Precision Machine
Design
ME 250
Errors & Compensation
Mark Sullivan
October 2, 2008
Precision Machine Design
Errors & Compensation
Agenda
• Terms
• Accuracy & Repeatability
• Errors
– Random (non-repeatable)
– Systematic (repeatable)
• Precision Engineering Considerations
• Compensation Methods
Sullivan
Oct 2, 2008
Page 2
Precision Machine Design
Errors & Compensation
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/
Sullivan
Oct 2, 2008
Page 3
Precision Machine Design
Errors & Compensation
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
Sullivan
Oct 2, 2008
Page 4
Precision Machine Design
Errors & Compensation
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
•
Sullivan
Oct 2, 2008
Engineering Metrology Toolbox
Page 5
Accuracy, Resolution, &
Repeatability
Precision Machine Design
Errors & Compensation
• 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
Sullivan
Oct 2, 2008
Precision engineers make use of the
difference between
accuracy, resolution, & repeatability
Page 6
Precision Machine Design
Errors & Compensation
Accuracy
• How well you achieve the goal
Sullivan
Oct 2, 2008
From “Multi-Scale System Design,” Culpepper
Page 7
Precision Machine Design
Errors & Compensation
Repeatability
• How well you perform a function multiple times
Sullivan
Oct 2, 2008
From “Multi-Scale System Design, 2.76,” Culpepper
Page 8
Accuracy, Repeatability
& Errors
Precision Machine Design
Errors & Compensation
• 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
Sullivan
Oct 2, 2008
From “Precision Engineering, ME 324,” DeBra & Beach
and “Multi-Scale System Design, 2.76,” Culpepper
Page 9
Accuracy, Repeatability
& Errors (2)
Sullivan
Oct 2, 2008
From “Multi-Scale System Design, 2.76,” Culpepper
Precision Machine Design
Errors & Compensation
Page 10
Random Errors occur
differently at different times
Precision Machine Design
Errors & Compensation
• 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
Sullivan
Oct 2, 2008
Page 11
Systematic Errors occur
the same way each time
Precision Machine Design
Errors & Compensation
• 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
Sullivan
Oct 2, 2008
Page 12
Rules for
Normally Distributed Data
Precision Machine Design
Errors & Compensation
• 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.
The statistical approach allows a
degree of confidence to be
attached to a measurement.
Sullivan
Oct 2, 2008
http://en.wikipedia.org/wiki/Central_limit_theorem
Page 13
Precision Machine Design
Errors & Compensation
Normal Variables
• Standard Deviation
Sullivan
Oct 2, 2008
http://hyperphysics.phy-astr.gsu.edu/Hbase/math/gaufcn.html#c2
Page 14
Deterministic System
Design Principles
Precision Machine Design
Errors & Compensation
• Inputs → System → Outputs
Sullivan
Oct 2, 2008
From “Multi-Scale System Design,” Culpepper
Page 15
Precision Machine Design
Errors & Compensation
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!
Sullivan
Oct 2, 2008
Page 16
Strategy for dealing
with Errors
Sullivan
Oct 2, 2008
From “Multi-Scale System Design, 2.76,” Culpepper
Precision Machine Design
Errors & Compensation
Page 17
Precision Machine Design
Errors & Compensation
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
Sullivan
Oct 2, 2008
Page 18
Precision Machine Design
Errors & Compensation
Error Sources (2)
Sullivan
Oct 2, 2008
From “Multi-Scale System Design, 2.76,” Culpepper
Page 19
Precision Engineering
Considerations
Precision Machine Design
Errors & Compensation
• 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.
Sullivan
Oct 2, 2008
From “Precision Engineering, ME 324,” DeBra & Beach
Page 20
Precision Engineering
Considerations (2)
Precision Machine Design
Errors & Compensation
• 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.
Sullivan
Oct 2, 2008
From “Precision Engineering, ME 324,” DeBra & Beach
Page 21
Precision Engineering
Considerations (3)
Precision Machine Design
Errors & Compensation
• 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.
Sullivan
Oct 2, 2008
From “Precision Engineering, ME 324,” DeBra & Beach
Page 22
Precision Engineering
Considerations (4)
Precision Machine Design
Errors & Compensation
• 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.
Sullivan
Oct 2, 2008
Page 23
Precision Machine Design
Errors & Compensation
Error Strategies
• Approaches for improving machine accuracy
– Error reduction
• Isolate error sources and eliminate them to the degree required by the
application.
– Error correction
1.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.
Sullivan
Oct 2, 2008
Page 24
Precision Machine Design
Errors & Compensation
Error Compensation
• Active
– Control systems
• Feedback
• Feedforward
• Passive
– e.g., Athermalized designs
Sullivan
Oct 2, 2008
Page 25
Precision Machine Design
Errors & Compensation
Control Systems
• Examples:
– Automotive cruise control
– DC servo motor position controller
– LODTM positioning system
Sullivan
Oct 2, 2008
Page 26
Precision Machine Design
Errors & Compensation
Control Systems (2)
Sullivan
Oct 2, 2008
Page 27
Precision Machine Design
Errors & Compensation
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
Sullivan
Oct 2, 2008
Page 28
Athermalization Example:
Actuators on Hex Mirror
Sullivan
Oct 2, 2008
Precision Machine Design
Errors & Compensation
Page 29
Precision Machine Design
Errors & Compensation
Piezoelectric Mirror Actuator
Isometric View
Sullivan
Oct 2, 2008
Isometric Section View
Page 30
Precision Machine Design
Errors & Compensation
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
Sullivan
Oct 2, 2008
SiC Mirror
Rib
Components
used to
athermalize
expansion
PZT Actuator
Page 31
Precision Machine Design
Errors & Compensation
Athermalized Actuator
PZT Actuator
Outer Cylinder
End Cap
Inner Cylinder
SiC Mirror
Sullivan
Oct 2, 2008
Page 32
One-Dimensional Spring Model
for End Cap
Precision Machine Design
Errors & Compensation
x1
• Springs represent the
stiffness of the
components
KCap
x3
KMirror
KOuter
KInner
x2
– Capture basic physics, but
with lower fidelity than
FEA
– Springs can expand based
on constituent material
CTE
• Broad temperature range
evaluation
KActuator
– Induced bondline stresses
– Unmatched contraction
– Added compliance
• Matlab model
Sullivan
Oct 2, 2008
Page 33
Candidate
Athermalization Materials
Precision Machine Design
Errors & Compensation
Actuator Athermalization
Max
Temperature, K
L = L  T
Combination 0
Material
, 10-6/K
L, mm
L, m
L = L L/L
Combination 1
Material
, 10-6/K
L, mm
L/L
L, m
Combination 2
Material
, 10-6/K
L, mm
L/L
L, m
Combination 3
Material
, 10-6/K
L, mm
L/L
L, m
Sullivan
Oct 2, 2008
T
Min
300
Mirror
SiC
2.4
25.0
-1.78E-05
Mirror
SiC
2.4
25.0
-0.000194
-4.85E-06
Mirror
SiC
2.4
25.0
-0.000194
-4.85E-06
Mirror
SiC
2.4
25.0
-0.000194
-4.85E-06
4
Actuator
PZT
1
13.0
-3.85E-06
Actuator
PZT
1
13.0
-0.0015
-1.95E-05
Actuator
PZT
1
13.0
-0.0015
-1.95E-05
Actuator
PZT
1
13.0
-0.0015
-1.95E-05
-296
Inner Cylinder
Outer Cylinder
Al 6061-T6
316 SS
23
16
-10.4
17.4
7.08E-05
-8.24E-05
End Cap
Invar
Inner Cylinder
Outer Cylinder
Al 6061-T6
Ti
23
8.6
-14.85
21.85
-0.0037
-0.0015
5.49E-05
-3.28E-05
End Cap
Ti
Inner Cylinder
Outer Cylinder
316 SS
Ti
16
8.6
-29.70
36.70
-0.0026
-0.0015
7.72E-05
-5.51E-05
End Cap
Ti
Inner Cylinder
Outer Cylinder
Vespel
Ti
45
8.6
-4.66
11.66
-0.0085
-0.0015
3.96E-05
-1.75E-05
End Cap
Ti
1.5
5.0
-2.22E-06
8.6
5.0
-0.0015
-7.50E-06
8.6
5.0
-0.0015
-7.50E-06
8.6
5.0
-0.0015
-7.50E-06
Sum
25.0
-1.77E-05
Sum
25.0
-4.83E-06
Sum
25.0
-4.83E-06
Sum
25.0
-4.88E-06
Difference
0.0
-88.8E-9
Must equal 0
Minimize
Difference
0.0
-20.0E-9
Must equal 0
Minimize
Difference
0.0
-20.0E-9
Must equal 0
Minimize
Difference
0.0
30.0E-9
Must equal 0
Minimize
Page 34
Thermal-Matching
End 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
Sullivan
Oct 2, 2008
Precision Machine Design
Errors & Compensation
SiC Mirror
Rib
Ti
W
Brass
PZT Actuator
Page 35
Precision Machine Design
Errors & Compensation
Thermal Expansion Data
• End cap uses large thermal expansion difference between tungsten
and brass
• Ti, W and brass readily available
Sullivan
Oct 2, 2008
Page 36
Precision Machine Design
Errors & Compensation
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
Sullivan
Oct 2, 2008
Page 37
Precision Machine Design
Errors & Compensation
Results (2 of 3)
• Mismatched
displacement must
be corrected by the
actuator
• Broad range with 0.5
microns
Sullivan
Oct 2, 2008
Page 38
Precision Machine Design
Errors & Compensation
Results (3 of 3)
• Tensile bondline
stresses are below
2ksi
• Compressive loads
may be limited by
buckling
Sullivan
Oct 2, 2008
Page 39
Precision Machine Design
Errors & Compensation
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
Sullivan
Oct 2, 2008
Page 40