ESD.33 -- Systems Engineering
Session #9
Critical Parameter Management &
Error Budgeting
Dan Frey

Plan for the Session
• Follow up on session #8
• Critical Parameter Management
• Probability Preliminaries
• Error Budgeting
– Tolerance
– Process Capability
– Building and using error budgets
• Next steps

Atish Banergee –
We first studied S-curves in technology strategy…The question remained why the S-curve has the peculiar shape. Well I found the answer in system dynamics. It is a general phenomenon and not restricted to technology.
It can be thought of as two curves:
1. The lower part of the curve is growth with acceleration....
2. The upper part of the s-curve is called a goal-seeking curve and can be thought of as growth with deceleration…

Trends in Compressor Performance
Wisler, D. C., 1998, Axial Flow Compressor and Fan Aerodynamics”, Handbook of Fluid Dynamics ,
CRC Press., ed. R. Johnson.

Cruise thrust
specific fuel consumption ( lb fuel/hr lb thrust
(
Evolution of Jet Engine
Performance
1.05
1.00
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
0.45
0.40
Ghost
JT3C
JT8D-9
JT8D-17
JT8D-209
JT9D-3
JT9D-3A
JT9D-7 JT8D-217A
JT9D-59A
JT9D-7A
CF6-50E/C
CF6-6D
JT8D-15
JT8D-17A
JT8D-219
TAY
CFM56-3
2037
CF6-80C2
RB211-524D4 UP
JT9D-7R4H1 CFM56-5B
V2500
4056
4156
4460
RB211-524D4D
4380
4168
CFM56-5C2
4083
Certification date
Adapted from Koff, B. L. "Spanning the World Through Jet Propulsion.” AIAA Littlewood Lecture. 1991.
P&W deHavilland
RR
GE

Plan for the Session
• Follow up on session #8
• Critical Parameter Management
• Probability Preliminaries
• Error Budgeting
– Tolerance
– Process Capability
– Building and using error budgets
• Next steps

• CPM provides discipline and structure
• Produce critical parameter documentation
– For example, a critical parameter drawing
• Traces critical parameters all the way through to manufacture and use
• Determines process capability ( C p or C pk
)
• Therefore, requires probabilistic thinking

Plan for the Session
• Follow up on session #8
• Critical Parameter Management
• Probability Preliminaries
• Error Budgeting
– Tolerance
– Process Capability
– Building and using error budgets
• Next steps

• Sample space – a list of all possible outcomes of an experiment
– Finest grained
– Mutually exclusive
– Collectively exhaustive
• Event - A collection of points in the sample space

• You roll 2 dice
• Give an example of a single point in the sample space?
• How might you depict the full sample space?
• What is an example of an “event”?

• Axioms
– For any event A , P ( A ) ≥ 0
– P ( U )=1
– If A ∩ B = φ , then P ( A U B )= P ( A )+ P ( B )
For the case of rolling two dice:
A = rolling a 7 and
B = rolling a 1 on at least one die
Is it the case that P ( A + B )= P ( A )+ P ( B )?

• A random variable that can assume any of a set of discrete values
• Probability mass function
– p x
( x o
) = probability that the random variable x will take the value x o
• Let’s build a pmf for rolling two dice
– random variable x is the total p x
( x ) x x =10

• Can take values anywhere within continuous ranges
• Probability density functions obey three rules
–
–
P
{
L
0 ≤
< f x x ≤
( x
U
}
= ∫
U
) for
L all f x x
( x ) d x f x
( x ) P
{
L < x ≤ U
}
–
−
∞
∫
∞ f x
( x ) d x = 1
L U x

• Expected value E ( g ( x )) = b
∫ a g ( x ) f x
( x ) d x
• Mean µ = E ( x )
• Arithmetic average
• Median
• Mode f x
( x )
1 n i n
= 1 x i x

• Variance VAR ( x ) = σ 2 = E (( x − E ( x )) 2 )
• Standard deviation
• Sample variance
• n th central moment
σ = E (( x − E ( x )) 2 )
S 2 = n
1
− 1 i n
∑
− 1
( x i
− x ) 2
E (( x − E ( x )) n )
• Covariance E (( x − E ( x ))( y − E ( y )))

• Average of the sum is the sum of the average (regardless of distribution and independence)
E ( x + y ) = E ( x ) + E ( y )
• Variance also sums iff independent
σ 2 ( x + y ) = σ ( x ) 2 + σ ( y ) 2
• This is the origin of the RSS rule
– Beware of the independence restriction!

• A bracket holds a component as shown.
The dimensions are independent random variables with standard deviations as noted. Approximately what is the standard deviation of the gap?
A) 0.011
B) 0.01
”
”
σ
σ
=
=
0
0 .
01 "
.
001 "
C) 0.001
” gap

• A reasonable (conservative) assumption when you know the limits of a variable but little else
σ = ( U − L ) 2 3
L
U

• I have two spinners
0 x= result of blue spinner y= result of red spinner z=x+y
0
1.5
0.75
0.25
0.5
0.5
1.0
• What are the pdfs for variables x , y , and z ?
P
{ a < x ≤ b
}
= ∫ b a f x
( x ) d x
−
∞
∫
∞ f x
( x ) d x = 1 0 ≤ f x
( x ) for all x

Simulation Can Quickly Answer the
Question trials=10000;nbins=trials/1000; x= random('Uniform',0,1,trials,1); y= random('Uniform',0,2,trials,1); z=x+y; subplot(3,1,1); hist(x,nbins); xlim([0 3]); subplot(3,1,2); hist(y,nbins); xlim([0 3]); subplot(3,1,3); hist(z,nbins); xlim([0 3]);

• If z is the sum of two random variables x and y z = x + y
• Then the probability density function of z can be computed by convolution p z
( z ) =
−
∫
∞ z x ( z − ζ ) y ( ζ ) d ζ

p z
( z ) =
− ∞
∫ z x ( z − ζ ) y ( ζ ) d ζ

p z
( z ) =
−
∫
∞ z x ( z − ζ ) y ( ζ ) d ζ

The mean of a sequence of n iid random variables with
– Finite
– E
( x i
−
µ
E ( x i
)
2 + δ
)
< ∞ δ > 0 approximates a normal distribution in the limit of a large n .

-6 σ
f x
( x ) =
σ
1
2 π e
−
( x − µ )
2
2 σ 2
-3 σ -1 σ µ
+1 σ +3 σ
+6 σ
68.3%
99.7%
1-2ppb

p =
( )
1 m
K exp
⎧
−
1
2
( x − µ ) T K − 1 ( x − µ )
• The lines of constant probability density are ellipsoids
• If the matrix K is diagonal, then the variables are uncorrelated and independent uncorrelated correlated

• Random variables x and y are said to be independent iff f xy x y = f x f y
• Or, knowledge of x provides no information to update the distribution of y

y ( E ( x ))
E ( y ( x ))
S=E ( y ( x ))y ( E ( x )) f y
( y(x ))
S
E(x)
Under utility theory (DBD),
S is a key difference between probabilistic and deterministic design y ( x ) x f x
( x )

Plan for the Session
• Follow up on session #8
• Critical Parameter Management
• Probability Preliminaries
• Error Budgeting
– Tolerance
– Process Capability
– Building and using error budgets
• Next steps

• A tool for predicting and managing variability in an engineering system
• A model that propagates errors through a system
• Links aspects of the design and its environment to tolerance and capability
• Used for tolerance design, robust design, diagnosis…

• Tolerance --The total amount by which a specified dimension is permitted to vary
(ANSI Y14.5M)
• Every component within spec adds to the yield ( Y ) p ( ( y ) )
Y
L U

W
> 25% W
Lead
Land

0.25
THIS ON A DRAWING
0.25 wide tolerance zone
MEANS THIS

For Individual
Features
For Individual or Related
Features
For Related
Features
Geometric Characteristic Symbols
Type of
Tolerance
Characteristic
Form
Profile
Orientation
Location
Straightness
Flatness
Circularity (Roundness)
Cylindricity
Profile of a Line
Profile of a Surface
Angularity
Perpendicularity
Parallelism
Position
Concentricity
Symbol
Runout
Circular Runout
Total Runout
Arrowhead(s) may be filled in.

• Most products have many tolerances
• Tolerances are pass / fail
• All tolerances must be met (dominance)
35

• Many noise factors affect the system
• Some noise factors affect multiple dimensions (leads to correlation)
36

p( q )
U − L
2
µ
L U + L
2
U q
• Process Capability Index
• Bias factor k ≡
µ −
U +
2
( U − L ) /
L
2
• Performance Index
C p
≡
(
U − L
)
/ 2
3 σ
C pk
≡ C p
( 1 − k )
37

• Motorola’s “6 sigma” programs suggest that we should strive for a C p of 2.0. If this is achieved but the mean is off target so that k =0.5, estimate the process yield.

C p
k
• By definition p ( q )
Y
FT
=
U
L
∫ ( )d
Y
L U q
• If Gaussian
Y
FT
=
1
2
⎡
⎢
⎣ ⎢ erf
⎝
3 2
2
C p
( 1 − k )
⎠ erf
⎝
3 2
2
C p
( 1 + k )
⎠
⎤
⎥
This function to maps C p and k to yield
39

C p
k
L ( q )
A o
L U + L
2
U q
Taguchi's quality loss function
ANSI's implied quality loss function
Quality Loss =
[
( U −
A o
L ) / 2
]
2 ⎝
⎜
⎛ d −
U +
2
L
⎠
⎟
⎞
2
E(Quality Loss) = A o k 2 +
1
9 C p
2

• What does a crankshaft do?
• How would you define the tolerances?
• How does variation affect performance?

• What does the second level connection do?
• How would you define the tolerances?
• How does variation affect performance?

C p
k
C p
= 0 82 k = 008
Y
FT
= 983%
150
100
50
0
250
200
LL UL
4

• Rolled throughput yield ( Y
RT
)--
The probability that all tolerances are met
• Motorola’s approach
Y
RT
= i m
∏
= 1
Y
FT i
• Assumes probabilistic independence
Motorola’s formula
Y
RT
= 368 =
Hughes’ data
Y
RT
= 66 7%
5

Side #4
20
Lead number
40
Side #3
20 40 60
Lead number
80 100 120
Side #2
Side #1
20
Lead number
40
20 40 60
Lead number
80 100 120
6

Plan for the Session
• Follow up on session #8
• Critical Parameter Management
• Probability Preliminaries
• Error Budgeting
– Tolerance
– Process Capability
– Building and using error budgets
• Next steps

• Kinematic errors
– Straightness
– Squareness
– Bearings
• Drive related errors
• Thermal errors
• Static loading
• Dynamics

Once per revolution lead error ( µ m)
1 revolution
Cumulative lead error ( µ m/mm)
Nominal travel (mm)

ε y
X
ε y z y x
OK, so you put the error in the model. Now what will happen when the machine moves?
X z y x
ε y z y x

60 mm
400 mm 500 mm
Θ z2
Θ z1
Z
300 mm
1000 mm z x y
Base
Point p

Error Description
ε z1
ε z2
δ z3
ε x3
Drive error of joint #1
Drive error of joint #2
Drive error of joint #3
Pitch of joint #3
ε y3
Yaw of joint #3 xp
2
Parallelism of joint 2 in the x direction
µ
0 rad
σ
0.0001 rad
0 rad 0.0001 rad
Z ·0.0001
0.01mm
0 rad 0.00005 rad
0 rad
0.0002 rad
0.00005 rad
0.0001 rad

• The matrices describe the intended motions and the errors
NOTE: These two should be swapped
0 T
3
=
⎡ 1
⎢
⎢
0
⎣
⎢
⎢
0
0
0
1
0
0
0
0
1
0
1000 mm ⎤
0
0 mm mm
⎥
⎥
1
⎥
⎥
⎦
⋅
⎢
⎢
⎣
⎢
⎢
⎡ cos( Θ sin z 1
0
0
Θ
+ z 1
ε z 1
) − sin cos( Θ z 1
Θ
+ z 1
ε
0 z 1
)
0
0
0
1
0
0
0
0 mm ⎤ mm
⎥
⎥ mm
1
⎥
⎥
⎦
⋅
⎢
⎢
⎣
⎢
⎢
⎡ 1
0
0
0
0
1 xp
2
0
0
− xp
2
1
0
500
0
60 mm mm
1 mm ⎤
⎥
⎥
⎦
⎥
⎥
⋅
⎢
⎢
⎣
⎢
⎢
⎡ cos( Θ sin z 2
Θ
0
0
+ z 2
ε z 2
) − sin cos( Θ z 2
Θ
+ z 2
ε
0 z 2
)
0
0
0
1
0
0 mm
0 mm
0 mm
1
⎥
⎥
⎤
⎥
⎥
⎦
⋅
⎢
⎢
⎡
⎢
⎢
⎣
1
0
0
0
0
1
0
0
0
0
1
0
400
0
0 mm ⎤ mm mm
⎥
⎥
1
⎥
⎥
⎦
⋅
⎢
⎢
⎡
⎢
⎢
⎣
−
1
0
ε y 3
0
0
1
ε
0 x 3
−
ε
ε y 3 x 3
1
0
−
0 mm
0 mm
Z −
1
δ z 3
⎥
⎥
⎤
⎥
⎥
⎦
• Can be applied to any point on the end effector
⎪
⎩
⎧
⎪ p p '
' x y p '
1 z
⎫
⎪
⎪
⎭
= 0
⎧
⎪
T
3
⎪
⎩
−
0
0
300
1
⎪
⎭
⎫
⎪

• Short answers on TRIZ and probability
• Error budgeting
– Two tasks are to be done with the robot
– Analyze the tasks
– Discuss changes to the system
• A Matlab file is available in the HW folder just so you don’t have to re-type the matrices

• You can download HW #5 Error Budgetting
– Due 8:30AM Tues 13 July
• See you at Thursday’s session
– On the topic “Design of Experiments”
– 8:30AM Thursday, 8 July
• Reading assignment for Thursday
– All of Thomke
– Skim Box
– Skim Frey