Lecture 19 - UKF[pdf] - CCAR

Statistical Orbit Determination 2 

Monday, April 9, 2012 

Lecture 19 

April 9, 2012 

ASEN 6080, Spring 2012


Unscented Kalman Filter 

(UKF) 


• 

• 

• 

Limitations of Linear Filters 

Restricted to Gaussian noise distributions (i.e. distribution 

completely described by first two moments) 

• 

• 

• 


EKF just linearizes non-linear models 

Unstable filter outside of linearity region 

Requires computation of Jacobians 

Linear Kalman Filter propagates mean & uncertainty linearly 

1st-order approximation → large errors when system is nonlinear 



Graphical Comparison of Linear & Non-Linear Sigma- 

Point Propagations 

Eric A. Wan and Rudolph van der Merwe, “The Unscented Kalman Filter for Nonlinear Estimation.” 

How can we better 

model statistics of nonlinear 

systems? 

One option: 

Unscented Kalman Filter 


UKF 

• 


Basic principal: a set of discretely 

sampled points can be used to 

parameterize the mean & 

covariance of a system 

• 

Propagate a (weighted) random 

sample of points & recompute 

mean & uncertainty (Unscented 

Transform) 

Julier & Uhlmann, 

Oxford, 1997 

(paper online) 

Mean & 

uncertainty 

captured precisely 

to 2 nd order 

Eric A. Wan and Rudolph van der Merwe, “The Unscented Kalman Filter for Nonlinear Estimation.” 


Weights capture distribution statistics. 

We’ll talk more about this next week. 

HW 8 weights selected for Gaussian 

distribution. 

L = Number of States 

(2) Initialize the UKF given: 

P ˆ , 

t! 

1 , X t! 

1 

Unscented Kalman Filter (2/4) 

R 

t 

W 

(1) Compute weights: 

m 

i 

W0 c 

W m 

0 

Inertial Measurements for Aerobraking Navigation (IMAN) 

( ! ) 

= ! / L + 

2 

( + # ) + ( 1 $ " ! ) 

= # / L 

+ 

{ 2( 

L + ) } i 1,........, 

2L 

c 

= W = 1 / ! = 

i 

( L + ) ! L 

2 

$ = # " 

\hat{X} denotes estimated state, not state deviation 

at t = 0, \hat{X} = a priori X 

where: 

1 e ! 4< 

" < 1 

Doctoral Research Defense May 27, 2005 

MKJ-5 


! = 3 - L 

Courtesy of 6 Moriba Jah 

(3) Compute Sigma Points (matrix) for t-1 

(4) Propagate Sigma Points through non-linear system 

note: this implies integrating 2L+1 state vectors at each time step (i.e. each column) 

( " ) 

" t F u 

/ ! 1 

! 1 , ! 1 

= t 

t t 

( ! ) 

" = L + 

! = 

# 

t! 

1 

2 

= 

“tuneable” parameters: 

λ is a scaling parameter 

α determines point spread about mean 

β incorporates a priori distribution knowledge 

[ X ˆ ˆ 

ˆ 

t! 

1 X t! 

1 + " Pt 

! 1 X t! 

1 ! " Pt 

! 1] 

L x ( 2L+ 

1) 

sqrtm(P) 

repmat(X,1,6) 

to save time, don’t loop through integrator, 

integrate L(2L+1) state vector 

and 

L = 6 

χ = (6x13)



(5) Perform a state and covariance time update 

X 

t 

= 

2L 

! 

i= 

0 


MKJ-6 


prediction of state & 

uncertainty includes 

process noise 

if system has no process 

noise, set Q = 0 

W 

m 

i 

# 

i, 

t / t" 

1 

( )( ) T 

2L 

c 

" Wi 

# i, 

t / t! 

1 ! X t # i, 

t / t 1 X t 

P t = Q + 

! ! 

i= 

0 

where: Q is the process noise covariance 

(6) Re-compute Sigma Points to incorporate effects of process noise 

# 

= 

[ X t X t + " Pt 

X t ! " Pt 

] L x ( 2L 

1) 

t / t! 

1 

+ 

(7) Compute the measurements associated with each Sigma Point vector and their weighted average 

Y " 

y 

/ ! 1 ( / ! 1 = t t G t t 

t 

= 

2L 

! 

i= 

0 

W 

m 

i 

Y 


) 

i, 

t / t" 

1 

\bar{X} = mean of weighted sigma points 

\bar{P} = covariance of weighted sigma points 

and



(8) Compute the innovation and cross-correlation covariances 

( )( ) T 

2L 

c 

" Wi 

Yi, 

t / t! 

1 ! yt 

Yi, 

t / t 1 yt 

P yy = R + 

! ! 

i= 

0 

( )( ) T 

2L 

c 

" Wi 

# i, 

t / t! 

1 ! X t Yi, 

t / t 1 yt 

P xy = ! ! 

i= 

0 

(9) Compute the Kalman Gain, the best estimate of the state and the a posteriori state covariance 

t 

K 

t 

= PxyP 

yy 

t 


MKJ-7 


where: yt 

Measurement Update 

! 1 

( y y ) 

X ˆ = X + K ! 

P t = Pt 

! 

(10) Replace t with t-1 and return to (3) until all the observations have been processed. 


t 

K 

t 

t 

P 

yy 

K 

t 

T 

t 

are the observations 

and

• 

• 

• 

• 

• 

Important Points 

UKF ~ combination of a Monte Carlo and an EKF 

Suitable for any dynamic model 

No Gaussian noise requirements (UKF can be tuned) 

No Jacobians required 

Performance equivalent to linear Kalman filter for linear 

systems 

• 2nd order accuracy for all non-linear systems 

• 3rd order accuracy for Gaussian distributions 




Scaled Unscented Transform 


• 

• 

• 

UKF Review 

UKF is a non-linear sequential filter 


Uses Sigma-Points to propagate mean & covariance more 

accurately 

Inertial Measurements for Aerobraking Navigat 

• 


W 

(1) Compute weights: 

Accurate to second-order 

What are α, β? 

m 

i 

W0 c 

W m 

0 

( ! ) 

= ! / L + 

( ) ( 2 

+ # + 1 $ " ! ) 

= # / L 

+ 

{ 2( 

L + ) } i 1,........, 

2L 

c 

= W = 1 / ! = 

i 

( L + ) ! L 

$ = # " 

where: 

2 1 e ! 4< 

" < 1 

! = 3 - L 

( ! ) 

" = L + 

! = 

2 


t − 1) = 

• 

• 


� 

ˆXt−1 ˆ Xt−1 + γ � Pt−1 ˆ Xt−1 − γ � � � 

1 

γ = Pt−1 

UKF Weights - Unscaled 

Sigma-points represent spread about 

mean: 

� 

1 

Weighting γ = requirement: 2L+1 

• 

• 

Sigma-points are not necessarily symmetrically 

distributed about mean 

• 

W 

Positive semi-definite P not guaranteed 

m i = W c W 

1 

>> 1 

i = 

m 0 = 

• 

2W m i 

� 

W 

i=0 

′ i =1 

W ′ i 1 

� 

Lx(2L+1) 

χ ′ i = ˆ Xi + 

� 

1 

2W ′ i 

Pi 

χi = χ0 + α(χi − χ0) 

W m′ 

Solution: Scale sigma-point distribution 

Wi = W ′ i 

� 

Lx(2L 


Scaled UT 

Julier & Uhlmann created a framework for controlling 

Sigma-point distance and preserving P.S.D. of covariance 


2002 paper: Scaled Unscented Transform with α 

Preserves second-order χ accuracy 

′ i = ˆ � 

1 

Xi + 

• 

Unscaled UT 

χ ′ i = ˆ Xi + 

� 

1 

2W ′ i 

Pi 

χi = χ0 + 

� 1 

Pi 

χi 2Wi = χ0 + 

Scaled UT 

� 1 

χi = χ0 + 

2Wi 

2W ′ i 

� 1 

χi = χ0 + α(χi − χ0) (10) 

Pi 

Pi 

χi = ˆ Xi + α(χi − ˆ Xi) 

W m 0 = 

m′ W0 (9) 

2Wi 

Pi 

ASEN 6080, Spring 2012 

1 

+ (1 − ) 

α2 α2

Scaled Weights α2 2L+1 � 

2L+1 � 

i=0 

� 

2L+1 

i=0 

• 


χ ′ i = ˆ Xi + 

• 

University of Colorado, Boulder, CO, 80309 

Wi = W ′ i 

Determine new weights based on constraints: 

2L+1 � 

W ′ Wi =1 i =1 

(14) 

χi = χ0 + α(χi − χ0) 

i=0 

(12) 

W c 0 = W m 0 + (1 − α 2 + β) (13) 

W 

2L+1 � 

Wi =1 

′ i (χ ′ i − ¯ X)(χ ′ i − ¯ � 

X)=Pxx 1 

χi = χ0 + α(χi − χ0) 

χ 

(15) 

′ i = ˆ � 

1 

Xi + 

2W ′ Pi 

i 

� 

2L+1 

i=0 

2L+1 

i=0 

W c 0 = W m 0 + (1 − α 2 Wi =1 + β) (14) 

� 

2L+1 

i=0 

Wi(χi − ¯ X)(χi − ¯ X)=Pxx 

γ = √ L + λ 

W m i = W c W 

1 

i = 

2(L + λ) 

′ i (χ ′ i − ¯ X)(χ ′ i − ¯ X)=Pxx 

(15) 

2L+1 � 

Wi(χi − ¯ X)(χi − ¯ Wi = 

X)=Pxx 

W ′ i 

α2 W 

m′ W0 1 

+ (1 − ) 

2 2 m m′ W0 1 

0 = + (1 − ) 

α2 α2 � 

Wi(χi − ¯ X)(χi − ¯ α X)=Pxx 

2 

i=0 

χi = χ0 + 

i=0 

Results in scaled weights to describe sigma-point 

distribution: 

� 

1 

2W ′ i 

χ ′ χi = 

i = χ0 + 

ˆ Xi + α(χi − ˆ Xi) (1 

W m 0 = 

2W ′ i 

Pi 

2L+1 � 

W 

i=0 

′ i (χ ′ i − ¯ X)(χ ′ i − ¯ W 

X)=Pxx (16) 

m 0 = W m 0 ′ 1 

χi = χ0 + α(χi − χ0) 

m′ 

+ (1 − ) 

α2 α2 W0 1 

+ (1 − ) (1 

α2 α2 W m 0 = 

� 

α 

Wi = W ′ i 

1 

2W ′ i 

Pi 

α 

W = W ′ i 

Pi 

(16) 


( 

(1

• 

Pxx,t = 

= 

2L+1 �i=0 

i=0 

i=0 

Scaled UT 

With these scaled weights, the Scaled unscented transform is 

given by: 

χt = f(χt−1) 

2L+1 � 

χt 

ˆXt = 

f(χt−1) 

Covariance 2L+1 � 


2L+1 � 

i=0 

• 

ˆXt = 

ˆXt = 

χt = f(χt−1) 

i=0 

2L+1 

� 

2L+1 � 

i=0 

Wiχt,i 

Wiχt,i 

Non-linear propagation of 

Sigma-points 

Wi(χt,i − i=0 

ˆ Xt)(χt,i − ˆ Xt) T + (1 − α 2 )(χt,0 − ˆ Xt)(χt,0 − ˆ Xt) T 

Wi(χt,i − ˆ Xt)(χt,i − ˆ Xt) T + (1 − α 2 ) ∗ (χt,0 − ˆ Xt)(χt,0 − 

Wi(χt,i − ˆ Xt)(χt,i − ˆ Xt) T + (1 − α 2 ) ∗ (χt,0 − ˆ Xt)(χt,0 − 

2 

Wiχt,i 

Mean 

1e-4 ≤ α ≤ 1 general guidelines for scaling to 

ensure P.S.D. covariance 2 & maintain 2nd-order 

accuracy 


• 

Scaled Xt UT: f(Xt−1) = f(Xt−1) β ≈ f( ( 

¯ Xt−1)+▽fδX + 1 

2 ▽2fδX 2 + 1 

3! ▽3 f(Xt−1) ≈ f(Xt−1)+▽fδX + ▽ fδX + ▽ fδX + ... 

2 3! 

¯Xt = E[Xt] =f( fδX 

¯ Xt−1)+ 1 

So that explains α, but what about β? 

Xt = f(Xt−1) 

Xt = f(Xt−1) ( 

−1) ≈ f( • Approximate to get propagated state: 

¯ Xt−1)+▽fδX + 1 

2 ▽2fδX 2 + 1 

3! ▽3fδX 3 ▽f = + ... ( 

∂f(Xt−1) 

▽f = ∂f(Xt−1) 

Pxx,t = E[(Xt − 

∂Xt−1 

¯ Xt)(Xt − ¯ Xt) T ] 

f(Xt−1) ≈ f( ¯ Xt−1)+▽fδX + 1 

≈ f( ¯ Xt−1)+▽fδX + 1 

2 ▽2fδX 2 + 1 

3! ▽3fδX 3 + ... ( 

▽f = ∂f(Xt−1) 

δX = Xt−1 − ( 

∂Xt−1 

¯ δX = Xt−1 − 

Xt−1 

¯ Xt−1 

• 


▽f = ∂f(Xt−1) 

▽f = 

∂Xt−1 

∂f(Xt−1) 

mean: 

∂Xt−1 

¯Xt = E[Xt] =f( ¯ Xt−1)+ 1 

2 ▽2fPxx,t−1 + 1 

3! ▽3fE[δX 3 ]+ 

∂Xt−1 

2 ▽2fδX 2 + 1 

3! ▽3fδX 3 + ... 

Take the expected value of the propagated state to get 

2 ▽2fPxx,t−1 + 1 

3! ▽3fE[δX 3 ]+... 


(

2 ▽2fPxx,t−1 + 1 

3! ▽3fE[δX 3 � 

W ]+... 

′ i (χ ′ i − ¯ X)(χ ′ i − ¯ X)=Pxx 

¯Xt = E[Xt] =f( ¯ Xt−1)+ 1 

¯Xt = E[Xt] =f( ¯ Xt−1)+ 1 

Scaled UT: Covariance 

i=0 

¯Xt = E[Xt] =f( ¯ Xt−1)+ 1 

• 

• 

2L+1 

2 ▽2fPxx,t−1 + 1 

3! ▽3fE[δX 3 ]+... 

� 

Wi(χi − ¯ X)(χi − ¯ X)=Pxx 

Pxx,t = E[(Xt − ¯ Xt)(Xt − ¯ Xt) T ] 

Compute covariance of propagated state: 

i=0 


χt = f(χt−1) 

Pxx,t = ▽fPxx,t−1(▽f) T + 1 

4 ▽2fE[δX 3 ](▽f) T + 1 

Pxx,t = 

Goal: Incorporate the higher order terms into: 

2L+1 � 

i=0 

2 ▽2fPxx,t−1 + 1 

3! ▽3fE[δX 3 ]+... 



,t = ▽fPxx,t(▽f) T + 1 

4 ▽2fE[δX 3 ](▽f) T + 1 

2 ▽fE[δX3 ](▽ 2 f) T 

+ 1 

2 ▽2 � 

f E[δX 4 ] − E[δX 2 Pxx,t] − E[Pxx,tδX 2 ]+P 2 � 

xx,t (▽ 2 f) T 

• Multiply expected value & substitute approximated 

terms: 


2L+1 � 

ˆXt = Wiχt,i 

+ i=0 

1 

2 ▽fE[δX3 ](▽ 2 f) T 

+ 1 

2 ▽2 � 

f E[δX 4 ] − E[δX 2 Pxx,t−1] − E[Pxx,t−1δX 2 ]+P 2 � 

xx,t−1 (▽ 2 f) T 

4th order 

3! ▽3 fE[δX 4 ](▽f) T + ... 

+ 1 

3! ▽3 fE[δX 4 ](▽f) T + ... 

Wi(χt,i − ¯ Xt)(χt,i − ¯ Xt) T + (1 − α 2 )(χt,0 − ¯ Xt)(χt,0 − ¯ Xt) T 

17 

ASEN ASEN 6080, 6080, Spring Spring 2011 2012

• 


Pxx,t = 

ˆXt = Wiχt,i 

i=0 

• Incorporate the higher order terms into: 

2L+1 � 

Wi(χt,i − ¯ Xt)(χt,i − ¯ Xt) T + (1 − α 2 )(χt,0 − ¯ Xt)(χt,0 − ¯ Xt) T 

δX = Xt−1 − ¯ 4 2 

+ 

Xt−1 

1 

2 ▽2 � 

f E[δX 4 ] − E[δX 2 Pxx,t−1] − E[Pxx,t−1δX 2 ]+P 2 xx,t−1 xx,t−1 xx,t−1 

2 

� 

xx,t−1 (▽ 

+ 1 


• 

i=0 

3! ▽3 fE[δX 4 ](▽f) T + ... 

¯Xt = E[Xt] =f( ¯ Xt−1)+ 1 

Recall approximation for mean state: 

3! ▽3fE[δX 4 ](▽f) T + ... 

+ 1 

2 

2 ▽2fPxx,t−1 + 1 

χt,0 − ¯ Xt = 1 

2 ▽2fPxx,t−1 + 1 

3! ▽3fE[δX 3 ]+... 

Pxx,t = E[(Xt − ¯ Xt)(Xt − ¯ Xt) T Compute square of difference: 

] 

3! ▽3 fE[δX 3 ]+... 

Compute difference between sigma-point mean & approximated 

mean: 

χt,0 − ¯ Xt = 1 

2 ▽2fPxx,t−1 + 1 

3! ▽3fE[δX 3 ]+... 

• 

(χt,0 − ¯ Xt)(χt,0 − ¯ Xt) T = 1 

4 ▽2 fP 2 xx,t(▽ 2 f) T + ... 

(χt,0 − ¯ Xt)(χt,0 − ¯ Xt) T = 1 

Weighting the mean more strongly allows 

inclusion of higher-order terms 

3 

4 ▽2 fP 2 xx,t(▽ 2 f) T + ... 

3rd order 

3rd order 


• 

Pxx,t = 

Pxx,t = E[(Xt − ¯ Xt)(Xt − ¯ Xt) T 2 3! 

] 

(χt,0 − ¯ Xt)(χt,0 − ¯ Xt) T = 1 

4 ▽2fP 2 xx,t(▽ 2 f) T Wi(χt,i − 

+ ... ( 

¯ Xt)(χt,i − ¯ Xt) T + (1 + β − α 2 )(χt,0 − ¯ Xt)(χt,0 − ¯ Xt) T 

Add extra weight to mean sigma-point to include higher-order effects: 

,t = ▽fPxx,t(▽f) 

• Recall approximated covariance: 

T + 1 

4 ▽2fE[δX 3 ](▽f) T + 1 

2 ▽fE[δX3 ](▽ 2 f) T 

+ 1 

2 ▽2 � 

f E[δX 4 ] − E[δX 2 Pxx,t] − E[Pxx,tδX 2 ]+P 2 � 

xx,t (▽ 2 f) T 


2L+1 � 

Pxx,t = Wi(χt,i − 

i=0 


2L+1 � 

Pxx,t = Wi(χt,i − 

i=0 

( 


(1 − β) ? 

Pxx,t = ▽fPxx,t−1(▽f) T + 1 

4 ▽2fE[δX 3 ](▽f) T + 1 

2L+1 � 3 

• 


� 

2L+1 


i=0 

error 

Pxx,t = 

+ 1 

2 ▽2 f 

If X Gaussian: 

error → 

2 ▽fE[δX3 ](▽ 2 f) T 

� 

E[δX 4 ] − E[δX 2 Pxx,t−1] − E[Pxx,t−1δX 2 ]+P 2 � 

xx,t−1 (▽ 2 f) T 

(1 − β) 

i=0 

+ 1 

3! ▽3fE[δX 4 ](▽f) T E[δX 

+ ... 

4 ] = 3P 2 xx,t 

+ 1 

Wi(χt,i − ¯ Xt)(χt,i − ¯ Xt) T + (1 + β − α 2 )(χt,0 − 

3! ▽3fE[δX 4 ](▽f) T E[δX 

+ ... 

4 ] = 3P 2 xx,t 

(2 − β)▽ 2 fP 2 xx,t(▽ 2 f) T + 1 

(1 − β) 

(2 − β)▽ 2 fP 2 xx,t(▽ 2 f) T + 1 

3! ▽3 fE[δX 4 ](▽f) T 

Error minimized for β = 2 under Gaussian assumption 


3! ▽3fE[δX 4 ](▽f) T


χ ′ i = χ0 + 

2W ′ 

Pi 

(9) 

χi = χ0 + α(χi − χ0) (10) 

i 

W m 0 = λ 

� 

L + λ 

χi = χ0 + α(χi − χ0) (10) 

Scaled UT Summary 

χi = χ0 + α(χi • 

− χ0) W (10) 

Summary of Scaled UT: 

m′ Wm′ W 0 1 

0 + (1 − 1 ) (11) 

α2 + (1 − 2 α2 m′ 

0 = 3α2 − L 

3α2 W c 0 = λ 

L + λ + (1 − α2 W 

+ β) 

m′ 

0 = 3α2 − L 

3α2 χi = ˆ Xi + α(χi − ˆ Xi) 

W m W 

0 = m 0 = 

W m 0 = 

α 

m′ W0 1 

+ (1 − ) (11) 

α2 α2 W m 0 = λ 

W L + λ 

m 0 = λ 

L + λ 

Wi = W ′ Wi = 

i 

W ′ i 

α2 Wi = W ′ i 

α2 (12) 

W c 0 = W m′ 

0 + (1 − α 2 + β) W (13) 

c 0 = λ 

L + λ + (1 − α2 W + β) 

4 

c 0 = λ 

L + λ + (1 − α2 + β) 

Note: these are the same weights we learned earlier 

W c 0 = W m 0 + (1 − α 2 + β) (13) 

2L+1 

α 2 

) (11) 

α2 (12) 

(12) 

W c 0 = W m 0 + (1 − α 2 + β) (13) 

What happens to Sigma-point scaling when α = 1? 

2L+1 � 

Wi =1 W (14) 

� 

Wi =1 (14) 

m 1 

i W = 

2(L + λ) 

m Wi = 1 

i = 

2(L + λ) 

W ′ i 

α2 2L+1 

i=0 

χ ′ i = ˆ Xi + 

W m 0 = 

m′ W0 1 

2W ′ i 

Pi 

W m i = 

1 

+ (1 − ) 

α2 α2 1 

2(L + λ) 



Scaled UT Summary 

• 

Basic concept of the scaled UT is to control the 

distribution of Sigma-points χ about the mean: 

′ 1 

i = χ0 + 

Unscaled UT 

χ ′ i = ˆ Xi + 

� 

1 

2W ′ i 

� 

2W ′ i 

Pi 

χi = χ0 + α(χi − χ0) 

Pi 

W m 0 = 

m′ W0 Changing the distribution would violate the statistics 

1 

+ (1 − ) 

α2 α2 Exercise: Show that χ′ = χ (L = 6) 

Wi = W ′ i 

α 2 

W c = W m + (1 − α 2 + β) 

W c 0 = L + λ + (1 − α 2 + β) 

W m 1 

i = 

Scaled 2(L UT + λ) 

(in our filters): 

χi = ˆ Xi + γ � χi = χ0 + α(χi − χ0) Pi (10) 

W m 0 = 

m′ W0 χ ′ i = ˆ Xi + 

� 

(9) 

0 1 

+ (1 − ) 

4α2 

α2 1 

+ (1 − ) (11) 

α2 α2 1 

2W ′ i 

Pi 

χi = ˆ Xi + α(χi − ˆ Xi) 

W m 0 = 

W m′ 

Wi = W ′ i 

2 


ˆXt = 

2L+1 � 

Scaled UT: Exercise 

Wiχt,i Solution 

i=0 

ˆXi + α(χi − ˆ Xi) = ˆ Xi + γ � Pi 

Exercise: Show that χ′ = χ 

γ = √ L + λ = � L + α 2 (L + κ) − L + L = � L + α 2 (L +3− L) =α √ 3 

ˆXi + α(χi − ˆ Xi) = ˆ Xi + γ � Pi 


α(χi − ˆ Xi) =γ � Pi 

α(χi − ˆ Xi) =α � 3Pi 



γ = √ L + λ = � L + α 2 (L + κ) − L + L = � L + α 2 (L +3− L) =α √ 3 ( 

� 

� 

� 

� 

1 

2W ′ i 

1 

1 

2W ′ i 

2W ′ i 

Pi 1 = 

2W ′ i 

Pi = 

Pi = 

Pi = 

� 1 

� 1 

2α 2 Wi 

� 1 

� 1 


2 


α(χi − ˆ Xi) =γ � Pi 

2 

α(χi − ˆ Xi) =α � � 

2(L + λ) 

Pi = 

2α 3Pi 

2 

Pi 

� 

2(L + λ) 

Pi = 

2α = 2 

Pi = 

2α 2 Wi 

Pi = 

2α2Wi 2α2 χi = ˆ 1 

Xi + 

2W ′ χi = Pi 

i 

ˆ 1 

Xi + 

2W ′ Pi = χ 

i 

′ i 

Pi = 

2α2 χi = 

Wi 

ˆ Xi + 

� 

2(L + λ) 

Pi � = 

� 

� 

2(L + λ) 

2α2 Pi 

� 

1 

Pi = 

2W ′ i 

� 3α 2 

� 3α 2 

α 2 Pi = � 3Pi 

� 3α 2 

α 2 Pi = � 3Pi 

α 2 Pi = � 3Pi 

� 3α 2 

α 2 Pi = � 3Pi 


( 

( 

(


Square-root UKF 


SR-UKF 

• Introduced by van der Merwe and Wan, 2001 

–Paper on D2L 

• Improve UKF by making a square-root 


formulation 

–Numerical stability 

–PSD covariance 

–Decrease computational burden if possible 

24 



UKF 



25 

QR Decomposition 

Cholesky update 

- method for updating 

Cholesky factor 

without redoing the 

whole thing 

- in Matlab as function 

“cholupdate” 




Conclusions: 

- same performance as UKF 

- more numerically stable 

- ensure PSD covariance 

- same computational efficiency as and EKF 

which is better than a UKF 

26 

Superior performance 

by UKFs vs EKF

Lecture 19 - UKF[pdf] - CCAR

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