Foundations — Converting between DCM, quaternions, Euler angles

3.5.10 · D1 · Physics › Guidance, Navigation & Control (GNC) › DCM, quaternions, Euler angles ke beech convert karna

Isse pehle ki aap representations ke beech convert kar sako, aapko raw ingredients mein fluent hona chahiye. Hum unhe dependency order mein banate hain: arrows → frames → do frames ko link karne wale numbers → un numbers ko package karne wali matrix → axis-angle idea → half-angle quaternion → angles wापस padhne ke trig tools.

1. Ek vector $\vec{v}$ — ek arrow jiske paas length aur direction hai

Figure s01 se kya samajhna hai: burnt-orange arrow $\vec{v}$ hai. Iska floor par shadow (teal dashes) aapko $v_x$ aur $v_y$ deta hai; iska vertical rise (plum dash) $v_z$ deta hai. Lesson yeh hai: woh teen shadows HI teen numbers hain — kuch bhi mysterious nahin, bas "har direction ke along kitna door".

Topic ko yeh kyun chahiye: rotation ka poora point yeh hai ki ek arrow ko dusre arrow mein turn karo, $\vec{v}\to {\vec{v}}^{\prime }$. Aage aane wali sab cheezein — DCM, quaternion, Euler angles — us ek turn ko describe karne ki machinery hain.

2. Ek unit vector $\hat{e}$ — exactly 1 length ka arrow

Topic ko yeh kyun chahiye: rotation axis $\hat{e}=(e_1,e_2,e_3)$ hamesha ek unit vector hota hai — hum sirf yeh care karte hain ki spin-axis kis taraf point karta hai, na ki hum use kitna lamba draw karte hain. Quaternion formula $(q_1,q_2,q_3)=\hat{e}\sin \frac{\Phi }{2}$ (§9 mein build hua) us pure direction ko ek number se multiply karta hai.

3. Do frames A aur B — reference axes ke do sets

Figure s02 se kya samajhna hai: teal arrows ka star frame $A$ hai (duniya). Plum star wahi origin hai lekin twist kiya hua — frame $B$ (body). Ek hi lesson: poora topic sirf yeh poochta hai ki "B, A ke relative kitna rotated hai?" — har representation is picture ka jawab hai.

Topic ko yeh kyun chahiye: DCM, quaternion, aur Euler angles teeno sirf arrows ke in do star-shaped bundles ke beech ke twist ko likhne ke teen tarike hain.

4. Dot product $\vec{a}\cdot \vec{b}$ — "do arrows kitna agree karte hain?"

Topic ko yeh kyun chahiye: ek direction cosine literally frame $A$ ke ek axis aur frame $B$ ke ek axis ke beech ka dot product hai. Direction Cosine Matrix naam ka matlab hai "un nau cosines ka table".

5. Cross product $\vec{a}\times \vec{b}$ aur skew matrix $[\hat{e}{]}_{\times }$

Topic ko yeh kyun chahiye: matrix $[\hat{e}{]}_{\times }$ Rodrigues' rotation ko ek clean matrix formula mein likhne deta hai instead of ek cross-product jo andar dabi ho. Yeh woh bridge hai jo "geometry" ko "ek table jo aap multiply kar sako" mein turn karta hai.

6. Ek matrix $C$ aur product $C\vec{v}$ — ek machine jo arrow khaati hai, arrow ugalti hai

Topic ko yeh kyun chahiye: DCM sabhi conversions ka "hub" hai. Iske 9 numbers jo 6 constraints ke saath hain (3 unit-length + 3 perpendicularity) woh honest, bulky encoding hai jise baaki sab compress karte hain. Dekhein Direction Cosine Matrix Properties.

7. Identity matrix $I$ — "kuch mat karo"

Topic ko yeh kyun chahiye: Rodrigues' formula $I\cos \Phi$ se shuru hota hai: $\Phi =0$ par, $\cos 0=1$, toh $C=I$ — zero rotation kuch nahin karta. Yeh woh sanity check hai jo poori formula ko anchor karta hai.

8. Axis–angle $(\hat{e},\Phi )$ — master key

Figure s03 se kya samajhna hai: plum arrow $\hat{e}$ object ke through skewer hai. Orange arrow $\vec{v}$ dark arc ke along swing karta hai teal arrow ${\vec{v}}^{\prime }$ tak; arc ka opening angle $\Phi$ (Greek capital phi) hai. Lesson: ek rotation sirf do cheezein mein fully captured hai — skewer $\hat{e}$ ki direction aur turn $\Phi$ ki miqdar.

Topic ko yeh kyun chahiye: DCM aur quaternion dono ke paas $(\hat{e},\Phi )$ se seedhe clean formulas hain. Isliye parent note dono ko is anchor se derive karta hai, phir conversions compose karta hai. Yeh master karo toh baaki sab bookkeeping hai. Dekhein Rodrigues Rotation Formula.

9. $\cos$, $\sin$, $\tan$ — aur half-angle $\frac{\Phi }{2}$

Topic ko yeh kyun chahiye: har conversion formula trig glue hai. Khaas taur par, $q_0=\cos \frac{\Phi }{2}$ (aage §10 mein define hua), aur identities $\sin \Phi =2\sin \frac{\Phi }{2}\cos \frac{\Phi }{2}$ aur $1-\cos \Phi =2{\sin }^2\frac{\Phi }{2}$ woh hain jo "quaternion" ko "DCM" mein turn karti hain.

10. Quaternion $q$ — ek 4-number rotation token

11. Euler angles $(\varphi ,\theta ,\psi )$ — teen human-readable tilts

Figure s04 se kya samajhna hai: teen curved arrows ke saath ek chhota aircraft — orange yaw (nose left/right swing karta hai vertical ke around), teal pitch (nose upar/neeche tip karta hai), plum roll (wings nose line ke around tilt karte hain). Lesson: Euler angles orientation batane ka human tarika hain, lekin kyunki har turn agle ke liye axes change karta hai, ulte order mein karne par alag result milta hai — aur $\theta =\pm 90°$ par do axes line up ho jaate hain (Gimbal Lock).

Topic ko yeh kyun chahiye: Euler angles woh hain jaise ek rotation kisi person ko report hoti hai. Inhe DCM se wापस nikalna (e.g. $\theta =-\arcsin C_{13}$, $\varphi =atan2(C_{23},C_{33})$) exactly isliye §12 ke inverse-trig tools chahiye.

12. $\arcsin$ aur $atan2$ — angle wापस padhna

Figure s05 se kya samajhna hai: do arrows jinka same ratio $y/x$ hai — orange wala quadrant I $(+,+)$ mein, teal wala quadrant III $(-,-)$ mein. Plain $\arctan (y/x)$ sirf ratio dekhta hai aur dono ko confuse karta hai; $atan2$ dono signs padhta hai aur har ek ko uske correct quadrant mein land karta hai. Lesson: hamesha dono numbers alag-alag feed karo.

Topic ko yeh kyun chahiye: DCM se Euler angles extract karne ke liye full-circle inverse trig chahiye, warna aapka reported roll/pitch/yaw silently flip ho jaayega. Yahi tool hai jahan Gimbal Lock bhi show up karta hai: pitch $=\pm 90°$ par arguments $atan2(0,0)$ ban jaate hain — undefined.

Prerequisite map

Neeche ka chain top-down padhein: raw arrows dot aur cross products spawn karte hain; woh dono DCM aur quaternion ke multiplication rule dono banate hain; axis-angle aur half-angle trig numbers fill karte hain; inverse trig Euler angles read out karta hai. Teeno representations topic par milte hain.

Equipment checklist

Khud test karo — parent note kholne se pehle har ek ka jawab de sako.