Rigid Part Mating
• Goals of this class
– understand the phases of a typical part mate
– determine the basic scaling laws
– understand basic physics of part mating for simple
geometries
– relate forces and motions arising from geometric errors
– compare logic branching and direct error-feedback part
mating strategies
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© Daniel E Whitney 2000
1
Basic Bandwidth Issues and Time-Mass-
Distance Scaling Laws
• Torque required to move a mass M at the end of an arm of length L in time T is proportional to – M L2/T2
• This implies that really fast motions must be really
small or use a small arm with small mass
• I estimated
– my hand’s mass = 250g, effective length = 10cm
– my arm + hand’s mass = 1700g, effective length = 35 cm
– ratio arm:hand of ML2 = T2 = 85
• Don’t forget: arm mass+payload mass=M
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© Daniel E Whitney 2000
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Main Phases of a Part Mating Event
contact
force
Approach
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Chamfer
Crossing
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One-point
Contact
© Daniel E Whitney 2000
Two-point
Contact
Line
Contact
3
Required Bandwidth for Chamfer Crossing
lateral motion
E
0.5E
E/2
time
E/2V
0.5E/V
τ
10E/V
20E/V
T
Fourier coefficient = 2 π T / (n2 π2 τ) sin (2 n πτ /T)
T = 20 E / V; τ = E / 2 V; T / τ = 40
Period = 2π = ωT=ω20E/V
ω = π V /10 E
f = V / 20 E
If V = 10 in/s and E = 0.05", f = 10 Hz
If 5th harmonic must be adhered to, bandwidth needed = 50Hz
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© Daniel E Whitney 2000
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Trapezoidal Wave Harmonics
Image removed for copyright reasons.
Source:
Figure 9-7 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development. New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
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Conclusions
• Gross motions can be (must be) done by large
arms that necessarily will move slowly
• No robot arm with practical reach can make fine
motion error removal adjustments at 50 Hz
• Fine motions can be fast if they are done by small
arms, and must be fast to absorb typical errors at
economical speeds
• Big tasks with big parts will take a long time
compared to small tasks with small parts
• What we see: small parts cycle times are ~5s while
big parts cycle times are ~ 60s.
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© Daniel E Whitney 2000
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Essentials of Part Mating Theory for Fine Motions
•
•
•
•
•
•
•
•
Quasi-static assumption
Geometry of pegs and holes
Applied forces
Normal reaction forces and friction reaction forces
Entry geometry limits
Wedging conditions
Jamming conditions
Alternate strategies for accomplishing fine motion
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The Basic Idea
• In gross motions, it pays to pre-plan to prevent
errors
• In fine motion, it does not pay to try to prevent
errors
• So the principle is to anticipate errors and figure
out how to make assembly happen anyway
• This requires us to understand three factors:
– Geometry
– Compliance
– Friction
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© Daniel E Whitney 2000
8
Geometry of Peg-Hole Mates
10
l/ D
=
c/
l /d = c /θ
C = 0.1
C = 0.01
1
ll /D
√
2 C
TYPICAL MACHINED
PARTS
θ
C = 0.001
θ
m
=
θ m = 2c
EA
CH
C
0.1
C = 0.0001
Image removed for copyright reasons.
Source:
Figure 10-15 in [Whitney 2004] Whitney, D. E.
0.01
0.001
Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development.
New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
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© Daniel E Whitney 2000
θ
m
FO
R
PRECISION
PARTS
0.01
0.1
1.0
θ RADIANS
9
Dimensioning Practice
Image removed for copyright reasons.
Source:
Figure 10-16 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development. New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
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Geometry Definitions
Insertion Direction
θo
r
c = (D-d)/D
d
α
εo
W
R
D
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Insertion History
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Insertion History
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Insertion History
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Insertion History
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Insertion History
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Insertion History
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Insertion History
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Insertion History
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Life Cycle of a Part Mate
Image removed for copyright reasons.
Source:
Figure 10-12 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development. New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
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© Daniel E Whitney 2000
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Model of a Compliant Support
Images removed for copyright reasons.
Source:
Figure 10-10 in [Whitney 2004] Whitney, D. E.
Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development.
New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
rigid part mating
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© Daniel E Whitney 2000
All support is assumed
concentrated at one point
and consists of one
lateral stiffness and one
angular stiffness
21
How Compliance Center Reacts to Force
Images removed for copyright reasons.
Source:
Figure 10-11 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development. New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
Force away from
C. C. causes rotation
and translation
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Force on C. C. causes only translation
© Daniel E Whitney 2000
22
Forces and Moments - Two Point Contact Case
Images removed for copyright reasons.
Source:
Figure 10-18 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development. New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
All applied and reaction
forces are expressed
in coordinates at peg’s tip
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Forces Applied During Two-point Contact
K
K
x
θ
When Lg >> 0
K
x
K
θ
K
θ
When Lg ~ 0
Lg
K
x
K
θ
Big
K
x
Small
Big
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Making Lg Small is Good
• How to do it?
• Active Robot Force Feedback
– Costly
– Slow •
•
•
•
•
Some way that acts by itself
It was invented almost 30 years ago
Called Remote Center Compliance
Reduces assembly force
Avoids one of two main failure modes
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© Daniel E Whitney 2000
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INSERTION FORCE Fz, NEWTONS
TWO-POINT
CONTACT
ONE-POINT
CONTACT
l1
l2
Insertion Force History
CHAMFER CROSSING
l*
INSERTION DEPTH l, mm
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Assembly Failure Modes
• Both occur during two-point contact
• Wedging sets up compressive forces inside the
parts
• Jamming results from incorrect insertion forces
• We can derive the requirements to avoid both of
these failure modes
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Wedging: Compressive Friction Forces Prevent Insertion Regardless of Insertion Force
d
d
φ == t an
-1
φ
µ
φ == t an
-1
φ
µ
f2
f
l
θ
1
rigid part mating
l
l - dθ
θ
D
Wedging can happen if θ > c/µ
when two-point contact occurs
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f
f2
1
l - dθ
D
Wedging can be avoided if µ is small
enough or if two-point contact occurs
deep enough in the hole
© Daniel E Whitney 2000
28
What’s a Friction Cone?
F
Friction cone
θ = tan-1 µ
FN
F
F
FT
µFN
Sliding will occur if FT > µFN
FT /FN = tan θ
So, sliding will occur if tan θ > µ
and F will lie on the boundary
of the cone
rigid part mating
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© Daniel E Whitney 2000
If F is inside the cone
then sliding will not happen
because FT < µFN
and F can be any value
29
Conditions for Avoiding Wedging
S =
Lg
2
L g + K / Kx
θ
Images removed for copyright reasons.
Source:
Figure 10-20 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development. New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
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Jamming: Insertion Force Directed the Wrong Way - Can’t Overcome Friction
COMPONENT NORMAL TO PEG AXIS
COMPONENT
PARALLEL TO
PEG AXIS
INSERTION
FORCE
COMPONENT NORMAL TO PEG AXIS
FRICTION
CORRESP.
TO NORMAL
COMPONENT
COMPONENT
PARALLEL TO
PEG AXIS
INSERTION
FORCE
REACTION
TO NORMAL
COMPONENT
Component of
Insertion force
Along insertion direction
Not big enough:
Peg Is Jammed
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© Daniel E Whitney 2000
Component of
Insertion force
Along insertion direction
Is big enough:
Peg Goes In
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Conditions for Avoiding Jamming Image removed for copyright reasons.
Source:
Figure 10-21 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development. New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
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Jamming Examples
COMPONENT NORMAL TO PEG AXIS
COMPONENT
PARALLEL TO
PEG AXIS
INSERTION
FORCE
COMPONENT NORMAL TO PEG AXIS
FRICTION
CORRESP.
TO NORMAL
COMPONENT
COMPONENT
PARALLEL TO
PEG AXIS
REACTION
TO NORMAL
COMPONENT
M
Fz
Fz
Fx/Fz is small.
M/rFz is small.
Fx/Fz is big.
M/rFz is big.
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M
Fx
Fx
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INSERTION
FORCE
© Daniel E Whitney 2000
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Target Expands as Depth Increases
M
rF
This point moves
As λ increases
l
λ = 2r
µ
z
2λ+1
λ
1/µ
1
F
-1
−1/µ
F
x
z
−λ
-( 2λ+1 )
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© Daniel E Whitney 2000
This point moves
As λ increases
34
Experimental Data
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Experimental Data -2
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Test Your Understanding
• Why does insertion force rise and then fall during
two-point contact?
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Experimental Data - 3
When
WhenLLgg==00
there
thereisisbarely
barely
any
anyinsertion
insertionforce.
force.
All
Allthat’s
that’sleft
leftisis
chamfer
chamfercrossing
crossing
force.
force.
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Test Your Understanding Again
• Why does the insertion force not rise after
chamfer-crossing is finished?
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Review of Force Feedback Strategy
• Create a coordinate frame at the “working point”
of the part or tool
• Separate lateral and angular sensing and response
motions in that frame
• Devise a response strategy • The Remote Center Compliance is a purely
passive implementation of one such strategy
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Simplified Explanation of the Remote Center Compliance (RCC)
Images removed for copyright reasons.
Source:
Figure 9-8 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development. New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
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RCC Response to External Loads
(d) RCC UNDER
LATERAL LOAD
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(e) RCC UNDER
ANGULAR LOAD
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(f) LINKAGE
RCC UNDER LATERAL
AND ANGULAR
DEFORMATION
42
(d) RCC UNDER
LATERAL LOAD
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(d) RCC UNDER
LATERAL LOAD
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(e) RCC UNDER
ANGULAR LOAD
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(e) RCC UNDER
ANGULAR LOAD
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First RCC Experiment - 1
r = 3°
Angular Error
47
© Daniel E Whitney 2000
First RCC Experiment - 2
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Commercial Remote Center Compliances
Images removed for copyright reasons.
Source:
Figure 9-9 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development. New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
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Assembly is a matter of geometry
Geometry contains errors
in parts
in equipment
But they are small
and usually obscured
Collisions
contain info
Collisions +
stiffness matrix
= force signals
Small stiffness →
small signals and
oscillations and
position uncertainty
But: friction limits
the available information
in force data
Large stiffness →
large signals but
possible instability
Active force feedback
Passive accommodation (incl small stiffness)
Will be stable
Can be smart
can be unstable
can be "strange"
can be slow
Engineered
Will avoid jamming
Accidental,
contextual
May not avoid jamming
With sensors → IRCC Without sensors → RCC
basically Lyapunov stable
taxonomy of fine motions
rigid part mating
Costs too much
Detect errors directly
by sensing them
geometrically
Detect errors indirectly
by sensing collisions
Collisions
are bad
Taxonomy of Fine Motions
Seek perfection
Accept errors
Ignore errors
(Matt Mason
approach)
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