IMU — integrated accelerometer + gyroscope: Question bank

3.5.15 · D5 · Physics › Guidance, Navigation & Control (GNC) › IMU — integrated accelerometer + gyroscope

Ye items IMU ke ideas pe focus karte hain — iske do sensors, strapdown chain, aur yeh kyun drift karta hai. Yahan kuch bhi calculator se nahi — woh D3/D4 ka kaam hai.

Notation & conventions jo is page pe use hote hain

Traps se pehle, hum har symbol aur frame convention lock kar lete hain. Agar inme se kuch naya lagta hai, toh yeh section hi definition hai — yahan kuch bhi kahin aur se assumed nahi hai.

Definition Symbols, ek baar aur hamesha ke liye

$f_{b}$ — body axes mein express kiya hua specific force vector (m/s²): jo teen accelerometers actually read karte hain.
$a_{w}$ — body ki true acceleration world axes mein express ki gayi (m/s²): jo hum ultimately chahte hain.
$g$ — world mein gravity, NED mein $(0, 0, + g)$ ( $+ z$ ke along yaani neeche point karta hai).
$R_{b}^{w}$ — rotation matrix jo body axes mein likhe vector ko world axes mein re-express karta hai: $v_{w} = R_{b}^{w} v_{b}$ . Notation padhein " $b$ se $w$ tak".
$ω_{b} = (ω_{x}, ω_{y}, ω_{z})$ — gyro ki angular rate body axes mein (rad/s).
$b_{g}$ — ek constant gyro bias (rad/s): ek fake steady rotation jo gyro report karta hai chahe bilkul still ho.
$ε_{a}$ — ek constant accelerometer error/bias (m/s²): ek fake steady specific-force offset.

Figure — IMU — integrated accelerometer + gyroscope

Intuition Cross product rotation kyun run karta hai — aur skew matrix kahan se aati hai

Figure s02 dekho. Ek spinning body pe position $r$ pe ek point ek circle sweep karta hai. Uski velocity spin axis $ω$ aur radius $r$ dono ke perpendicular hoti hai, size $ω r_{⊥}$ ke saath. Sirf woh vector operation jo "dono ke perpendicular, aise scaled" output deta hai woh hai cross product $ω \times r$ — dot product toh sirf ek number deta, direction of motion nahi. Skew matrix $[ω]_{\times}$ bas yahi cross product hai jo matrix ke roop mein pack kiya gaya hai taaki woh saare axes pe ek saath act kar sake: $[ω]_{\times} v = ω \times v, [ω]_{\times} = 0 ω_{z} - ω_{y} - ω_{z} 0 ω_{x} ω_{y} - ω_{x} 0 .$ $R_{b}^{w}$ ka har column world mein dekha hua ek body axis hai; ek chhoti time $d t$ mein woh axis $ω \times (axis)$ se move karta hai, jo exactly $R_{b}^{w} [ω_{b}]_{\times} d t$ hai. Teeno columns pe sum karo aur kinematic law $\dot{R}_{b}^{w} = R_{b}^{w} [ω_{b}]_{\times}$ milti hai: "har body axis spin ke saath sideways drag hota hai." Yahi us ODE ka poora geometric meaning hai.

True or false — justify karo

Ek accelerometer vehicle ki acceleration directly measure karta hai.

False. Yeh specific force

f = a - g

measure karta hai — spring push per unit mass — toh gravity hamesha baked in rehti hai aur true

a

paane se pehle use add back karna padta hai.

Ek table pe rakhaa hua stationary accelerometer zero read karta hai.

False.

a = 0

ke saath yeh

f_{b} = - g = (0, 0, - g)

read karta hai, magnitude

\approx 9.81

m/s²; kyunki

+ z

neeche hai, yeh vector upar point karta hai, table ki normal force sense karta hai, net (zero) acceleration nahi.

Ek bilkul free-fall mein girta hua accelerometer saare axes pe zero read karta hai.

True. Free fall mein

a = g

, toh

f = a - g = 0

— proof mass pe koi spring force act nahi karta; yeh exactly elevator/weightlessness case hai.

Gyroscope se akele "kaun sa direction neeche hai" pata kar sakte hain.

False. Gyro sirf rate of turning sense karta hai; gravity ki direction accelerometer se aati hai rest mein, jiska reading

f_{b} = - g

neeche ke opposite point karta hai.

Gyro ki angular rate ko ek baar integrate karne se orientation milti hai.

Principle mein True —

\dot{R}_{b}^{w} = R_{b}^{w} [ω_{b}]_{\times}

har body axis ko spin se drag karta hai aur rate ko attitude mein integrate karta hai — lekin koi bhi bias

b_{g}

ek growing angle error

b_{g} t

ki tarah leak hoti hai.

Gyroscope vehicle ka heading angle directly measure karta hai.

False. Yeh angular velocity

ω_{b}

(rad/s) measure karta hai; heading sirf us rate ko integrate karne se milti hai, yahi reason hai ki heading offset time ke saath accumulate hoti hai.

Ek stationary tilted IMU ka accelerometer output change ho jaata hai chahe woh accelerate nahi kar raha.

True. Fixed gravity reaction

- g

bas tilted body axes ke andar differently point karta hai, toh teeno readings redistribute ho jaate hain jabki magnitude

\approx g

rehti hai.

Constant accelerometer bias se position error time ke saath linearly badhta hai.

False. Ek constant

ε_{a}

ko do baar integrate karne se

\frac{1}{2} ε_{a} t^{2}

milta hai — quadratic — yahi reason hai ki ek tiny offset bhi ek minute baad bada ho jaata hai.

Gyro-tilt position error long times pe accelerometer-bias error se bhi faster badhta hai.

True. Gyro bias

b_{g}

frame ko tilt karta hai (

\propto t

), gravity ramp leak karta hai; do baar integrate karne se

\frac{1}{6} g b_{g} t^{3}

milta hai — cubic eventually quadratic ko beat karta hai.

Ek high-quality IMU bina kisi external help ke ghanton tak accurately navigate kar sakta hai.

False. Errors

t^{2}

(

ε_{a}

) aur

t^{3}

(

b_{g}

) ki tarah badhte hain, toh excellent units bhi ek minute mein meters drift kar jaate hain aur inhe external references ke saath fuse (GPS-Denied Navigation) karna padta hai.

Rotation matrix

R_{b}^{w}

aur gravity vector irrelevant hain agar sirf relative motion chahiye.

False. Bina

R_{b}^{w}

f_{b}

rotate kiye aur

g

subtract kiye, ek merely-tilted still sensor accelerating lagta hai, jo relative tracks ko bhi corrupt kar deta hai.

Error dhundho

"Position paane ke liye, bas accelerometer output ko do baar integrate karo — calculus guarantee karta hai

\iint a d t = p

Output specific force

f_{b}

hai,

a

nahi; pehle isse

R_{b}^{w}

se rotate karna hoga aur gravity add back karni hogi (

a_{w} = R_{b}^{w} f_{b} + g

), warna ek resting sensor

g

pe "accelerate" karta dikhega.

"IMU world frame mein acceleration read karta hai, toh koi rotation needed nahi."

Yeh

f_{b}

apne body frame mein read karta hai (Reference Frames — Body vs World); world-frame acceleration ke liye

a_{w} = R_{b}^{w} f_{b} + g

current attitude use karke chahiye.

"Ek resting tilted sensor ka

f_{b}

rotate back hoke

(0, 0, g)

ke alawa kuch deta hai, jo prove karta hai ki woh move kar raha hai."

Ek still sensor ke liye

f_{b} = - R_{b}^{w ⊤} g

hai, toh

R_{b}^{w} f_{b} = - g = (0, 0, - g)

hamesha hoga; kuch aur milna matlab galat

f_{b}

use kiya, yeh nahi ki box move kiya.

"Kyunki gravity constant hai, isse rotate karne se pehle body frame mein subtract kar sakte hain."

Gravity world frame mein constant hai

(0, 0, + g)

; tilted body frame mein iske components attitude ke saath change hote hain, toh pehle

f_{b}

ko world mein rotate karo, phir world gravity subtract karo. (Concretely:

3 0^{\circ}

tilt karo aur ek still box

f_{b} = (g sin 30, 0, - g cos 30)

read karta hai; sirf

R_{b}^{w}

ke baad hi yeh

(0, 0, - g)

pe collapse karta hai.)

"Skew matrix

[ω]_{\times}

ek dot product likhne ka fancy tarika hai."

Yeh cross product encode karta hai

[ω]_{\times} v = ω \times v

(Figure s02 dekho), jo rigid-body rotation ki perpendicular velocity produce karta hai — dot product ek scalar deta, motion nahi.

"0.01°/s gyro bias negligible hai — yeh basically zero hai."

Cubic

\frac{1}{6} g b_{g} t^{3}

ke through yeh

b_{g}

ek minute mein tens of meters ka position error produce karta hai, toh yeh decidedly not negligible hai.

"Hum gyro skip karke sirf accelerometer use kar sakte hain orientation jaanne ke liye jab move kar rahe hain."

Accelerating ke waqt,

f_{b} = a - g

— motion term gravity direction ko contaminate kar deta hai, toh aap ab isse "down" ke roop mein trust nahi kar sakte; gyro-driven

R_{b}^{w}

yeh gap fill karta hai.

Why questions

Accelerometer upar point karne wala vector kyun report karta hai jab woh accelerate nahi kar raha?

Kyunki woh spring/normal force sense karta hai jo proof mass ko gravity ke against hold karti hai:

f = a - g

mein

a = 0

hone par

- g

bachta hai, aur

- g

neeche ke opposite (upar) point karta hai.

Strapdown chain mein gyroscope data ko accelerometer data se pehle kyun process karna padta hai?

Attitude

R_{b}^{w}

jaanna zaroori hai taaki pata chale ki accelerometer ke body axes world mein kahan point karte hain, taaki

f_{b}

sahi se rotate ho sake aur gravity sahi direction mein add ho sake (Figure s03).

Cross product (koi aur operation nahi) rotation ki natural language kyun hai?

Position

r

pe ek spinning body ka point velocity

ω \times r

se move karta hai — axis aur radius dono ke perpendicular (Figure s02) — jo exactly rigid rotation hai (Rotation Matrices & Quaternions).

\dot{R}_{b}^{w} = R_{b}^{w} [ω_{b}]_{\times}

geometrically kyun hold karta hai?

R_{b}^{w}

ka har column world mein dekha hua ek body axis hai;

d t

mein spin us axis ko

ω \times (axis)

se drag karta hai, aur teeno drags ko ek matrix mein pack karne se exactly

R_{b}^{w} [ω_{b}]_{\times}

milta hai.

Gyro-tilt position error

t^{3}

ki tarah kyun scale karta hai jabki accelerometer bias

t^{2}

ki tarah?

Gyro bias

b_{g}

tilt ko linearly badhata hai, gravity ramp (

\propto t

) leak karta hai; ek integration se

t^{2}

velocity error aur doosre se

t^{3}

position error milta hai — constant

ε_{a}

se ek power zyaada.

IMU ko "inertial" aur self-contained kyun kaha jaata hai?

Yeh sirf internal proof masses aur rate sensors pe rely karta hai jo motion pe react karte hain, kisi external signal ki zaroorat nahi — aur yahi reason hai ki yeh drift karta hai, kyunki iske paas correct karne ke liye kuch nahi.

IMU ko GPS ya camera ke saath kyun fuse karte hain sirf better sensors use karne ki jagah?

Koi bhi sensor bias-free nahi hota (

ε_{a}, b_{g} \neq = 0

), aur integration kisi bhi residual bias ko unbounded

t^{2}

t^{3}

growth mein turn kar deta hai; ek external absolute reference ise Kalman filter ke zariye bound karta hai.

IMU ke saath Dead Reckoning seconds ke liye trustworthy kyun hai lekin minutes ke liye nahi?

Chhote spans mein drift terms (

t^{2}

t^{3}

) tiny hote hain, lekin inki steep growth jaldi dominate karti hai, toh IMUs short-term mein superb hain aur long-term mein hopeless.

Edge cases

Perfectly straight line mein constant speed se move karne wali body ke liye gyro kya read karta hai?

Zero angular rate (

ω_{b} = 0

) — bina kisi turning ke translation koi gyro signal produce nahi karta, chahe body clearly move kar rahi ho.

Usi straight, constant-speed motion ke liye accelerometer kya read karta hai?

Bas

- g

(jaise rest mein), kyunki

a = 0 \Rightarrow f = - g

; constant velocity proof mass ke liye standstill se indistinguishable hai, kyunki net force uski spring ko load nahi karta.

Ek ideal accelerometer projectile ke arc ke bilkul top pe kya read karta hai?

Saare axes pe zero — poori flight free fall hai (

a = g

), toh

f = a - g = 0

aur proof mass har instant pe unloaded hai (koi spring force nahi), apex included.

Kya ek accelerometer "pahaad pe khadi gaadi" aur "flat ground pe accelerating gaadi" mein fark kar sakta hai?

Akele nahi — tilt

- g

ko body axes ke across redistribute kar deta hai, aur horizontal acceleration same

f_{b}

produce kar sakta hai; sirf gyro-tracked

R_{b}^{w}

(ya fusion) ambiguity resolve karta hai.

Agar gyro momentarily exactly zero read kare jabki body actually spin kar rahi ho toh attitude ka kya hoga?

Integrated

R_{b}^{w}

freeze ho jaata hai aur true orientation se peeche reh jaata hai; woh attitude error phir har baad ke step mein

g

ko mis-point karta hai, toh

a_{w} = R_{b}^{w} f_{b} + g

leaked-gravity error inherit karta hai.

Equator vs pole pe, ek perfect stationary IMU identical gravity read karta hai?

Bilkul exactly nahi — Earth ki spin aur shape equator pe

g

ko slightly smaller banate hain aur tiny rotation-rate terms add karte hain, toh resting

f_{b} = - g

subtly differ karta hai aur ultra-precise units ko ise model karna padta hai.

Deep space mein, kisi bhi mass se door, ek hypothetical zero-bias, zero-noise IMU drop kiya gaya toh uska reading kya hoga?

Dono sensors zero read karte hain — koi spring force nahi (koi gravity ya contact nahi, toh

f = 0

) aur koi rotation nahi (

ω_{b} = 0

) — dead reckoning ko jo bhi initial velocity seed ki gayi thi woh preserve karne dete hain.

Recall Quick self-test

Teen drift growth rates jo kabhi confuse nahi karne chahiye ::: angle error $\propto t$ (gyro bias $b_{g}$ ), position-from-accel $\propto t^{2}$ (accel bias $ε_{a}$ ), position-from-tilt $\propto t^{3}$ (gyro-tilt leaking gravity). Woh ek sentence jo zyaadatar IMU mistakes fix kar deta hai ::: attitude pehle → $R_{b}^{w}$ se $f_{b}$ ko world mein rotate karo → gravity add back karo → do baar integrate karo.