Foundations — IMU — integrated accelerometer + gyroscope

3.5.15 · D1 · Physics › Guidance, Navigation & Control (GNC) › IMU — integrated accelerometer + gyroscope

Parent note mein ek bhi equation padhne se pehle, aapko woh alphabet chahiye jisme woh equations likhi hain. Yeh page har symbol aur idea introduce karta hai, ek aisi tartib mein jahan har ek cheez sirf pehle waalon se bani ho. Kuch bhi assume nahi kiya gaya. Agar ek smart 12-saal-ka baccha line one se padhe, toh usse kabhi wall nahi milni chahiye.

1. Vectors — ek arrow jisme length bhi hai aur direction bhi

Parent note ki sabse pehli cheez hai ${\mathbf{f}}_b$, bold mein. Bold letters ordinary numbers nahi hote. Ek plain number jaise $9.81$ sirf ek size hai. Ek bold letter ek vector hai: ek arrow jisme length (kitna) aur direction (kis taraf) dono hote hain.

IMU ko yeh kyun chahiye. Ek push ka direction hota hai — aapko sideways ya neeche push kiya ja sakta hai. Gravity ek specific taraf point karti hai (neeche). "Upar kaun sa taraf hai" ek direction hai. Yeh koi bhi single number nahi ho sakta; yeh sab arrows hone chahiye. Isliye poora subject vectors mein likha gaya hai.

2. Axes aur right-hand rule — teen directions ko naam dena

"$x$-axis ke along" kehne ke liye hume teen fixed reference directions chahiye jo ek doosre ke relative mein nahi hilte. Hum inhe $x$, $y$, $z$ kehte hain.

Mirror-image confusion se bachne ke liye (kya "upar" $+z$ hai ya $-z$?), engineers right-hand rule se handedness fix karte hain: apni right index finger ko $x$ ki taraf point karo, middle finger ko $y$ ki taraf, aur thumb $z$ ki taraf point karega. Yahi haath baad mein positive spin direction bhi batata hai, isliye ise ek baar yahan seekh lo.

IMU ko yeh kyun chahiye. Sensor har reading mein teen numbers report karta hai. Agreed axes ke bina, "pehla number" meaningless hai. Parent note mein har ${\mathbf{\omega }}_b=({\omega }_x,{\omega }_y,{\omega }_z)$ ka matlab hai "$x$ ke baare mein spin, $y$ ke baare mein spin, $z$ ke baare mein spin."

3. Teen cheezein jo ek number yahan measure kar sakta hai: force, acceleration, velocity, position

Parent note mein $\mathbf{f}$, $\mathbf{a}$, $\mathbf{v}$, $\mathbf{p}$, $\mathbf{g}$ freely mix hote hain. Yeh chaar alag-alag tarah ki quantities hain. Inhe confuse karna poore topic mein #1 error hai.

IMU ko yeh kyun chahiye. Poora navigation trick ek chain hai: jo aap feel karte hain ($\mathbf{f}$) use true acceleration ($\mathbf{a}$) mein badlo, use velocity ($\mathbf{v}$) mein add karo, us ko position ($\mathbf{p}$) mein add karo. Chaar alag quantities, chaar alag meanings.

4. Rate of change aur dot — $\dot{R}$ aur "$\int$" ka kya matlab hai

Parent note mein ${\dot{R}}_b^w$, ${\dot{\mathbf{v}}}_w$, aur integral signs $\int$ likhte hain. Yeh ek hi idea ke do halve hain.

IMU ko yeh kyun chahiye. Ek gyro ek rate deta hai (angle ka $\dot{}$). Ek accelerometer position ki rate of rate feed karta hai. Orientation aur position paane ke liye, box ko integrate karna padta hai — chhote pieces ko add karna. Dono symbols unavoidable hain.

5. Reference frames — body frame vs world frame

Subscripts $b$ aur $w$ (jaise ${\mathbf{f}}_b$, ${\mathbf{p}}_w$, $R_b^w$ mein) yahan ka sabse tricky idea hai, isliye hum dhheere chalte hain.

IMU ko yeh kyun chahiye. Sensor "body" bolti hai. Navigation answer "world" hona chahiye. Dono ke beech translate karna rotation object $R_b^w$ ka ek hi kaam hai (agla section). Yeh translation skip karo aur ek sirf-tilted, stationary sensor aisa lagega jaise woh ud raha ho — exactly woh trap jo parent note ke Worked Example 2 mein hai.

6. Rotation matrix $R_b^w$ — frames ke beech translator

IMU ko yeh kyun chahiye. Yeh object gyro ko integrate karke banta hai, aur yahi cheez hume felt push ${\mathbf{f}}_b$ ko world acceleration mein rotate karne deti hai. Yeh poore method ka hinge hai.

7. Angular velocity $\mathbf{\omega }$ aur cross product $\times$

IMU ko yeh kyun chahiye. Gyro ke spin-rate ko "orientation kaise change hoti hai" mein turn karna hi cross product hai, parent note mein skew matrix $[\mathbf{\omega }{]}_{\times }$ ke roop mein packaged.

8. Radians aur degrees — angle ko honestly measure karna

Parent dono $30^{\circ }$ aur rad/s use karta hai, aur $0.01°/\text{s}=1.75\times 10^{-4}$ rad/s convert karta hai.

9. Bias aur error — kyun kuch bhi perfect nahi hai

Parent $b_g$ (gyro bias) aur ${\epsilon }_a$ (accel error) introduce karta hai.

Yeh foundations topic ko kaise feed karte hain

Equipment checklist

Khud ko test karo — aap parent note ke liye ready hain sirf tab jab har reveal wahi ho jo aap kehte.