Sensor fusion — complementary filter (simple), Kalman filter (optimal): Foundations

3.5.20 · D1 · Physics › Guidance, Navigation & Control (GNC) › Sensor fusion — complementary filter (simple), Kalman filter

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1. Angle $θ$ aur uski rate of change $\dot{θ}$

Socho ek drone aage ki taraf tip ho raha hai. $θ$ woh angle hai jo body level ground ke saath banati hai. $\dot{θ}$ us tipping ki speed hai — bada $\dot{θ}$ matlab hai woh tezi se palat raha hai, $\dot{θ} = 0$ matlab hai woh jis bhi tilt par hai wahan ruka hua hai.

Figure — Sensor fusion — complementary filter (simple), Kalman filter (optimal)

Figure 1 — Drone pitch $θ$ (orange arc) se level ground ke upar tilted hai, aur tipping speed $\dot{θ}$ (plum arrow) rotation ki direction ke roop mein dikhaya gaya hai. Nose-up ko positive draw kiya gaya hai, jo humara sign convention visually fix karta hai.

2. Integration — rate ko angle mein badalna

Socho ek car ka speedometer ( $\dot{θ}$ , rate) aur uska odometer ( $θ$ , total). Tum seedha speedometer se distance nahi padhte — tum speed ko time ke saath accumulate karte ho. Yahan bhi aise hi: ek rate sensor tumhe tipping speed batata hai; tum use accumulate karte ho tilt jaanne ke liye.

3. Do sensors: gyroscope aur accelerometer

Figure 2 — 10 seconds mein: true angle (ink), gyro estimate dheere dheere bhatak raha hai kyunki uski bias integrate ho rahi hai (teal drift), aur accelerometer truth ke aas paas noisily bounce kar raha hai lekin kabhi drift nahi karta (orange jitter). Yeh picture kyun fusion exist karta hai — har sensor ki weakness doosre ki strength hai.

Intuition Unki errors opposite jagahon par rehti hain

Gyro: smooth aur instant, lekin dheere dheere drift karta hai (kyunki hum uski bias integrate karte hain — item 2 dekho). Fast / short-term changes ke liye acha hai.
Accel: kabhi drift nahi karta (gravity ek fixed reference hai), lekin uski reading kisi bhi non-gravitational linear acceleration se corrupt ho jaati hai. Accelerometer gravity aur vehicle ki apni acceleration mein fark nahi kar sakta — isliye speed badhana, turn karna, ya vibrate karna sab tilt ki tarah dikhta hai. Bumps aur vibrations iska fast version hain; ek sustained maneuver iska slow version hai. Kul milake yeh sirf slow / long-term truth ke liye trustworthy hai, jab averaged acceleration sirf gravity ho. Yahan "Fast" aur "slow" ka matlab frequency hai — agla symbol.

Agar yeh shaky lagte hain toh prerequisite reading: Attitude estimation (AHRS) aur Inertial Navigation Systems dikhate hain ki yeh sensors ek real vehicle mein kahan baithe hain.

4. Frequency, low-pass aur high-pass

5. Blend weight $α$ aur uska matlab

$α$ near $1$ : "zyaadatar smooth gyro par believe karo."
$α$ near $0$ : "zyaadatar drift-free accel par believe karo."

Yahan $τ$ (tau) ek time constant hai — "kitne seconds ka gyro trust karun pehle accel ko mujhe wapas kheenchne dun."

Formula

$α = \frac{τ}{τ + Δ t}$ kyon (derived, guessed nahi) Low-pass path ek first-order smoother hai jo $τ \overset{y}{˙} = θ_{a} - y$ follow karta hai: estimate $y$ , accel reading $θ_{a}$ ki taraf $τ$ seconds ki lag ke saath leak karta hai. Ise ek tick $Δ t$ par $\overset{y}{˙} \approx \frac{y _{k} - y _{k - 1}}{Δ t}$ use karke discretize karo: $τ \frac{y _{k} - y _{k - 1}}{Δ t} = θ_{a} - y_{k} .$ $Δ t$ se multiply karo aur $y_{k}$ terms ikatha karo: $τ y_{k} + Δ t y_{k} = τ y_{k - 1} + Δ t θ_{a} \Rightarrow y_{k} = \frac{τ}{τ + Δ t} y_{k - 1} + \frac{Δ t}{τ + Δ t} θ_{a} .$ Carried-forward (gyro-predicted) value par coefficient exactly $α = \frac{τ}{τ + Δ t}$ hai, aur accel coefficient $1 - α = \frac{Δ t}{τ + Δ t}$ hai — yeh automatically 1 mein sum hote hain. Toh $α$ koi free guess nahi hai: yeh forced hai cutoff time constant $τ$ wala low-pass chahne se. Bada $τ$ (slower leak, lower cutoff frequency) $α \to 1$ push karta hai; chhota $τ$ (faster leak, higher cutoff) $α \to 0$ push karta hai.

Figure 3 — Trust knob action mein: jaise $α$ 0 se 1 tak slide karta hai, weight smoothly accel (orange) se gyro (teal) ki taraf shift hoti hai. Do weights mirror images hain jo hamesha 1 mein add hote hain, jo $H_{lp} + H_{hp} = 1$ ka discrete echo hai.

6. Uncertainty ek number ke roop mein: variance aur Gaussian

Upar ki har cheez ne ek fixed trust knob use kiya. Kalman filter isko upgrade karta hai apni uncertainty ko ek number ke roop mein track karke. Iske liye humein teen ideas chahiye.

Figure 4 — Do bell curves fuse ho rahi hain: ek wide "unsure prior" (teal) aur ek tighter "measurement" (orange) milakar ek bell (plum) banate hain jo dono se taller aur narrower hai. Narrower = zyada confident — yahi "precisions add" ka geometric matlab hai.

7. Estimate hats aur gain $K$

Kyun do Gaussians multiply karna aur minimum-variance blend chunna "optimal" hai uski deeper theory Bayesian estimation se aati hai; multi-dimensional bookkeeping State-space models se aati hai; aur non-straight-line systems ka version Extended Kalman Filter hai.

Yeh foundations topic ko kaise feed karte hain

Equipment checklist

Right side cover karo aur khud test karo. Agar koi bhi answer surprising lage, toh upar woh item dubara padho.

\dot{θ}

mein dot ka kya matlab hai?

Rate of change per second — angle kitni tezi se badal raha hai.

θ

ke liye hamara sign convention kya hai?

θ = 0

level hai, nose-up positive hai, nose-down negative hai — aur

\dot{θ}

usi sign ko share karta hai.

Gyro integrate karne se drift kyun hoti hai?

Ek tiny constant bias har tick mein add hoti hai aur errors pile up hoti rehti hain cancel hone ki bajaye.

Kaun sa sensor absolute, drift-free angle deta hai, aur use kya corrupt karta hai?

Accelerometer, gravity ki fixed direction ke zariye; koi bhi non-gravitational linear acceleration (bumps, turns, thrust) tilt reading corrupt karta hai.

Gyro kis frequency band mein trustworthy hai?

High frequency (fast, short-term changes).

Complementary filter ke do filters ko kaunsa rule satisfy karna chahiye?

Unhe 1 mein sum hona chahiye (unity DC gain, no double-counting).

α = τ / (τ + Δ t)

kahan se aata hai?

First-order low-pass

τ \overset{y}{˙} = θ_{a} - y

ko ek tick

Δ t

par discretize karne se — carried-forward coefficient exactly wahi

α

hai.

P

P^{-}

Q

R

mein se har ek kya measure karta hai?

P

= post-fusion estimate variance,

P^{-}

= prior (pre-measurement) variance,

Q

= process noise,

R

= measurement noise — sab variances hain.

Bada

R

matlab sensor par zyada trust karo ya kam?

Kam — bada

R

= noisier sensor = chhota gain.

Kalman filter mein

x

kya hai, aur hat ka kya matlab hai?

x

woh general state hai jo track ho rahi hai (yahan angle); hat ise hamara estimate mark karta hai, true value nahi.

Kalman gain

K

α

se alag kaise hai?

K

har step par current uncertainty se recompute hota hai;

α

frozen hota hai.