# Modal Analysis, Metrology, and Error Budgeting of a Precision Motion Stage

^{*}

## Abstract

**:**

## 1. Introduction

- The nano-positioner utilizes a single shaft engaged to air bushings, instead of the more commonly used double shaft arrangement. The roll resistance is provided by the air bearing at the bottom. This way, the self-aligning property of the air bushings, which are held in the housings using O-rings, is exploited to the greatest extent, making the manufacturing and assembly of the stage easy and low-cost.
- Actuation is provided by two voice coil actuators (VCA) operating in moving magnet mode in complementary double configuration.

- A modified ‘peak-picking’ approach has been developed which directly utilizes the accelerance measurement, through its modified formulations for the estimation of modal parameters. This way, the bias at low frequency, observed in the receptance plots obtained from accelerance via double integration, is avoided. Searching the modal analysis literature [3,4,5], the authors were not able to find a peak picking method which directly works with accelerance. With this study, we hope to fill in this gap.
- Modal testing, laser interferometric metrology, and error budgeting, although being established methods, have been applied to a novel precision motion stage design [2] for the first time. Outcome from these tests has allowed an in-depth evaluation of several design features regarding the overall accuracy and applicability of the motion stage in the micro-machining framework.
- A hybrid error budget has been compiled which combines the commonly considered quasi-static geometric, thermal, and steady-state servo errors with the dynamic component due to cutting forces. The 3-axis harmonic deflections due to machining forces, at the workpiece level of the stage, could only be predicted using the spatial modal testing results.

## 2. Vibratory Dynamics

#### 2.1. Predicted Vibration Modes

#### 2.2. Summary of Modal Testing Methods

#### 2.3. Modal Analysis Method 1 (Modified Peak-Picking Approach)

#### 2.3.1. Impact and Measurement Points

#### 2.3.2. Method of Analysis

- Setting $\frac{\partial {G}_{r}}{\partial \omega}=0$ yields two positive roots as, ${\omega}_{r,1}={\omega}_{r}/\sqrt{1+2{\zeta}_{r}}$ and ${\omega}_{r,2}={\omega}_{r}/\sqrt{1-2{\zeta}_{r}}$. Hence, in our approach, the frequency values coinciding with the minimum and maximum real components of accelerance around the natural frequency (${\omega}_{r}$) are used to determine the damping ratio as,$${\zeta}_{r}=\frac{\left({\omega}_{r,2}^{2}-{\omega}_{r,1}^{2}\right){\left({\omega}_{r}\right)}^{2}}{4{\omega}_{r,1}^{2}{\omega}_{r,2}^{2}}$$

- ii.
- Setting $\frac{\partial {H}_{r}}{\partial \omega}=0$ yields only ${\omega}_{\mathrm{max}}={\omega}_{r}\sqrt{2\sqrt{{\zeta}_{r}^{4}-{\zeta}_{r}^{2}+1}+2{\zeta}_{r}^{2}-1}$ as the positive root. For ${\zeta}_{r}<<1$, ${\omega}_{r}\approx {\omega}_{\mathrm{max}}$ can be assumed. Hence, the imaginary peak/dip location is used to identify the natural frequency (${\omega}_{r}$). For receptance, the imaginary peak location is obtained as,$${\omega}_{\mathrm{max}}^{\ast}={\omega}_{r}\sqrt{\frac{2}{3}\sqrt{{\zeta}_{r}^{4}-{\zeta}_{r}^{2}+1}-\frac{2}{3}{\zeta}_{r}^{2}+\frac{1}{3}}$$

- iii.
- Limit $\omega /{\omega}_{r}$$\to $$\infty $ yields ${G}_{r}$$\to $$\frac{{\omega}_{r}^{2}\text{}{\psi}_{r}^{i}{\psi}_{r}^{k}}{{k}_{r}}$ and ${H}_{r}$$\to $ 0. Hence, real part of accelerance has residues from the lower frequency modes, and using the horizontal axis crossing of ${G}_{r}$ for natural frequency estimation would be inaccurate.
- iv.
- Limit $\omega /{\omega}_{r}$$\to $ 0 yields ${G}_{r}$$\to $ 0 and ${H}_{r}$$\to $ 0. Hence, in accelerance, higher frequency modes typically do not have an influence on their lower frequency counterparts. In the case of receptance, the situation is reversed in which the higher frequency modes affect the real part only, and lower frequency modes exert very little influence.

#### 2.4. Modal Analysis Method 2 (Software Package)

#### 2.4.1. Impact and Measurement Points

^{®}was used to acquire and view the FRF’s.

#### 2.4.2. Method of Analysis

^{®}. The proprietary ‘PolyMAX’ algorithm carries out a similar operation to the commonly used least-squares time domain complex exponential method, in the frequency domain [18]. The resulting stabilization diagram is interpreted for natural frequencies and damping ratios. For constructing the mode shapes, these identified parameters are used in the least-squares frequency domain (LSFD) algorithm [19], which finds the best fit to the modal displacement vector based on the agreement between the measured and fitted FRF’s. The software package allows either complex or real mode shapes to be fit. In this paper, complex mode shapes are enabled to test the proportional damping assumption. Complexity of mode shapes is rated using ‘modal phase collinearity ($MPC$)’ and ‘mean phase deviation ($MPD$)’ [19]. $MPC$ and $MPD$ rate the complexity of the mode on a scale 0 to 100%, and 0°–90°, respectively. Having obtained a minimum $MPC$ of 96.5%, and a maximum $MPD$ of 12° in the set of identified mode shape vectors, the proportional damping assumption used in method 1 (Section 2.3) is observed to be justifiable. Identified mode shapes can be animated as a 3D video, and screenshots of the animated mode shapes are presented in Section 2.6.

#### 2.5. Measurements from the Encoder

^{®}DS3002 encoder interface board, were fed to CutPRO

^{®}’s MalTF interface, as a position measurement, using the DS2102 digital to analog converter. The boards (DS3002, DS2102) ran at sampling frequency of 20 kHz. The same impact points in three planes mentioned for method 1 (Figure 4) were used, with the accelerometer replaced by the encoder. Receptances acquired this way did not yield any vibratory modes in the 0–2000 Hz range. Eventually, position control bandwidth in the X-axis could be increased up to 650 Hz without experiencing any interactions with vibratory modes, affirming these results. The bandwidth was mainly limited by the phase advance that can be contributed by the control scheme at the desired cross-over frequency, while ensuring that amplification of measurement noise through feedback did not display a significantly deteriorating effect.

#### 2.6. Comparative Results and Discussion

^{®}) are presented in Figure 9, along with natural frequency predictions made at the design phase (Section 2.1). It is observed that the identified natural frequencies for methods 1 and 2 are rather close, except for a slightly larger deviation for the first mode. Damping ratios are also observed to be close. On the other hand, large discrepancies between the experimentally identified and predicted natural frequencies can be noticed. This discrepancy is especially critical in the case of the first mode (roll). While methods 1 and 2 have measured 65 Hz and 79 Hz, respectively, the initial theoretical prediction was 538 Hz. The roll motion is only constrained by the flat air bearing at the bottom, and such bearings are usually not rated for rotational stiffness.

- For a quick and straightforward assessment of the most prominent vibration modes (starting from the first mode) the ‘peak-picking’ procedure can be applied, as it is shown to yield sufficient accuracy. This information can be incorporated in the determination of the control bandwidth.
- For further investigation of the vibratory dynamics, typically with >5 modes, as well as for the representation of mode shapes in three dimensions, a software package similar to the one utilized in this paper can be used.
- Results from the two methods can be combined in the assessment of design features, like the magnitude and geometry of the compliances of the stage and bearings, as exemplified in this study by the less than ideal roll resistance observed through modal testing.

## 3. Laser Interferometric Metrology

#### 3.1. Methodology of Measurements

#### 3.2. Evaluation of the Results

#### 3.3. Measurement Results

#### 3.3.1. Linear Positioning Error (EXX)

^{®}LIP501 R encoder scale is graded for ±1 μm accuracy [25], which corresponds to PV 2 μm possible error. The repeatability is evaluated as $R$ = 0.7 μm, which is poorer than expected for the typical optical encoder. Considering the approximately 300 mm dead path, the repeatability corresponds to about ±1 ppm deviation, which can be attributed to environmental disturbances on the laser measurement, such as the turbulence of the ambient air. Altogether, the accuracy is evaluated as $A$ = 1.8 μm.

#### 3.3.2. Straightness Error in Y (EYX)

#### 3.3.3. Straightness error in Z (EZX)

#### 3.3.4. Yaw Error (EBX)

#### 3.3.5. Pitch Error (ECX)

## 4. Error Budget

#### 4.1. Predicted Error Budget

- The PV error due to position sensor resolution is given by ${E}_{res}$ = 0.97 nm, derived from the 4096 times arctangent interpolation of the 4 μm measurement signal period.
- The linear encoder scale is rated for ±1 μm grating error [25], hence the PV error due to encoder grating defects is given by ${E}_{grating}$ = 2000 nm.
- The main shaft of the motion stage acts as the guideway for the air bushings. It is manufactured to a cylindricity tolerance of 5 μm as shown in Figure 17. Assuming that roughly half of the possible errors due to the errors in the shaft dimension are cancelled by its realignment in the air gap, PV errors in Y and Z directions can be assumed to be ${E}_{Y}$ = 2.5 μm, and ${E}_{Z}$ = 2.5 μm, respectively.

#### 4.2. Actual Error Budget

#### 4.2.1. Geometric Component

#### 4.2.2. Thermal Component

#### Thermal Disturbance Sensitivities

- In the case of thermal expansion along the X-axis, combined effect of the expansion of the encoder scale and the top plate needs to be considered. If the encoder scale is thought of as fixed at its center to the top plate, the deviation in X-positioning would be represented by,$${\delta}_{x}=\left({\alpha}_{al}-{\alpha}_{enc}\right)\cdot {L}_{2}\cdot \Delta T$$$${\delta}_{x}={\alpha}_{al}\cdot {L}_{2}\cdot \Delta T$$
- Along the Y-axis, thermal expansion of the stage would push the point of interest upwards by$${\delta}_{y}=\frac{{\alpha}_{al}\cdot {L}_{1}\cdot \Delta T}{2}$$
- Thermal expansion along the Z axis does not affect the point of interest.The thermal sensitivity in X and Y axes can be defined as,$${\gamma}_{x}=\frac{{\delta}_{x}}{\Delta T}={\alpha}_{al}\cdot {L}_{2},\text{}{\gamma}_{y}=\frac{{\delta}_{y}}{\Delta T}=\frac{{\alpha}_{al}\cdot {L}_{1}}{2}$$The total linear thermal sensitivity of positioning can be expressed as,$${\gamma}_{T}=\sqrt{{\gamma}_{x}^{2}+{\gamma}_{y}^{2}}$$

#### Thermal Disturbances and the Resulting Thermal Error

^{2}(which is slightly short of the maximum achievable acceleration of 6480 mm/s

^{2}), and the resulting high feedrate, F = 200 mm/s, constrained by the stroke length limitation. With a 0.05 s dwell period prescribed at the far end of the stroke, the period of back and forth motions is determined as 0.35 s. Thermocouples were mounted at the adjustment ring near the VCA core and the top plate on the main shaft. The stage was run for 8 h, for which the collected temperature readings are presented in Figure 19.

#### 4.2.3. Machining Force Component

_{xy}can be used as ${G}_{Fy}$, and the FRF measured between F1-A

_{xz}can be used as ${G}_{Fz}$. As the FRF’s are measured in terms of accelerance, conversion to receptance needs to be carried out for the position response. Instead of directly dividing the accelerance magnitudes by the square of the frequency (as suggested by the mathematical definition), receptances are synthesized from modal parameters which were fitted using peak-picking, in order to prevent distortions in the low-frequency region. Resulting ${G}_{Fx}$, ${G}_{Fy}$, ${G}_{Fz}$ are presented in Figure 21a–c. The compliance in the Z-direction is about an order of magnitude higher than the others, due to the low effective roll resistance provided by the flat air bearing, as discussed in Section 2.6.

#### 4.2.4. Compilation of the Error Budget

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 6.**Comparison of using different peak-picking formulations: (

**a**) Damping ratio estimation; (

**b**) Natural frequency estimation.

**Figure 8.**Experimental procedure for method 2: (

**a**) Photograph of the impact hammer and the tri-axis accelerometer at the A4 position; (

**b**) Impact and measurement points.

**Figure 9.**Comparative modal testing results from the two methods and predictions at the design phase. Dimensions in mm.

**Figure 10.**Machine tool coordinate system and error motions (copied by Waterloo University with the permission of the Standards Council of Canada (SCC) on behalf of ISO) [20].

**Figure 11.**Locations of the measurement optics with respect to the center of gravity: (

**a**) Retroreflector; (

**b**) Wollaston prism. Dimensions in mm.

**Figure 12.**EXX results: (

**a**) Raw measurement; (

**b**) Abbe compensated with straight line fit; (

**c**) ISO presentation after slope removal.

**Figure 13.**EYX results: (

**a**) Raw measurement; (

**b**) Abbe compensated with straight line fit; (

**c**) ISO presentation after slope removal.

**Figure 14.**EZX results: (

**a**) Raw measurement; (

**b**) Abbe compensated with straight line fit; (

**c**) ISO presentation after slope removal.

**Figure 18.**Diagram of the stage with key variables relevant to thermal disturbance sensitivities. Dimensions in mm.

**Figure 21.**Frequency domain functions: (

**a**–

**c**) Transfer function between forces and deflections in the X-, Y-, Z-directions; (

**d**) Total deflection as a function of frequency.

Property | Symbol | Value |
---|---|---|

Air bushing axial stiffness | ${k}_{1yz}$ | 23 N/μm |

Air bushing rotational stiffness | ${k}_{1bc}$ | 2.8 Nm/mrad |

Air bearing axial stiffness | ${k}_{2y}$ | 35 N/μm |

Air bearing roll stiffness (estimated) | ${k}_{2a}$ | 4.7 Nm/mrad |

Air bearing pitch stiffness (estimated) | ${k}_{2c}$ | 7.3 Nm/mrad |

Air bearing length along the X-axis | ${L}_{2x}$ | 50 mm |

Air bearing length along the Z-axis | ${L}_{2z}$ | 40 mm |

Distance between the middle of the air bushings along the X-axis | ${L}_{x}$ | 180 mm |

Property | Symbol | Value |
---|---|---|

Mass | $m$ | 1.318 kg |

Moment of inertia in A (Roll) | ${I}_{a}$ | 409 kg mm^{2} |

Moment of inertia in B (Yaw) | ${I}_{b}$ | 17,439 kg mm^{2} |

Moment of inertia in C (Pitch) | ${I}_{c}$ | 17,454 kg mm^{2} |

Direction | Expression | Natural Frequency (Hz) |
---|---|---|

Y (Vertical) | ${\omega}_{y}=\sqrt{\frac{2{k}_{1yz}+{k}_{2y}}{m}}$ | 1248 |

Z (Horizontal) | ${\omega}_{z}=\sqrt{\frac{2{k}_{1yz}}{m}}$ | 940 |

A (Roll) | ${\omega}_{a}=\sqrt{\frac{{k}_{2a}}{{I}_{a}}}$ | 538 |

B (Yaw) | ${\omega}_{b}=\sqrt{\frac{2{k}_{1bc}+2{\left(\frac{{L}_{x}}{2}\right)}^{2}{k}_{1yz}}{{I}_{b}}}$ | 741 |

C (Pitch) | ${\omega}_{c}=\sqrt{\frac{2{k}_{1bc}+2{\left(\frac{{L}_{x}}{2}\right)}^{2}{k}_{1yz}+{k}_{2c}}{{I}_{c}}}$ | 748 |

Feature | Method 1 (Modified Peak-Picking) | Method 2 (Software Package) |
---|---|---|

Frequency response function (FRF) acquisition system | CutPRO^{®} MalTF moduleby Manufacturing Automation Laboratory (MAL), Inc. | LMS Test.Lab^{®}by Siemens-PLM Software |

Testing procedure | Roving hammer | Roving accelerometer |

Accelerometer type | Dytran^{®} 3035AG(1-channel) | PCB Electronics^{®} 356A02(3-channel) |

Impact hammer type | Dytran^{®} 5800SL | Dytran^{®} 5800SL |

Identification of the natural frequency and damping ratio | Modified peak-picking method | PolyMAX [18] |

Identification of the mode shape vectors | Modified peak-picking method | Least-Squares Frequency Domain (LSFD) [19] |

Presentation of the mode shapes | Manual 2D drawings | Automated 3D animations |

i | ${\mathit{P}}_{\mathit{i}}$ (mm) | i | ${\mathit{P}}_{\mathit{i}}$ (mm) | i | ${\mathit{P}}_{\mathit{i}}$ (mm) |
---|---|---|---|---|---|

1 | 1.509 | 7 | 7.548 | 13 | 13.545 |

2 | 2.508 | 8 | 8.501 | 14 | 14.509 |

3 | 3.507 | 9 | 9.549 | 15 | 15.533 |

4 | 4.510 | 10 | 10.515 | 16 | 16.547 |

5 | 5.548 | 11 | 11.526 | 17 | 17.549 |

6 | 6.501 | 12 | 12.543 | 18 | 18.505 |

**Table 6.**Error motion parameters [21].

Parameter | Definition | Formula |
---|---|---|

$B$ | Reversal value | ${B}_{i}={\overline{d}}_{i}\uparrow -{\overline{d}}_{i}\downarrow $, $B=\mathrm{max}\left\{{B}_{i}\right\}$ |

$\overline{B}$ | Mean reversal value | $\overline{B}=\frac{1}{m}{\displaystyle \sum _{i=1}^{m}{B}_{i}}$ |

$M$ | Range mean bidirectional positional deviation | $M=\mathrm{max}\left\{{\overline{d}}_{i}\right\}-\mathrm{min}\left\{{\overline{d}}_{i}\right\}$ |

$E$ | Systematic positional deviation | $E=\mathrm{max}\left\{{\overline{d}}_{i}\uparrow ;\text{}{\overline{d}}_{i}\downarrow \right\}-\mathrm{min}\left\{{\overline{d}}_{i}\uparrow ;\text{}{\overline{d}}_{i}\downarrow \right\}$ |

$R$ | Repeatability of positioning | ${R}_{i}\uparrow =4{s}_{i}\uparrow $, ${R}_{i}\downarrow =4{s}_{i}\downarrow $, ${R}_{i}=\mathrm{max}\left\{2{s}_{i}\uparrow +2{s}_{i}\downarrow +\left|{B}_{i}\right|;\text{}{R}_{i}\uparrow ;\text{}{R}_{i}\downarrow \right\}$, $R=\mathrm{max}\left\{{R}_{i}\right\}$ |

$A$ | Accuracy | $A=\mathrm{max}\left\{{\overline{d}}_{i}\uparrow +2{s}_{i}\uparrow ;\text{}{\overline{d}}_{i}\downarrow +2{s}_{i}\downarrow \right\}$ − $\mathrm{min}\left\{{\overline{d}}_{i}\uparrow -2{s}_{i}\uparrow ;\text{}{\overline{d}}_{i}\downarrow -2{s}_{i}\downarrow \right\}$ |

EXX (Linear) | EYX (Vertical) | EZX (Horizontal) | |||||||
---|---|---|---|---|---|---|---|---|---|

(μm) | $\downarrow $ | $\uparrow $ | bi | $\downarrow $ | $\uparrow $ | bi | $\downarrow $ | $\uparrow $ | bi |

$B$ | N/A | N/A | 0.1 | N/A | N/A | 0.2 | N/A | N/A | 0.1 |

$\overline{B}$ | N/A | N/A | 0.0 | N/A | N/A | 0.1 | N/A | N/A | 0.0 |

$M$ | N/A | N/A | 1.2 | N/A | N/A | 2.1 | N/A | N/A | 0.8 |

$E$ | 1.2 | 1.2 | 1.2 | 2.1 | 2.2 | 2.3 | 0.8 | 0.8 | 0.9 |

$R$ | 0.7 | 0.7 | 0.7 | 0.4 | 0.3 | 0.5 | 0.8 | 0.9 | 0.9 |

$A$ | 1.6 | 1.7 | 1.8 | 2.3 | 2.4 | 2.5 | 1.6 | 1.5 | 1.6 |

EBX (Yaw) | ECX (Pitch) | |||||
---|---|---|---|---|---|---|

(μm/m) | $\downarrow $ | $\uparrow $ | bi | $\downarrow $ | $\uparrow $ | bi |

$B$ | N/A | N/A | 0.7 | NA | NA | 2.0 |

$\overline{B}$ | N/A | N/A | 0.0 | NA | NA | −1.0 |

$M$ | N/A | N/A | 4.5 | NA | NA | 163.4 |

$E$ | 4.4 | 4.5 | 4.5 | 163.7 | 163.2 | 163.8 |

$R$ | 3.2 | 3.4 | 3.4 | 6.6 | 4.5 | 6.6 |

$A$ | 6.6 | 7.2 | 7.2 | 167.0 | 167.2 | 167.5 |

Error Components | PV Magnitude (nm) |
---|---|

Position sensor resolution (${E}_{res}$) | 0.97 |

Position sensor grating error (${E}_{grating}$) | 2000 |

Y straightness (${E}_{Y}$) | 2500 |

Z straightness (${E}_{Z}$) | 2500 |

Arithmetic sum | 7001 |

RMS sum | 1173 |

Mean | 4087 |

Component | Accuracy (A) | Repeatability (R) | Units |
---|---|---|---|

EXX | 1.8 | 0.7 | (µm) |

EYX | 2.5 | 0.5 | (µm) |

EZX | 1.6 | 0.9 | (µm) |

EBX | 7.2 | 3.4 | (µm/m) |

ECX | 167.5 | 6.6 | (µm/m) |

Quantity | Symbol | Value |
---|---|---|

Thermal coefficient of expansion of Aluminum 6061 | ${\mathsf{\alpha}}_{al}$ | 23.5 ppm/K |

Thermal coefficient of expansion of the glass encoder scale | ${\mathsf{\alpha}}_{enc}$ | 8 ppm/K |

Thickness of the moving body | ${L}_{1}$ | 33.7 mm |

Distance between the center of the top plate and the encoder scale | ${L}_{2}$ | 10 mm |

Thermal sensitivity along the X-axis | ${\mathsf{\gamma}}_{x}$ | 235 nm/K |

Thermal sensitivity along the Y-axis | ${\mathsf{\gamma}}_{y}$ | 396 nm/K |

Total thermal sensitivity | ${\mathsf{\gamma}}_{T}$ | 460 nm/K |

PV Magnitude (nm) | |||
---|---|---|---|

Repeatable Errors Conserved | Repeatable Errors Subtracted | Estimated at Design Phase | |

Linear (EXX) | 3964 | 675 | 2001 |

Straightness | |||

Vertical (EYX) | 2503 | 486 | 2500 |

Horizontal (EZX) | 1620 | 936 | 2500 |

Angular | (included in Linear) | - | |

Servo | 30 | 30 | - |

Thermal | 92 | 92 | - |

Machining force | 1748 | 1748 | |

Total Error | |||

Arithmetic Sum | 9957 | 3967 | 7001 |

RMS Sum | 1518 | 621 | 1173 |

Mean | 5738 | 2294 | 4087 |

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## Share and Cite

**MDPI and ACS Style**

Okyay, A.; Erkorkmaz, K.; Khamesee, M.B. Modal Analysis, Metrology, and Error Budgeting of a Precision Motion Stage. *J. Manuf. Mater. Process.* **2018**, *2*, 8.
https://doi.org/10.3390/jmmp2010008

**AMA Style**

Okyay A, Erkorkmaz K, Khamesee MB. Modal Analysis, Metrology, and Error Budgeting of a Precision Motion Stage. *Journal of Manufacturing and Materials Processing*. 2018; 2(1):8.
https://doi.org/10.3390/jmmp2010008

**Chicago/Turabian Style**

Okyay, Ahmet, Kaan Erkorkmaz, and Mir Behrad Khamesee. 2018. "Modal Analysis, Metrology, and Error Budgeting of a Precision Motion Stage" *Journal of Manufacturing and Materials Processing* 2, no. 1: 8.
https://doi.org/10.3390/jmmp2010008