Foundations — Kalman filter derivation — predict step, update step

3.5.21 · D1 · Physics › Guidance, Navigation & Control (GNC) › Kalman filter derivation — predict step, update step

Is page mein assume kiya gaya hai ki aapko kuch nahi pata. parent topic mein predict/update equations touch karne se pehle, aapko symbols ki ek chhoti si pile mein fluent hona chahiye. Hum unhe ek-ek karke build karte hain, har ek ko ek picture se, har ek agla earn karta hai.

0. Ek single picture jo dimaag mein rakhni hai

Sab kuch ek bump ke baare mein hai. "Guess with doubt" ko ek bell curve ke roop mein draw kiya jaata hai: peak aapke best guess par baiThi hoti hai, aur width batati hai ki aap kitne unsure hain. Narrow bump = confident. Wide bump = clueless.

1. Scalars, vectors, aur state $x$

Jo hidden cheez hame care karni hai woh hai state $x$. GNC mein yeh usually kai physical quantities ko bundle karta hai:

$x=\left[\begin{array}{c}p\\ v\end{array}\right]⟸\text{position }p\text{ and velocity }v,\text{ stacked}$

$x_k$ ka matlab hai "time step $k$ par state". Woh chhota $k$ ek clock tick hai: $k-1$ pichla instant hai, $k$ abhi hai, $k+1$ agla hai. Hum State-space representation ko bilkul isi indexing se build karte hain.

2. Hat $\hat{x}$ — guess versus truth

Yeh distinction poore subject mein sabse zyada miss ki jaane wali idea hai. Har baar jab aap hat dekhein, yaad dilaiye khud ko: yeh ek belief hai, koi fact nahi.

3. Variance ${\sigma }^2$ aur bump ki width

4. Covariance matrix $P$

Ek number (${\sigma }^2$) ek quantity ke baare mein doubt capture karta hai. Lekin $x$ ek vector hai, isliye humein har entry ke baare mein doubt chahiye aur unke doubts kaise linked hain. Woh bookkeeping matrix $P$ hai.

Symbol $\mathbb{E}[\cdot ]$ ka matlab expectation hai — agar aap experiment ko forever repeat karte toh long-run average. Picture: noisy duniya ko ek million baar run karo, result average karo. $\mathbb{E}$ ka hamesha yahi matlab hota hai.

Superscript $\top$ (jaise $A^{\top }$ mein) transpose hai: matrix ko uske diagonal ke upar flip karo, rows ko columns mein badlo. Picture: ek grid jo uski top-left-se-bottom-right line ke paas reflect hui ho. Humein ise isliye chahiye kyunki ek column times ek row ek poori grid build karta hai — bilkul woh outer product jo $P$ ke andar hai.

5. Gaussian $\mathcal{N}(0,Q)$ aur noise

Parent par do noises aati hain:

• $w_k\sim \mathcal{N}(0,Q)$ — process noise. Picture: chhote random jhaTke jo physics model predict karne mein fail karta hai (wind gusts, unmodelled forces). $Q$ = woh jhaTke kitne baDe hain.
• $v_k\sim \mathcal{N}(0,R)$ — measurement noise. Picture: sensor reading par jitter. $R$ = sensor kitna shaky hai.

6. Transforming matrices $F$, $B$, $H$

Yeh teen matrices woh "machines" hain jo states, controls, aur measurements ko connect karti hain.

7. Measurement $z$, innovation $y$, aur gain $K$

Superscript minus, jaise ${\hat{x}}_k^{-}$ aur $P_k^{-}$ mein, ek prior mark karta hai — measurement fold in hone se pehle ("predicted, not yet corrected"). Bina minus ke = posterior, correction ke baad. Picture: minus = "coasting", bina-minus = "just snapped to the sensor".

8. Do calculus tools jo derivation borrow karta hai

Parent best gain dhundhta hai minimise karke doubt ko. Do tools woh karte hain:

Yeh sab topic ko kaise feed karta hai

Jab yeh foundations solid ho jaayein, do baDe steps (predict, update) bas yahi machinery do baar apply karti hai: bump ko $F$ ke saath aage push karo ($Q$ se wider karo), phir $K$ ke saath sensor ki taraf squeeze karo. Yahan se aap Extended Kalman Filter (EKF) aur Unscented Kalman Filter (UKF) par bhi chaRh sakte hain nonlinear worlds ke liye, sab Bayesian inference par resting.

Equipment checklist

Khud test karo — right side cover karo aur answer do: