conjugate unscented transformation “o q · 2012. 4. 23. · approximate calculation of multiple...

CONJUGATE UNSCENTED TRANSFORMATION –“OPTIMAL QUADRATURE”

Nagavenkat Adurthi, Puneet Singla, Tarunraj SinghDepartment of Mechanical & Aerospace Engineering

University at Buffalo, Buffalo, NY-14260

SIAM Conference on Uncertainty Quantification

April 2–April 5, 2012

Collaborators: M. Bursik, M. Jones, A. Patra, E. B. Pitman, P. Scott, T. SinghH. Bjornsson, S. Carn, K. Dean, M. Pavolonis, M. Ripepe, P. Webley

INTRODUCTIONPROBLEM STATEMENT

Stochastic System: ẋ = f (Θ ,x)+g(Θ ,x)η(t)Source of Uncertainties: system parameters, initial conditions,input to the system, modeling error.

Robust modeling of the propagation of these uncertainties isimportant to accurately quantify the uncertainty in the solution atany future time.

X3

f(X2 η2 θ)ηi

X

X1

X2f(X0,η0,θ)

f(X1,η1,θ)f(X2,η2,θ)ηi

(X t θ)X0 p(X3,t3,θ)

p(X1,t1,θ)

p(X2,t2,θ)

p(X0,t0,θ)

p( 2, 2, )

Figure: State and pdf transition.

CONJUGATE UNSCENTED TRANSFORMATION LAIRS.ENG.BUFFALO.EDU 2 / 28

http://lairs.eng.buffalo.edu

INTRODUCTIONMOTIVATION

(a) (b)Approximate Solution to exact problem: Kolmogorov Equation,Monte Carlo Methods, generalized Polynomial Chaos (gPC).Exact solution to approximate problem: Gaussian Closure,Equivalent Linearization, and Stochastic Averaging.All these methods involves the evaluation of expectationintegrals:

E[ f (x)] =∫ ∫

· · ·∫

w(x) f (x)dx≈n

∑i=1

wi f (xi)



INTRODUCTIONMOTIVATION

(c) (d)Approximate Solution to exact problem: Kolmogorov Equation,Monte Carlo Methods, generalized Polynomial Chaos (gPC).Exact solution to approximate problem: Gaussian Closure,Equivalent Linearization, and Stochastic Averaging.All these methods involves the evaluation of expectationintegrals:

E[ f (x)] =∫ ∫

· · ·∫

w(x) f (x)dx≈n

∑i=1

wi f (xi)



INTRODUCTIONREVIEW OF QUADRATURE RULES

METHODSMonte Carlo Methods

Gaussian Quadrature

Sparse Grid Quadrature

Unscented Transform

Minimal Cubature rules





Gaussian Quadrature


Unscented Transform


E[ f (x)] =1n

n

∑i=1

f (xi)

Involves the random sampling of the pdfw(x).Error decrease as a factor of 1√n .





Gaussian Quadrature


Unscented Transform


E[ f (x)] =n

∑i=1

wi f (xi)

The weights wi and points xi aredeterministic.

1-D integrals: m quadrature points arerequired to reproduce the expectationintegrals involving 2m−1 degreepolynomial functions.

N-D integrals: Tensor product leads toexponential growth of points (mN).

Nested quadratures: Clenshaw-Curtis.





Gaussian Quadrature


Unscented Transform


E[ f (x)] =n

∑i=1

wi f (xi)


The curse of dimensionality involved inGaussian Quadratures is avoided byconsidering a sparse tensor product of1-Dimensional quadrature rules.

Example: Smolyak Quadratures.

Some weights can be negative.





Gaussian Quadrature


Unscented Transform


E[ f (x)] =n

∑i=1

wi f (xi)

The weights wi and points xi aredeterministic.1

2N +1 points are required to integrateall polynomials up to degree 3.

GH2

SmolyakGH

UT23

45

6

0

10

20

30

40

50

60

70

Method

Dimension

Nu

mb

er o

f p

oin

ts

Figure: Comparison of 3rd degree rules

1S.J. Julier, J.K. Uhlmann, and H.F. Durrant-Whyte. “A New Method for the Nonlinear Transformation of Means andCovariances in Filters and Estimators”. In: IEEE Transactions on Automatic Control 45.3 (2000), pages 477–482.





Gaussian Quadrature


Unscented Transform


E[ f (x)] =n

∑i=1

wi f (xi)


Minimum cubature rules try to evaluatethe expectation integrals with as fewpoints as possible.

A extensive work was carried out byStroud1, where a number of minimalcubature rules of various order anddimension are documented.

1A. H. Stroud. Approximate Calculation of Multiple Integrals. Englewood Cliffs, NJ: Prentice Hall, 1971.



INTRODUCTIONREVIEW OF FEW POPULAR METHODS

TABLE: Minimal Cubature Rules for Gaussian Integralsa

Degree Points Valid for All positive weights3 2N all dimensions Yes3 2N all dimensions Yes5 N2 +N +2 2≤ N ≤ 7b Yes5 2N +2N all dimensions Yes5 2N2 +1 all dimensions Only for N ≤ 37 2N +2N2 +1 N = 3,4,6,7c Only for N 6= 77 (4N3 +8N +3)/3 N = 3,4,5,6 No9 (2N4−4N3 +22N2−8N +3)/3 N = 3,4,5,6 No

N→ is the dimension of the integralaA. H. Stroud. Approximate Calculation of Multiple Integrals. Englewood Cliffs, NJ: Prentice Hall, 1971.bNo solution exists for other dimensionscNo solution/Imaginary solution for other dimensions



INTRODUCTIONOBJECTIVE

CUBATURE RULE: E[ f (x)] =∫ ∫· · ·∫

w(x) f (x)dx≈ ∑ni=1 wi f (xi)Most of the non-product cubature rule possess certainsimilarities:

exploit the symmetry of the pdf, assume a structure for thecubature points, solve a system of nonlinear equations.

Drawbacks: inconsistency of the set of nonlinear equations, thepresence of negative weights/complex roots and the inability toprovide a generalized solution that works for any dimension.



INTRODUCTIONOBJECTIVE

CUBATURE RULE: E[ f (x)] =∫ ∫· · ·∫

w(x) f (x)dx≈ ∑ni=1 wi f (xi)Most of the non-product cubature rule possess certainsimilarities:

exploit the symmetry of the pdf, assume a structure for thecubature points, solve a system of nonlinear equations.

Drawbacks: inconsistency of the set of nonlinear equations, thepresence of negative weights/complex roots and the inability toprovide a generalized solution that works for any dimension.

PRIMARY OBJECTIVETo find a fully symmetric reduced sigma/cubature point set withall positive weights that sum up to one and that is equivalent tothe Gaussian quadrature product rule of same order.



THE UNSCENTED TRANSFORMATION

3rd order rule with 2N +1 points.

Can integrate all odd degreemonomials due to symmetry of thepoints chosen.

This rule cannot be extended tohigher degree since pointsconstrained to lie on principal axescannot capture any cross moment.

The method provides the insight: ‘Tochoose additional axes that are ableto capture higher order moments’

UNSCENTED TRANSFORM 2

X0 = (0, · · · ,0) W0 = κ/(N +κ)

Xi =√(N +κ)Ii Wi = 1/[2(N +κ)]

Xi+N =−√(N +κ)Ii Wi+N = 1/[2(N +κ)]

Figure: Selection of points in 2D

2Julier, Uhlmann, and Durrant-Whyte, “A New Method for the Nonlinear Transformation of Means and Covariancesin Filters and Estimators”.



CONJUGATE UNSCENTED TRANSFORMATIONDEFINITIONS

MOMENT CONSTRAINT EQUATIONS(MCEs):

n

∑i=1

wi{x(i,1)n1x(i,2)n2 · · ·x(i,N)nN}= E[xn11 xn22 · · ·x

nNN ]

A FULLY SYMMETRIC SETA set of points is called fully symmetric if it is closed under allcoordinate and sign permutations.

For example consider the set X = {x1,x2, · · ·} to be a fullysymmetric set and if xi = [a,b]T ∈ X, then{[b,a]T , [−a,b]T , [a,−b]T ,[−a,−b]T , [−b,a]T , [b,−a]T , [−b,−a]T} ∈ X.




σ : Represents the Principal axes (or orthogonal axis) in the cartesiancoordinate space. Each point on the principal axis is denoted as σi

TABLE: Fully Symmetric set of pointsType Sample Point No. of points

σ (1,0,0, · · · ,0) 2N

cM (1,1, · · · ,1︸︷︷︸M

,0,0, · · · ,0︸︷︷︸N−M

) 2M(

NM

)sN (h) (h,1,1, · · · ,1) N2N




cM: Represents the Mth Conjugate axes (fully symmetric set) and thecorresponding points are enumerated as cMi


σ (1,0,0, · · · ,0) 2N

cM (1,1, · · · ,1︸︷︷︸M

,0,0, · · · ,0︸︷︷︸N−M

) 2M(

NM

)sN (h) (h,1,1, · · · ,1) N2N




sN(h): Represents the Scaled Conjugate axes and the points aredenoted as sNi (h). The parameter h is a scaling factor.


σ (1,0,0, · · · ,0) 2N

cM (1,1, · · · ,1︸︷︷︸M

,0,0, · · · ,0︸︷︷︸N−M

) 2M(

NM

)sN (h) (h,1,1, · · · ,1) N2N



CONJUGATE UNSCENTED TRANSFORMATIONGRAPHICAL VISUALIZATION OF THE POINTS/AXES


3Dpoints1.aviMedia File (video/avi)

2Dpoints.aviMedia File (video/avi)


CONJUGATE UNSCENTED TRANSFORMATIONCUBATURE POINTS OF 2nd /3rd ORDER

GAUSSIAN PDFThe moments up to 2nd order are:

n

∑i=1

wi = 1, E[x2i ] = 1

Select a set of points on σ at a distance of r1 andweight w1 .

MCEs are:

2Nw1 = 1, 2r21w1 = 1

The solution is:

w1 =1

2N, r1 =

√N

Choosing r1 = N +κ gives the UT

UNIFORM PDFThe moments up to 2nd order are:

n

∑i=1

wi = 1, E[x2i ] = 1/3

Select a set of points on σ at a distance of r1 andweight w1 .

The solution is:

w1 =1

2N, r1 =

√N3

Above dimension 3, the points go out of the supportΩ = [−1,1]Alternatively, choose cN instead of σ for which thesolution of MCEs are:

w1 =1

2N, r1 =

1√3

2A. Nagavenkat, P. Singla, and T. Singh. “The Conjugate Unscented Transform-An Approach to EvaluateMulti-Dimensional Expectation Integrals”. In: American Control Conference, Montreal, Canada (2012).



CONJUGATE UNSCENTED TRANSFORMATIONCUBATURE POINTS FOR 4th /5th AND 6th /7th DEGREE

TABLE: CUT4Position Weights

1≤ i≤ 2N Xi = r1σi Wi = w11≤ i≤ 2N Xi+2N = r2cNi Wi+2N = w2

Central weight X0 = 0 W0 = w0n = 2N +2N (+1)

GH3

SmolyakGH

CUT42

3

4

5

6

0

100

200

300

400

500

600

700

800

Method

Dimension

Num

ber o

f poi

nts

TABLE: CUT6, (N ≤ 6)Position Weights


1≤ i≤ 2N(N−1) Xi+2N+2N = r3c2i Wi+2N+2N = w3

Central weight X0 = 0 W0 = w0n = 2N2 +2N +1

GH4

SmolyakGH

CUT62

3

4

5

6

0

500

1000

1500

2000

2500

3000

3500

4000

4500

Method

Dimension

Num

ber o

f poi

nts

2Nagavenkat, Singla, and Singh, “The Conjugate Unscented Transform-An Approach to Evaluate Multi-DimensionalExpectation Integrals”.



CONJUGATE UNSCENTED TRANSFORMATIONCUBATURE POINTS FOR 8th /9th DEGREE

TABLE: Sigma Points for CUT8, (2≤ N ≤ 6)Position Weights


1≤ i≤ 2N(N−1) Xi+2N+2N = r3c2i Wi+2N+2N = w3

1≤ i≤ 2N Xi+2N+2N+2N(N−1) = r4cNi Wi+2N+2N+2N(N−1) = w4

1≤ i≤ N1 Xi+2N+2N+2N(N−1)+2N = r5c3i Wi+2N+2N+2N(N−1)+2N = w5

1≤ i≤ N2N Xi+2N+2N+2N(N−1)+2N+N1= r6sNi Wi+2N+2N+2N(N−1)+2N+N1

= w6

Central weight X0 = 0 W0 = w0n = 2N +2N +2N(N−1)+N1 +2N +N2N +1 , {N1 = 4N(N−1)(N−2)/3}

GH5SmolyakGH

CUT823

45

6

0

2000

4000

6000

8000

10000

12000

14000

16000

Method

Dimension

Nu

mb

er o

f p

oin

ts



CONJUGATE UNSCENTED TRANSFORMATIONCOMPARISON OF CUBATURE RULE

MINIMAL RULES FOR GAUSSIAN INTEGRALSDegree Points Valid for All positive weights

3 2N all dimensions Yes3 2N all dimensions Yes5 N2 +N +2 2≤ N ≤ 7a Yes5 2N +2N all dimensions Yes5 2N2 +1 all dimensions Only for N ≤ 37 2N +2N2 +1 N = 3,4,6,7b Only for N 6= 77 (4N3 +8N +3)/3 N = 3,4,5,6 No9 (2N4−4N3 +22N2−8N +3)/3 N = 3,4,5,6 No

aNo solution exists for other dimensionsbNo solution/Imaginary solution for other dimensions

THE CONJUGATE UNSCENTED TRANSFORMDegree Points Valid for All positive weights

5 2N +2N all dimensions Yes7 2N2 +2N +1 N ≤ 6 Yes7 2N +2N +N1 +1 7≤ N ≤ 9 Yes7 2N +2N +N1 +1 10≤ N ≤ 13 No9 2N2 +2N +2N(N−1)+N1 +2N +N2N +1 N ≤ 10 Yes

N1 = 4N(N−1)(N−2)/3



NUMERICAL EXPERIMENTSEXAMPLE: E[(

√1+xT x)α ] =

∫(√

1+xT x)α N (x : 0,0.1I)dx

100 101 102 103 104 105 1060

0.25

0.5

0.75

1

1.25

1.5

No. of points (log−scale)

% re

lativ

e er

ror

2D3D4D5D6D

(a) Gauss-Hermite Convergence

100 101 102 1030

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

No. of points (log−scale)

% re

lativ

e er

ror

2D3D4D5D6D

(b) CUT Convergence

GH4

CUT823

45

6

0

500

1000

1500

2000

2500

3000

3500

4000

4500

MethodDimension

Num

ber o

f poi

nts

(c) Number of points required to achieve 0.5% Rel. error



NUMERICAL EXPERIMENTSRE-ENTRY VEHICLE DYNAMICS

RE-ENTRY VEHICLEDYNAMICS 3

ẋ1(t) = x3(t)

ẋ2(t) = x4(t)

ẋ3(t) = D(t)x3(t)+G(t)x1(t)+ν1(t)ẋ4(t) = D(t)x4(t)+G(t)x2(t)+ν2(t)ẋ5(t) = ν3(t)

D(t) =−β (t)exp{ [R0−R(t)]H0

}V (t)

G(t) =− Gm0R3(t)

β (t) = β0exp(x5(t))

R(t) =√

x21(t)+ x22(t)

V (t) =√

x23(t)+ x24(t)

µ0 =

6500.4349.14−1.8093−6.79670.6932

,P0 =

0.1 0 0 0 00 0.1 0 0 00 0 0.1 0 00 0 0 0.1 00 0 0 0 1

6340 6360 6380 6400 6420 6440 6460 6480 6500 65200

50

100

150

200

250

300

350

400

450

xy

Earth

Vehicle Trajectory

Figure: Typical Trajectory

3Julier, Uhlmann, and Durrant-Whyte, “A New Method for the Nonlinear Transformation of Means and Covariancesin Filters and Estimators”.




0 5 10 15

x 104

0

0.5

1

1.5

2

2.5x 10

−3

number of samples

supr

emum

of %

rel

ativ

e er

ror

in 2

−no

rm o

f mea

ns

Figure: Convergence of MC Mean

0 5 10 15

x 104

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

number of samples

supr

emum

of %

rel

ativ

e er

ror

in fr

oben

ius−

norm

of C

OV

mat

rix

Figure: Convergence of MC Cov

0 20 40 60 80 100 120 140 160 180 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10

−3

t(s)

% e

rror

in m

ean

of x

1

ckfutcut4cut6cut8gh

Figure: % error in the mean of x1

0 20 40 60 80 100 120 140 160 180 2000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t(s)

% e

rror

in m

ean

of x

3

ckfutcut4cut6cut8gh

Figure: % error in the mean of x3




0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

t(s)

% e

rror

in v

aria

nce

of x

1

ckfutcut4cut6cut8gh

Figure: % error in the var of x1

0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

t(s)

% e

rror

in v

aria

nce

of x

3

ckfutcut4cut6cut8gh

Figure: % error in the var of x3

TABLE: % Maximum Rel. error in Frobenius norms of Cov matrix

method CKF UT CUT4 CUT6 CUT8 GH5 MCNo. of points 10 11 42 83 355 3125 100000

% error 11.5306 17.6092 1.9398 1.0396 0.9391 0.9369 0.2019



NUMERICAL EXPERIMENTSICELAND VOLCANO (EYJAFJALLAJÖKULL) ERUPTION

The BENT integral eruption column model was used to produceeruption column parameters (mass loading, column height, grainsize distribution) given a specific atmospheric sounding andsource conditions.

BENT takes into consideration atmospheric (wind) conditions asgiven by atmospheric sounding data.Plume rise height is given as a function of volcanic source andenvironmental conditions.

The PUFF Lagrangian model was used to propagate ash parcelsin a given wind field (NCEP Reanalysis).

PUFF takes into account dry deposition as well as dispersion andadvection.

Polynomial chaos quadrature (PCQ) was used to select samplepoints and weights in the uncertain input space of vent radius,vent velocity, mean particle size and particle size variance.




TABLE: Eruption source parameters based on observations of Eyjafjallajökull volcano andinformation from other similar eruptions.

Parameter Value range PDF Comment

Vent radius, b0 , m 65-150 Uniform Measured from radar image of summit ventsVent velocity, w0 ,m/s

Range: 45-124 Uniform M. Ripepe, Geneva, Switzerland, 2010, pre-sentation

Mean grain size,Mdϕ

2 boxcars: 1.5-2and 3-5

Multi-Modal Uni-form

Woods and Bursik (1991), Table 1, vulcanianand phreatoplinian. A. Hoskuldsson, Icelandmeeting 2010, presentation

σϕ 1.9±0.6 Uniform Woods and Bursik (1991), Table 1, vulcanianand phreatoplinian.




105 106 1070

10

20

30

40

50

60

70

80

90

100

Number of Particles

% C

hang

e in

Con

cent

ratio

n

Height = 0 2kmHeight = 2 4kmHeight = 4 6km

Figure: Concentration (52N 13.5E) vs. Number of PUFF Particles




nAsh 105 5×105 106 2×1064×106

8×106 107

Conc. 7.40×10−5 1.17×10−4 1.07×10−4 1.12×10−41.09×10−4

1.15×10−4 1.10×10−4



MEAN ASH TOP HEIGHTICELAND VOLCANO (EYJAFJALLAJÖKULL) ERUPTION

Figure: 94 Clenshaw Curtis Runs Figure: 161 CUT Runs


meanmaxheightPCQ.aviMedia File (video/avi)

meanmaxheightCUT.aviMedia File (video/avi)


STANDARD DEVIATION OF ASH TOP HEIGHTICELAND VOLCANO (EYJAFJALLAJÖKULL) ERUPTION

Figure: 94 Clenshaw Curtis Runs Figure: 161 CUT Runs


stdmaxheightPCQ.aviMedia File (video/avi)

stdmaxheightCUT.aviMedia File (video/avi)


COMPARING CUT WITH CLENSHAW CURTISICELAND VOLCANO (EYJAFJALLAJÖKULL) ERUPTION

Figure: Mean Figure: Standard Deviation


meanmaxheightcutPCQ.aviMedia File (video/avi)

stdmaxheightCUTpcq.aviMedia File (video/avi)


PROBABILITY OF ASH TOP HEIGHTICELAND VOLCANO (EYJAFJALLAJÖKULL) ERUPTION

(A) 0041-04-16 00:00:00Z

Model: probability, outer contour 0.2, inner 0.7

Data: ash top height, m

Figure: Pr(Top Ash Height) ≥ 0km

(A) 0041-04-16 00:00:00Z




(A) 0041-04-16 00:00:00Z




(A) 0041-04-16 00:00:00Z





probabilityheightlevel1.aviMedia File (video/avi)





PROBABILITY OF ASH TOP HEIGHTICELAND VOLCANO (EYJAFJALLAJÖKULL) ERUPTION

(A) 0041-04-16 00:00:00Z



(a) 00 hrs

(B) 0041-04-16 06:00:00Z



(b) 06 hrs

(C) 0041-04-16 00:00:00Z



(c) 12 hrs

(D) 0041-04-16 18:00:00Z



(d) 18 hrs



CONCLUDING REMARKSWHAT NEEDS TO BE DONE

Efficient quadrature scheme is developed for the accuratedetermination of high dimension expectation integrals involvingsymmetric pdfs.

non-product cubature rule.generalization for any pdf.

Numerical examples illustrates the effectiveness of the newquadrature scheme.

We were able to predict ash footprint with good confidence eventhough source uncertainty is large.Convergence in concentration value is slow.Any other models for ash dispersion can be included in theuncertainty analysis.Effect of uncertainty in wind data still needs to be analyzed.

Question arise: Can we estimate the distribution of sourceparameters using Satellite imagery or other sensory data?

Likelihood functions: validating the sensor data.uniqueness of the parameters.



ACKNOWLEDGEMENT

This work is jointly supported through National Science Foundationunder Awards No. CMMI- 1054759, CMMI-1131074 and AFOSRFA9550-11-1-0336.



AIR TRAFFIC SCENARIO

PROCESS DYNAMICS AND SENSORMODEL4

xk =

1 sin(ΩT )Ω 0 −

1−cos(ΩT )Ω 0

0 cos(ΩT ) 0 −sin(ΩT ) 00 1−cos(ΩT )Ω 1

sin(ΩT )Ω 0

0 sin(ΩT ) 0 cos(ΩT ) 00 0 0 0 1

xk−1 +νk−1[

rkθk

]=

[√(ξk)2 +(ηk)2

tan−1(ηkξk

)

]+ωk

Where xk = [ξk, ξ̇k, ηk, η̇k, Ωk]T is the state vector,ξk and ηk are the x and y coordinates respectively.The process and measurement noise are modelled as

νk−1 ∼N (νk−1 : 0,Qk−1)ωk ∼N (ωk : 0,Rk)

Qk−1 = L1

T 33

T 22 0 0 0

T 22 T 0 0 0

0 0 T3

3T 22 0

0 0 T2

2 T 00 0 0 0 L2L1

T

Rk =

[σ2r 00 σ2θ

]

−6000 −4000 −2000 0 2000 4000 6000 8000−12000

−10000

−8000

−6000

−4000

−2000

0

2000

xy

Airplane TrajectoryRadar Location

Position Measurment

Figure: Typical Trajectory

4I. Arasaratnam and S. Haykin. “Cubature Kalman Filter”. In: IEEE transactions on Automatic Control 54.6 (2009),pages 1254 –1269.



SIMULATION PARAMETERS

FILTERS USEDCKF- with 10 points UKF - with 11 pointsCUT6- with 83 points Particle Filter (PF)- with 10000 samples

The initial condition uncertainty is

x0 =

1000 m300 m/s1000 m0 m/s

− 3π180 rad/s

, P0/0 =

100 m2 0 0 0 00 10 m2/s2 0 0 00 0 100 m2 0 00 0 0 10 m2/s2 00 0 0 0 100×10−6rad2/s2



% 2-NORM RMSEpos VS FREQ OF MEASUREMENT

1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14x 10

4

Time gap between meas updates

2−no

rm o

f the

RM

SE

err

or in

pos

Bearing std dev 0.18119degrees

ckfukfcut6PF−ModePF−Mean

(a) Bearing std dev σθ = 0.1811o1 2 3 4 5 6 7 8 9 10

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

5


2−no

rm o

f the

RM

SE

err

or in

pos



(b) Bearing std dev σθ = 0.5435o

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10x 10

5


2−no

rm o

f the

RM

SE

err

or in

pos



(c) Bearing std dev σθ = 0.90593o1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7x 10

6


2−no

rm o

f the

RM

SE

err

or in

pos



(d) Bearing std dev σθ = 1.993o



% 2-NORM RMSEpos VS FREQ OF MEASUREMENT

1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5x 10

6


2−no

rm o

f the

RM

SE

err

or in

pos



(e) Bearing std dev σθ = 1.2683o1 2 3 4 5 6 7 8 9 10

0

0.5

1

1.5

2

2.5

3

3.5

4x 10

6


2−no

rm o

f the

RM

SE

err

or in

pos



(f) Bearing std dev σθ 1.6307o

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9x 10

6


2−no

rm o

f the

RM

SE

err

or in

pos



(g) Bearing std dev σθ = 2.3554o1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12x 10

6


2−no

rm o

f the

RM

SE

err

or in

pos



(h) Bearing std dev σθ = 2.7178o



POLAR TO CARTESIAN

POLAR TO CARTESIAN

E[

xy

]=∫ ∫ [

rcos(θ)rsin(θ)

]N (

[rθ

]:[

µrµθ

],

[σ2r 00 σ2θ

])drdθ

−0.3 −0.2 −0.1 0 0.1 0.2 0.30.9

0.92

0.94

0.96

0.98

1

1.02

1.04

x

y

MCUTCKFGH3GH4CUT4CUT6

µr = 1m µθ = 90 degσr = 0.022m2 σθ = 152 deg2

34 36 38 40 42 44 46 48 50 52 54−25

−20

−15

−10

−5

0

5

10

15

20

25

x

y

MCUTCKFGH3GH4CUT4CUT6

µr = 50m µθ = 0 degσr = 0.022m2 σθ = 302 deg2



introductionConcluding RemarksAppendix

conjugate unscented transformation “o q · 2012. 4. 23. · approximate calculation of multiple...

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