Kalman filter derivation — predict step, update step: Visual walkthrough

3.5.21 · D2 · Physics › Guidance, Navigation & Control (GNC) › Kalman filter derivation — predict step, update step

Humein sirf ek running story chahiye: "Mera drone kahan hai, ek dimension mein?" Ek number line, metres mein marked. Bus itna hi.

Step 1 — Ek guess ek hill hai, point nahi

KYA. Mujhe lagta hai mera drone $10$ m par hai. Lekin mujhe yakeen nahi. Toh number line par ek dot ki jagah, main ek hill draw karta hoon — jahan mujhe lagta hai wahan oonchi, jahan doubt hai wahan neech. Yeh hill ek Gaussian hai (ek bell curve).

Hill kyun, dot kyun nahi? Kyunki ek dot perfect knowledge ka claim karta hai, jo mere paas nahi hai. Hill ek saath do baatein carry karti hai: uski peak (mera best guess) aur uski width (mera doubt). Hum inhe naam denge.

PICTURE. Figure s01 dekho. Peak $a = 10$ m par hai — yeh mera best guess hai (mean). Width $σ_{a}$ se set hoti hai — standard deviation, literally "kitne metres ki wiggle room." Moti hill = sure nahi; patli spike = confident.

Figure — Kalman filter derivation — predict step, update step

Exponential kyun, say, triangle kyun nahi? Kyunki jab tum do independent evidence combine karte ho, probabilities multiply hoti hain, aur exponential woh ek aisi shape hai jo multiply karne ke baad bhi wahi shape rehti hai (ek hill times ek hill abhi bhi ek hill hai). Hum Step 3 mein iska faida uthate hain. Dekho Covariance matrices and Gaussian distributions.

Step 2 — Ek measurement ek doosri hill hai

KYA. Mera GPS ab $b = 14$ m report karta hai, aur GPS bhi fuzzy hai, apni width $σ_{b}$ ke saath. Toh main ek doosri hill draw karta hoon, $14$ par peaked.

Ise same tarike se kyun draw karein? Taaki do pieces of evidence ek hi picture par rahein aur fairly compare ho sakein — dono position ki hills, dono honest width ke saath.

PICTURE. Figure s02: cyan hill mera prior guess hai (peak $10$ ), amber hill measurement hai (peak $14$ ). Notice karo ki woh overlap karti hain. Sahi position shayad us overlap mein kahin hai — exactly kisi bhi peak par nahi.

Step 3 — Hills multiply karo; answer ek aur hill hai

KYA. Do independent pieces of evidence combine hote hain apni hills ko multiply karke, number line par point by point. Jahan dono oonchi hain, product ooncha hai; jahan koi bhi near zero hai, product squash hokar zero ho jaata hai.

Multiply kyun? Yeh Bayesian inference ek move mein hai: independent evidence multiply hoti hai. Yahi reason hai ki fused hill do peaks ke beech sit karti hai aur dono se thinner hoti hai.

PICTURE. Figure s03: cyan hill × amber hill = white hill. Teeno cheezein padhni hain:

Uski peak $10$ aur $14$ ke beech hai — ek compromise.
Woh jis original hill ki taraf lean karti hai woh thinner hoti hai (zyada trusted).
Woh dono parents se narrower hai — evidence combine karna zyada sure banata hai.

Do exponentials multiply karne ka matlab hai unke exponents add karna: $- \frac{( x - a ) ^{2}}{2 σ _{a}^{2}} - \frac{( x - b ) ^{2}}{2 σ _{b}^{2}} .$

Har term :: ek hill ki "penalty" agar candidate $x$ us hill ki peak se door hai.
Woh $+$ :: kyunki $exp (P) \cdot exp (Q) = exp (P + Q)$ — hills multiply karna = exponents add karna.

Step 4 — Nayi peak dhundho (yahi fused estimate hai)

KYA. Fused hill ki peak wahan hai jahan upar wali penalties ka sum sabse chhota ho (kam penalty = sabse oonchi hill). $x$ ko slide karo jab tak slope flat na ho — woh hill ka top hai.

Derivative kyun use karein? Derivative woh tool hai jo slope measure karta hai. "Peak" ka matlab hai "slope $= 0$ ." Toh hum exponent ko differentiate karte hain aur zero set karte hain — woh ek jagah hai jahan hill chadhna band karke girna shuru karti hai. Poore page par yahi ek calculus hai, aur yeh exactly isi reason ke liye hai: maximum locate karna.

PICTURE. Figure s04: white fused hill apni peak $\overset{x}{^}$ par ek flat tangent line ke saath. Dono taraf ke do grey slopes dikhate hain ki derivative left par positive hai, right par negative, exactly top par zero.

Exponent differentiate karo, $0$ set karo: $\frac{x - a}{σ _{a}^{2}} + \frac{x - b}{σ _{b}^{2}} = 0.$

$\frac{x - a}{σ _{a}^{2}}$ :: prior hill kitni zor se $x$ ko $a$ ki taraf "pull" karti hai; patli hill (chhota $σ_{a}^{2}$ ) zyada zor se pull karti hai.
$\frac{x - b}{σ _{b}^{2}}$ :: measurement $b$ ki taraf wahi pull.
$= 0$ :: dono pulls balance ho jaate hain — peak wahan hoti hai jahan tug-of-war barabar hota hai.

$x$ ke liye solve karo (ise $\overset{x}{^}$ kaho, fused estimate): $\overset{x}{^} = \frac{σ _{b}^{2} a + σ _{a}^{2} b}{σ _{a}^{2} + σ _{b}^{2}} .$

Cross-weighting notice karo: $a$ ko doosri hill ki variance $σ_{b}^{2}$ se multiply kiya gaya hai. Chhota $σ_{a}^{2}$ (main confident hoon) → $a$ ki weight $σ_{b}^{2}$ dominate karti hai → answer $a$ ki taraf lean karta hai. Sahi hai: patli hill jeetti hai.

Step 5 — "Purana guess + ek nudge" ke roop mein rewrite karo: gain $K$ se milo

KYA. Upar wala formula symmetric aur sundar hai, lekin ek filter online chalta hai: uske paas ek purana guess hota hai aur woh use nayi data ki taraf nudge karna chahta hai. Toh hum algebraically usi $\overset{x}{^}$ ko rearrange karte hain $\overset{x}{^} = a + K (b - a), K = \frac{σ _{a}^{2}}{σ _{a}^{2} + σ _{b}^{2}} .$

Yeh form kyun? Kyunki $b - a$ surprise hai (measurement mera guess se kitni door jaati hai), aur $K$ woh fraction hai jitne surprise par main actually act karta hoon. Yeh real Kalman gain ka seed hai.

PICTURE. Figure s05: number line mein $a$ at $10$ , $b$ at $14$ ; $a$ se $b$ tak length $(b - a) = 4$ ka ek full arrow; $K (b - a)$ length ka ek chhota amber arrow $\overset{x}{^}$ par land karta hua. Gain $K$ literally woh fraction hai jitna arrow tum travel karte ho.

Yeh $K$ exactly $K_{k} = P_{k}^{-} H^{⊤} (H P_{k}^{-} H^{⊤} + R)^{- 1}$ hai parent se, jahan $σ_{a}^{2} \to P^{-}$ , $σ_{b}^{2} \to R$ , aur $H = 1$ . Same idea, vectors ke liye dressed — dekho State-space representation.

Step 6 — Nayi width: confidence hamesha kyun badhti hai

KYA. Fused hill thinner hai. Uska variance: $σ^{2} = (1 - K) σ_{a}^{2}, equivalently \frac{1}{σ ^{2}} = \frac{1}{σ _{a}^{2}} + \frac{1}{σ _{b}^{2}} .$

Width kyun shrink karti hai? Kyunki $K \in [0, 1]$ , factor $(1 - K)$ $0$ aur $1$ ke beech hai, toh $σ^{2} < σ_{a}^{2}$ hamesha. Doosra form punchline hai: precisions (1/variance) add hoti hain. Har hill apni sharpness contribute karti hai, aur sharpnesses stack hoti hain.

PICTURE. Figure s06: parent hills background mein faint, fused white hill solid drawn aur visibly dono se taller aur narrower. Uski half-width $σ$ marked hai aur $σ_{a}$ aur $σ_{b}$ dono se chhoti hai.

Step 7 — Do degenerate cases (kabhi nahi chhodne chahiye)

KYA. Dials ko extreme par push karo taaki koi reader kabhi surprised na ho.

Case A — ek perfect sensor ( $σ_{b}^{2} \to 0$ ). Measurement hill ek infinitely thin spike ban jaati hai. $K \to 1$ . Fused hill $b$ par collapse ho jaati hai: $\overset{x}{^} \to b$ , $σ^{2} \to 0$ . Hum sensor par poora bharosa karte hain.

Case B — ek useless sensor ( $σ_{b}^{2} \to \infty$ ). Measurement hill ek pancake mein flat ho jaati hai jo koi information carry nahi karti. $K \to 0$ . Fused hill exactly wahan rehti hai jahan prior tha: $\overset{x}{^} \to a$ , $σ^{2} \to σ_{a}^{2}$ . Hum garbage data ignore karte hain. (Yahi wajah hai ki parent mein $Q = 0$ set karna ek trap hai: yeh prior ko infinitely-thin wala banata hai, toh filter hamesha ke liye har real measurement ignore karta hai.)

Yeh kyun dikhayein? Kyunki ek formula jise tum limits par stress-test nahi kar sakte woh aisa formula hai jo tumhe samajh nahi aaya. Dono extremes common sense se agree karte hain — yahi hamara proof hai ki yeh sahi hai.

PICTURE. Figure s07: do side-by-side panels. Left: ek spike measurement (amber) fused peak ko apne upar kheenchti hai. Right: ek flat measurement (amber) fused hill ko prior par baitha rehne deti hai.

Ek-picture summary

Figure s08 poori derivation ko ek frame mein stack karta hai: prior hill (cyan) $a$ par, measurement hill (amber) $b$ par, aur fused hill (white) dono ke beech baithe — dono se thinner — gain-fraction arrow $K (b - a)$ ke saath jo peak ko $a$ se $\overset{x}{^}$ tak le jaata hai. Page par har symbol is ek picture mein kahin na kahin rehta hai.

Recall Feynman retelling — ise zor se bolo

Socho tum guess kar rahe ho ki tumhara drone kahan hai. Tumhara pehla guess number line par ek hill hai: tall jahan tumhe lagta hai, wide agar sure nahi ho. Phir GPS ek doosri hill kahin aur deta hai. Do honest opinions combine karne ke liye, tum hills multiply karte ho — aur ek hill times ek hill ek aur, skinnier hill hoti hai dono ke beech baithe hue. Uski peak tumhara naya best guess hai; woh jis hill ki taraf lean karti hai woh skinnier hoti hai (jis par tumne zyada trust kiya). Woh peak dhundhne ke liye tum tab tak slide karte ho jab tak slope flat na ho — calculus ek chhota sa kaam kar raha hai: top locate karna. Rearrange karo, answer kehta hai "apne purane guess se shuru karo, phir measurement ki taraf $K$ fraction step lo." Woh fraction $K$ tumhara apna doubt divided by total doubt hai: khud par sure nahi → door step lo; confident → barely budge. Aur nayi hill hamesha dono se thinner hoti hai — kyunki data dekhna tumhari ignorance sirf reduce kar sakta hai. Yahi poora Kalman update hai. Matrix version yahi kaam sirf kai dimensions mein karta hai.

Recall Quick self-test

Do Gaussians multiply karne se Gaussian kyun milta hai? ::: Kyunki exponentials multiply karna unke exponents add karta hai, aur $x$ mein do quadratics ka sum abhi bhi ek single quadratic hai — ek Gaussian ki pehchaan. $\overset{x}{^} = a + K (b - a)$ mein, $b - a$ kya represent karta hai? ::: Innovation / surprise — measurement land karti hai wahan se kitni door jahan main ne predict kiya tha. Agar $K = 0$ ho, toh kya hua? ::: Sensor useless tha (ya prior perfect tha); estimate measurement ignore karti hai aur $a$ par rehti hai. Fused variance hamesha $σ_{a}^{2}$ se chhoti kyun hoti hai? ::: Kyunki $σ^{2} = (1 - K) σ_{a}^{2}$ aur $K \in [0, 1]$ , toh $(1 - K) < 1$ — evidence combine karna sirf sharpen kar sakta hai.

Yeh aage kahan jaata hai: wahi teen-hill picture, non-linear banai jaaye, Extended Kalman Filter (EKF) aur Unscented Kalman Filter (UKF) ban jaati hai; "precisions add" idea Recursive Least Squares ka engine hai; aur alag instruments se hills fuse karna exactly IMU and GPS sensor fusion hai.