IMU — integrated accelerometer + gyroscope

3.5.15 · Physics › Guidance, Navigation & Control (GNC)

IMU KYA hai?

YEH DONO SAATH KYUN? Gyro batata hai ki box kis orientation mein hai (yeh jaanna zaroori hai ki "down" aur "forward" kis direction mein hain). Accelerometer batata hai ki box ko kaise push kiya ja raha hai. Acceleration ko world-frame motion mein convert karne se pehle tumhe orientation chahiye — isliye dono sensors ek-doosre ke bina kaam nahi kar sakte.

Accelerometer specific force kyun measure karta hai, acceleration nahi

Yeh sabse zyada galat samjha jaane wala point hai. Ek accelerometer ek proof mass on a spring hota hai. Yeh woh force per unit mass report karta hai jo spring ko mass ko jagah par rakhne ke liye lagani padti hai.

Aao isse derive karte hain. Ek inertial frame mein proof mass yeh follow karta hai: $m a = F_{spring} + m g .$ Sensor specific force $f \equiv F_{spring} / m$ report karta hai. Solve karo: $f = a - g$

Gyroscope data orientation kaise banta hai (strapdown mechanization)

Gyro body-frame rate $ω_{b} = (ω_{x}, ω_{y}, ω_{z})$ deta hai. Hum rotation $R_{b}^{w}$ chahte hain jo body vectors ko world vectors mein map karta hai, aur yeh time ke saath evolve hota hai.

Propagation derive karo. $d t$ ke upar axis $ω$ ke around ek chota rotation kisi bhi body-fixed vector ko rotate karta hai. Rotation matrix ke liye kinematic law hai: $\dot{R}_{b}^{w} = R_{b}^{w} [ω_{b}]_{\times},$ jahaan skew matrix cross product ko encode karta hai $[ω]_{\times} v = ω \times v$ : $[ω]_{\times} = 0 ω_{z} - ω_{y} - ω_{z} 0 ω_{x} ω_{y} - ω_{x} 0 .$

Cross product kyun? Position $r$ par ek rotating body ka point $ω \times r$ velocity se move karta hai — yahi rigid-body rotation hai. Matrix ODE bas yeh sab axes ke liye ek saath pack karta hai.

Ek chote step ke liye, $R_{k + 1} = R_{k} exp ([ω_{b}]_{\times} Δ t) \approx R_{k} (I + [ω_{b}]_{\times} Δ t) .$

Poora IMU position dead-reckon kaise karta hai (strapdown integration)

Teen integrations ko chain karo:

gyro ω_{b} \int R_{b}^{w} a_{w} = R_{b}^{w} f_{b} + g v_{w} = \int a_{w} d t p_{w} = \int v_{w} d t

Figure — IMU — integrated accelerometer + gyroscope

Errors kyun explode karte hain: integration drift

Ek chota constant gyro bias $b_{g}$ (rad/s) ek angle error $b_{g} t$ ban jaata hai — time mein linear. Woh angle error gravity ko galat direction mein point karta hai, toh $g$ ka ek component $a_{w}$ mein leak ho jaata hai. Ek constant acceleration error $ε_{a}$ ko do baar integrate karne par position error $\frac{1}{2} ε_{a} t^{2}$ milta hai.

Worked Examples

Worked example 2 — True acceleration recover karna

Ek bilkul stationary IMU $y$ ke around $3 0^{\circ}$ tilt hai. Tilted body frame mein woh phir bhi gravity ke reaction ko hi sense karta hai, $f_{b} = R^{⊤} (0, 0, g) = (g sin 30, 0, g cos 30) = (4.905, 0, 8.496)$ m/s². World acceleration nikalo. World mein rotate karo: $f_{w} = R f_{b}$ . $30°$ about $y$ ke liye, $R = cos 30 0 - sin 30 010 sin 30 0 cos 30$ . $f_{w} = (4.905 \cdot 0.866 + 8.496 \cdot 0.5, 0, - 4.905 \cdot 0.5 + 8.496 \cdot 0.866) = (8.5, 0, 4.905)$ . Gravity add karo: $a_{w} = f_{w} + g = (8.5, 0, 4.905) + (0, 0, - 9.81) = (8.5, 0, - 4.905)$ . Ruko — yeh zero nahi hai! Kyun? Chosen $f_{b}$ ek truly-still tilted sensor ki correct reading nahi tha; correct still-reading hai $f_{b} = R^{⊤} (0, 0, g) = (g sin 30, 0, g cos 30) = (4.905, 0, 8.496)$ — jo vapas $f_{w} = (0, 0, 9.81)$ mein rotate hota hai aur $a_{w} = 0$ deta hai. Lesson: agar tum rotation skip karke naively $f_{b}$ ko world acceleration ki tarah read karo, toh ek merely-tilted-but-still sensor aisa lagta hai jaise woh accelerate kar raha hai. Hamesha pehle rotate karo, phir gravity subtract karo.

Common Mistakes

Flashcards

Ek accelerometer asal mein kya physical quantity measure karta hai?

Specific force

f = a - g

(spring force per mass), acceleration nahi.

Table par rest karte waqt accelerometer kya read karta hai?

+ g

upar ki taraf (magnitude 9.81 m/s²), kyunki yeh normal force sense karta hai.

Free fall mein accelerometer kya read karta hai?

Zero — proof mass par koi spring force act nahi karta.

IMU mein do core sensors aur har ek kya output karta hai?

Accelerometer → specific force

f_{b}

; Gyroscope → angular rate

ω_{b}

Accelerometer data se pehle gyro data process kyun karna padta hai?

Tumhe orientation

R_{b}^{w}

chahiye taaki specific force ko world frame mein rotate kar sako aur gravity vapas add kar sako.

Gyro rate se attitude propagation ODE?

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

(skew-symmetric cross-product matrix).

IMU se true world acceleration nikalne ki equation?

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

Constant gyro bias time ke saath position error ko kaise affect karta hai?

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

ki tarah badhta hai (tilt ek gravity-ramp leak karta hai, double-integrated).

Constant accelerometer bias position error ko kaise affect karta hai?

\frac{1}{2} ε_{a} t^{2}

ki tarah badhta hai.

IMUs ko hamesha GPS/vision ke saath kyun fuse kiya jaata hai?

Inertial drift unboundedly badhta hai (

t^{2}

t^{3}

); bahari references error ko bound karte hain.

Recall Feynman: 12-saal ke bachche ko explain karo

Socho ek phone ek blindfolded box mein hai. Uske paas do feelings hain. Ek feeling ("gyro") hai "main kitni tezi se ghoom raha hoon?" Doosri feeling ("accelerometer") hai "koi mujhe kitni zor se push kar raha hai?" — jaise woh push jo tumhe apni back mein lagti hai jab car speed up karti hai. Spins ka running total rakh ke, box hamesha jaanta hai woh kis taraf face kar raha hai. Phir woh jaanta hai kaunsa push sirf zameen usse thaam rahi hai (gravity) aur kaunsa push real movement hai. Real pushes add karne se pata chalta hai woh kitni tezi ja raha hai, aur unhe add karne se pata chalta hai woh kahan gaya — sab aankhein band karke! Catch yeh hai: har choti galti add hoti rehti hai, toh kuch der baad woh kho jaata hai, isliye kabhi-kabhi GPS se peek karta hai.

Connections

Sensor Fusion & Kalman Filter — IMU ke drift ko GPS/vision se bound karta hai
Rotation Matrices & Quaternions — attitude representation $R_{b}^{w}$
Dead Reckoning — double-integration idea generalized
Coriolis Effect — MEMS gyros physically $ω$ kaise sense karte hain
Reference Frames — Body vs World — woh rotation jo $f_{b}$ ko $a_{w}$ se jodta hai
GPS-Denied Navigation — jahaan IMUs indispensable hain