uncertainties, error types and error propagation …...random errors and systematic errors. random...

Prof. Dr.-Ing. habil. Hermann Lödding

Prof. Dr.-Ing. Wolfgang Hintze

©

PD Dr.-Ing. habil. Jörg Wollnack

EP.1 30.04.2015

Uncertainties,

Error Types

and

Error Propagation Analysis



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EP.2 30.04.2015

Physical measured values

are in principle

occupied with uncertainties!

Problem Definition



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EP.3 30.04.2015

If one measures e.g. the lengths of a production series

of lengths scales, then coincidental errors can have

been submitted to the set of scales

(coincidental production errors),

coincidental errors of the individual measuring

(Measurement noise or surface roughness at the individual) and

a systematic error of all individuals

(object temperatures differ constantly from the standard temperature).

Didactic Example for Uncertainties



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EP.4 30.04.2015

An error is a bound on the precision and accuracy

of the result of a measurement.

These can principal be classified into two types:

Random Errors and Systematic Errors.

Random Errors are caused by inherently unpredictable

fluctuations in the readings of a measurement apparatus,

whereas Systematic Errors are always constant under the same

physical conditions (repeatability).

If the values of the systematic errors can be identified,

then it can be eliminated,

if the (sensor) system model is complete.

We typically know only the ranges

within the true values lies.

Uncertainties I



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EP.5 30.04.2015

The central limit theorem states that if the sum of

independent identically distributed random variables has a

finite variance, then it will be approximately normally

distributed (Gaussian distribution).

Formally, a central limit theorem is any of a set of weak-

convergence results in probability theory.

They all express the fact that any sum of many independent

and identically-distributed random variables will converge to

be distributed according to a particular "attractor distribution“

(normally distribution).

The Central Limit Theorem I



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EP.6 30.04.2015

if are randomly independent variables with existing mean

average value and variance, then the standardized sum

Xk

U

X

X

n

k X

k

n

k X

k

n

k

k

1

2

1

d i

converges for to a normalized normal distribution

(under weak restrictions1 to the distributions).

n

1 to below restricts variance und up limited absolute moments 3rd order

The Central Limit Theorem II



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EP.7 30.04.2015

xI

xS

Saturation behavior Dead zone

Sensor Characteristics

I := true Value

S:= Sensor Value ideale Kennlinie ideal characteristic line Hysteresis behavior

Neukurve Initial curve

Stationary

curve

x ySensor/System

physikalischeGröße

Sensor-Meßwert

Physical

Value

Sensor

measured

value



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EP.8 30.04.2015

xI

xS

stationary

Model

characteristic

true

stationary

characteristic

systematic

Error

Systematic Error



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EP.9 30.04.2015

Additives Random Errors

xI

xS

stationary

Model

characteristic

true stationary

characteristic

K - KB

K +

- +

x

pK

K -

KB

K +

- +

x

pK



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EP.10 30.04.2015

xI

xS

x0I

xMin xMax

regression

straight

line

lower and upper

regression

straight lines

Regression Straight Line as Linear Approximation I

lower and upper

straight lines



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EP.11 30.04.2015

x x x x x x x x xS I S I SL I IMin I IMaxMax ( ) ( ) ,m rSystematic Error

x x x m x xSL I S S I 0I( ) 0 d iRegression line

The sensor behaviour and with that the parameters of the

parametric sensor model are generally dependent on

further physical influence quantities.

Regression Straight Line as Linear Approximation II



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EP.12 30.04.2015

• Temperature

• Air pressure

• Humidity

• Supply voltage

• Tolerances caused by production of function defining

sensor components

• Electromagnetic Fields

• Mechanical forces etc..

Sensor Influence Effects

This influence effects not

contributes insignificantly

to the systematic and

random measuring errors of

a sensor.



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EP.13 30.04.2015

The transient response

of a sensor Systems generally differs

from the ideal linear, time-invariant behaviour.

• Linearity and offset error,

• Temperature drift with zero point and gain drift,

• Transient characteristic with the frequency and phase response model

or under assumption of a defined transient characteristic (e.g. low-pass

of n-th order) we use the group delay and the impulse rise time model,

• Noise behavior and dynamics of the system.

One distinguishes the following error classes:

Sensor/System Response Characteristic



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EP.14 30.04.2015

: Input noise (random error)Xn

: systematic input errorx

A : Output noise (random error)ynA : systematic output errory

Uncertainties I

x=xI

xI

xM

y=yI

yMSystem

Eingangs-größen

Ausgangs-größen

nA

y ΔyA

+ +

nx

Δx

++

Input Output I := true Value

S:= Sensor Value



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EP.15 30.04.2015

M I xx x x n

A A

M I yy y y nA A

M M y( , ) xy f p x x n y n

A A

M y( , ) xy f p x x n y n

( , ) ( , ) y xf p x y n f p x x n

( , ) ( , ) y xy n f p x x n f p x

Uncertainties

Structure Uncertainties

Incremental Approach

Uncertainties II



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EP.16 30.04.2015

( , ) ( , ) y xy n f p x x n f p x

Incremental Approach

( ) xy J x x

Linear approximation for systematic errors

Worst-Case-Analysis (x Values and signs unknown)

Known systematic errors can be corrected

if the model is known. Therefore only the unknown

systematic and coincidental deviations

of interest are.

Max

,m mn ny xJ p x x

Uncertainties III



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EP.17 30.04.2015

Max

,m mn ny xJ p x x

Uncertainties IV

Max

Max

0 , {1,..., } ,

,

m m

m mn n

n

y y m M

y

xJ p x x

Using the triangle inequality of the norm we get:

In a compact matrix notation we get for the worst case error:

, for

, for

M N

mn mn i j

M

m mn i

a a

b b

A A

b b



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EP.18 30.04.2015

Worst-Case-Analysis (x, p Values and signs unknown)

Max

, ,m mn n mn ny p xJ p x p J p x x

Uncertainties V



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EP.19 30.04.2015

( , , ) ( , , ) ( , , ) p x yJ p x y p J p x y x J p x y y 0

Linear approximation of the implicit model

f p x y 0( , , )

( , , ) ( , , ) y xJ p x y y J p x y x

1

( , , ) ( , , )

y xy J p x y J p x y x

p 0

1

( , , ) from left

yJ p x y

1

, with = ( , , ) ( , , )

y xy Q x Q J p x y J p x y

Uncertainties Implicit Model I



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EP.20 30.04.2015

1

, with = ( , , ) ( , , )

y xy Q x Q J p x y J p x y

Linear approximation for delta values

implicit Model

, with ( , ) xy Q x Q J p x explicit Model

Conclusion?

An implicit model can be transformed local (in linear approximation)

structurally to the explicit model.

Uncertainties Implicit Model II



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EP.21 30.04.2015

Probability

of the

Worst Case Condition



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EP.22 30.04.2015

f f x x f xn i

i

n

( ) ( , , ) ( )x 1

1

FHG

IKJ

f ek

x

k

nk Xk

Xk( )x1

2

1

2

1

2

FHGIKJ

F

HGIKJ

1

2

1

1

1

2

1

2

b gn kk

nx

k

n

e

k Xk

Xk

FHGIKJ

FHG

IKJ

1

2

1

1

1

2

2

1

b gn kk

nx

e

i Xi

Xii

n

f q eX X n

kk

n q

k k

i

n

( , , )

FHGIKJ

1

2

1

1

1

2

2

1

b gq n q

k

n2

1

2

Worst-Case-Condition I

Central limit theorem

x qk X Xk k

At the

Worst Case border

Statistic independence of the values



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EP.23 30.04.2015

FHGIKJ

f q eX X n

kk

nnq

k k( , , )

1

2

1

1

1

2

2

b g e eab a b

c h

The probability of the appearance of the worst case condition fells

with a power law. Therefore the worst case condition can already

get truly at a relatively low number of influence variables only with

an "astronomically" low probability.

This is at least economically a far too expensive criterion!

f q eX X n

kk

n n q

k k( , , )

FHGIKJ

1

2

1

1

2

b g d i

Worst-Case-Condition II



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EP.24 30.04.2015

Because of the low probability of the

worst case conditions frequently it is practicable,

to compute instead of the worst case error propagation

the Gaussian error propagation.

One assumes in the end that the systematic errors of a series are

influenced from a random process.

If this assumption does not make sense, then we can determine

only the systematic error and correct it or

we analyse it for the worst case condition.

Worst-Case-Bedingung III



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EP.25 30.04.2015


of the

Uncertainties

of

Random Errors



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EP.26 30.04.2015

The probability of the occurrence of a

continuous signal value characterizes the

density function.

Random Signal

x t( )

t

stochastisch

f(x)

x

Dichtefunktionrandomly density function



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EP.27 30.04.2015

n - dimensional density function f f x xn( ) ( , , )x 1

Multidimensional Distribution and Density Function

n - dimensional distribution function

P X r X r F f f x x x xn n n n

rr n

( , , ) ( ) ( ) ( , , )1 1 1 1

1

z zz r x x

X r

d d d

1 1( ) d ( , , ) d d 1n

n nf f x x x x

x xMain topic



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EP.28 30.04.2015

( ) d ,n

n n

X XXE X x f x x n

Central moments of n-th order

Central Moments

X E X x f x x

za f ( ) d

Expected value / Average value



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EP.29 30.04.2015

1 1

1 1

( , , ) d dn n

k k k k n n

k k

E q X q x f x x x x

zz q x f x x x xk k n n

k

n

( , , )1 1

1

d d

zzq x f x x x xk k n n

k

n

( , , )1 1

1

d d

Expected value / Average value

1 1

n n

k k k k

k k

E q X q E X



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EP.30 30.04.2015

1 1

1 1

( , , ) d dn n

k k n n

k k

E X x f x x x x

zz x f x f x x xk n n

k

n

( ) ( )1 1

1

d d

Product Distribution of Independent Events

1 1

( ) dn n

k k k k

k k

E X x f x x



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EP.31 30.04.2015

Covariance of two Values

X X X XE X Xs t s ts t d i d i E X X X XX X X Xs t s tt s s t

E X X E X E X EX X X Xs t s tt s s t

E X X E X E Xx x x xs t s tt s s t

X X x xE X Xs t s ts t



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EP.32 30.04.2015

E q X q XX X1 1 2 2

2

1 2

LNM

OQP e j d ie j

q E X q q E X X q E XX X X X1

2

1

2

1 2 1 2 2

2

2

2

1 1 2 22 d i d id i d i

E Y Y q X q XX X

2

1 1 2 21 2

, e j d i

Covariance of a Superposition of two Values I

YY X X X X X Xq q q q 1

2

1 2 2

2

1 1 1 2 2 22 0

positive definit



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EP.33 30.04.2015

y q A q A A t t,

FHG

IKJFHGIKJq q

a a

a a

q

q1 2

11 12

21 22

1

2

b g

FHG

IKJq q

a q a q

a q a q1 2

11 1 12 2

21 1 22 2

b g

a q a q q a q q a q11 1

2

12 1 2 21 1 2 22 2

2 a q a q q a q11 1

2

12 1 2 22 2

22

YY X X X X X Xq q q q 1

2

1 2 2

2

1 1 1 2 2 22 0

YY

X X X X

X X X X

FHG

IKJq q

t 1 1 1 2

2 1 2 2

positive definit

Covariance of a Superposition of two Values II



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EP.34 30.04.2015

A twice steadily differentiable implicit vector

mapping is available.

( , ) f x y 0

The Input have systematic and random errors

according to the super position: N R x x Δx x

Since each random process can be divided

in the kind: Rz

R R R R, with and z zz μ Δz μ E z E Δz 0

Therefore we get the relation:

N R N R( , ) x yf x Δx μ Δx y Δy μ Δy 0

Second order Statistics of Linear Vector Maps I



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EP.35 30.04.2015

Second order Statistics of Linear Vector Maps II

Via Taylor series round the central quantities we get:

0 0 R R R R( , ) ( , ) , withy xf x y J Δx J Δy R Δx Δy 0

0 0 0 0 0 N

0 N

( , ) , ( , ) , and

x y x

y

f fJ x y J x y x x Δx μ

x y

y y Δy μ

It is , so we further get : ( , ) f x y 0

R R R R( , )y xJ Δx J Δy R Δx Δy 0

If exist the Jacobian matrix , we get for the output: 1

yJ

1 1

R R R R( , ) , with yx y yx y xΔy Q Δx J R Δx Δy Q J J



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EP.36 30.04.2015

With the output second order statistics:

t

R R

t1 1

R R R R R R( , ) ( , ) yx y yx y

y y

Q Δx J R Δx Δy Q Δx J R Δx Δy

we get the average value operator after application

of the super position law to:

tt

R R R R

t1

R R R

t 1 t

R R R

1 t 1t

R R R R

( , )

( , )

( , ) ( , )

yx yx

y yx

yx y

y y

y y Q Δx Q Δx

J R Δx Δy Q Δx

Q Δx R Δx Δy J

J R Δx Δy R Δx Δy J

Second order Statistics of Linear Vector Maps III



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EP.37 30.04.2015

Since the residual term has a square convergence behavior,

we can neglect the fourth term opposite the second and

third term.

With the statement and the triangle inequality

of the norm we get the inequality for the criterion of first order:

t t tA B B A

1 t t t 1 t

R R R R

1 t t t 1 t

R R R R

t t

R R

( ) ( )

( ) ( )

y yx yx y

y yx yx y

yx yx

J R Δx Δx Q Q Δx R Δx J

J R Δx Δx Q Q Δx R Δx J

Q Δx Δx Q

Second order Statistics of Linear Vector Maps IV



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EP.38 30.04.2015

If this inequality is true, then we can describe the output

covariance matrix in linear approximation via:

t t t

R R R R yx yxy y Q Δx Δx Q

Because of the super position law of the average value

operator we get the transformation of the input and output

covariance's in linear approximation to:

t t t

R R R R yx yxy y Q Δx Δx Q

Second order Statistics of Linear Vector Maps V



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EP.39 30.04.2015

We can note this compactly via:

t

1

N N N N

t t

R R R R

, with

( , ) ( , ) and

,

y yx x yx

yx x y x y

y x

S Q S Q

f fQ x Δx μ y μ x Δy μ y μ

y x

S y y S Δx Δx

We use the fact that the norm is transposition invariant,

so we can transform the approximation criterion into

the relation:

1 t t t t

R R R R

1( )

2

y yx yx yxJ R Δx Δx Q Q Δx Δx Q

Second order Statistics of Linear Vector Maps VI



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EP.40 30.04.2015

Linear approximation of the vector mapping

permitted and residual term is negligible.

t ,Y yx X yxS Q S Q SZ

F

HGGG

I

KJJJ

Z Z Z Z

Z Z Z Z

m

m m m

1 1 1

1

s E Z Zmn m Z n Zm nZc he j d id io te je j

yxy Q x

Error Propagation of the Uncertainties I

Worst Case Analysis of the systematic errors

Max

m mn ny yxQ x

positive definit

Covariance Analysis of the random errors



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EP.41 30.04.2015

SZ Z

s Z Z

Z Z E Z Z

mn m n

m n m Z n Zm n

c he j b gb g d id io te je j

cov , ,

cov ,

Covariance Matrix

tY yx X yxS Q S Q

Generalized Gauss Error Propagation Law

Error Propagation of the Uncertainties II



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EP.42 30.04.2015

1

= ( , , ) ( , , )

yx y xQ J p x y J p x y

( , )yx xQ J p x explicit Model

implicit Model

Correlation-free input errors (Special case: Gauss error propagation law)

E Y q E Xf

xE Xi Y k k k X

k

ni

k

k X

k

n

i k k

FHGIKJ

d i d i d i22

2

1

22

1

Yi

k

X

k

n

i

f

x

2

2

2

1

FHGIKJ

Error Propagation of the Uncertainties III



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EP.43 30.04.2015

Systematic errors determine the absolute errors

Random errors determine repetition accuracy

This applies to sensors and actuators as well as

machines.

Influence of the Uncertainties



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EP.44 30.04.2015

a) The procedure is consistent, since the output values can be

used with compositions as inputs. Second order statistics

serve thereby for the description the uncertainty of the values.

The density functions can be arbitrary, as long as their

average values and second order statistics exist.

b) For infinitesimal cross correlations of the uncertainties the

generalized error propagation law changes into the Gauss

error propagation law.

Generalized Gauss Error Propagation Law I

The generalized Gauss error propagation law has the

following important characteristics:



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EP.45 30.04.2015

( ) , with ( ) y f u u g x

( )( ) ( ( )) ,f g x f g x

Q J J Jx

f g

u

f

x

g

( )f g

x

f

u

g

x

Composition

Generalized chain rule

Jacobian Matrices

Generalized Gauss Error Propagation Law II



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EP.46 30.04.2015

Correlation Coefficient and Covariance’s I

i j

X X

X X X X

i j

i j

i X j X

i X j X

i j

i i j j

i j

i j

X X E X X

E X E X

LNMOQP

cov ,c h d i e jd i e j2 2

Correlation coefficient

1 1 i j

Characteristics

0 , If between and

no deterministic relation exists.

i j i jx x



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EP.47 30.04.2015

S s rmn mn m nb gc h b gc h

rs

s s

smn

mn

mm nn

mn

m n

Evidence for linear dependences

of strongly correlated

output quantities only delivers

the cross correlation coefficients

if input covariance matrix

is free from cross correlations.

Correlation Coefficient and Covariance’s II

Characteristics



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EP.48 30.04.2015

The complete covariance matrices are not always known

in practice. The empirical average values and the variances

of the input quantities can, however, be determined.

We assume that the errors do not correlate with each other.

If there are, however, correlations exist,

then the calculated variance indicates an upper bound

for the variance appearing really.

We consequently lie at this assumption on the safe side.

Practical Procedure



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EP.49 30.04.2015

Estimated values are not always completely free of

subjective influences.

This is acceptable as long as no additional information is

available or these has to be determined only with

economically not acceptable effort.

Approaches for the Determination of Input Data I

• Repeatable, statistically independent measuring

The empirical average values and covariance’s are used as

estimated values.



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EP.50 30.04.2015

• Values from manufacturer data

Using manufacturer's indications values the nominal values should

be average values and the worst case values should be 3X ranges

of the probability density adopted as normal distributed.

This assumption is allowed since the input values are determined

by a large number of statistically independent effects and

therefore the validity of the central limit theorem is given.

Approaches for the Determination of Input Data II



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EP.51 30.04.2015

• Input values to which only upper and lower limits are

known

If the distributions and central moments are not known, then

upper and lower limits in which the true values are should

defined. Moreover, it is assumed that the input data are

uncorrelated.

Approaches for the Determination of Input Data III



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EP.52 30.04.2015

• Single Values

The case can appear for single values that there are merely

literature values, values of a previous measuring or empirical

values are known (nature constants, material constants etc.).

These values then have to be accepted together with the

possibly existing details on the variance as estimated values.

An insignificant correlation is assumed with other data.

Approaches for the Determination of Input Data IV



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EP.53 30.04.2015

• Covariance’s

Data from different independent experiments being able to be

accepted as uncorrelated approximately.

• Random Disturbances and Series Characteristics

At the characterization of the behavior of a series we assume

that the random disturbances and series characteristic quantities

are free from correlations provided that no information about

possible correlations are known.

Approaches for the Determination of Input Data V



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EP.54 30.04.2015

• Inputs with Small Influence

Insensitive or opposite other very exact input values can be

selected as constant.

Approaches for the Determination of Input Data VI



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EP.57 30.04.2015

B

TCP

B

TCP

cos( ) cos( )

sin( ) sin( )

i i i

i i ii

i

x c

y a

z

G a G

G a G

TCP

TCP

B

0

B

0

0

360

xi

i yi

zi i i i

F

F

F G a F

F

Kinematic Forward Transform of a SCARA



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EP.58 30.04.2015

0

cos( ) cos( )

sin( ) sin( )

0

0

360

x

y

z

x c

y a

z z

G a G

G a G

F

F

F G a F

Kinematic Forward Transform of a SCARA (kompakt)



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EP.59 30.04.2015

cos( ) cos( )

sin( ) sin( )

sin( ) sin( ) 0 0

cos( ) cos( ) 0 0

0 0 1 0

0 0 0 0

0 0 0 0

0 0 0 1

x y z

x y z

x y z

c a

x y z

z

G a GF F F

G a G

G a G

G a G

F F F

G a F

p

x

J

0 0

J

Jacobian Matrices of a SCARA



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EP.60 30.04.2015

cos( ) cos( )

sin( ) sin( )

sin( ) sin( ) 0 0

cos( ) cos( ) 0 0

0 0 1 0

0 0 0 0

0 0 0 0

0 0 0 1

x

y

z

x

y

z c

a

z

G a G

G a GF

F

F

G a G

G a G G

a

F

0 0

-Error Propagation of a SCARA



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EP.61 30.04.2015

cos( ) cos( )

sin( ) sin( )

sin( ) sin( ) 0 0

cos( ) cos( ) 0 0

0 0 1 0

0 0 0 0

0 0 0 0

0 0 0 1

x

y

z

x

y

z c

a

z

G a G

G a G

F

F

F

G a G

GG a G

a

F

0 0

Worst Case Error Propagation of a SCARA I



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EP.62 30.04.2015

-1000-600-20020060010004000-400-8000,00E+001,00E-012,00E-013,00E-01

400

-400

0

-600

-600

0

0,3

0

mm

mm

mm

Worst Case Error Propagation of a SCARA II



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EP.63 30.04.2015

t ,Y yx X yxS Q S Q

sin( ) sin( ) 0 0

cos( ) cos( ) 0 0

0 0 1 0

0 0 0 0

0 0 0 0

0 0 0 1

G a G

G a G

yx xQ J

Covariance Error Propagation of a SCARA

SZ Z

s Z Z

Z Z E Z Z

mn m n

m n m Z n Zm n

c he j b gb g d id io te je j

cov , ,

cov ,

uncertainties, error types and error propagation …...random errors and systematic errors. random...

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