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

3.5.25 · Physics › Guidance, Navigation & Control (GNC)

UKF exist kyun karta hai?

Iska engine hai Unscented Transform (UT).

Unscented Transform (scratch se derivation)

KAYA chahiye: ek discrete set of weighted points ${X_{i}, W_{i}}$ jo $x$ ke mean aur covariance ko exactly reproduce kare, taaki unhe transform karne ke baad $y$ ki statistics ka achha estimate mile.

KAISE sigma points banate hain — first principles se.

Hume aise points chahiye jinका weighted mean $\overset{x}{ˉ}$ ke barabar ho aur weighted spread $P$ ke barabar ho: $\sum_{i} W_{i} X_{i} = \overset{x}{ˉ}, \sum_{i} W_{i} (X_{i} - \overset{x}{ˉ}) (X_{i} - \overset{x}{ˉ})^{⊤} = P .$

Ye constraint kyun? Kyunki agar point cloud ka sahi mean aur covariance ho, toh uski low-order statistics second order tak Gaussian se match karti hain — aur do moments match karna exactly wahi hai jo ek Kalman filter care karta hai.

Points ko $P$ ki "shape" ke along spread karne ke liye, hume uska matrix square root $S$ chahiye jahan $S S^{⊤} = P$ . Hum $P$ use karte hain (usually Cholesky factor). Hum place karte hain:

X_{0} X_{i} X_{i + n} = \overset{x}{ˉ} = \overset{x}{ˉ} + ((n + λ) P)_{i}, = \overset{x}{ˉ} - ((n + λ) P)_{i}, i = 1, \dots, n i = 1, \dots, n

Toh hume milte hain $== 2 n + 1 ==$ sigma points: ek center pe, aur $P$ ke har principal axis ke dono taraf ek-ek. Factor $(n + λ)$ control karta hai ki points kitni door baithte hain.

KAISE weights chunte hain. Hume do moment conditions chahiye (mean, covariance), isliye hum do weight sets allow karte hain:

W_{0}^{(m)} = \frac{λ}{n + λ}, W_{0}^{(c)} = \frac{λ}{n + λ} + (1 - α^{2} + β)

W_{i}^{(m)} = W_{i}^{(c)} = \frac{1}{2 ( n + λ )}, i = 1, \dots, 2 n

Do weight sets kyun? Mean weight $W^{(m)}$ aur covariance weight $W^{(c)}$ sirf center point pe differ karte hain, isliye hum Gaussian-tuning term $(1 - α^{2} + β)$ covariance mein inject kar sakte hain bina mean ko disturb kiye. Clever bookkeeping, kuch nahi.

Outer weights sahi sum kyun karte hain? Check karo: $W_{0}^{(m)} + \sum_{i = 1}^{2 n} W_{i}^{(m)} = \frac{λ}{n + λ} + 2 n \cdot \frac{1}{2 ( n + λ )} = \frac{λ + n}{n + λ} = 1.$ ✔ (unbiased mean).

Nonlinearity ke through push karo, phir statistics recover karo:

Y_{i} = f (X_{i})

\overset{y}{ˉ} = i = 0 \sum 2 n W_{i}^{(m)} Y_{i}, P_{y} = i = 0 \sum 2 n W_{i}^{(c)} (Y_{i} - \overset{y}{ˉ}) (Y_{i} - \overset{y}{ˉ})^{⊤}

Ye mean aur covariance ko Gaussian inputs ke liye 3rd order tak accurate capture karta hai (EKF ke 1st order ke muqable mein). Yahi poora payoff hai.

Figure — Unscented Kalman Filter (UKF) — sigma points, better for nonlinear

Poora UKF cycle

Diya gaya nonlinear dynamics $x_{k} = f (x_{k - 1}) + w$ (process noise cov $Q$ ) aur measurement $z_{k} = h (x_{k}) + v$ (noise cov $R$ ):

1. Predict — $(\overset{x}{^}_{k - 1}, P_{k - 1})$ se sigma points generate karo, $f$ ke through propagate karo:

X_{i}^{-} = f (X_{i}), \overset{x}{^}_{k}^{-} = i \sum W_{i}^{(m)} X_{i}^{-}, P_{k}^{-} = i \sum W_{i}^{(c)} (X_{i}^{-} - \overset{x}{^}_{k}^{-}) (\dots)^{⊤} + Q

2. Measurement update — predicted sigma points ko $h$ ke through push karo:

Z_{i} = h (X_{i}^{-}), \overset{z}{^}_{k} = i \sum W_{i}^{(m)} Z_{i}

P_{z z} = i \sum W_{i}^{(c)} (Z_{i} - \overset{z}{^}_{k}) (\dots)^{⊤} + R, P_{x z} = i \sum W_{i}^{(c)} (X_{i}^{-} - \overset{x}{^}_{k}^{-}) (Z_{i} - \overset{z}{^}_{k})^{⊤}

3. Kalman gain & correction:

K = P_{x z} P_{z z}^{- 1}, \overset{x}{^}_{k} = \overset{x}{^}_{k}^{-} + K (z_{k} - \overset{z}{^}_{k}), P_{k} = P_{k}^{-} - K P_{z z} K^{⊤}

Worked Examples

Common Mistakes (Steel-manned)

Active Recall

Recall Reveal karne se pehle khud test karo

$n$ -dim state ke liye kitne sigma points? $2 n + 1$ kyun?
Sigma set ko exactly kaunsi do statistics reproduce karni chahiye?
Gaussian-tuning $β = 2$ kahan enter hota hai?
UKF, EKF ko $y = x^{2}$ at $x = 0$ pe kyun beat karta hai?
Gain mein Jacobian $H$ ki jagah kya aata hai?

UKF Jacobians kyun avoid karta hai?

Ye deterministic sigma points ko true nonlinear function ke through propagate karta hai aur unse statistics estimate karta hai, isliye koi linearization/derivative ki zaroorat nahi.

n-dimensional state ke liye kitne sigma points?

2 n + 1

— ek center point aur

P

ke principal axes ke along har dimension mein do (±).

Sigma points ko define karne wali do moment conditions kya hain?

Unka weighted mean

\overset{x}{ˉ}

ke barabar aur unka weighted covariance exactly

P

ke barabar hona chahiye.

Sigma point formula do.

X_{0} = \overset{x}{ˉ}

X_{i} = \overset{x}{ˉ} \pm ((n + λ) P)_{i}

, with

λ = α^{2} (n + κ) - n

Mean weight

W_{0}^{(m)}

kya hai?

W_{0}^{(m)} = λ / (n + λ)

; baaki sab

1/ [2 (n + λ)]

hain.

W_{0}^{(c)}

aur

W_{0}^{(m)}

mein kya fark hai?

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

; sirf center covariance weight Gaussian tuning term carry karta hai.

β

kya hai aur uski optimal Gaussian value kya hai?

Ek parameter jo prior distribution knowledge encode karta hai;

β = 2

Gaussian priors ke liye optimal hai.

Linear Kalman gain ke

P H^{⊤}

ki jagah kya aata hai?

Sigma-point cross-covariance

P_{x z}

; gain

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

UT Gaussians ke liye EKF ke muqable mein kis order tak accurate hai?

UT: 3rd order; EKF: 1st order.

x \sim N (0, σ^{2})

aur

y = x^{2}

ke liye UKF kya mean deta hai?

Exactly

σ^{2}

(EKF galat tarike se 0 deta hai kyunki origin pe Jacobian 0 hai).

Additive UKF mein

Q

aur

R

kahan add hote hain?

Q

predicted covariance

P_{k}^{-}

mein;

R

innovation covariance

P_{z z}

mein.

Recall Feynman: 12-saal ke bachche ko explain karo

Socho tumhare paas dots ka ek dhundhla cloud hai jo dikhata hai ki rocket kahan ho sakta hai. Tumhe jaanna hai ki ek curved, twisty motion ke baad wo kahan hoga. Purana trick (EKF) ye dakhavta hai ki twisty path ek seedhi line hai — sasta lekin tez modon pe galat. UKF trick: bas kuch "scout" dots chuno — ek beech mein aur ek-ek dono taraf — har scout ko real twisty path ke through bhejo, aur dekho wo kahan utarte hain. Jahan scouts aaye, wahan se naya cloud banao. Kyunki tumne real path use kiya (seedhi-line guess nahi), tumhara naya cloud bahut zyada accurate hai — aur tumhe sirf muthi bhar scouts move karne pade, na ki ek million.

Connections

Extended Kalman Filter (EKF) — linearization approach jise UKF improve karta hai.
Kalman Filter (linear) — parent algorithm; UKF uski gain/update structure reuse karta hai.
Particle Filter — bahut saare random samples use karta hai; UKF kuch deterministic ones use karta hai.
Cholesky Decomposition — sigma-point placement ke liye $P$ compute karta hai.
Taylor Series Expansion — wo tool jis par EKF rely karta hai aur UKF sidestep karta hai.
State Estimation in GNC — jahan UKF deploy hota hai (attitude, orbit determination).
Nonlinear Systems — wo regime jahan UKF ka advantage dikhta hai.