Question bank — Modified Rodrigues parameters — singularity-free, compact

False. Inki ek genuine singularity hai $\Phi =360^{\circ }$ par (jahan scalar part $q_0=-1$ hota hai); baat sirf yeh hai ki yeh avoidable hai sets switch karke, jabki quaternions aur DCMs mein koi singularity hi nahin hoti.

False. Yeh classical Gibbs vector hai. MRPs quarter-angle use karte hain, isliye $\Vert \mathbf{\sigma }\Vert =\tan (\Phi /4)$ — yahi extra halving precisely hai jo blow-up ko $180^{\circ }$ se badhakar $360^{\circ }$ tak le jaati hai.

False. Har attitude ke do MRP sets hote hain (original $\mathbf{\sigma }$ aur uska shadow ${\mathbf{\sigma }}^S$), kyunki $q$ aur $-q$ same rotation describe karte hain aur har ek apne MRP par map hota hai.

False. Switch sirf coordinates ka ek change hai: $\mathbf{\sigma }$ aur ${\mathbf{\sigma }}^S$ identical physical rotation encode karte hain, bas do quaternion sign branches se. Kuch bhi physical move nahin hota.

False. Yeh $\frac{1}{4}$ hai. Quaternions half-angle carry karte hain ($\frac{1}{2}$); MRPs quarter-angle carry karte hain, isliye ek doosri halving $\dot{\mathbf{\sigma }}=\frac{1}{4}B(\mathbf{\sigma })\mathbf{\omega }$ mein appear hoti hai.

True. MRPs 3 numbers use karte hain jabki DCM ke 9 (6 constraints ke saath) aur quaternion ke 4 (unit-norm constraint ke saath), jo MRPs ko ek poore turn par sabse compact smooth description banata hai.

True. $\mathbf{\sigma }=0$ par identity term ka factor $(1-{\sigma }^2)=1$ hota hai, aur cross-product term $2[{\mathbf{\sigma }}^{\times }]$ aur outer-product term $2\mathbf{\sigma }{\mathbf{\sigma }}^{\top }$ dono vanish ho jaate hain (unke paas $\mathbf{\sigma }=0$ ka factor hai), jo $B={\mathbb{I}}_3$ chhod jaata hai.

True. $\Vert \mathbf{\sigma }\Vert =1$ matlab $\Phi =180^{\circ }$ hai, jo $360^{\circ }$ blow-up tak bilkul aadha raasta hai; $1$ par ya neeche rehna (sets switch karke) tumhe well-behaved region mein rakhta hai.

True. Kyunki ${\mathbf{\sigma }}^S$ ka magnitude $\tan ((360^{\circ }-\Phi )/4)$ hai, $\Phi =300^{\circ }$ par ek attitude ko chhoti rotation $60^{\circ }$ ke saath re-describe kiya jaata hai.

Error: $\Vert \mathbf{\sigma }\Vert =10$ par tum $\Phi =4\arctan (10)\approx 337^{\circ }$ par ho, jo $360^{\circ }$ singularity aur numerical blow-up ke khatarnaak qareeب hai. Tumhe shadow set par switch kar lena chahiye tha jab $\Vert \mathbf{\sigma }\Vert$ $1$ se zyada ho gaya.

Error: Gibbs vector $\hat{e}\tan (\Phi /2)$ hai ($180^{\circ }$ par singular); MRP $\hat{e}\tan (\Phi /4)$ hai ($360^{\circ }$ par singular). Alag half-angle, alag singularity — dekho Classical Rodrigues parameters (Gibbs vector).

Error: $B$ nearly orthogonal hai, $B^{\top }B=(1+{\sigma }^2{)}^2\mathbb{I}$, isliye $B^{-1}=B^{\top }/(1+{\sigma }^2{)}^2$ — ek closed-form transpose-and-scale, koi numerical matrix inversion needed nahin.

Error: $\Vert \mathbf{\sigma }\Vert >1$ simply matlab hai $\Phi >180^{\circ }$ — ek bilkul valid rotation. Yeh ek cue hai sets switch karne ka, bad data ka sign nahin.

Error: Euler angles gimbal lock mein phans jaate hain jab middle angle $\pm 90^{\circ }$ ho, jo normal operating range ke andar hi hai. MRPs apni (avoidable) singularity ko poore $360^{\circ }$ tak push kar dete hain — kaafi better deal hai.

Error: ye sab rotations ko cover karte hain sivaay $\Phi =360^{\circ }$ ke, aur covering two-to-one hai (original plus shadow), one-to-one nahin. "Teen unconstrained numbers" ka matlab global chart nahin hota.

Scalar part $q_0$ se divide karne par Gibbs vector milta hai, jo $\Phi =180^{\circ }$ par blow up ho jaata hai jahan $q_0=0$ hota hai. $(1+q_0)$ sirf $q_0=-1$ par vanish hota hai, yaani $\Phi =360^{\circ }$ par, jo trouble ko do guna door push kar deta hai.

Tangent tab blow up hota hai jab uska argument $90^{\circ }$ tak pahunch jaata hai. $\tan (\Phi /2)$ ke liye yeh $\Phi =180^{\circ }$ hai; $\tan (\Phi /4)$ ke liye yeh $\Phi =360^{\circ }$ hai. Argument ko aadha karne se blow-up tak pahunchne wala angle do guna ho jaata hai.

Yeh $-\mathbf{q}$ (doosra quaternion branch) ko MRP definition mein daalne se aata hai, jo algebraically $-\mathbf{\sigma }/({\mathbf{\sigma }}^{\top }\mathbf{\sigma })$ deta hai. Plain $-\mathbf{\sigma }$ ek genuinely alag rotation hogi.

$\Vert \mathbf{\sigma }\Vert =1$ par original aur shadow magnitudes equal hote hain, isliye wahan switch karna tumhe strictly $\Vert \cdot \Vert \le 1$ region mein rakhta hai $360^{\circ }$ blow-up se maximum margin ke saath.

MRP space curved hai; identity se door ek raw $\mathbf{\omega }$ ko local MRP coordinates mein rotate aur scale karna padta hai. $B$ exactly yahi karta hai — simple $\frac{1}{4}\mathbf{\omega }$ sirf $\mathbf{\sigma }=0$ ke paas hold karta hai.

Ye minimal hain (3 numbers, koi constraint enforce nahin karna), poore $\pm 180^{\circ }$ working range par smooth hain, aur inki kinematics closed form mein invert hoti hai — isliye control errors aur feedback laws practice mein simple aur singularity-free rehte hain.

Do signs do MRP sets dete hain; ek annoyance hone ki jagah, yahi pair exactly hai jo tumhe shadow set par hop karne aur singularity dodge karne deta hai.

$\mathbf{\sigma }=0$, kyunki $\tan (0/4)=0$; axis undefined hai par irrelevant hai kyunki ise zero se multiply kiya jaata hai.

$\Vert \mathbf{\sigma }\Vert =\tan (45^{\circ })=1$ (finite aur well-behaved), jabki Gibbs vector $\tan (90^{\circ })$ blow up ho jaata hai — yahi woh case hai jahan MRPs clearly jeette hain.

$\Vert \mathbf{\sigma }\Vert =\tan (\Phi /4)\to \infty$ (singularity), jabki shadow $\Vert {\mathbf{\sigma }}^S\Vert =\tan ((360^{\circ }-\Phi )/4)\to 0$ — isliye tum attitude ko hamesha finite shadow se describe kar sakte ho.

Nahin — $\Phi =360^{\circ }$ same hai jaise $\Phi =0^{\circ }$ (ek full turn identity hai), isliye shadow ${\mathbf{\sigma }}^S=0$ par land karta hai, perfectly finite. Practice mein tum original ko singularity tak pahunchne hi nahin dete.

Yeh undefined hai ($\Vert \mathbf{\sigma }{\Vert }^2=0$ se division), jo scalar part $q_0=-1$ ke saath correspond karta hai — shadow branch ka singular attitude. Yeh harmless hai kyunki tum sirf tab switch karte ho jab $\Vert \mathbf{\sigma }\Vert >1$ ho, kabhi $0$ par nahin.

Yeh ${\mathbb{I}}_3$ reduce ho jaata hai, isliye $\dot{\mathbf{\sigma }}\approx \frac{1}{4}\mathbf{\omega }$ — linearized, small-angle limit jo quarter-angle intuition se match karta hai.

Ye shadow partners hain jo same physical attitude describe karte hain; raw numbers compare karna galti se ek bada error suggest karega, isliye ek controller ko yeh pairing recognize karni chahiye.