precision machine design me 250 errors & compensation mark sullivan october 2, 2008

Precision Machine DesignME 250

Errors & Compensation

Mark Sullivan

October 2, 2008

Precision Machine Design


SullivanOct 2, 2008

• Terms• Accuracy & Repeatability• Errors

– Random (non-repeatable)

– Systematic (repeatable)

• Precision Engineering Considerations• Compensation Methods

Agenda



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Acknowledgements

• Text and figures in these lecture notes are taken from the following sources:

– DeBra, D., Beach, D., “Precision Engineering, ME 324,” Stanford University.– Culpepper, M., “Multi-Scale System Design, 2.76,” MIT.– Furman, B., “Precision Machine Design, ME 250,” San Jose State University.– Hale, L. C., “Precision Engineering Principles,” ASPE Tutorial, Monterey,

2006.– Smith, S. T., Chetwynd, D. G., Foundations of Ultraprecision Mechanism

Design, Taylor & Francis, 1994.– Hale, L. C., “Principles and Techniques for Designing Precision Machines,”

UCRL-LR-133066, Lawrence Livermore National Laboratory, 1999. (http://www.llnl.gov/tid/lof/documents/pdf/235415.pdf)

– Slocum, A. H., Precision Machine Design, SME, 1992.– Slocum, A. H., FUNdaMENTALs of Design, MIT, 2008.– Precision Engineering Research Group, MIT

• http://pergatory.mit.edu/

• http://pergatory.mit.edu/kinematiccouplings/



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Metrology Terms

• Range– The extent over which the machine functions within specification

• Resolution– Smallest discernable change that can be registered by machine

• Repeatability– Scatter of results obtained when a machine tries to exactly

reproduce a given operation

• Accuracy– Worst case deviation of measured result from true value

• Ex: Difference between commanded move and actual move



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More Metrology Terms

• Precision– Three definitions

1. Synonym for Repeatability

2. Resolution / Range

3. Better accuracy or smaller precision than typically obtained

• Error– Amount an assumed value deviates from true value

– Random Error vs. Systematic Error

• Traceability– Enables “legal metrology” in any machine shop or lab

– Relates practical measurements to international standards

– NIST• Engineering Metrology Toolbox



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Accuracy, Resolution, & Repeatability

• Accuracy is the ability to tell the truth.• Repeatability is the ability to tell the same story each time.• Resolution is the detail to which you tell a story.

– Alexander Slocum, MIT

Precision engineers make use of the difference between

accuracy, resolution, & repeatability



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Accuracy

• How well you achieve the goal

From “Multi-Scale System Design,” Culpepper



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Repeatability

• How well you perform a function multiple times

From “Multi-Scale System Design, 2.76,” Culpepper



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Accuracy, Repeatability & Errors

• Repeatability is the fundamental limit on accuracy• All errors are either non-repeatable or systematic

– Systematic errors occur the same way each time (reproducible)

– Non-repeatable errors occur differently at different times• Non-repeatable errors are often called random errors

From “Precision Engineering, ME 324,” DeBra & Beachand “Multi-Scale System Design, 2.76,” Culpepper



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Accuracy, Repeatability & Errors (2)




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Random Errors occur differently at different times

• Random Error– Non-repeatable

– Scatter of results• Treated statistically

– Where does the scatter come from?• Influences inherent in the design where effects that have not been

controlled by the designer.– e.g., Scatter in results from repeated moves in a translation stage.

» Possible causes: backlash, hysteresis, frictional effects, temperature, vibration.

• Can quantify the repeatability for normally distributed phenomena– e.g., Multiple runs and averaging may improve precision of value

– Magnitude of random errors judged by results of repeated measurements



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Systematic Errors occur the same way each time

• Systematic Error– Repeatable

– Inherent to the system

– Occur in the same way with every measurement• e.g., Mis-calibration

– Can be mapped / calibrated / corrected

– Reduction of systematic errors is one of the principal jobs of precision engineers

– Note: Systematic errors are why accuracy is more expensive than precision



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Rules for Normally Distributed Data

• Central Limit Theorem: The probability distribution function of a combination of random variables tends toward a normal (Gaussian) distribution as the number of variables increases.

http://en.wikipedia.org/wiki/Central_limit_theorem

The statistical approach allows a degree of confidence to be attached to a measurement.



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Normal Variables

• Standard Deviation

http://hyperphysics.phy-astr.gsu.edu/Hbase/math/gaufcn.html#c2



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Deterministic System Design Principles

• Inputs → System → Outputs

From “Multi-Scale System Design,” Culpepper



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Determinism

• Systems transform inputs into outputs• Desire a one-to-one relationship between inputs & outputs• Deterministic relationship = one relationship• Closed-form modeling is then possible!



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Strategy for dealing with Errors




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Error Sources

• Thermal– Absolute temperature changes– Temperature gradients– Changes in temperature gradients

• Compliance– Statics– Dynamics

• Constraint

• Measurement– Abbé Error– Cosine Error– Metrology Loop affected by Structural Loop

• Manufacturing– Tolerances

• e.g., backlash, stiction

• Wear



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Error Sources (2)




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Precision EngineeringConsiderations

• System Errors– Minimize system errors.– Repeatability is the fundamental limit on accuracy.– All errors are either non-repeatable or systematic. If you make a

model (geometric, thermal, etc.) with which one can reduce the error in a repeatable way, the error is systematic.

– Employ techniques to minimize the sensitivity to error. Some of these are:

• Minimize Abbé offset• Maximize stiffness• Minimize coefficients of thermal expansion• Maximize diffusivity

– Error correction involves:• Understanding the functional dependence on phenomena that can be

measured and subtracting out the calculated effect, or• Measuring repeatable error relative to a standard and subtracting it out.

– Error compensation is a design technique for introducing an element that has the opposite effect as the error in question (e.g., athermalization of pendulum clock or optic mount).

– Reversal can be used for self-calibration of right angles, flats, straightedges, and mechanism spindles.

From “Precision Engineering, ME 324,” DeBra & Beach



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Precision EngineeringConsiderations (2)

• Thermal– Keep heat out of a precision system.– Minimize the coefficient of thermal expansion (CTE).– Maximize thermal diffusivity.– Work at the international standard temperature of 20°C.

• Kinematic Design and Elastic Accommodation– Apply kinematic design for repeatability and to avoid stress

propagation.– An unconstrained rigid body has six degrees of freedom.– The number of contact points between any two perfectly rigid

bodies is equal to the number of constraints.– Stiffness and robustness require large surface areas in contact that

is inconsistent with the point contacts of kinematic design.– As the rules of kinematic constraint are compromised,

manufacturing tolerances must become more exacting if systems are to function in a satisfactory manner. A divergence from pure kinematic design results in increased manufacturing costs.




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• Materials Selection– Separate geometry effects from the anticipated effects of materials

properties and optimize each separately. Evaluate materials based on a performance function that includes all of the requirements.

– Use geometry first. Materials properties cannot substitute for proper scaling and best use of form.

• Vibration– Reduce system response by increasing structural natural frequency,

selection of support points, geometry, and material.

– Passive isolation can be increased by lowering the natural frequency of the support or using multiple stages.

– For an isolated system, make the natural frequency of the isolators small and the natural frequency of the structure to be isolated as high as possible. This will yield the best vibration isolation.

– Damping can reduce the vibration amplitude. It cannot change the natural frequency of a system.




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• Metrology– Minimize Abbé offset.

• When measuring the displacement of a specified point, it is not sufficient to have the axis of the probe parallel to the direction of motion, the axis should also be aligned with (pass through) the point.

– Separate measurement and structural loops as far as possible.• Keep the measurement loop short and unstressed. Any changes that

occur to components in a measurement loop will result in changes in measured results that are indistinguishable from the measurement.



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Error Strategies

• Approaches for improving machine accuracy– Error reduction

• Isolate error sources and eliminate them to the degree required by the application.

– Error correction1.Understand functional dependence on phenomena that can be measured,

and subtract out the calculated effect.

2.Measure repeatable error relative to a standard and subtract it out.

– Error compensation• Introduce an element which responds to the error source with the

opposite effect than the uncompensated system

– Self-calibration using reversal• Can be used for the calibration of squares, levels, straightedges,

spindles, and more.



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Error Compensation

• Active– Control systems

• Feedback• Feedforward

• Passive– e.g., Athermalized designs



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Control Systems

• Examples:– Automotive cruise control

– DC servo motor position controller

– LODTM positioning system



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Control Systems (2)



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Athermalized Systems

Four Approaches

1. Control instrument temperature

2. Build from single material– e.g., Be, SiC

3. Use materials with:– Matching ΔL/L to operating temperature and– Matching CTE at operating temperature

• Invar 36 & Fused Silica (SiO2)• Invar 39 & SiC• Super Invar & Zerodur

4. Athermalization compensation



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Athermalization Example:Actuators on Hex Mirror



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Piezoelectric Mirror Actuator

Isometric View Isometric Section View



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Thermal-Matching End Cap

Approach

• End cap contains components that match the thermal expansion difference of the mirror and actuator

• End cap effectively has a negative thermal expansion coefficient

• End cap is made relatively stiff to maintain actuator authority

Advantages

• Compensation approach with broad range

• Simple construction

• Large design space with many candidate materials

Issues

• Proper sizing of components critical

SiC Mirror Rib

PZT Actuator

Components used to

athermalize expansion



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Athermalized Actuator

SiC Mirror

PZT Actuator

Inner Cylinder

Outer Cylinder

End Cap



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One-Dimensional Spring Modelfor End Cap

KOuter

KActuator

x1

x3

KMirror

x2

KInner

• Springs represent the stiffness of the components

– Capture basic physics, but with lower fidelity than FEA

– Springs can expand based on constituent material CTE

• Broad temperature range evaluation

– Induced bondline stresses– Unmatched contraction– Added compliance

• Matlab model

KCap



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CandidateAthermalization Materials

Actuator Athermalization

Max Min TTemperature, K 300 4 -296

L = L T

Combination 0 Mirror Actuator Inner Cylinder Outer Cylinder End Cap Sum DifferenceMaterial SiC PZT Al 6061-T6 316 SS Invar, 10-6/K 2.4 1 23 16 1.5L, mm 25.0 13.0 -10.4 17.4 5.0 25.0 0.0 Must equal 0L, m -1.78E-05 -3.85E-06 7.08E-05 -8.24E-05 -2.22E-06 -1.77E-05 -88.8E-9 Minimize

L = L L/L

Combination 1 Mirror Actuator Inner Cylinder Outer Cylinder End Cap Sum DifferenceMaterial SiC PZT Al 6061-T6 Ti Ti, 10-6/K 2.4 1 23 8.6 8.6L, mm 25.0 13.0 -14.85 21.85 5.0 25.0 0.0 Must equal 0L/L -0.000194 -0.0015 -0.0037 -0.0015 -0.0015L, m -4.85E-06 -1.95E-05 5.49E-05 -3.28E-05 -7.50E-06 -4.83E-06 -20.0E-9 Minimize

Combination 2 Mirror Actuator Inner Cylinder Outer Cylinder End Cap Sum DifferenceMaterial SiC PZT 316 SS Ti Ti, 10-6/K 2.4 1 16 8.6 8.6L, mm 25.0 13.0 -29.70 36.70 5.0 25.0 0.0 Must equal 0L/L -0.000194 -0.0015 -0.0026 -0.0015 -0.0015L, m -4.85E-06 -1.95E-05 7.72E-05 -5.51E-05 -7.50E-06 -4.83E-06 -20.0E-9 Minimize

Combination 3 Mirror Actuator Inner Cylinder Outer Cylinder End Cap Sum DifferenceMaterial SiC PZT Vespel Ti Ti, 10-6/K 2.4 1 45 8.6 8.6L, mm 25.0 13.0 -4.66 11.66 5.0 25.0 0.0 Must equal 0L/L -0.000194 -0.0015 -0.0085 -0.0015 -0.0015L, m -4.85E-06 -1.95E-05 3.96E-05 -1.75E-05 -7.50E-06 -4.88E-06 30.0E-9 Minimize



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Thermal-MatchingEnd Cap Materials

Approach

• Low thermal expansion Tungsten outer cylinder sleeve and end cap

• High thermal expansion brass inner cylinder sleeve creates large thermal contraction effect during cooldown

• Titanium actuator endcap CTE matches PZT for bondline low shear stress (TBV)

• W endcap CTE matches SiC for low bondline shear stress (TBV)

• All components stiffer than PZT. W is 6 times stiffer

• Brass sleeve is slotted into the Ti and W and creates a strong interference fit during cooldown

SiC Mirror Rib

PZT Actuator

W

Brass

Ti



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Thermal Expansion Data

• End cap uses large thermal expansion difference between tungsten and brass

• Ti, W and brass readily available



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Results (1 of 3)

• Need to match the blue curve with SiC mirror contraction

• Match obtained with a 10mm long outer sleeve and a 7.2 mm long inner sleeve



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Results (2 of 3)

• Mismatched displacement must be corrected by the actuator

• Broad range with 0.5 microns



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Results (3 of 3)

• Tensile bondline stresses are below 2ksi

• Compressive loads may be limited by buckling



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Conclusions

• Material combination are available that athermalize PZT actuator relative to SiC mirror

– Spreadsheet calculations done for 4 Kelvin case– Matlab model to solve for material combinations

• Collect L/L information for candidate materials at different temperatures• Minimize (L/L)actuator vs. (L/L)mirror over temperature range

– Will match actuator and mirror stiffnesses (Kactuator and Kmirror)

• Cryogenic material properties references identified– Thermal Expansion, Metallic Elements and Alloys, Touloukian, et al.– NIST publications– NIRCam cryo materials library

• Athermalized actuator appears simple in construction– Fits within mirror webs– Parts have a simple geometry– Fabrication appears straightforward

precision machine design me 250 errors & compensation mark sullivan october 2, 2008

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