Question bank — Euler angles — roll φ, pitch θ, yaw ψ; rotation sequence (3-2-1 convention)

False. Naam axes ko neeche count karta hai: 3 (yaw about $z$) pehle apply hota hai, phir 2 (pitch about $y$), phir 1 (roll about $x$). Roll last apply hota hai.

False. Different axes ke baare mein rotations commute nahi karte: $R_x{R}_y{R}_z\ne R_z{R}_y{R}_x$. Physically, yawing-then-pitching nose ko alag jagah le jaata hai, jaisi ki pitching-then-yawing se hoti hai.

Is sense mein True ki dono same rotation describe karte hain, lekin ye opposite directions mein map karte hain: $R_n^b$ nav→body leta hai, jabki $(R_n^b{)}^T=R_b^n$ body→nav leta hai. Galat wala use karne se rotation sense flip ho jaata hai.

False. Orthogonality (columns orthonormal) 6 constraints impose karta hai, sirf 3 degrees of freedom bacha ke — isliye 3 Euler angles kaafi hain.

False. $\theta =\pm 90^{\circ }$ par (gimbal lock) infinitely many $(\varphi ,\psi )$ pairs same orientation dete hain — sirf sum ya difference $\varphi \pm \psi$ determine hota hai, isliye wahan map one-to-one nahi hai.

False. Ye ek doosre ke transposes hain. Passive (frame rotate karna) $-\sin$ ko lower-left mein rakhta hai; active (vector rotate karna) $-\sin$ ko upper-right mein rakhta hai. Same angle, opposite off-diagonal sign.

False. Rotation angles mein nonlinear hai — $R(2\varphi ,2\theta ,2\psi )$ generally nahi hai $R(\varphi ,\theta ,\psi {)}^2$, kyunki teen spins different, sequentially-defined axes ke baare mein hain.

Galat direction. $R_n^b$ nav→body map karta hai. Ek body vector ko nav mein bhejna ho toh $R_b^n=(R_n^b{)}^T$ chahiye. Yahan $R_n^b$ apply karne se reverse rotation milti hai.

Galat. $\arcsin$ sirf $[-90^{\circ },+90^{\circ }]$ mein values return karta hai. Ye actually pitch ke liye sahi hai (physically pitch usi range mein rehta hai), lekin roll ya yaw ke liye $\arcsin$ use nahi kar sakte — unke liye $atan2$ chahiye jo chaaon quadrants cover kare.

Galat. $R_{13}=-\sin \theta$ lekin $R_{33}=\cos \varphi \cos \theta$ hai, isliye wo ratio $\varphi$ se contaminated hai. Pitch cleanly sirf $R_{13}$ akele se aata hai: $\theta =-\arcsin (R_{13})$.

Galat jab signs matter karte hain. Plain $\arctan$ quadrants collapse karta hai (har $180^{\circ }$ par repeat karta hai), isliye ye $\varphi$ aur $\varphi +180^{\circ }$ mein distinguish nahi kar sakta. $atan2(R_{23},R_{33})$ use karna zaroori hai, jo doono signs alag se padhta hai.

Galat. Gimbal lock Euler parametrization ki ek structural singularity hai: $\theta =\pm 90^{\circ }$ par do axes physically coincide ho jaate hain aur ek DOF genuinely khatam ho jaata hai. Koi bhi precision help nahi karti; representation switch karna padta hai (quaternions/DCM).

Product mein galat order. Applied-first right side par jaata hai taaki wo vector ko pehle hit kare: $R_n^b=R_x(\varphi )R_y(\theta )R_z(\psi )$. Intrinsic 3-2-1 sequence, extrinsic product mein collapse hoti hai jisme $R_z$ rightmost hota hai.

Kyunki teen angles minimal, human-readable, aur pilot intuition se match karte hain (heading, nose-up, bank). Matrix redundant hai (9 numbers, 6 constraints); angles compact 3-DOF core hain.

Kyunki ye mirror karta hai ki ek vehicle ko naturally kaise describe kiya jaata hai: pehle nose point karo (yaw/heading), phir usse uthao (pitch), phir wings bank karo (roll). Baad ka har spin us axis ke baare mein hai jo pehle ke spins ne already establish kar diya.

Kyunki frame ko $+\alpha$ se rotate karne par ek fixed vector ke coordinates, active (vector) rotation ke transpose se transform hote hain. Transpose karne se off-diagonal sines swap hote hain, negative sign diagonal ke neeche chala jaata hai.

Har ratio ke numerator aur denominator dono mein ye common hai: $R_{23}/R_{33}=(s\varphi c\theta )/(c\varphi c\theta )=\tan \varphi$, aur similarly yaw ke liye bhi. Shared $\cos \theta$ divide out ho jaata hai — siwa jab $\cos \theta =0$, jo exactly gimbal lock hai.

Quaternions rotation ko ek smooth 4-number sphere par parametrize karte hain jisme koi coordinate singularities nahi hain, isliye koi bhi orientation kabhi DOF lose nahi karta. Euler angles teen coordinate charts hain jo inevitably $\theta =\pm 90^{\circ }$ par fold ho jaate hain.

Agile vehicles (fighters, rockets, drones doing loops) ke liye seedha upar point karna ek real, required attitude hai, koi edge nahi jise dodge kiya jaye. Usse faithfully represent karna zaroori hai, jo quaternions ya DCM jaisi singularity-free scheme ko force karta hai.

Identity: body frame perfectly aligned nav frame ke saath ($R_n^b=I$). Nose North, right wing East, belly Down.

Roll axis ($x_b$) aur yaw axis ($z_n$) align ho jaate hain, isliye $\varphi$ mein change aur $\psi$ mein change same physical motion produce karte hain. Sirf unka combination observable hai — ek DOF khatam ho jaata hai (gimbal lock).

Haan — angles $360^{\circ }$ period ke saath periodic hain, isliye identical rotation matrices dete hain. Ye periodicity (gimbal lock nahi) ek harmless, expected non-uniqueness hai.

Ye different angle paths se same physical attitude represent kar sakte hain — Euler angles ki ek known non-uniqueness jo gimbal lock se door bhi hoti hai. Isliye DCM se angles extract karte waqt ek fixed convention chahiye (jaise $\theta \in [-90^{\circ },90^{\circ }]$).

Ye body ka Down axis nav coordinates mein express hua hai; 1 se kam $z$-component matlab vehicle level se tilted hai. Ek orthogonal matrix ki row ke roop mein ye ek unit vector hai, isliye uski magnitude hamesha exactly 1 hoti hai.

Horizontal plane mein North aur East ke beech mein. Koi pitch/roll nahi hone se nose level rehta hai, aur $+45^{\circ }$ yaw usse East ki taraf half a right angle se swing karta hai.