A cylindrical scanning setup with an optical distance probe is non-contact, universal and fast. With a probe with 5 mm range, circular tracks on freeform surfaces can be measured rapidly with minimized system dynamics. By applying a metrology frame relative to which the position of the probe and the product are measured, most stage errors are eliminated from the metrology loop. Because the probe is oriented perpendicular to the aspherical best-fit of the surface, the sensitivity to tangential errors is reduced. This allows for the metrology system to be 2D. The machine design consists of the motion system, the metrology system and the non-contact probe.
2.1 Conceptual considerations
Based on the review of existing measurement methods, some fundamental choices can be made with regard to the characteristics of the measurement method.
Method
Probe type
Several principles can be used for probing the surface, as summarized in (Schwenke et al., 2002; Weckenmann et al., 2006). Optical probes are truly non-contact and especially single point sensors allow for scanning speeds in the order of meters per second. Care must be taken when applying an optical technique, since measurement relies on reflection of the surface under test, which generally is sensitive to imperfections such as dust and scratches.
Semi-contact scanning probe microscopy principles, such as Atomic Force Probes, can be used to measure the distance to a surface by keeping the stylus at constant distance or using it tapping mode. Since the tip is extremely fragile, scanning speeds are much smaller than one millimeter per second, which is similar to contact probes. Capacitive probes are capable of high-accuracy, but can only be applied to metallic surfaces and are thus not suitable for transmission optics. The same goes for other non-contact sensors such as Eddy-current sensors. Ultrasonic sensors have not been found to provide the required resolution.
Only optical probes are truly non-contact, have sufficient accuracy and allow for high scanning speeds. Hence, they are the probe type of choice. The global surface slope can amount up to 45° or even 90°. This requires the optical probe to be oriented perpendicular to the surface by the scanning system to recollect the beam that is reflected by the surface.
Measurand
When scanning a surface point by point, the position, slope, slope difference, curvature or a combination thereof can be measured at each point (Figure 2.1). Slope, slope difference and curvature have to be integrated once and twice, respectively, to obtain the surface form. Optically, the slope is the functional parameter since this changes the propagation direction of light by refraction or reflection. In manufacturing practice, however, the surface form is specified. Form is also required to determine the amount of material that is to be removed during the iterative corrective machining process.
Figure 2.1: Measurement of position, slope, slope difference and curvature
2.1 Conceptual considerations
Measuring position (i.e. surface coordinates) gives similar measurement versatility as current CMMs. It allows for true universal measurement, also for discontinuous parts, reference markers and quick alignment checks. Further, each individual measurement point can be directly traced back to a point on the surface, which is not the case when measuring other measurands.
Optically measuring distance is relatively difficult. It generally requires a spot to be focused onto the surface. To obtain high axial resolution, a high NA and thus a small spot (micrometer order) is necessary. This makes it very sensitive to local defects. Furthermore, measuring position requires the position of the sensor relative to the product to be known at similar accuracy level, which yields high requirements for the scanning system.
Instead of measuring slope, the slope difference between two points (Geckeler and Weing?rtner, 2002) or the local curvature at a point (Schulz and Weing?rtner, 2002) can also be measured. These properties are intrinsic properties of a surface, which can be measured independently of the object’s orientation relative to an outside reference. This relieves the positioning requirements of the sensor relative to the product. By dual integration, the surface form can be calculated. Due to the required integration, these methods suffer from the same disadvantages as slope measurement. Further, curvature ranges from convex to concave with local radii of curvature down to some tens of millimeters. No curvature sensor yet has sufficient dynamic range to universally satisfy these demands. The curvature sensor commonly used is a miniature interferometer using a CCD, which does not allow for continuous scanning.
Traceability
Regarding the above, measuring position is believed to result in the most versatile and traceable measurement method. Therefore, a distance measuring optical probe will be scanned over the surface.
Setup
To position the probe over the surface, an orthogonal, cylindrical or polar setup can be used. Conventional coordinate measurement machines use an orthogonal setup as shown in Figure 2.2. The probe is translated in 3 directions (x, y and z). To orient the probe perpendicular to the surface, this setup can be extended with two rotations (?andψ)
2.1 Conceptual considerations
Since the surfaces are more or less rotationally symmetric (within 5 mm PV), a cylindrical setup appears to be the obvious choice. The product can be mounted on a continuously rotating spindle (θ), while the probe is positioned in radial (r) and vertical (z) direction. To align the probe to the global surface slope, one rotation (ψ)
Figure 2.3: Cylindrical setup
Figure 2.4: Polar setup for convex and concave surfaces
Concluding from the above, the cylindrical setup of Figure 2.3 is the obvious choice. It is truly universal, can be fast with relatively little system dynamics, uses only four axes of motion and it has a 2? D metrology loop.
Stage layout
Some variations are possible to the cylindrical setup of Figure 2.3, similarly to the considerations of (Vermeulen, J., 1999). The spindle can be oriented vertically and horizontally. The vertical orientation is preferred due to the large product mass and diameter of 500 mm. This way the accessibility and visibility for loading, aligning and fixing of a product is most convenient. Further the spindle is loaded axially, preventing mass dependent bending forces and eccentricity of the spindle rotor. The ψ-rotation can best be done with the probe, since it is the smallest mass and thus is least affected by the changing direction of gravity. The remaining 2 translations (r and z) can be performed with the spindle, the probe or a combination of the two. Due to the large spindle and product mass (~200 kg), moving the lightest component (i.e. the probe) will result in the best dynamic behaviour. Furthermore, this setup will lead to the shortest metrology loop as will be shown in Chapter 4. The cylindrical setup of Figure 2.3 is thus the preferred stacking sequence of the stages.
2.2 Machine concept
The chosen cylindrical concept is shown in Figure 2.5. The product is mounted on an air-bearing spindle, which is rotating continuously at for instance 1 rev/s. An optical distance probe is mounted on a rotation axis (Ψ-axis) which positions it perpendicular to the rotationally symmetric best-fit of the product. The probe can be moved in radial and vertical direction by an R and Z-stage, respectively.
The probe will be positioned on a circular track on the product and the stages will be locked. This improves the positioning stability of the stages since the electronic noise of the motors and encoders is cancelled. The track can be measured multiple times with little extra effort, to obtain redundant data, for instance for averaging. After this, the stages are moved about 1 mm further to the next track and the process is repeated. This allows for measurement times in the order of 15 minutes for large surfaces.
The largest contribution is expected to be determined by the metrology loop between the probe and the product. In Figure 2.5, this loop was equal to the structural loop and thus included the stages, bearings, base etc. By applying a metrology frame relative to which the probe and product position are measured as directly as possible, most stage errors can be eliminated from the metrology loop, as shown in Figure 2.6.
The spindle is intended to rotate at 1 rev/s. The worst case deformation of the surface under test due to the centrifugal force was estimated using FEM for a ?500 mm flat, concave and convex surface with material properties similar to aluminium or glass. Depending on the boundary conditions (e.g. fixed backside, three-point mount, fixed centre), the deformation varied between 3 and 40 nm. Part of this deformation is of the less-sensitive type (see section 2.3.2). by modelling the effect in FEM, this deformation can be compensated for in the data-processing.
This concept can meet the design goals. The Ψ-axis enables application of an optical probe. By increasing the measurement range of this probe to 5 mm and combining it with a cylindrical machine setup, freeforms can be measured universally with high scanning speeds and minimal dynamics. The separate 2D metrology loop somewhat
2.2 Machine concept
2.3 Error budget
To emphasize the cylindrical nature of the machine, an r,y,z coordinate system has been adopted (Figure 2.8). The origin of the system is located at the spindle surface centre, the z-direction is coaxial with the spindle centre line and the r-direction (‘radial’) is parallel to the plane of motion of the probe. This plane is also referred to as the ‘measurement plane’. The rotations around the r, y and z
As will become clear, it is convenient to also define a local coordinate system that rotates with the probe. In this system, the c-direction is aligned with the probe axial direction and the b-direction is parallel to the y-direction of the global coordinate system. Rotations around the a, b and c axes are called α, β and γ, respectively. The β-direction coincides with the ψ-direction. The origin is located at the Ψ-axis centre line.
The metrology loop is composed of three main elements: the distance measured by the probe, the measured position of the probe relative to the metrology frame and the measured position of the product relative to the metrology frame. To model the sensitivity of errors in these three elements, they are interpreted as displacements of the probe relative to the product. The error in the distance measured by the probe as a result of this displacement δ is the measurement error Δ..
n with its resulting vector in normal direction n of the surface, both result in a direct measurement error n
n) and tangential (t
A position measurement error tin tangential direction t of the surface, results in an error c that is dependent on local surface geometry. Since the surfaces are smoothly curved and have low roughness, only local curvature and tilt have to be taken into account here. When measuring rotationally symmetric surfaces (flats, spheres and aspheres), the probe is always nominally perpendicular to the surface (Figure 2.9 B). Therefore only local curvature can cause an error here. A position measurement error in tangential direction only results in a second order error c. Figure 2.10 (left) shows this error up to an Rc of 10 mm and a t of 2 μm, to be only 0.2 nm. Errors due to local curvature are thus negligible.
Local surface slopes of 5° are expected (section 1.1.3), which is also the maximum misalignment between probe and surface (Figure 2.9 C). The resulting distance measurement error is linearly dependent to this local slope, as shown in Figure 2.10 (right). This error can be substantial for large local slopes and large tangential errors. A sub-micrometer out-of-plane error is therefore desirable.
When the sensitivity of 0.087 is compared to the sensitivity of 1 for errors in normal direction, over an order of magnitude reduced sensitivity to tangential errors is seen. For surfaces with less local slope, this reduction is even better. A clear distinction between normal and tangential errors can thus be concluded.
With the metrology frame as a reference, 13 degrees of freedom can be distinguished. These are the 6 DOFs of the product, plus the 6 DOFs of the probe, plus the distance measured by the probe. For each of these, the resulting error (normal and/or tangential) between probe and product can be calculated as a function of position measurement error and surface geometry. Further, the measurement position has to be taken into account. When measuring a flat surface for instance, a radial error has no influence, and tilting of the product linearly influences the measurement error as a function of the radial measurement position. The resulting measurement error will thus be measurement-task specific.
As explained in Figure 2.10, the errors resulting from tangential errors on a curved surface are negligible. For aspheres, the measurement uncertainty is therefore predominantly determined by errors in normal direction. Only six degrees of freedom cause errors in normal direction, as shown in Figure 2.11. These degrees of freedom determine the basic measurement uncertainty for aspheres and mild freeforms, and are called the ‘sensitive directions’. The other seven directions are referred to as the ‘lesssensitive
When the error sources of the 13 individual degrees of freedom are assumed uncorrelated and normally distributed, the resulting uncertainty in a, b and c direction can be calculated with (2.1). Here, σi,j is the rms error of part i in direction j, where the part is the spindle (S) or the probe (P). The probe angle is ψ, R and Z are the probe tip position and LP
The uncertainty in normal direction is equal to the uncertainty in c-direction, while the uncertainty in tangential direction is the resulting vector of the a and b
With the maximum error values chosen as shown in Table 2.1, the resulting form measurement uncertainty is shown in Figure 2.12. The uncertainty is task-specific, but is mainly dependent on the diameter of the part and the local slope. Figure 2.12 was calculated with ψ = 22.5°, Z = 50 mm and a probe length LP of 100 mm. The gray area at the base plane shows the surfaces for which the 30 nm expanded uncertainty objective is met. The uncertainty increases to 55 nm for heavily freeform surfaces.
2.4 Machine design overview
An overview of the main concepts that have been applied in the machine design is given in the following section, to aid better understanding of the detailed subsystem designs in the coming chapters.
Structural loop
Figure 2.13: Z-stage with 3 bearings to a vertical plane
To minimize distortions and hysteresis, separate preload and position frames will be applied in both the R and Z-stage. Due to the cylindrical setup and the long range probe, the stages can be stationary while measuring a circular track. To increase the stiffness and stability of the stages in the actuated directions, mechanical brakes will be applied that clamp the stages. This prevents stage motion from encoder, amplifier and EMC noise, and may provide higher stiffness compared to the achievable servo
2.4 Machine design overview
stiffness. Only the degree of freedom in the motion direction of the stage will be clamped to prevent stage deformation.
Metrology loop
To satisfy the Abbe principle, the measurement systems should be aligned to the probe focal point. When measuring concave optics, however, the edge of the optic is blocking the horizontal line of sight between the outside reference and the probe focal point. Moreover, the vertical and horizontal position of the probe focal point changes with ψ-rotation, whilst the stages remain stationary. This would require a metrology system that tracks the probe-tip position independently of the stage position.
The setup is less-sensitive to tangential errors of the probe focal point relative to the product. As shown in the error budget calculations, this also includes the ψ-rotation of the probe. The Abbe point can therefore be shifted from the probe focal point to the Ψ-axis centre. The measurement systems can now be aligned with the Ψ-axis centre, and move along with the R and Z-stages. Due to the reduced tangential sensitivity, the Abbe principle is still satisfied.
To measure the displacement of the Ψ-axis rotor (1 in Figure 2.14) relative to the metrology frame (2), an interferometry system is applied. Hereto the Ψ-axis rotor has been made reflective such that the measurement beam (3) can be focused onto it by a lens (4). The beam layout is such that the measurement beam also reflects on a reference mirror (5), directly measuring the displacement between the Ψ-axis rotor and the reference mirror. A similar setup is employed in horizontal direction (6). Reaction forces of the probe servo system (7) on the Ψ-axis bearing, or pressure variations in the stage bearings will cause displacement of the probe. With this setup, the position errors of the whole motion system in r- and z
Although the acquired spindle (8) is specified to have an error motion of only 25 nm and 0.1 μrad, the position of the rotor needs to be determined to 15 nm and 0.1 μrad. To determine the position of the spindle rotor relative to the metrology frame, capacitive probes (9) are therefore added. Principally, three capacitive probes are sufficient to measure the three sensitive directions r, z and ψ. Seven probes are however applied, to separate the roundness error from the error motion with the multiprobe method.
The displacement of the probe and spindle are measured relative to the metrology frame. This frame is constructed out of Silicon Carbide. The high specific stiffness results in high eigenfrequencies, and the high thermal diffusivity combined with low expansion coefficient minimize the sensitivity to thermal gradients. Further, the material is hard enough to polish the reference mirrors directly onto the beams, avoiding separate strip mirrors of limited stiffness and stability and connection elements.
With this metrology loop concept, all six sensitive directions of Figure 2.11 are measured directly. When measuring a freeform surface, only the probe focusing mechanism is moving dynamically while measuring a circular track. The static and dynamic displacements that occur during this measurement are recorded by the metrology system and can be compensated for in the (off-line) data-processing.
Non contact probe
2.4.2 Machine design overview
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