Visual walkthrough — Mechanization equations — integrating IMU to get position, velocity, attitude

3.5.16 · D2 · Physics › Guidance, Navigation & Control (GNC) › Mechanization equations — integrating IMU to get position, v

Kisi bhi equation se pehle, aao hum un do shabdon par agree kar lete hain jis par poora subject tika hua hai.

Step 1 — Proof mass: "specific force" kaisa dikhta hai

KYA. Har accelerometer ke andar ek tiny weight hota hai (ek proof mass) jo springs se hold hota hai. Hum spring stretch ko padhte hain.

KYUN yahan se shuru karein. Baad ki sab cheez ek fact par bani hai — accelerometer spring squeeze report karta hai, kabhi gravity nahi. Agar tum yeh galat samjho, toh har baad ka integral kharab ho jaata hai. Isliye hum mass ko teen situations mein dekhte hain aur uski spring padhte hain.

PICTURE. Teen panels dekho.

• Left (table par): table mass ko upar push karta hai. Spring upar compress hoti hai. Reading upar point karti hai size $g$ ke saath. Bhale hi kuch move nahi kar raha, accelerometer ek force report karta hai.
• Middle (free fall): mass ko sirf gravity touch karti hai, jo mass aur case dono ko saath kheenchti hai. Spring relax hoti hai → reading zero hai, bhale hi box neeche tezi se accelerate kar raha ho.
• Right (thrust): engine case ko aage push karta hai; mass peeche reh jaata hai, spring ko backward squeeze karta hai. Reading aage point karti hai.

Step 2 — Teen frames of reference (woh stage jis par drama hota hai)

KYA. Hum teen coordinate systems ("frames") naam dete hain. Ek frame sirf teen arrows ka set hai jo kisi cheez se fixed hain, jiske against hum vectors measure karte hain.

KYUN. Gyro body axes mein bolta hai, gravity nav axes mein rehti hai, aur Earth inertial frame mein spin karti hai. Hum alag-alag frames mein measured numbers add nahi kar sakte — pehle hume translate karna hoga. Isliye hume unhe naam dena hoga.

PICTURE. Teen sets of axes:

Step 3 — Gyro rate ko ek spinning matrix mein convert karna

KYA. Gyro ek angular rate deta hai. Hume "main kitni tezi se turn kar raha hoon" ko "meri orientation matrix kaise change ho rahi hai" mein convert karna hoga.

KYUN ek matrix aur sirf teen angles nahi? Rotations ordinary numbers ki tarah add nahi hote — right-then-up turn karna up-then-right se alag hota hai. Orientation ek curved space par rehta hai, isliye hum poori $3\times 3$ matrix $C_b^n$ track karte hain aur poochte hain uske har column kaise move karta hai.

PICTURE. Ek single body vector $\mathbf{u}$ jo rotation se carry hota hai:

Red arrow $\mathbf{u}$ dekho. Angular rate $\mathbf{\omega }$ (green) par spin karte hue, $\mathbf{u}$ ki tip ek circle par move karti hai. Iska velocity $\mathbf{\omega }$ aur $\mathbf{u}$ dono ke perpendicular hai — exactly wahi jo ek cross product produce karta hai:

$\dot{\mathbf{u}}=\mathbf{\omega }\times \mathbf{u}$

• $\dot{\mathbf{u}}$ — arrow ki tip kitni tezi se move karti hai (blue tangent).
• $\mathbf{\omega }\times \mathbf{u}$ — cross product: ek vector deta hai jo $\mathbf{\omega }$ aur $\mathbf{u}$ ke plane ke perpendicular hai, length $|\mathbf{\omega }||\mathbf{u}|\sin \theta$ ke saath. Isliye hum cross product choose karte hain: yeh woh ek operation hai jiska output ek spinning arrow ka sideways swing hai.

$C_b^n$ ka har column nav se dekha ek fixed body axis hai, isliye har column iska palan karta hai. Teen columns ko stack karte hue:

Dekho Direction Cosine Matrix and Quaternions ki kyun matrix ek valid rotation rehti hai, aur Strapdown vs Gimbaled INS ki kyun hum software mein integrate karte hain ek spinning platform ki jagah.

Step 4 — Kyun raw gyro rate woh nahi hai jo hum chahte hain

KYA. Gyro rotation taaron ke versus measure karta hai (${\mathbf{\omega }}_{ib}^b$), lekin Step 3 ko nav frame ke versus rotation chahiye (${\mathbf{\omega }}_{nb}^b$). Hume frame ki apni motion subtract karni hogi.

KYUN. NED frame khud do reasons se turn kar raha hai: Earth tumhare neeche spin karti hai, aur "down" tilt hoti hai jab tum round planet par travel karte ho. Agar hum yeh ignore karein, toh equator par ek bilkul still IMU slowly roll karte hue dikhega.

PICTURE. Nav frame quietly rotate karta hua:

${\mathbf{\omega }}_{nb}^b={\mathbf{\omega }}_{ib}^b-C_n^b(\underset{\text{Earth spin}}{\underbrace{{\mathbf{\omega }}_{ie}^n}}+\underset{\text{transport}}{\underbrace{{\mathbf{\omega }}_{en}^n}})$

• ${\mathbf{\omega }}_{ib}^b$ — raw gyro (body vs inertial).
• ${\mathbf{\omega }}_{ie}^n$ — Earth ki spin, ${\Omega }_e\approx 7.292\times 10^{-5}\text{rad/s}$, nav axes mein dekha gaya.
• ${\mathbf{\omega }}_{en}^n$ — transport rate: "down" ka tilt jab tum fly karte ho. Dekho Radii of Curvature of the Earth Ellipsoid.
• $C_n^b(\cdots )$ — hum un nav-frame rates ko body axes mein rotate karte hain taaki unhe body-frame gyro se subtract kiya ja sake. Kyun $C_n^b$? Kyunki tum sirf same frame mein likhe vectors subtract kar sakte ho.

Step 5 — Velocity banana: felt force ko rotate karo, phir use heal karo

KYA. Ab hamare paas $C_b^n$ hai. Hum felt force ko nav axes mein rotate karte hain aur do cheezein fix karte hain: gravity (jo accelerometer se miss hui) aur yeh fact ki hamar frame rotate kar raha hai.

KYUN. Newton ka law sirf inertial frame mein clean hai. Use rotating nav frame mein rewrite karne par correction terms add ho jaate hain — physically kuch naya nahi ho raha, hum sirf use ek spinning viewpoint se describe kar rahe hain.

PICTURE. Force assembly line:

Step 6 — Static sanity check (degenerate case)

KYA. Box ko flat aur still rakho. Kuch move nahi hona chahiye.

KYUN. Ek correct equation ko zero motion ke liye zero motion dena chahiye. Yahi acid test hai.

PICTURE. Down-axis ke do arrows cancel karte hue:

NED mein, "Down" $+z$ hai. Table push ${\mathbf{f}}^b=(0,0,-9.81)$ register karti hai (support upar hai, isliye specific force upar hai = negative Down). Gyros $0$ read karte hain, velocity $0$ hai isliye Coriolis vanish hota hai:

${\dot{\mathbf{v}}}^n=(0,0,-9.81)+(0,0,+9.81)=0✓$

• Blue arrow: rotated felt force $C_b^n{\mathbf{f}}^b=(0,0,-9.81)$ (upar point karta hai).
• Green arrow: ${\mathbf{g}}^n=(0,0,+9.81)$ (neeche point karta hai).
• Yeh exactly cancel hote hain → vehicle wahan ka wahan rehta hai. Attitude bhi hold karta hai kyunki $\dot{C}=C[0\times ]=0$.

Step 7 — Curved Earth par Position

KYA. Velocity ko changing latitude, longitude, height mein convert karo.

KYUN radius se divide karte hain? Ek arc length radius times angle ke barabar hoti hai, isliye angle rate = arc-length rate ÷ radius. 1 km north move karne se latitude zyada change hoti hai agar Earth choti hai, kam agar badi hai — isliye division.

PICTURE. North speed ek circle par latitude change ban rahi hai:

Ek-picture summary

Bayi taraf se do raw feelings enter karti hain; teen integrations rightward flow karte hain; gravity aur Earth-rotation corrections exactly wahan inject ki jaati hain jahan zaroori hain.