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

3.5.20 · D5 · Physics › Guidance, Navigation & Control (GNC) › Sensor fusion — complementary filter (simple), Kalman filter

Shuru karne se pehle, un chaar numbers ki ek quick reminder jo neeche sab kuch decide karti hain, taaki koi symbol undefined na rahe:

True or false — justify

Do sensors ko fuse karna tab hi worth hai jab ek doosre se zyada accurate ho

False — fuse karna tab worth hai jab unki errors alag frequency bands mein ho (slow drift vs fast noise); do equally-bad sensors bhi help karte hain agar unki weaknesses overlap nahi karti.

Complementary weights

α

aur

1 - α

ka sum exactly 1 hona chahiye

True — yeh

H_{lp} + H_{hp} = 1

ka discrete echo hai, jo ek constant (DC) input par unity gain guarantee karta hai; warna ek steady angle galat scale ho jaayega aur bias chhod jaayega.

Ek bada measurement-noise variance

R

Kalman filter ko sensor par zyada trust karne ke liye kehta hai

False —

R

noise hai, toh bada

R

matlab ek jittery sensor; gain

K = \frac{P ^{-}}{P ^{-} + R}

shrink hota hai, toh filter measurement par kam trust karta hai.

Steady state par Kalman filter ek complementary filter ki tarah behave karta hai

True — jab

P

ek constant par converge ho jaata hai,

K

change karna band kar deta hai, toh Kalman update ek fixed-weight blend ban jaata hai, exactly ek complementary filter jiska gain frozen ho.

Kalman gain

K

aur complementary weight

α

ek hi role play karte hain

Roughly —

K

correspond karta hai

(1 - α)

se, yeh measurement par weight hai; fark yeh hai ki

K

har step mein adapt karta hai jabki

α

frozen hai.

Gyro ko integrate karne se drift-free absolute angle milta hai

False — ek tiny constant bias ko integrate karte rehne se woh unboundedly badhta hai (drift); gyro sirf short-term (high frequency) par hi trustworthy hai.

Accelerometer angle moment-to-moment instant aur reliable hota hai

False — yeh ek absolute reference deta hai bina drift ke, lekin har vibration fast noise add karta hai, toh yeh sirf long-term (low frequency) par hi trustworthy hai.

Jab do Gaussian estimates fuse hoti hain toh Precisions (inverse variances) add hote hain

True —

\frac{1}{P _{k}} = \frac{1}{P _{k}^{-}} + \frac{1}{R}

; information add hoti hai, yahi wajah hai ki fused uncertainty dono inputs se chhoti hoti hai.

Kalman filter posterior variance ko dono inputs se bada bana sakta hai

False — do independent Gaussians ko fuse karna hamesha ek aisi variance deta hai jo dono se chhoti hoti hai, kyunki precisions add hoti hain aur har precision positive hoti hai.

Complementary filter ko angle ka explicit numerical derivative chahiye

False — high-pass factor mein ek

s

hota hai, lekin gyro pehle se hi rate

\dot{θ}

supply karta hai, toh noisy data ka koi differentiation kabhi nahi hota.

Spot the error

Q = 0

set karo; model exact hai toh hum computation bachate hain."

Q = 0

ke saath predicted

P^{-} = P_{k - 1}

kabhi badhta nahi, toh

P \to 0

K \to 0

, aur filter sensors sunna band kar deta hai — yeh filter divergence hai jab reality model se inevitably alag ho jaati hai.

α

ko 1 ke paas choose karo taaki accelerometer noise completely khatam ho jaaye."

α

1 ke paas accel noise suppress karta hai, lekin yeh absolute reference ki taraf wapas bhi barely kheechta hai, toh gyro drift unbounded wapas aa jaata hai;

α

ek trade-off hai, noise switch nahi.

"Kyunki ek achha measurement aaya,

R

badhao taaki hum uspar lean karein."

Ulta hai — ek achha (low-noise) measurement matlab chhota

R

, jo

K

badhata hai; tum ek reading par trust karne ke liye

R

inflate nahi karte.

"Update ke baad,

P

ko agले predict mein unchanged carry karo."

Galat — predict mein process noise add karni chahiye,

P_{k}^{-} = P_{k - 1} + Q

;

+ Q

skip karna filter ko falsely certain aur sensors ke liye deaf bana deta hai.

"Blend weight

α

ki units seconds hain."

α = \frac{τ}{τ + Δ t}

do times ka ratio hai, toh yeh dimensionless hai aur

(0, 1)

mein rehta hai; seconds cancel ho jaate hain.

"Kalman ko fixed gain chahiye kyunki complementary filters ek use karte hain."

Kalman ka pura point ek time-varying gain hai jo uncertainty track karta hai; ek fixed gain sirf steady state par optimal hai.

"Bada Kalman gain hamesha ek better estimate matlab hai."

Bada

K

sirf matlab hai "measurement ki taraf lean karo"; yeh appropriate sirf tab hai jab prior sensor se zyada uncertain ho — transients ke dauran ya reset ke baad — universally achha nahi hai.

Why questions

Do transfer functions ka sum 1 se kisi bhi constant pe kyun nahi hona chahiye?

1 pe sum karna unity DC gain force karta hai: ek constant input full scale par pass through hota hai, toh ek truly steady angle na amplify hota hai na shrink — koi bhi doosra sum steady-state bias introduce karta hai.

Kalman gain

P

shrink hone par kyun shrink hota hai?

K = \frac{P ^{-}}{P ^{-} + R}

: jaise hum confident hote hain (chhota

P^{-}

), numerator girta hai, toh hum naye measurements ko kam weight dete hain — hum pehle se bahut jaante hain, toh har reading kam matter karti hai.

Predict step uncertainty ko fixed kyu nahi rakhta, badhata kyun hai?

Measurements ke beech true state process noise

Q

se wander kar sakta hai; independent variances add karne se

P^{-} = P + Q

milta hai, jo honestly hamare belief ko wide karta hai taaki agla measurement ab bhi suna jaaye.

Do imperfect sensors akele better wale se better kyun hote hain?

Kyunki unki errors alag bands mein hoti hain — accel gyro ke slow drift ko cover karta hai aur gyro accel ke fast noise ko cover karta hai — toh har ek exactly doosre ke weak spot ko patch karta hai.

High-pass path gyro par aur low-pass accelerometer par kyun apply hota hai, ulta kyun nahi?

Gyro high frequency par trustworthy hai (short-term smoothness) aur accel low frequency par (long-term truth); filters har signal ko us band mein route karte hain jahan woh reliable hai.

Posterior variance minimize karna

K

choose karne ka sahi criterion kyun hai?

Yeh directly estimate ke expected squared error ko minimize karta hai;

P = (1 - K)^{2} P^{-} + K^{2} R

par

\frac{d P}{d K} = 0

set karne se unique variance-minimizing, hence optimal-linear, gain milta hai.

Edge cases

Jab sample time

Δ t \to 0

aur

τ

fixed ho toh

α

ka kya hoga?

α = \frac{τ}{τ + Δ t} \to 1

— infinitely fast sampling matlab har tick gyro par almost fully trust karta hai, kyunki ticks ke beech barely koi drift accumulate hoti hai.

Jab

τ \to 0

ho toh kya hoga?

α \to 0

, toh estimate raw accelerometer angle par collapse ho jaata hai — low-pass cutoff itna high move ho jaata hai ki koi bhi gyro smoothing survive nahi karti.

Jab sensor noiseless ho

R \to 0

toh

K

kya hoga?

K = \frac{P ^{-}}{P ^{-} + 0} = 1

— ek perfect measurement completely believe kiya jaata hai, aur posterior variance

P = (1 - 1) P^{-} = 0

certainty par collapse ho jaati hai.

Jab sensor useless ho

R \to \infty

toh

K

kya hoga?

K \to 0

— ek infinitely noisy reading ignore ki jaati hai, aur

P

P^{-}

par rehta hai; update pure prediction mein reduce ho jaata hai.

Kalman filter pehle step par ek huge initial

P_{0}

ke saath kya karta hai?

Bada

P_{0}

K \approx 1

banata hai, toh yeh pehle measurement par almost fully snap karta hai — sensible hai, kyunki humne maana ki shuru mein hum kuch nahi jaante the.

Agar

Q

bahut bada set kiya jaaye, toh filter har naye reading par kaise respond karta hai?

P^{-} = P + Q

bada ho jaata hai,

K

1

ki taraf push karta hai, toh filter har measurement chase karta hai (fast lekin jittery) — state ko highly unpredictable model karte hue.

Jab dono inputs ki equal variance

P^{-} = R = σ^{2}

ho toh posterior variance kya hogi?

\frac{1}{P} = \frac{1}{σ ^{2}} + \frac{1}{σ ^{2}}

P = \frac{σ ^{2}}{2}

milta hai — do equally-trusted sources variance halve kar dete hain, yeh sabse clear sign hai ki information add hoti hai.

Recall One-line self-test

Agar koi kahe "bas Kalman gain freeze kar do, yeh simpler hai," toh ek-sentence mein rebuttal kya hai? ::: Yeh sirf steady state par optimal hai; transients ke dauran ek frozen gain ya toh ek stale prior par over-trust karta hai ya ek fresh measurement ignore karta hai, toh tum complementary filter re-invent kar rahe ho aur woh adaptivity kho rahe ho jisne Kalman ko effort ke layak banaya tha.