measured lengths Li. From the above discussion of Fig 4 and equation (5):
where γi is a term relating to the difference between the real error and the modelled error, and
Here is the estimated unit direction vector of the
measurement, since should in theory equal Li, and
in practice is close to Li because the errors are small. From E4:
Substituting E8 into E6 and multiplying out:
where:
ΔLi = L i meas . L i (E10 )
Consider m distance measurements:
or as given in equation (9) above:
ΔL = X β + γ (E12)
The computer solves this equation by linear regression as
described above, using the data for the m measurements.
In the case of measuring an artefact with a constant
unknown length βL, β can be presented as:
The actual length of the artefact is defined as:
L = Lassumed + βL (E14)
Equation E10 can now be described as:
(E15) ΔLi = Limeas - (Lassumedi + βL)
In order to estimate βL, and thus relate the estimated error model to the absolute length, at least one measurement of known length has to be made.
REFERENCES
The following are incorporated herein by reference, items [2], [3], and [4] being referred to in the above
description of the method of the invention:
[1] U.K. Patent Applications Nos. 9121687.9, 9121686.1 and 9121685.3, from which the present application claims priority.
[2] Smith P.L. "Splines as a Useful and Convenient
Statistical Tool",
The American Statistician, 33(2):57-62, 1979.
[3] Soons J.A., Theeuws F.C., and Schellekens P.H.
"Modeling the errors of multi-axis machines: A General Methodology". Precision Engineering, 14(l):5-19, 1992.
[4] Montgomery D.C. Peck E.A. "Introduction to Linear
Regression Analysis" John Wiley & Sons, New York, 1982,

Claims

1. A method of measuring the accuracy of a multi-axis machine, the machine including a relatively fixed member and a component which is movable in at least two axes with respect to the fixed member, and having transducers for measuring the movement in said axes, the method comprising: mounting reflecting means on said component,
mounting light beam orientation means at a location which is fixed relative to the fixed member,
directing a light beam at said reflector via the light beam orientation means and receiving a return beam
therefrom,
measuring the difference in path length of said light beam between two positions of said component lying along a first orientation of the light beam, and comparing said path difference with values derived from the transducers of the machine, to produce an error value,
varying the orientation of the light beam using the orientation means to at least one further orientation, and measuring the difference in path length of said light beam between two positions of said component lying along said further orientation of the light beam, and comparing said path difference with values derived from the
transducers of the machine, to produce a further error value.
2. A method according to claim 1, wherein said path length different measuring steps each include measuring a
plurality of path length differences between more than two positions of said component, to produce a corresponding plurality of error values.
3. A method according to claim 1 or claim 2 wherein the path length difference measuring step is repeated at a plurality of different said further orientations.
4. A method according to any one of the preceding claims in which the orientation means is a two-axis orientation means capable of varying the orientation of the beam over a three-dimensional working volume of the machine.
5. A method according to any one of the preceding claims wherein the difference in path length is measured using an interferometer.
6. A method according to any one of the preceding claims in which the orientation means automatically tracks the movement of the reflector.
7. A method according to any one of the preceding claims including the further step of using a plurality of the error values derived at a plurality of said orientations to calculate parametric errors relating to the error
components of the machine.
8. A method according to any one of the preceding claims, including a preliminary step of determining said
orientations of the light beam, using said transducers of the machine.
9. Apparatus for measuring the accuracy of a multi-axis machine, the machine including a relatively fixed member and a component which is movable in at least two axes with respect to the fixed member, and having transducers for measuring the movement in said axes, the apparatus
comprising:
reflecting means provided on said component,
light beam orientation means provided at a location which is fixed relative to the fixed member,
means for directing a light beam at said reflector via the light beam orientation means and receiving a return beam therefrom,
means for varying the orientation of the light beam to at least one further orientation, means for measuring the difference in path length of said light beam between two positions of said component lying along a first orientation of the light beam, and comparing said path difference with values derived from the transducers of the machine, to produce an error value; and for measuring the difference in path length of said light beam between two positions of said component lying along a further orientation of the light beam, and comparing said path difference with values derived from the transducers of the machine, to produce a further error value.
10. Apparatus according to claim 9 in which the orientation means is a two-axis orientation means capable of varying the orientation of the beam over a three-dimensional working volume of the machine.
11. Apparatus according to claim 9 or claim 10 wherein the means for measuring the difference in path length is an interferometer.
12. Apparatus according to any one of claims 9 to 11, wherein the means for varying the orientation is at least partially controlled automatically.
13. Apparatus according to claim 12, including means for causing the orientation means to automatically track the movement of the reflector.
EP92921141A 1991-10-12 1992-10-12 Measuring the accuracy of multi-axis machines Withdrawn EP0607240A1 (en)

Applications Claiming Priority (7)

Application Number Priority Date Filing Date Title
GB919121686A GB9121686D0 (en) 1991-10-12 1991-10-12 A method to estimate a multi axis machine's accuracy using length measurements
GB9121685 1991-10-12
GB9121686 1991-10-12
GB9121687 1991-10-12
GB919121687A GB9121687D0 (en) 1991-10-12 1991-10-12 A system to measure the volumetric positioning distance accuracy of multi axis machines
GB919121685A GB9121685D0 (en) 1991-10-12 1991-10-12 The efficiency of artifact based procedures to calibrate coordinate measuring machines
PCT/GB1992/001871 WO1993008449A1 (en) 1991-10-12 1992-10-12 Measuring the accuracy of multi-axis machines

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