3.5.9 · D5 · HinglishGuidance, Navigation & Control (GNC)
Question bank — Quaternion kinematics — q̇ = ½ Ξ(q) ω
3.5.9 · D5· Physics › Guidance, Navigation & Control (GNC) › Quaternion kinematics — q̇ = ½ Ξ(q) ω
True or false — justify karo
mein half-angle factor wahan se aata hai jahan se kinetic energy mein aata hai.
False. Iska energy se koi lena-dena nahin. Quaternion store karta hai; half-angle ko chain rule se differentiate karne par bahar aata hai. Yeh purely trigonometry hai, dynamics nahin.
Rotation matrix ke liye kinematics mein nahin hai, isliye quaternion version "galat" hai aur use correct karna padega.
False. Dono correct hain; woh same rotation ko alag-alag tarike se parametrise karte hain. full angle store karta hai, store karta hai. exactly aur ko differentiate karne ka farq hai.
hamesha ke perpendicular hota hai (4-vectors ke roop mein).
True. Humne dikhaya ki kyunki . Perpendicular velocity ka matlab hai ki tum unit sphere ke saath slide karte ho, kabhi uss se door nahin jaate, isliye 1 rehta hai.
Agar ho toh , yaani attitude frozen hai.
True. kisi bhi ke liye. Koi angular velocity nahin matlab orientation mein koi change nahin — physically obvious hai, aur equation isko respect karta hai.
ek square matrix hai isliye se recover karne ke liye use invert kiya ja sakta hai.
False. hai (chaar rows, teen columns) — rectangular hai, invertible nahin. Tum ko se recover karte ho, use karke.
Kyunki aur same physical rotation represent karte hain, unka same hona chahiye.
False. , isliye . Dono trajectories sphere par mirror images hain; woh har instant par same attitude represent karte hain lekin quaternion space mein genuinely alag points/velocities hain (double cover).
Equation fixed par mein linear hai.
True. ke roop mein likhne par yeh clearly mein linear hai — ek linear ODE jisme (time-varying) coefficient matrix hai. Yahi linearity hai jo EKF exploit karta hai.
Equation fixed par mein bhi linear hai.
True. ke roop mein likhne par yeh mein linear hai. Dono statements hold karte hain; jis bhi variable ko tum "freeze" karo, tak ka map ek matrix multiply hai.
Error dhundho
"Body-frame gyro rates ke saath main integrate karta hoon."
Galat order. Body-frame ko post-multiply karna chahiye: . Left-multiplication inertial-frame ke liye hai. Inhe swap karne se tum silently galat attitude integrate karte ho — yeh ek classic flight-software bug hai.
"Quaternion product basically dot product plus cross product hai, isliye yeh commute karta hai."
Non-commutative. Vector part mein hai, aur cross product anti-symmetric hai. aur ko swap karne se us term ka sign flip ho jaata hai, isliye — bilkul waise jaise rotations ko alag order mein compose karo.
"Main ko RK4 se integrate karunga aur kabhi renormalize nahin karunga, kyunki math prove karta hai ki ."
Sirf continuous time mein exact hai. RK4 finite steps leta hai, isliye chhoti-chhoti errors ko sphere se thoda door push kar deti hain aur accumulate hoti hain. Tumhe har step mein renormalize karna chahiye.
"Angular-velocity quaternion hai."
Scalar part galat hai. Yeh pure quaternion hai — scalar part zero, vector part poora (unit axis nahin). Pure quaternion precisely woh hota hai jiska scalar part zero ho.
" hamesha zero hota hai kyunki norm preserve hoti hai."
Sirf identity par. ; yeh tab zero hota hai jab (identity ) ya jab , lekin generally nonzero hota hai. Norm preservation hai, jo poore 4-vector par ek condition hai, sirf par nahin.
" aur same size ke hain, isliye yeh interchangeable hain."
Alag shapes aur roles. hai aur se multiply karta hai; hai aur se multiply karta hai. Woh same dete hain () lekin interchangeable matrices nahin hain.
"Kyunki par depend karta hai, kinematics nonlinear hai aur linear filter mein use nahin ki ja sakti."
Galat variable freeze kar raha hai. Filter ke prediction step mein state hai aur ek (measured) input hai, isliye state mein linear hai. ki state-dependence yeh nahin todti.
Why questions
ek pure quaternion kyun hai, full unit quaternion kyun nahin?
Kyunki ek infinitesimal rotation describe karta hai. Iska half-angle hota hai, isliye aur identity se farq sirf vector part mein hai — scalar part first order mein kuch contribute nahin karta.
unit sphere ke tangent kyun hona chahiye?
Valid attitude ke liye zaroori hai ki har waqt rahe. Sphere par rehne wali motion ka radial velocity zero hona chahiye, yaani — warna drift karke sphere se door ho jaata hai aur phir rotation represent nahin karta (SO(3) membership lost ho jaati hai).
Hum yahan Euler angles ki jagah quaternions kyun use karte hain?
Euler-angle kinematics certain orientations par explode kar jaata hai (gimbal lock — ek division by jo zero ho jaata hai). Quaternion equation ek smooth polynomial hai, sphere par har jagah singularity-free.
Relation ko Hamilton product chhod kar matrix ke roop mein kyun package kiya jaata hai?
Hamilton product mein linear hai, aur har linear map ek matrix hota hai. Ise likhne se GNC code ko ek clean matrix–vector multiply milta hai jo seedha linear-algebra libraries aur Jacobians mein fit ho jaata hai.
kyun hold karta hai — dono middle terms ko cancel karne wali cheez kya hai?
Notation box se block form use karte hue, (yeh cancel ho jaate hain) aur saath mein , kyunki koi bhi vector khud se cross karke zero deta hai. Yahi algebraic identity norm preservation guarantee karti hai.
Pure- spin par nahin balki frequency par oscillate kyun karta hai?
Kyunki half-angle track karta hai: . Physical rotation abhi bhi par advance karta hai; quaternion bas uski aadhi rate par spin karta hai — yahi double-cover hai in action.
Edge cases
Jab identity par ho aur ho toh kya hoga?
reduce ho kar ban jaata hai, isliye . Scalar part wahan ka wahan rehta hai aur vector part ke saath badhna shuru ho jaata hai — rotation "turn on" ho raha hai.
Agar numerical error se drift karke, say, ho jaaye toh kinematics ka kya hoga?
Kyunki , map ke saath linearly scale karta hai, aur abhi bhi hold karta hai — isliye drift ka radial component equation khud kabhi correct nahin karta. Yeh jo bhi (galat) norm ho use preserve karta hai — isliye tumhe renormalize karna hi padega.
Kya formula bahut bade angular rates ke liye bhi kaam karta hai?
Continuous equation kisi bhi ke liye exact hai — yeh bas ek linear ODE hai. Dikkat numerical hai: bade rates ka matlab hai bade per-step rotation angles, isliye fixed-step integrators accuracy kho dete hain aur tumhe chhote steps ya closed-form exponential update chahiye.
Jab current rotation axis ke anti-parallel ho jaaye us instant mein kaisa dikhega?
aur ke saath, milta hai: scalar part badh raha hai, matlab body identity ki taraf un-rotate ho rahi hai. Math correctly motion ko reverse karta hai.
Kya unit sphere par koi aisa hai jahan kinematics singular ya undefined ho?
Koi nahin. chaar components mein ek polynomial hai — har jagah defined aur finite hai. Euler angles ke unlike, quaternion kinematics mein par kaheen bhi koi coordinate singularity nahin hai.
Agar do bodies same se opposite spins aur par hoon, toh unke kaise compare karenge?
Woh exact negatives honge: , mein linearity se. Ek sphere par aage slide karta hai, doosra same great-circle tangent ke saath peeche.
Recall Quick self-test
Woh ek fact jo sab kuch self-consistent banata hai ::: , isliye aur preserve hoti hai. Woh ek fact jo log code mein sabse zyada galat karte hain ::: Multiplication order — body rates (post-multiply) use karte hain, aur tumhe har integration step mein renormalize karna zaroori hai.