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

3.5.25 · D1 · Physics › Guidance, Navigation & Control (GNC) › Unscented Kalman Filter (UKF) — sigma points, better for non

Isse pehle ki tum parent note ki ek bhi line padh sako, tumhe ek vocabulary chahiye. Yeh page har symbol ko build karta hai jo woh use karta hai, ek smart-but-blank slate se shuru karke. Yahan yeh assume nahi kiya gaya ki tumne pehle probability, matrices, ya filters dekhe hain.

1. "State" kya hota hai? Symbol $x$

Ise picture karo: ek single point jo ek kamre mein float kar raha hai. Agar kamre mein $n$ directions hain jisme tum move kar sako, toh point ko pin karne ke liye $n$ numbers chahiye.

Topic ko yeh kyun chahiye: poore filter ka kaam hai ki time ke saath $x$ ka achha estimate rakha jaye. Baaki sab — noise, covariance, sigma points — sirf yeh answer karne ke liye exist karta hai ki "$x$ actually kahan hai?"

• $n$ ::: $x$ mein entries ki sankhya, jise dimension kehte hain. 2D position → $n=2$.
• $\bar{x}$ (padho "x-bar") ::: $x$ ke liye hamara single best guess — average, cloud ka center.
• $\hat{x}$ (padho "x-hat") ::: filter ka $x$ ka estimate. Chhoti hat ka matlab hai "estimated, true value nahi."

2. Uncertainty ek cloud hai, point nahi

Hum $x$ ko exactly kabhi nahi jaante. Humara knowledge actually $\bar{x}$ ke center par ek fuzzy cloud of possibilities hai.

Upar wali bell-shaped fuzziness Gaussian (ya "normal") distribution hai. Yeh "roughly yahan, give or take" describe karne ka sabse common tarika hai.

• Gaussian $\mathcal{N}(\bar{x},P)$ ::: shorthand hai "ek bell-shaped cloud $\bar{x}$ par centered, spread $P$ ke saath." Nonlinear Systems par baad mein build karo; yahan sirf haze picture karo.

3. Spread measure karna: variance ${\sigma }^2$ aur covariance $P$

Hum "haze kitna wide hai" par number kaise lagate hain?

Picture karo: $\sigma$ 1D mein haze ki radius hai — middle se ek step bahar mostly cloud ($68\%$ ke aas paas) pakad leta hai.

Lekin ek state ke kai directions hote hain, aur uncertainty ek direction mein wide, doosre mein narrow, aur tilted ho sakti hai. Ek number kaam nahi karega.

Topic ko yeh kyun chahiye: UKF ka slogan hai "function nahi, distribution ko push karo." Distribution hai hi $(\bar{x},P)$ — ek center aur ek ellipse. Aage ka sab kuch is ellipse ko transform karne ke baare mein hai.

• $P$ ::: covariance = state ki uncertainty ellipse.
• $P_y$ ::: transform karne ke baad output $y$ ki covariance.
• $Q$ ::: har step mein extra uncertainty add hoti hai kyunki physics khud imperfect hai (process noise).
• $R$ ::: sensor readings mein uncertainty (measurement noise).
• $A^{\top }$ (padho "A-transpose") ::: ek matrix ko uske diagonal ke across flip karo; column ko row mein turn karta hai. Chahiye kyunki $(\mathcal{X}-\bar{x})(\mathcal{X}-\bar{x}{)}^{\top }$ woh tarika hai jisse ek single offset se covariance build karte hain — ek outer product jo ek number nahi balki full grid produce karta hai.

4. Nonlinear map: $f$, $h$, aur "nonlinear" kyun hurt karta hai

Topic ko yeh kyun chahiye: UKF sirf isliye exist karta hai kyunki $f$ aur $h$ curve karte hain. Agar woh straight hote, toh plain Kalman Filter (linear) already perfect hota.

5. Woh tool jis par EKF lean karta hai: derivative aur Taylor series

Curve ko straighten karne ke liye, EKF calculus use karta hai.

Topic ko yeh kyun chahiye (as a foil): Jacobian samajhna zaroori hai taaki punchline appreciate kar sako — UKF kabhi ek bhi compute nahi karta. Jahan $f$ ka koi clean derivative nahi hota (e.g. $h=\Vert x\Vert$ origin par), EKF ruk jaata hai aur UKF chalता rehta hai.

6. Matrix square root: $\sqrt{P}$ aur Cholesky

UKF apne sample points ellipse $P$ ki shape ke saath rakhta hai. Har direction mein "ek ellipse-radius" step karne ke liye, use matrix square root chahiye.

Picture karo: 1D variance ${\sigma }^2$ ke liye, square root sirf $\sigma$ hai — ek standard-deviation step. $n$ dimensions mein, $\sqrt{P}$ ellipse ke tilted, stretched directions mein "$\sigma$" ko generalize karta hai.

Topic ko yeh kyun chahiye: sigma-point formula ${\mathcal{X}}_i=\bar{x}\pm (\sqrt{(n+\lambda )P}{)}_i$ literally matrix square root ke bina likha hi nahi ja sakta. Uske columns woh arrows hain jo uncertainty ellipse ke har principal axis ke saath point karte hain.

7. Show ke stars: sigma points, weights, Greek tuning knobs

• ${\mathcal{X}}_i$ (calligraphic X, padho "curly-X sub i") ::: $i$-th sigma point (ek input sample).
• ${\mathcal{Y}}_i$ ::: woh point $f$ se push hone ke baad: ${\mathcal{Y}}_i=f({\mathcal{X}}_i)$.
• ${\mathcal{Z}}_i$ ::: measurement model $h$ se push kiya gaya sigma point.
• $W_i^{(m)}$ ::: point $i$ ka mean weight — $\bar{y}$ find karne ke liye average karte time kitna count karta hai.
• $W_i^{(c)}$ ::: point $i$ ka covariance weight — spread $P_y$ build karte time kitna count karta hai. Sirf center point par $W^{(m)}$ se differ karta hai.

Woh Greek knobs jo decide karte hain ki points kitne door baithe hain aur weights kaise set hote hain:

• $\lambda$ (lambda) ::: ek scaling number, $\lambda ={\alpha }^2(n+\kappa )-n$, jo points ki reach set karta hai.
• $\alpha$ (alpha) ::: chhota (e.g. $10^{-3}$); spread control karta hai — chhota $\alpha$ points ko mean ke paas rakhta hai.
• $\kappa$ (kappa) ::: ek secondary scaling number, usually $0$ ya $3-n$.
• $\beta$ (beta) ::: cloud ki shape ki prior knowledge inject karta hai; Gaussian ke liye $\beta =2$ best hai.

Topic ko yeh kyun chahiye: yeh saat symbols hi Unscented Transform hain. Is list ko master karo aur parent ke boxed formulas plain sentences ki tarah padhenge.

8. Sab kuch kaise connect hota hai

Ise upar se neeche padho: ek state ek uncertainty ellipse ($P$) carry karta hai; uska square root sigma points rakhta hai, Greek knobs se weighted; inhe nonlinear maps se push karna Unscented Transform hai, jise filter har cycle run karta hai — us Jacobian ko sidestep karte hue jis par EKF depend karta hai.

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