Unscented Kalman Filter (UKF) — sigma points, better for nonlinear: Question bank

3.5.25 · D5 · Physics › Guidance, Navigation & Control (GNC) › Unscented Kalman Filter (UKF) — sigma points, better for non

Definition Is page par baar baar aane wale symbols

Traps pakadne se pehle yeh symbols achhe se yaad kar lo.

$x$ — state (jo hum estimate karte hain), dimension $n$ ; $\overset{x}{ˉ}$ uska mean, $P$ uska covariance.
$f$ — state-transition function: yeh state ko ek step aage map karta hai, $x_{k} = f (x_{k - 1}) + w$ . Yeh nonlinear ho sakta hai (yahi wajah hai ki hum UKF use karte hain).
$h$ — measurement function: yeh state ko woh value map karta hai jo sensor padhta, $z_{k} = h (x_{k}) + v$ , yeh bhi possibly nonlinear.
$w$ — random process noise jiska covariance $Q$ hai: yeh is fact ko model karta hai ki $f$ kabhi bhi reality ka perfect description nahi hota (unmodelled forces, disturbances).
$v$ — random measurement noise jiska covariance $R$ hai: yeh sensor ki imperfection ko model karta hai. $Q$ aur $R$ do "kitna trust karoon" wale knobs hain; inhe sigma-point sums ke baad wapas add karna zaroori hai kyunki points sirf prior uncertainty carry karte hain.

Do words aur bhi hain jo yaad rakhne chahiye:

sigma points ::: $2 n + 1$ deterministically chosen sample points ( $n$ = state dimension) jinke weighted mean aur spread input mean $\overset{x}{ˉ}$ aur covariance $P$ ko exactly reproduce karte hain.
Unscented Transform (UT) ::: yeh procedure hai jisme sigma points ko true nonlinear function ( $f$ ya $h$ ) ke through push kiya jaata hai aur output mean aur covariance transformed points se read ki jaati hai.

True ya false — justify karo

UKF nonlinear function

f

ko ek polynomial se approximate karta hai.

False. UKF distribution ko approximate karta hai (sigma points se) aur true, un-approximated

f

ko har point par apply karta hai — ek arbitrary function approximate karne se distribution approximate karna aasaan hai.

UKF mein Jacobian matrix compute karni padti hai.

False. Yeh kabhi linearize nahi karta; yeh estimate karta hai ki uncertainty kaise propagate hoti hai sigma points ko directly transform karke, toh

f

h

ke koi bhi derivatives kabhi nahi lete.

Sigma points Monte Carlo samples ki tarah randomly draw kiye jaate hain.

False. Yeh deterministic aur minimal (

2 n + 1

) hote hain, pehle do moments exactly match karne ke liye place kiye jaate hain; randomness sirf sampling noise add karti, accuracy nahi.

Ek purely linear system ke liye, UKF aur ordinary Kalman Filter (linear) ka result same hota hai.

True. Jab

f

aur

h

linear hain toh sigma-point mean aur covariance exact linear propagation reproduce karte hain, isliye UKF standard Kalman recursion mein collapse ho jaata hai.

Mean weights

W_{i}^{(m)}

aur covariance weights

W_{i}^{(c)}

har sigma point ke liye identical hote hain.

False. Yeh sirf center point $X_{0}$ par alag hain:

W_{0}^{(c)} = W_{0}^{(m)} + (1 - α^{2} + β)

. Baaki

2 n

outer points dono ke liye same weight share karte hain.

UKF ek Bayesian sampling method hai jo Particle Filter ki tarah zyada particles se improve hota hai.

False. Particle filter bahut saare random particles use karta hai aur arbitrary (non-Gaussian) distributions represent karta hai; UKF ek fixed minimal set use karta hai jo assumed Gaussian ke sirf do moments match karta hai.

UT mein weights sab positive hone chahiye.

False.

W_{0}^{(m)} = λ / (n + λ)

negative ho sakta hai jab

λ < 0

ho (jaise bade

n

ke saath

κ = 0

), phir bhi weighted moment conditions hold karti hain — negative weights yahan valid hain.

UT kisi bhi nonlinear function ke mean ke liye exact hai.

False. Yeh Gaussian inputs ke liye mean ke liye 3rd-order accurate hai — sirf uss order tak ke polynomials ke liye (aur

y = x^{2}

jaise special cases ke liye) exact hai, arbitrary

f

ke liye nahi.

Error dhundo

f

ke through sigma points propagate karne ke baad, predicted covariance hai

P_{k}^{-} = \sum_{i} W_{i}^{(c)} (X_{i}^{-} - \overset{x}{^}_{k}^{-}) (X_{i}^{-} - \overset{x}{^}_{k}^{-})^{⊤}

+ Q

missing hai. Sigma points sirf prior uncertainty carry karte hain; additive process-noise covariance

Q

add karna zaroori hai, warna filter overconfident ho jaata hai aur diverge kar sakta hai.

"Main innovation covariance

P_{z z}

banane ke liye

W_{0}^{(m)}

use karunga."

Galat weight hai. Covariance sums

W_{i}^{(c)}

use karte hain; sirf mean

\overset{z}{^}_{k}

W_{i}^{(m)}

use karta hai. Inhe swap karne se Gaussian-tuning (

β

) correction corrupt ho jaati hai jo

W_{0}^{(c)}

ke andar hoti hai.

"Measurement covariance hai

P_{z z} = \sum_{i} W_{i}^{(c)} (Z_{i} - \overset{z}{^}_{k}) (Z_{i} - \overset{z}{^}_{k})^{⊤}

+ R

missing hai. Measurement-noise covariance

R

additively

P_{z z}

mein inject hoti hai; iske bina gain

K

sensor par zyada trust karta hai.

P

ka matrix square root koi bhi

S

ho sakta hai jiska

S^{⊤} S = P

ho."

Humein

S S^{⊤} = P

chahiye:

S

ke columns woh axis-directions ban jaate hain jinke saath sigma points place kiye jaate hain, isliye un columns ko

\sum_{j} S_{j} S_{j}^{⊤} = P

satisfy karna chahiye. Practice mein lower-triangular Cholesky Decomposition factor use hota hai;

S^{⊤} S = P

lene se geometry transpose ho jaati hai aur axes galat jagah place ho jaate hain.

"Kyunki

α = 1 0^{- 3}

sigma points ko almost mean par rakhta hai, covariance estimate tiny ho jaata hai."

Nahi. Shrink exactly

(1 - α^{2} + β)

term in

W_{0}^{(c)}

aur bade

1/ (n + λ)

outer weights se cancel ho jaata hai, isliye recovered covariance sahi rehta hai — yahi scaled UT ka point hai.

λ = α^{2} (n + κ)

Subtraction missing hai: correct definition hai

λ = α^{2} (n + κ) - n

- n

bhool jaane se har sigma point galat jagah ho jaata hai aur moment match toot jaata hai.

"UKF mein Kalman gain hai

K = P H^{⊤} (H P H^{⊤} + R)^{- 1}

UKF mein koi

H

nahi hota. Gain hai

K = P_{x z} P_{z z}^{- 1}

, jisme sigma-point cross-covariance

P_{x z}

use hoti hai

P H^{⊤}

ki jagah — poora point yahi hai ki koi Jacobian

H

exist nahi karta.

Why questions

UKF sharply curved

f

par Extended Kalman Filter (EKF) ko kyun beat karta hai?

EKF Taylor Series Expansion ka sirf linear term rakhta hai aur curvature discard karta hai; UKF true

f

ko spread-out points par evaluate karta hai, isliye woh 2nd/3rd-order effects capture karta hai jo EKF throw away karta hai.

Exactly

2 n + 1

sigma points kyun hote hain, zyada kyun nahi?

Ek center point mean fix karta hai aur do points per dimension (har principal axis of

P

ke saath plus/minus) covariance fix karte hain — yeh minimal set hai jo dono moments exactly match karta hai.

Matrix square root

(n + λ) P

ki zaroorat kyun hai?

Points ko uncertainty ellipsoid ki shape aur orientation ke saath spread karne ke liye; square root ke columns principal-axis directions hain jo unke standard deviations se scale kiye hue hain.

Ek weight set ki jagah

W^{(m)}

aur

W^{(c)}

mein kyun split karte hain?

Taaki Gaussian-tuning term

(1 - α^{2} + β)

covariance mein inject ho sake (

W_{0}^{(c)}

ke through) bina mean shift kiye, kyunki yeh sirf center weight ko touch karta hai.

β = 2

Gaussian inputs ke liye optimal kyun maana jaata hai?

β

sirf

W_{0}^{(c)}

ke through enter hota hai, isliye yeh covariance expansion ke 4th-moment (kurtosis) term mein center point ke contribution ko re-weight karta hai;

β = 2

set karne se Gaussian ke liye leading kurtosis error exactly cancel ho jaati hai, jiska kurtosis known hai — isliye "Gaussians ke liye optimal."

UKF

h (x) = ∥ x ∥

jaisi measurement ko kyun handle kar leta hai jahan EKF origin par struggle karta hai?

EKF ka Jacobian

x^{⊤} /∥ x ∥

x = 0

par undefined hai, lekin UKF ka

P_{x z}

evaluated points ka ek finite weighted sum hai — derivative-free, isliye koi singularity nahi.

"Distribution ke do moments match karna" Kalman-style filter ke liye kyun kaafi hai?

Ek Kalman filter sirf mean aur covariance propagate karta hai; agar sigma cloud woh do moments reproduce kare, toh yeh recursion ko sab kuch feed karta hai jo woh use karta hai — higher moments update ke liye irrelevant hain.

Edge cases

Agar

f

exactly linear ho toh kya hoga?

UT exact transformed mean aur covariance return karta hai (koi approximation error nahi), isliye UKF us step par linear Kalman Filter (linear) ke identical ho jaata hai.

Ek scalar

x \sim N (0, σ^{2})

aur

y = x^{2}

ke liye, EKF kya mean predict karta hai versus truth?

EKF Jacobian

2 x ∣_{x = 0} = 0

use karta hai aur

\overset{y}{ˉ} = 0

predict karta hai; truth (aur UT) hai

\overset{y}{ˉ} = σ^{2}

— EKF ek symmetric extremum par completely fail karta hai.

Agar

λ

negative aaye aur

W_{0}^{(m)}

negative ho jaaye toh?

Bilkul valid hai: moment conditions phir bhi hold karti hain, though negative center weight kabhi kabhi non-positive-definite

P

estimate de sakta hai — yahi ek known reason hai scaled/square-root UKF forms prefer karne ka.

Agar prediction mein additive

Q

bhool jaao aur yeh kaafi steps tak chale toh?

Covariance bina bound ke shrink hoti jaayegi, filter overconfident ho jaayega, naye measurements ignore karega, aur aakhirkar true state se diverge kar jaayega.

Kya UKF tab exact hota hai jab input non-Gaussian ho?

Nahi. Sigma points aur

β = 2

Gaussian ke liye tuned hain; heavy-tailed ya multimodal inputs ke liye two-moment match structure miss karta hai — yahan Particle Filter prefer kiya jaata hai.

Sabse chhoti state dimension kaunsi hai jahan UKF

f

ko sirf mean par evaluate karne se alag result deta hai?

Koi bhi

n \geq 1

n = 1

ke liye bhi

\pm (n + λ) σ

par do outer points woh curvature sense karte hain jo ek single center evaluation nahi kar sakta — yahi exactly

y = x^{2}

wala case hai.

Agar sigma points galat scaling factor se place kiye jaayein (jaise

(n + λ) P

ki jagah

P

), toh kya toota?

Recovered covariance galat scale hogi — points bahut paas ya bahut door honge, isliye

P_{y}

true propagated covariance se match nahi karega even for linear

f