In order to study nonlinearity of a capacitive probe moving laterally relative to a spherical surface, an experiment is conducted shown in Figure 1(a) where the probe moves laterally every time by some micrometers while a same probe is used to measure the lateral displacement. The data is shown in Figure 1(b) and a quadratic curve is used to fit the experimental data. It is obvious that the nonlinearity exists when the probe moves laterally above the spherical surface. This leads to two problems when measuring radial error of axis of rotation. One is whether the linear gain will change when the probe targeting spherical surface at different lateral offsets. The other is the lateral component of eccentric movement of artifact ball may lead to additional reading error of a capacitive probe which will be discussed later in section 2.2. To investigate the former problem one experiment is made in which readings of a probe approaching the artifact ball at different lateral offsets and the results are shown in Figure 1(c). It is concluded that the linear gain remains constant when lateral displacement varying between 0 and 40 μm. When measure error motion, the lateral displacement is always keep minimum by adjusting the probe in lateral direction to approach the highest point of the artifact ball and this will ensure the linear gain is constant.

A misalignment between the artifact and axis of rotation leads to eccentric error in the probe signals. Two primary methods exist to eliminate this effect, such as the least quadratic circle and the Fourier analysis to remove the fundamental frequency. However, little attention is given to the fact that lateral component of eccentric movement vector of artifact ball may lead to additional reading error of capacitive probe. Set the eccentric error to be e. At angular position θ , the lateral and the radial components of eccentric error are e ∗ cos θ and e ∗ sin θ respectively. Assuming the initial lateral offset of the probe e₀ relative to the ball is shown in Figure 2.

According to 2.1, output of the probe e lateral_effect caused by lateral offset χ can be presented by

e lateral_effect = a χ 2 = a ( e 0 + e cos θ ) 2 (1)

where, α -the identified coefficient and in this paper a = 0.00082 μm⁻².

The total contribution to the probe output caused by eccentric error is expressed by

e eccentric_effect = e lateral_effect + e sin θ = α ( e 0 + e cos θ ) 2 + e sin θ = α e 0 2 + 1 2 α e 2 + 1 2 α e 2 cos 2 θ + 2 α e 0 e cos θ + e sin θ (2)

From this formula, second order and first order errors will be included in the probe output and when the eccentric error e = 5 μm the second order error will be up to 10 nm which will be an unacceptable error and be impossible to be eliminated by mathematical method. The last two 1st order components in this formula can be removed by Fourier analysis to remove the fundamental frequency.

When considering radial error motion, one of the important error sources is attributed to misalignment between the capacitive probe and the artifact ball as is shown in Figure 3. The probe output is affected by four error motion components of which two are in the error sensitive direction and the other two in the error insensitive direction. The components have the following relationship:

e a = e a n + e a t (3)

e r = e r n + e r t (4)

where, e a t and e r t are error motion components in the error sensitive direcion, e r n and e a n in the error insensitive direction, and e a is the axial error motion and e r the radial error motion. Accordingly, the output m 1 of probe can be expressed as the combination effects of two parts, namely

m 1 = S x + E (5)

where S x and E are radial error motion in X direction and the error induced by misalignment, respectively. We have

E = e r n + e a t + f ( e a n + e r t ) − e r (6)

Substituting e r n = e r cos φ and e a t = e a sin φ in to (6) yields

E = e r ( cos φ − 1 ) + e a sin φ + f ( e a n + e r t ) (7)

where function f ( ⋅ ) corresponds to the lateral offset effects which is detailed in section 2.1 and φ is the tilt angle. Considering the lateral offset e a n and e r t are much smaller relative to the initial distance e 0 , we have

f ( e a n + e r t ) = a ( e 0 + e a n + e r t ) 2 − a e 0 2 (8)

f ( e a n + e r t ) ≈ 2 a e 0 ( e a n + e r t ) (9)

when φ is small enough, we have

E ≈ e a φ + 2 a e 0 ( e a n + e r t ) (10)

It can be concluded from (2) and (10) that the error motion in radial and axial direction and eccentric rotation of artifact ball will both induce lateral offset which causes a fake output of probes. When axial error motion is 0.4 μm and the initial lateral offset e₀ is 20 μm, the maximum error due to lateral offset effects is up to 13 nm, which is a large measurement error in calibration of an ultra-precision spindle.

Let the angular positioning error of artifact after reversal be φ which is shown in Figure 4(a), then the measurement error induced by angular error after reversing the artifact is derived as

E ( θ ) = R ( θ ) − R ( θ + φ ) 2 ≈ − φ 2 R ′ ( θ ) (11)

where R ( θ ) is roundness of the artifact.

The Donaldson reversal method needs to rotate the probe by 180 degrees relative to the rotor of the spindle measured at the same time. Angular position error of the probe will be introduced into the measurement signal. This kind of error is illustrated in Figure 4(b). The corresponding measurement error is derived as

M 2 ( θ ) = R ( θ − φ ) − S x ( θ ) cos φ + S y ( θ ) sin φ (12)

E ( θ ) = M 1 ( θ ) − M 2 ( θ ) 2 − S x ( θ ) (13)

E ( θ ) = 1 2 [ R ( θ ) − R ( θ − φ ) + S x ( cos φ − 1 ) − S y sin φ ] (14)

E ( θ ) ≈ 1 2 [ φ ( R ′ ( θ ) − S y ) − 1 2 S x ( θ ) φ 2 + ο ( φ ) + ο ( φ 4 ) ] (15)

where S x and S y are error motion components in X and Y directions respectively. If φ is efficiently small and the measurement error will be simplified as

E ( θ ) ≈ 1 2 [ φ ( R ′ ( θ ) − S y ) + ο ( φ ) ] (16)

When using multi-position method to separate roundness of the artifact and rotating the artifact by a constant angle φ , an angular error Δ i exists, as is shown in Figure 5(a). According to the principle introduced in [4] , the measurement error induced by angular error Δ k is derived as

E ( θ ) = 1 N ∑ k = 0 N − 1 R ( θ + k φ + Δ k ) ≈ 1 N ∑ k = 0 N − 1 [ R ( θ + k φ ) + R ′ ( θ + k φ ) Δ k + ο ( Δ k ) ] (17)

As roundness of the artifact can be expressed as Fourier series and when N is an even integer, we have ∑ k = 0 N − 1 R ( θ + k φ ) = 0 . If Δ k is small enough, we have

E ( θ ) ≈ 1 N ∑ k = 0 N − 1 R ′ ( θ + k φ ) Δ k (18)

Three-probe method is detailed in [5] , here gives only the formulas. Define M ( θ ) as linear combination of m 1 ( θ ) , m 2 ( θ ) and m 3 ( θ ) with coefficients a, b and c respectively, namely

{ M ( θ ) = m A ( θ ) + b m B ( θ ) + c m C ( θ ) b = − sin β [ sin ( β − α ) ] c = sin α [ sin ( β − α ) ] (19)

where m 1 ( θ ) , m 2 ( θ ) and m 3 ( θ ) are outputs of sensors, and

b = − sin β / [ sin ( β − α ) ] , c = sin α / [ sin ( β − α ) ] . Applying discrete Fourier transformation (DFT) to formula (19) yields

M ( k ) = ( 1 + b e − j k α + c e − j k β ) R ( k ) (20)

when three-probe method is used, let angular position errors of probes at positions α and β be δ α and δ β respectively, as is shown in Figure 5(b). We have the measurement error of roundness of artifact in frequent domain

E ( k ) = ( 1 W 1 − 1 W ) M ( k ) − 1 W 1 [ C 1 S x ( k ) + C 2 S y ( k ) ] (21)

where W 1 ( k ) = 1 + b e − j k ( α + δ α ) + c e − j k ( β + δ β ) , W ( k ) = 1 + b e − j k α + c e − j k β , C 1 = 1 + bcos ( α + δ α ) + ccos ( β + δ β ) , C 2 = bsin ( α + δ α ) + csin ( β + δ β ) By inverse Fouries transformation we have the measurement error e ( θ ) = I D F T ( E ( k ) ) .