Kinematics Error Compensation for a Surface Measurement ... - MDPI

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Kinematics Error Compensation for a Surface Measurement Probe on an Ultra-Precision Turning Machine Duo Li *, Xiangqian Jiang *, Zhen Tong and Liam Blunt Engineering and Physical Sciences Research Council (EPSRC) Future Metrology Hub, University of Huddersfield, Queensgate, Huddersfield HD1 3DH, UK; [email protected] (Z.T.); [email protected] (L.B.) * Correspondence: [email protected] (D.L.); [email protected] (X.J.); Tel.: +44-148-447-1674 (D.L.) Received: 30 May 2018; Accepted: 28 June 2018; Published: 2 July 2018







Abstract: In order to enhance the measurement availability for manufacturing applications, on-machine surface measurement (OMSM) is integrated onto the machine tools, which avoids the errors caused by re-positioning workpieces and utilizes the machine axes to extend the measuring range as well. However, due to the fact that measurement probe actuation is performed using the machine tool axes, the inherent kinematics error will inevitably induce additional deviations onto the OMSM results. This paper presents a systematic methodology of kinematics error modelling, measurement, and compensation for OMSM on an ultra-precision turning lathe. According to the measurement task, a selective kinematics error model is established with four primary error components in the sensitive measurement direction, based on multi-body theory and a homogeneous transformation matrix (HTM). In order to separate the artefact error from the measurement results, the selected error components are measured using the reversal method. The measured error value agrees well with the machine tool’s specification and a kinematics error map is generated for further compensation. To verify the effectiveness of the proposed kinematics error modelling, measurement, and compensation, an OMSM experiment of an optically flat mirror is carried out. The result indicates the OMSM is the superposition of the sample surface form error and the machine tool kinematics error. With the implementation of compensation, the accuracy of the characterized flatness error from the OMSM improves by 67%. Keywords: on-machine surface measurement; kinematics error model; error measurement; error compensation; ultra-precision machine tool

1. Introduction Ultra-precision manufacturing has developed over the decades to produce highly demanding surfaces for optical, electronic, or mechanical applications [1]. In order to enhance the availability of metrology for such applications, there has been a shift in the metrology approach from offline, lab-based solutions towards the use of metrology within manufacturing platforms [2]. The advantage of on-machine surface measurement (OMSM) is that the coordinate system between the machining and measurement processes is kept consistent, which avoids the errors caused by re-positioning workpieces. Moreover, the machine axes are utilized to extend the measuring range [3]. However, due to the fact that measurement probe actuation is performed on the machine, the machine tool kinematics error will inevitably induce additional errors to OMSM results. Therefore, it is necessary to model, measure, and compensate the machine tool kinematics errors for OMSM in order to improve the measurement accuracy.

Micromachines 2018, 9, 334; doi:10.3390/mi9070334

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Modelling and compensation of kinematics errors for ultra-precision machine tools have received much interest [4–7]. Kong et al. [8] developed a kinematics error model for two-axis ultra-precision turning machines. Software error compensation was carried out by the modification of the ideal tool path in the numerical control program. Flat and tapered machining experiments have been conducted to verify the established theoretical kinematics model and higher surface accuracy was achieved with the implementation of the proposed compensation method. Chen et al. [9] studied the modelling of volumetric error and its sensitivity analysis for a five-axis ultra-precision machine. The volumetric error model, including 37 error components, was established based on rigid body kinematics and a homogeneous transformation matrix. In addition a sensitivity analysis was carried out for the purposes of accuracy design and manufacture specification of the machine tool. Yu et al. [10] analysed the main sources of machining errors and their effect on diamond-turned micro-structured surfaces. Sliding error, as well as the dynamic error of the fast tool servo mechanism, were identified and compensated for by means of tool path modification. Machining experiments of typical micro-structured surfaces demonstrated the effect of the component errors on profile accuracy and verified the effectiveness of the proposed compensation methods. Gao et al. [11] studied the measurement and compensation of error motions of a diamond turning machine for nano-fabrication of large sinusoidal metrology grids. The out-of-straightness of the X slide, and the axial and angular motion of the spindle, were measured, respectively. The experiment results indicated that the out-of-flatness of the workpiece was reduced from 0.27 to 0.12 µm after compensation machining by utilizing a fast tool servo unit. Liu et al. [12] classified the machining errors into five categories for a three-axis turning machine according to the coordinate distortions direction. The author pointed out that the effect of kinematic errors on the coordinate distortion and form accuracy are dependent on the surface type. A one plane-spherical surface was fabricated to identify the main machining errors on an ultra-precision turning machine. Both simulation and machining experiments were implemented to verify the effectiveness of the kinematic error model and compensation strategy. Most previous work focused on machine tool error modelling for the improvement of machining accuracy. There is relatively little research on the comprehensive investigation of kinematics error modelling, measurement, and compensation for on-machine surface measurement applications. To address this issue, this paper will firstly develop a selective kinematics error model according to the measurement task and the configuration of the machine tool. The measurement processes for selective kinematics errors in the sensitive direction are also presented. Next, a kinematics error map for OMSM compensation is generated using the established error model and error measurement results. Finally, an OMSM experiment of an optically flat mirror is carried out to verify the effectiveness of the proposed modelling, measurement, and compensation. 2. Overview of OMSM System In this work, an optical interferometric probe, as the on-machine surface measurement instrument, is integrated onto a three-axis ultra-precision turning lathe (Nanoform 250 Ametek Precitech, Keene, NH, USA), equipped with two linear hydrostatic axes and one rotational air bearing spindle. The probe, termed dispersed reference interferometry (DRI) [13], works on the principle of a modified Michelson interferometer with chromatic dispersion purposefully added in the reference arm, resulting in a wavelength-dependent optical path length. The dynamic design enables its measuring capacity to nanometre resolution (0.6 nm) and millimetre vertical range (800 µm). The use of a low coherence light source lends itself to an optical fibre-based implementation, giving the potential for remote configuration and miniaturization in the manufacturing environment. The DRI probe is mounted beside the diamond tool holder on the Z slide and aligned coaxially to the spindle axis before OMSM operations. The system configuration is illustrated in Figure 1.

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Figure On-machinesurface surface measurement measurement system Figure 1. 1. On-machine systemconfiguration. configuration.

For on-machineFigure measurement, the DRI probe is carried by the machine tool axes to scan over the 1. On-machine surface measurement system configuration. For on-machine measurement, theinherent DRI probe is carried by the machine tool to scan sample surface. However, due to the mechanical imperfections, wear of theaxes machine toolover the sample surface. However, due to the inherent wear of path, the machine elements and stage misalignments [14], and deviation from programmed scanning will For on-machine measurement, the DRI probe is mechanical carried by theimperfections, machine tool axes to scan over the toolsample elements and However, stage misalignments [14], and deviation from programmed scanning induce undesirable errors in the measurement results. Hence, the the machine kinematics error surface. due to the inherent mechanical imperfections, weartool of the machine toolpath, needs to be modelled, measured, compensated forresults. OMSM results. The flowchart of the proposed and stage misalignments [14], and deviation from the programmed scanning path, will willelements induce undesirable errors in and the measurement Hence, the machine tool kinematics methodology illustrated inthe Figure 2. According to Hence, thefor measurement task machine tool induce errors in measurement results. the machine tooland kinematics error error needsundesirable to beismodelled, measured, and compensated OMSM results. The flowchart of the configuration, a selective kinematics error modelling measurement will carried out. tool needs to be modelled, and for and OMSM Theprocess flowchart ofbe the proposed proposed methodology ismeasured, illustrated in compensated Figure 2. According to results. the measurement task and machine The machine tool kinematics in2.modelling the scanning is consequently in order to The methodology is illustrated in error Figure According toregion the measurement taskmapped andbemachine tool configuration, a selective kinematics error and measurement process will carried out. compensate for the OMSM result. To validate the and proposed methodology, result is configuration, a selective kinematics error modelling measurement processthe willOMSM be carried out. machine tool kinematics error in the scanning region is consequently mapped in order to compensate compared with offline measurement results. The machine tool kinematics error in the scanning region is consequently mapped in order to for the OMSM result. To validate the proposed methodology, the OMSM result is compared with compensate for the OMSM result. To validate the proposed methodology, the OMSM result is offline measurement results. compared with offline measurement results.

Figure 2. Flowchart of the kinematics error compensation for on-machine surface measurement (OMSM). Figure 2. Flowchart of kinematics the kinematics compensation for on-machine surface measurement Figure 2. Flowchart of the errorerror compensation for on-machine surface measurement (OMSM). (OMSM).

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3. Kinematics Error Modelling 3. Kinematics Error Modelling Kinematics error modelling for multi-axis machine tools is based on the multi-body system Kinematics error modelling for multi-axis machine tools is based on the multi-body system theory [15]. The multi-body system theory offers a comprehensive description of general mechanical theory [15]. The multi-body system theory offers a comprehensive description of general mechanical systems utilizing the lower-order body topological structure. Using a homogeneous transformation systems utilizing the lower-order body topological structure. Using a homogeneous transformation matrix (HTM), the relationship between multi-coordinates can be established and spatially-distributed matrix (HTM), the relationship between multi-coordinates can be established and single error components are consequently synthesized as a volumetric error model. For the machine spatially-distributed single error components are consequently synthesized as a volumetric error tool configuration in the current work, there are two kinematics error chains, as illustrated in Figure 3. model. For the machine tool configuration in the current work, there are two kinematics error chains, One is from the machine base to the machining surface, and the other is from the machine base to the as illustrated in Figure 3. One is from the machine base to the machining surface, and the other is DRI probe. from the machine base to the DRI probe.

Figure 3. Kinematics error chain for an an on-machine on-machine surface surface measurement measurement system. system.

The overall overallconfiguration configurationofof machine coordinate systems is shown in Figure 4. The The thethe machine tooltool coordinate systems is shown in Figure 4. The spatial spatial relationship between adjacent coordinate systems can be mathematically described using the relationship between adjacent coordinate systems can be mathematically described using the homogeneous transformation matrix. For example, the transformation matrix 𝑇 describes the i homogeneous transformation matrix. For example, the transformation matrix j T describes the coordinatetransformation transformation from coordinate j to coordinate i. By sequential coordinate from coordinate systemsystem j to coordinate system i. Bysystem sequential multiplication i between multiplication of four transformation terms, the comprehensive transformation matrix 𝑇 of four transformation terms, the comprehensive transformation matrix j T between two adjacent bodies twobe adjacent bodies can formulated as:can be formulated as: i i i i i i j T loce j T i= T jT loc T iT j TimTj Time T j

j loc j loce j m j me

(1) (1)

where ij T loc is the location transformation matrix, ij T loce is the location error transformation matrix, is the location error transformation matrix, where 𝑇 is the location transformation matrix, 𝑇 i T is the motion (translation or rotation) transformation matrix, and i T the motion (translation j 𝑇m is the motion (translation or rotation) transformation matrix, and j 𝑇me is is the motion (translation or rotation) error transformation matrix. According to the kinematics chain structure illustrated or rotation) error transformation matrix. According to the kinematics chain structure illustrated in in Figure 3, the comprehensive transformation matrices between adjacent coordinates can, thus, Figure 3, the comprehensive transformation matrices between adjacent coordinates can, thus, be be derived and listed in the following Table 1. derived and listed in the following Table 1. By transferring to a common machine base coordinate system from the two chains, we have: T= 0 3T

∏ ij T

= 01 T 12 T 23 T

0 5T

= 04 T 45 T

(2) (3) (4)

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Figure Figure 4. Configuration Configuration of of the machine tool coordinate systems. Table Table 1. Kinematics transformation matrices between adjacent coordinates.

Symbol Symbol

Kinematics Transformation Between Adjacent Coordinates Kinematics TransformationMatrices Matrices Between Adjacent Coordinates

1 10 00 0p1x  0 01 10 0p1y    0 0 1 p1z 000 00 1 1  p2x 0 0 1 0  p2y  0  0 p2zp2 1 x − EBOC   0 1 0 p 0 

00T 1

T

1

 1T 2

1 2

T

2T 3

1  0   01 00

0 0 1 0 001 010

 0 0 1  0 0 0 

0T 4

2 3

T

4T 5

1  0   0 0

2y   p2 z    EBOC  1  0

0 1 0 0

0 0 1 0

p1x  11 0 0 0 00  0 0 1 1 0 00 p 1 y  0 0 1 0 p1z  0 0 0 0 0 11  1 0  0 E0BOC0 1 0 E AOC 01

EAOC 0

 p4x 1  p4y  0  p4z  − EBOZ 0 1

0 1 0 0 x 1 −EECX EEBX EE XX  1 0 0 x 1 BX XX CX  0 00 11 0 0 0 0 EECX − EEYX 11  EEAX    CX AX YX   0 0 1 0  − EBX E AX 1 EZX  0  00 00 0 1 1 0    E0BX E0AX E1ZX  01    (θ ) 0 0   1  01   0 cos − E 0(θ )0 −1sin 0 0 0 1  EBC EXC CC   sin(θ )    ECC − E AOC 0  cos θ 0 0 1 − E E ( ) AC YC    E1BOC 0 0 E1BC EE   0  − E1BC EE cos0   sin 0  1 00  CC AC ZCXC   0  0  0 0 1 0 1 0 0 1 1  E AOC  0  sin   E AC EYC    cos 0 0   ECC 1 0 0 p3x   1  1 EZC  1 0    EBC E AC  00 1 00 p3y  0  0  0 1 p3z    0 10  0 00 1 0 0 1  0 0 1  0    0 EBOZ 10 0 1 0 0 p0 0 1 − ECZ EBZ EXZ  0 1 03 x 0  ECZ  1 0 0  1 − E E AZ YZ    00 1 0 0 0 p13 y z  −EBZ EAZ 0 1 1 EZZ   0 0  1 0 0 0 1 01 00 00 1 0p p03z 1 5x    0 1 0 p5y  1  0 0 0  0 0 1 p5z    ECZ EBZ E XZ  0 EBOZ0 00  01 10 0 0   1

   

   

 1 0 0 p4 x   1 0 1 0 p   0  E AZ EYZ  1 0 0  0 1 0 0   ECZ 1 4y   0  T 4    EBOZ  0 0 displacement 0 0 vector, 1 p4 zwhich 0 1 the0relative 1 z    EBZ deviation E AZ 1 between EZZ  the DRI The volumetric error describes      0 1 can, 0 1  as: 0 0 0 1  0 0 0 1   0 0 surface   0thus, 0be derived probe and the workpiece 0 0 p 5x     1    p E p 0 1 0 p x 3x 5x 5 y 4  E      p  5T  5y  0  0 1 0 p  y  0  p3y 5z (5)   = T  − 5T     Ez  3  p3z   0 0 0 1  p5z 

1 1 1 from the two chains, we have: By transferring to a common machine base coordinate system i All the error variables presented above follow convention according to the ISO 230-1 [16]. T  the (2) jT For three-axis turning machines there are 21 error components in the established kinematics model, including 18 error terms for individual axes (6 0× 3) and three error terms describing the relative location T  01T 21T 23T (3) 3 between the three axes. However, it is time-consuming and unnecessary to model and measure all the

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error components. More attention should be paid to the primary error components in the sensitive direction because they directly influence the surface measurement accuracy. In the current work, the Z direction illustrated in Figure 1 is considered as the measurement-sensitive direction according to the measurement task. Four error components are selected as primary factors affecting the on-machine measurement results in the sensitive direction. They are, respectively, the X axis straightness in the Z direction, EZX ; the squareness error between the X and C axes, EBOC ; the C axis axial error, EZC ; and the C axis tilt error, EBC . These four error components will be measured, synthesized, and employed to generate the kinematic error map in Section 4. Furthermore, the selective kinematics error model in the Z direction can be simplified as follows: EZ = z − EZC − EZX + x EBC + x EBOC

(6)

From the formula above, it can be seen that the synthesized error in the Z direction comprises the four selective error components. With the derived selective kinematics error model, the individual and combined effect of these errors on OMSM results are numerically simulated and illustrated as 3D error maps in Figure 5. The X axis straightness error in the Z direction EZX will cause the wavy pattern along the radial direction, while the squareness error EBOC between the C and X axes in the X-Z plane results in the cone-shaped surface. The C axis motion errors, including axial motion EZC and tilt error EBC , will induce several circumferential ripples, whose number depends on the spindle motion error characteristics. It can also be inferred that the squareness error and C axis tilt error tends to exaggerate the motion error in the Z direction with increasing sample radius. Compensation of the error components EBOC and EBC should receive more attention for the on-machine measurement of Micromachines 2018, 9, x 7 of 16 large-scale surfaces.

Figure Figure 5. 5. Simulation Simulation of of the the kinematics kinematics error error effect effect on the OMSM results.

4. 4. Kinematics Kinematics Error Error Measurement Measurement Ultra-precision machinetools toolsare areequipped equipped with high-precision linear hydrostatic guideways Ultra-precision machine with high-precision linear hydrostatic guideways and and air-bearing spindles. The motion errors of the linear and rotational axes often lie in the air-bearing spindles. The motion errors of the linear and rotational axes often lie in the sub-micrometre, sub-micrometre, even in the nanometer, range [17–19]. Without the influence of the inherent surface even in the nanometer, range [17–19]. Without the influence of the inherent surface form error on the form error onseparation the artefact, error separation techniques have been widely adopted for precision artefact, error techniques have been widely adopted for precision measurement of error measurement of error motions on ultra-precision machine tools [20–22]. Among them, the reversal method is considered simple and accurate for the measurement of part features without reference to an externally-calibrated artefact [23]. This section proposes a simple scheme for machine tool kinematics error measurement at the nanometric level, with capacitance probes (Lion Precision C8, Minnesota, MN, USA) and flat artefacts. The maximum sampling frequency of the capacitance probes used is up to 1 kHz and the displacement measurement resolution is 0.08 nm. The testing of the

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motions on ultra-precision machine tools [20–22]. Among them, the reversal method is considered simple and accurate for the measurement of part features without reference to an externally-calibrated artefact [23]. This section proposes a simple scheme for machine tool kinematics error measurement at the nanometric level, with capacitance probes (Lion Precision C8, Minnesota, MN, USA) and flat artefacts. The maximum sampling frequency of the capacitance probes used is up to 1 kHz and the displacement measurement resolution is 0.08 nm. The testing of the capacitance probe does not indicate drifting issues, which is included in Appendix A. Furthermore, the 2 mm spot size also automatically filters out short wavelength errors on the target surface so that the artefact surface finish will not affect the measurement. In the following part, the measurement process for the X axis straightness in the Z direction EZX , C axis axial error EZC , C axis tilt error EBC , and squareness error between X and C axes’ EBOC will be respectively described. 4.1. X Axis Straightness Error The schematic diagram and experimental setup of the EZX measurement using the reversal method are respectively shown in Figures 6 and 7. A metal flat mirror was mounted on the Z axis stage and kept stationary. The capacitance probe was carried on the X slide and scanned over the mirror for 38 mm at 10 mm/min. Subsequently, the mirror was rotated 180◦ using a manual rotational stage and the mirror was scanned again after the reversal operation. The forward and reversal scanning were carried Micromachines out three times the averaged output was used for error separation. 2018, 9, and x 8 of 16 Micromachines 2018, 9, x 8 of 16

(a)

(b)

(a)

(b)

Figure 6. Schematic diagram of the EZX measurement. (a) Before reversal operation, (b) After reversal 6. Schematic diagram ofofthe EZX measurement. (a) Before (b) After ZXmeasurement. operation. Figure 6. Schematic diagram the E (a) Before reversalreversal operation,operation, (b) After reversal

Figure reversaloperation. operation.

Figure 7. Experimental setup of the EZX measurement using the reversal method.

The two measurements are respectively denoted as M1 and M2. According to the reversal Figure 7. Experimental setup of the EZXmeasurement measurement using the reversal method. Figure 7. Experimental setup of the EZX using theof reversal method. principle, the straightness error E ZX can be separated from the surface error the flat mirror Eflat and calculated by:

The two measurements are respectively denoted as M1 and M2. According to the reversal EZX the surface error of the flat mirror Eflat and principle, the straightness error EZX can be  Mseparated 1  E flat from (7)  calculated by: M E E



2

flat

ZX

 M1  E flat  EZX   1 (M  M ) EZX M  22  E2flat  1EZX

(8)

(7)

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The two measurements are respectively denoted as M1 and M2 . According to the reversal principle, the straightness error EZX can be separated from the surface error of the flat mirror Eflat and calculated by: ( M1 = E f lat − EZX (7) M2 = E f lat + EZX EZX =

1 ( M2 − M1 ) 2

(8)

The error separation results are shown in Figure 8. As shown in the upper plot, the straightness error of the X axis EZX is 52.6 nm over a 38 mm measurement range, in accordance with the machine tool specification (50 nm over 25 mm range). It should be noted that the measuring probe was set up on the Z axis at the same height as the spindle axis. Thus, the Abbé errors would be included in the measurement results. In fact, under such circumstances, the measured and separated errors (using the reversal method) includes the pure X axis straightness in its carriage axis line and the Abbé errors. However, it is not an issue for OMSM compensation in this work. As in the OMSM process, the DRI probe was also set up on the Z axis at the same height with the spindle axis. Therefore, this combined Micromachines 2018, 9, x 9 of 16 error can be compensated directly.

Figure 8. Error separation results of straightness error ZX and artefact profile error Figure 8. Error separation results of straightness error EZXEand artefact profile error EflatE.flat.

4.2. C Axis Axial and Tilt Error 4.2. C Axis Axial and Tilt Error For axis error measurement, facial reversal method is utilized measure axial and For CC axis error measurement, thethe facial reversal method is utilized to to measure thethe axial and tilttilt motion error [24]. Facial error motion, which is parallel to the rotational axis, is the superposition motion error [24]. Facial error motion, which is parallel to the rotational axis, is the superposition of the tilttilt error. TheThe schematic diagram and experimental setupsetup of theof facial of the theaxial axialerror errorand and the error. schematic diagram and experimental the reversal facial measurement is, respectively, illustrated in Figures 9 and 10. Two capacitance probes were reversal measurement is, respectively, illustrated in Figures 9 and 10. Two capacitance probes wereset MM 1 and M2), the flat mirror was setseparately separatelyatatthe thedistance distanceL.L.After Afterthe theforward forwardmeasurement measurement(output (output 1 and M2 ), the flat mirror rotated 180° relative to the C axis and the two probes were moved according to Figure Next,9b. the ◦ was rotated 180 relative to the C axis and the two probes were moved according to 9b. Figure reversal measurement (output (output M3 and M M4)and wasMperformed. The rotational speed was set as 20 Next, the reversal measurement 3 4 ) was performed. The rotational speed was revolutions per minute for forward and reversal measurements, and and the the measurement was set as 20 revolutions per minute for forward and reversal measurements, measurement performed over the measurement measurement outputs outputsMM1and andMM4are arethethe was performed over2020revolutions. revolutions.ItItisisnoted noted that that the 1 4 flat, the tilt error EBC, and the axial error EZC. combination of the flat form error E combination of the flat form error E , the tilt error E , and the axial error E . flat

BC

ZC

separately at the distance L. After the forward measurement (output M1 and M2), the flat mirror was rotated 180° relative to the C axis and the two probes were moved according to Figure 9b. Next, the reversal measurement (output M3 and M4) was performed. The rotational speed was set as 20 revolutions per minute for forward and reversal measurements, and the measurement was performed over 20 revolutions. It is noted that the measurement outputs M1 and M4 are the Micromachines 2018, 9, 334 9 of 15 combination of the flat form error Eflat, the tilt error EBC, and the axial error EZC.

(a)

(b)

Figure 9. Schematic diagram of the facial reversal method. (a) Before reversal operation, (b) After Figure 9. Schematic diagram of the facial reversal method. (a) Before reversal operation, (b) After Micromachines 2018, 9,operation. x reversal reversal operation.

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Figure Experimentsetup setup of reversal measurement. Figure 10.10. Experiment ofthe thefacial facial reversal measurement.

According to the facial reversalprinciple, principle, the error EflatE, flat the tilt error EBC , and axial According to the facial reversal theform form error , the tilt error EBC,the and theerror axial error E can be separated by: ZC  EZC can be separated by:   M1 = EFlat + L × EBC + EZC L× + EZCE M4 =EE L EBCEBC Flat −  M ZC   M 1= M Flat  2 3 = EZC M 4  EFlat  L  EBC  EZC ( M2 + M3 )   EZC =  ME2  =M(3M1−2 ME4ZC ) BC 2L   (M +M ) EFlat = 1 2 4 − EZC



M

M

(9)

(9) (10)



2 3 Figure 11 illustrates the error separation  EZC  results of the C axis measurement. Axial error EZC is measured to be 4.4 nm, which is within the range of 2 the machine tool specification (less than 15 nm).  It is noticed that the tilt error EBC shows a two-lobe pattern, as shown in the polar plot (Figure 11b).

  M1  M 4   EBC  2L    M1  M 4   E  EFlat  ZC 2 

(10)

Figure 11 illustrates the error separation results of the C axis measurement. Axial error EZC is measured to be 4.4 nm, which is within the range of the machine tool specification (less than 15 nm). It is noticed that the tilt error EBC shows a two-lobe pattern, as shown in the polar plot (Figure 11b).

   M1  M 4   E  EFlat  ZC 2  Figure 11 illustrates the error separation results of the C axis measurement. Axial error EZC is 10 of 15 measured to be 4.4 nm, which is within the range of the machine tool specification (less than 15 nm). It is noticed that the tilt error EBC shows a two-lobe pattern, as shown in the polar plot (Figure 11b). Micromachines 2018, 9, 334

(a)

(c)

(b)

Figure 11. Error separation of C axis axial error EZC, tilt error EBC, and artefact profile error Eflat. (a) Figure 11. Error separation of C axis axial error EZC , tilt error EBC , and artefact profile error Eflat . Axial error EZC, (b) Tilt error EBC, (c) Artefact profile error Eflat. (a) Axial error EZC , (b) Tilt error EBC , (c) Artefact profile error Eflat .

4.3. Squareness Error between X and C Axes 4.3. Squareness Error between X and C Axes

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The squareness error EBOC tends to induce a linear trend deviation on the surface measurement The squareness error EBOC tends induce a linear trend deviation on the surface results. The schematic diagram andto experimental setup of the squareness error EBOCmeasurement measurement results. The schematic diagram and experimental setup of the squareness error E BOC measurement are respectively illustrated in Figures 12 and 13. The same flat mirror was mounted on the C axisare and respectively illustrated inaFigures 12 and 13. The flat mirror mounted on the C axisthe and the the measurement using capacitance probe wassame performed by Xwas directional scanning over mirror measurement using a capacitance probe was performed X directional overdescribes the mirror surface. Linear slope β can be calculated by linear fittingby of the acquisitionscanning data, which the surface. Linear slope β canXbe calculated by linear fitting of as theillustrated acquisition which describes angle between the linear axis motion and the flat mirror, in data, Figure 12. Then, the Cthe axis angle between180 the◦ linear X axis motion andthe the X flat mirror, as illustrated in Figure 12. Then,and thereversal C axis was rotated and the scanning along axis was performed again. The forward was rotated 180° and the scanning along the X axis was performed again. The forward and reversal measurement were carried out three times and the averaged outputs were used for the squareness measurement were carried out three times and the averaged outputs were used for the squareness error calculation. error calculation.

(a)

(b)

Figure 12. Schematic diagram of EBOC measurement. (a) Before reversal operation, (b) After reversal Figure 12. Schematic diagram of EBOC measurement. (a) Before reversal operation, (b) After operation. reversal operation.

The averaged measurement results are shown in Figure 14. The squareness error between the X axis and the C axis EBOC can be derived by: EBOC =

1 × ( β 1 + β 2 ) − Etilt 2

(11)

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(a)

(b)

Figure of EBOCderived measurement. (a) Before reversal operation, After reversal where β1 12. andSchematic β2 are thediagram fitted angles respectively from the fitting of the(b) two measurement operation. data sets. The squareness error EBOC is calculated to be 0.08 arc sec.

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Figure setupof ofEEBOC measurement. measurement. Figure13. 13. Experimental Experimental setup BOC

The averaged measurement results are shown in Figure 14. The squareness error between the X axis and the C axis EBOC can be derived by:

E BOC 

1   1   2   Etilt 2

(11)

where β1 and β2 are the fitted angles derived respectively from the fitting of the two measurement data sets. The squareness error EBOC is calculated to be 0.08 arc sec.

Figure 14. Squareness error E

measurement result.

Figure 14. Squareness errorBOC EBOC measurement result. 4.4. Kinematics Error Map Generation

4.4. Kinematics Error Map Generation

Based on the established kinematics error model and error measurement results presented above,

Based on the kinematics error model and measurement results the machine toolestablished kinematics error was numerically mapped. Theerror kinematics error can be stored presented as above, the machine kinematics errorofwas numerically mapped. The kinematics error can be a look-up table fortool further compensation on-machine measurement results. It can be observed Figure 15 that table the kinematics errorcompensation map is dominated by a 2 upr (undulations per revolution) storedin as a look-up for further of on-machine measurement results. It can be component along the circumferential direction, which mainly results from the C axis tilt error motion observed in Figure 15 that the kinematics error map is dominated by a 2 upr (undulations per EBC , corresponding to the measurement result shown in Figure 11b. revolution) component along the circumferential direction, which mainly results from the C axis tilt error motion EBC, corresponding to the measurement result shown in Figure 11b.

Based on the established kinematics error model and error measurement results presented above, the machine tool kinematics error was numerically mapped. The kinematics error can be stored as a look-up table for further compensation of on-machine measurement results. It can be observed in Figure 15 that the kinematics error map is dominated by a 2 upr (undulations per Micromachines 2018, 9, 334 12 of 15 revolution) component along the circumferential direction, which mainly results from the C axis tilt error motion EBC, corresponding to the measurement result shown in Figure 11b.

Figure 15. Machine tool kinematics error map. Figure 15. Machine tool kinematics error map.

5. OMSM Experiment and Analysis 5. OMSM Experiment and Analysis In order to verify the proposed kinematics error modelling, measurement, and compensation, In order to verify the proposed kinematics error modelling, measurement, and compensation, an OMSM experiment of an optically flat mirror (Edmund Optics, Barrington, NJ, USA) was carried an OMSM experiment of an optically flat mirror (Edmund Optics, Barrington, NJ, USA) was carried out. The optical mirror used in the experiment was a superpolished surface. In this way, we can better out. The optical mirror used in the experiment was a superpolished surface. In this way, we can better investigate the effect of the machine kinematics error on the OMSM form measurement without other investigate the effect of the machine kinematics error on the OMSM form measurement without other factors’ effects, such as turning marks, burrs, and defects. The use of a flat surface in the experiment factors’ effects, such as turning marks, burrs, and defects. The use of a flat surface in the experiment was also intended to minimize the effect of linearity errors from the DRI probe on the measurement was also intended to minimize the effect of linearity errors from the DRI probe on the measurement results. The mirror was mounted on the spindle chuck and scanned by the DRI probe in a spiral path results. The mirror was mounted on the spindle chuck and scanned by the DRI probe in a spiral (the measurement radius was 10 mm). For comparison, the flat mirror was also measured offline by path (the measurement radius was 10 mm). For comparison, the flat mirror was also measured a calibrated Twyman–Green interferometer (Fisba FS10, FISBA AG, St. Gallen, Switzerland). The offline by a calibrated Twyman–Green interferometer (Fisba FS10, FISBA AG, St. Gallen, Switzerland). offline result was regarded as an accurate representation of the flat surface form. On-machine and Micromachines 9, xwas regarded as an accurate representation of the flat surface form. On-machine 13 of 16 The offline 2018, result and offline measurements were performed three times to evaluate the repeatability. The results of the offline measurements were performed three evaluate the repeatability. Theshown resultsinofFigure the DRI DRI measurement, kinematics error map, andtimes Fisbatomeasurement are respectively 16. measurement, kinematics error map, and Fisba measurement are respectively shown in Figure 16. The kinematics error used in this experiment was interpolated from the modelled kinematics error The error used mapkinematics shown in Figure 15. in this experiment was interpolated from the modelled kinematics error map shown in Figure 15.

(a)

(b)

(c)

Figure 16. Dispersed reference interferometry (DRI) on-machine measurement (a), kinematics error Figure 16. Dispersed reference interferometry (DRI) on-machine measurement (a), kinematics error map (b), and Fisba measurement of the optically flat mirror (c). map (b), and Fisba measurement of the optically flat mirror (c).

By means of combination of the kinematics error map and the Fisba measurement, the results in By17means of combination the kinematics error map and the Fisbaofmeasurement, the mirror results Figure indicate that the DRI of on-machine surface measurement result the optically flat in Figure 17 the indicate that the DRI on-machine surface measurement result the optically flat mirror agrees with superposition of the machine tool kinematics error and theofsample form error. With agrees with the superposition of the machine tool kinematics error and the sample form error. Withwas the the aid of kinematics error mapping established above, the on-machine probing data aid of kinematics mapping established above, theThe on-machine probingflatness data was compensated compensated by error subtracting the kinematics error. characterized from the DRI measurement reduced from 17.3 nm to 11.4 nm (with standard deviation σ = 2.3 nm), compared with the results of the offline measurement of 8.7 nm (with standard deviation σ = 1.2 nm). After compensation, the averaged flatness characterization accuracy from the DRI on-machine measurement was improved by 67%. It is noted that the offline measurement needs to be aligned to perform the comparison and the alignment process would inevitably result in some deviation

By means of combination of the kinematics error map and the Fisba measurement, the results in Figure 17 indicate that the DRI on-machine surface measurement result of the optically flat mirror agrees with the superposition of the machine tool kinematics error and the sample form error. With the aid of kinematics error mapping established above, the on-machine probing13 ofdata was Micromachines 2018, 9, 334 15 compensated by subtracting the kinematics error. The characterized flatness from the DRI measurement reduced from 17.3 nm to 11.4 nm (with standard deviation σ = 2.3 nm), compared with by subtracting the kinematics error. The characterized flatness from the DRI measurement reduced the results of the offline measurement of 8.7 nm (with standard deviation σ = 1.2 nm). After from 17.3 nm to 11.4 nm (with standard deviation σ = 2.3 nm), compared with the results of the offline compensation, the averaged flatnessdeviation characterization accuracy from thethe DRI flatness on-machine measurement of 8.7 nm (with standard σ = 1.2 nm). After compensation, averaged measurement was improved by 67%. It is noted that the offline measurement needs to be aligned to characterization accuracy from the DRI on-machine measurement was improved by 67%. It is noted perform comparison and needs the alignment would inevitably in some deviation thatthe the offline measurement to be alignedprocess to perform the comparison andresult the alignment process would result in someasdeviation between the17. two measurements, as shown in Figure 17. between the inevitably two measurements, shown in Figure

Figure 17.

DRI on-machine measurement vs.

the combination of the kinematics error and

FigureFisba 17. measurement. DRI on-machine measurement vs. the combination of the kinematics error and Fisba measurement. 6. Conclusions

6. Conclusions This paper presents kinematics error modelling, measurement, and compensation for on-machine surface measurement. Both theoretical and experimental work has been conducted to generate the machine tool kinematics error map for compensation of the OMSM results. Conclusions of the present work can be summarized as follows: (1)

(2)

A selective kinematics error model for a three-axis ultra-precision turning machine was established, based on multi-body theory and homogeneous transformation matrix (HTM). Selected error components EZX , EZC , EBC , and EBOC in the OMSM sensitive direction were measured using the reversal method. From the generated error map, it can be seen that the machine tool kinematics error was dominated by the C axis tilt error EBC with a 2 upr component. The OMSM experiment of an optically flat mirror has shown the DRI measurement result comprises the sample surface form error and the machine tool kinematics error. After the proposed kinematics error compensation, the flatness characterization accuracy from the DRI on-machine measurement was improved by 67%.

Author Contributions: Conceptualization: D.L. and X.J.; investigation: D.L.; methodology: D.L.; resources: Z.T., X.J. and L.B.; supervision: X.J.; writing—original draft: D.L.; and writing—review and editing: Z.T., X.J., and L.B. Funding: This work is supported by the UK’s Engineering and Physical Sciences Research Council (EPSRC) funding (grant ref: EP/P006930/1, EP/I033424/1 and EP/K018345/1) and the China Scholarship Council (CSC). Conflicts of Interest: The authors declare no conflict of interest.

Author Contributions: Conceptualization: D.L. and X.J.; investigation: D.L.; methodology: D.L.; resources: Z.T., X.J. and L.B.; supervision: X.J.; writing—original draft: D.L.; and writing—review and editing: Z.T., X.J., and L.B. Funding: This work is supported by the UK’s Engineering and Physical Sciences Research Council (EPSRC) funding (grant ref: EP/P006930/1, EP/I033424/1 and EP/K018345/1) and the China Scholarship Council (CSC). Micromachines 2018, 9, 334

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Conflicts of Interest: The authors declare no conflict of interest.

AppendixAA Appendix ordertotocheck checkthe the drift drift issue, issue, the the testing is is described in in thisthis InInorder testing of of the theused usedcapacitance capacitanceprobe probe described section. In the experiment, the capacitance probe was mounted on the machine tool Z axis and section. In the experiment, the capacitance probe was mounted on the machine tool Z axis and measured the mirror at a fixed position, as shown in Figure 13. The duration of the measurement measured the mirror at a fixed position, as shown in Figure 13. The duration of the measurement was was 60 seconds. The high-frequency electronic noise was filtered out using a Gaussian low-pass filter 60 seconds. The high-frequency electronic noise was filtered out using a Gaussian low-pass filter (60 Hz). After low-pass filtration, the noise level can be as low as 0.6 nm (root mean square). The result (60 Hz). After low-pass filtration, the noise level can be as low as 0.6 nm (root mean square). The shown in Figure A1 does not indicate the drift of the used capacitance probe.

result shown in Figure A1 does not indicate the drift of the used capacitance probe.

Figure A1. Static test of the capacitance probe. Figure A1. Static test of the capacitance probe.

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