Foundations — Mechanization equations — integrating IMU to get position, velocity, attitude

3.5.16 · D1 · Physics › Guidance, Navigation & Control (GNC) › Mechanization equations — integrating IMU to get position, v

Is parent note Mechanization equations ko padhne se pehle, tumhe har woh symbol apna banana hoga jo woh tumhare samne phenkta hai. Yeh page har ek ko scratch se build karta hai, us order mein jisme woh ek doosre par depend karte hain.

0. Sabse basic idea: ek frame (directions ka ek set)

Navigation mein har cheez teen arrows ka ek set hai: "forward kaunsi taraf hai, right kaunsi taraf hai, neeche kaunsi taraf hai". Frame bas un teen arrows ka ek named choice hai.

Hum chaar frames use karte hain. Har ek ko arrows ki ek choti tripod samjho:

• Inertial frame ($i$) — ek tripod jo Earth ke saath spin nahi karta. Yeh door ke taaron ke relative fixed rehta hai. Yeh "honest" frame hai jahan Newton ka law $\text{force}=\text{mass}\times \text{acceleration}$ bina corrections ke kaam karta hai.
• Earth frame ($e$) — ek tripod jo planet se chipka hua hai; yeh Earth ke saath din mein ek baar spin karta hai.
• Navigation frame ($n$) — tumhari current jagah par ek tripod jo North, East, Down (NED) ki taraf point karta hai, ek right-handed set (N → E → D). Jab tum travel karte ho, yeh saath saath chalta hai aur dheere dheere tilt hota hai kyunki zameen curve karti hai.
• Body frame ($b$) — vehicle se bolted ek tripod: naak (forward), right wing, pait (neeche), yeh bhi right-handed. IMU yahan rehta hai.

1. Sub/superscript code padhna

Parent note ${\mathbf{\omega }}_{ib}^b$ aur $C_b^n$ jaise symbols se bhara hua hai. Yeh scary lagta hai; yeh actually ek tidy filing system hai.

2. Vectors aur unke components

Bold letters jaise $\mathbf{v}$, $\mathbf{f}$, $\mathbf{g}$ vectors ko denote karte hain. Navigation mein hum inhe zyaadatar teen ki stack mein likhte hain: ${\mathbf{v}}^n=\left[\begin{array}{c}{v}_N\\ v_E\\ v_D\end{array}\right]\text{(velocity North, East, Down).}$

Is topic ko vectors ki zaroorat hai kyunki velocity, force, aur gravity sabke paas size aur direction dono hain — ek akela number kabhi nahi keh sakta "North-East ki taraf aur thoda neeche 20 m/s".

3. Angular rate $\mathbf{\omega }$ — gyro ki language

Spinning ke liye vector kyun? Ek spin ko ek axis (ek direction) aur ek speed dono chahiye. Dono ko ek arrow mein pack karne se hum rotations par arithmetic kar sakte hain. Ek gyroscope ek sensor hai jo exactly yahi output karta hai: ${\mathbf{\omega }}_{ib}^b$, body ka inertial space ke relative spin, body axes mein likha gaya.

Ek radian natural angle unit hai: radius ke barabar ek arc sweep karo, woh 1 radian hai ($\approx 57.3^{\circ }$). Hum radians use karte hain kyunki arc-length $=$ radius $\times$ angle sirf radians mein cleanly kaam karta hai — aur wahi fact Layer 3 (position) ka poora base hai.

4. Specific force $\mathbf{f}$ — accelerometer ki language

Yahi akela fact hai ki parent note ko "software mein gravity wapas add" kyun karni padti hai. Sensor $\mathbf{f}$ deta hai; sahi motion ke liye $\mathbf{f}+\mathbf{g}$ chahiye.

5. Gravity vector $\mathbf{g}$

6. Cross product $\times$ — "ek spin point ko kaise move karta hai"

Parent note ka core rule $\dot{\mathbf{u}}=\mathbf{\omega }\times \mathbf{u}$ hai. Ise padhne ke liye, tumhe cross product chahiye.

Is topic ko yeh kyun chahiye: ek rotating rigid body apne saath attached har arrow ko circle mein le jaata hai. Cross product exactly woh formula hai jo batata hai ki us arrow ka har point kitni tezi se move karta hai. Isliye attitude propagation (Layer 1) isi se bani hai.

7. Skew-symmetric matrix $[\mathbf{\omega }\times ]$

Hum cross product ko ek saath teen arrows par apply karna chahte hain (rotation matrix ke teen columns). Ek neat trick "$\mathbf{\omega }$ ke saath cross" ko ordinary matrix multiply mein badal deti hai.

Bother kyun karo? Kyunki $\dot{C}=C[\mathbf{\omega }\times ]$ phir attitude matrix ke saare teen axes ko ek clean line mein handle karta hai.

8. Direction-cosine matrix $C_b^n$ (attitude)

Picture: $C_b^n$ ka har column ek body axis hai (naak, wing, pait) NED numbers mein drawn. Columns padho aur tumhe pata chal jaata hai ki vehicle exactly kaise tilt hua hai.

9. Derivative aur integral — "rate abhi" vs "sab kuch add karo"

Do calculus ideas poori recipe ko jodti hain. Dono simple pictures hain.

Yeh mechanization ka dil kyun hai: IMU sirf rates report karta hai (spin rate, aur — gravity fix karne ke baad — acceleration). Orientation, velocity, aur position paane ke liye hum integrate karte hain: rate → total, teen baar. Har integration parent note ki ek "layer" hai (Layer 1, 2, 3).

10. Earth ki spin aur transport rate

Do chote rotation vectors ek spinning, curved planet par rehne ke liye correct karte hain.

• ${\mathbf{\omega }}_{ie}$ — Earth-spin rate, ${\Omega }_e\approx 7.292\times 10^{-5}\text{rad/s}$ (ek sidereal day mein ek chakkar). Isliye honest inertial frame aur Earth frame agree nahi karte.
• ${\mathbf{\omega }}_{en}$ — transport rate: jab tum round Earth par travel karte ho, tumhara local "Down" dheere dheere tip hota hai Earth ke centre ki taraf point karte rehne ke liye. Nav tripod khud rotate karta hai. Iska size speed aur Earth ke radii of curvature par depend karta hai.

Milke yeh Coriolis / transport corrections produce karte hain jo tumhe velocity equation (Layer 2) mein dikhti hain. Deeper story: Coriolis and Centrifugal Effects. IMU box bolted hai ya gimbal par, yeh affect karta hai ki yeh kaise enter hote hain — dekho Strapdown vs Gimbaled INS.

Prerequisite map

Equipment checklist

Khud test karo — right side cover karo aur zor se jawab do.