3.5.3 · D1Direction cosine matrix (DCM) — construction from Euler angl
Do observers ek hi space mein arrow dekh rahe hain, lekin dono uske liye teen numbers ki alag-alag list likhte hain — kyunki unhone apne measuring axes alag-alag align kiye hain. Ek Direction Cosine Matrix (DCM) numbers ka woh chhota sa table hai jo ek observer ki list ko doosre ki list mein convert karta hai — aur ise scratch se banana ke liye sirf teen tools chahiye: unit vectors, dot product, aur ek pair of axes ki 2D rotation.
Is page par kuch bhi assume nahi kiya gaya hai . Parent note the DCM topic padhne se pehle, aapko woh har symbol apna banana hoga jo woh silently use karta hai. Hum unhe order mein banate hain, har ek pichle ke upar.
Ek vector ek arrow hai: uski ek length aur ek direction hoti hai. Physically yeh "star tracker jis direction mein dekh raha hai" ya "gravity jahan kheenchti hai" ho sakta hai. Hum ise ek tail aur ek head wale arrow ki tarah draw karte hain.
Definition Reference frameEk reference frame teen mutually perpendicular measuring directions (axes) ka set hai jo kisi observer se chipka hua hai. Har axis ki taraf point karo, poochho "arrow is direction mein kitni door tak pahunchta hai?", aur tumhe teen numbers milenge — woh us frame mein arrow ke components hain.
Poora topic isliye exist karta hai kyunki do frames hain:
N inertial (navigation) frame. Socho: taaron / aasman se fixed. Guidance aur orbits yahan describe kiye jaate hain.B body frame. Spacecraft se chipka hua. Sensors aur thrusters yahan rehte hain.
Intuition Do frames kyun hote hain?Ek gyroscope rotation body mein measure karta hai. Lekin tumhein jaanna hai ki tum taaron ke relative kahan point kar rahe ho. Usi physical arrow ke numbers B N
Ek unit vector woh arrow hai jis ki length exactly 1 n ^ n point karna hai; uski length koi information carry nahi karti.
Parent note frame axes ko n ^ 1 , n ^ 2 , n ^ 3 b ^ 1 , b ^ 2 , b ^ 3 1 , 2 , 3
Teen length-1
Humein iska kyun zaroorat hai: DCM ki entries bilkul isi se bani hain ki yeh hatted arrows ek doosre ke saath kaise align hote hain.
Yeh sabse important tool hai, isliye hum ise dhyan se banate hain.
Do arrows a b dot product a ⋅ b a ⋅ b = ∣ a ∣ ∣ b ∣ cos θ ∣ a ∣ a θ beech ka angle hai.
Intuition Dot product "kitna aligned?" kyun answer karta hai?cos θ 1 taraf point kar rahe hain, θ = 0 0 θ = 9 0 ∘ − 1 θ = 18 0 ∘ agreement measure karta hai.
Ab key special case, jo poore topic ko "direction cosine " naam dene ki wajah hai:
Intuition DCM se connectionParent kehta hai entry C ij = b ^ i ⋅ n ^ j = cos θ ij i j har entry ek "direction cosine" hai .
Common mistake "Dot product ke liye coordinates chahiye"Galat feeling: aap a ⋅ b a 1 b 1 + a 2 b 2 + a 3 b 3 Fix: woh component formula ∣ a ∣∣ b ∣ cos θ equal hai — yeh same quantity ko do tarike se dekha gaya hai. Geometric view (cos θ
Har rotation ek angle se measure hoti hai, aur machinery ko us angle ke sin cos
Definition Right triangle par Sine aur cosineEk right triangle draw karo. Ek non-right corner choose karo aur uske angle ko θ cos θ = hypotenuse adjacent side , sin θ = hypotenuse opposite side .
Intuition Woh picture jo inhe ek idea banati haiRadius 1 θ cos θ sin θ θ cos 2 θ + sin 2 θ = 1 1
Topic ko iska kyun zaroorat hai: elementary rotation matrices R 1 , R 2 , R 3 poori tarah rotation angle ke cos sin
Parent ki derivation cos ( ϕ − θ ) sin ( ϕ − θ )
Intuition Yeh kyun appear hote hainAxes ko + θ point ko − θ ϕ ϕ − θ cos ( ϕ − θ ) sin ( ϕ − θ ) R 3 ( θ ) poora origin hai.
DCM ek 3 × 3 matrix hai. Yahan bare minimum hai, zero se.
Definition Matrix aur matrix–vector productEk 3 × 3 matrix 9 3 3 C ij row i j mein hoti hai.
Matrix C v = [ v 1 , v 2 , v 3 ] T ek row aur v hota hai.
( C v ) i = C i 1 v 1 + C i 2 v 2 + C i 3 v 3 .
T v T C T diagonal ke across flip karo : row i j j i T C − 1 = C T free hai (grid ko doosre taraf se bas re-read karo), isliye yeh itna bada computational win hai.
Definition Matrix productA B A B i j A i B j commutative nahi hai: A B = B A order kyun matter karta hai.
Topic ko iska kyun zaroorat hai: teen rotations compose karna (R 1 R 2 R 3 matrix multiplication hi hai, aur transpose inverse rotation deta hai.
Parent DCM se angles wapas nikalne ke liye inhe use karta hai.
arcsin ( x ) "kis angle ka yeh sine hai?" Yeh sin − 9 0 ∘ + 9 0 ∘ 15 0 ∘ 3 0 ∘
atan2 ( y , x ) "kis angle ka vertical part y x Crucially yeh correct quadrant (chaaron mein se) choose karne ke liye y x dono ke signs use karta hai. Isliye parent arctan atan2 atan2
arctan ( y / x ) Galat feeling: "ratio ka arctan Yeh kyun fail hota hai: ratio y / x arctan atan2 atan2
Parent ka small-angle example I − [ α × ]
Definition Skew-symmetric matrixI identity matrix hai (1 0 [ α × ] α = ( α 1 , α 2 , α 3 ) 3 × 3 anti-symmetric hai: ise diagonal ke across flip karo toh har entry negate ho jaati hai ([ α × ] T = − [ α × ] [ α × ] = 0 α 3 − α 2 − α 3 0 α 1 α 2 − α 1 0
Tumhein abhi yeh master nahi karna — iska apna note hai — bas shape pehchaan lo jab parent ki linearization appear ho.
Angle difference identity
Matrix product order matters
Khud ko test karo — tum parent note ke liye ready ho jab tum har question bina rukke answer kar sako.
n ^ Uski length exactly 1
Do unit vectors ka dot product sirf cos θ Kyunki a ^ ⋅ b ^ = ∣ a ^ ∣∣ b ^ ∣ cos θ 1
Right triangle par cos θ Adjacent side divided by hypotenuse.
Kin quadrants mein cos θ Quadrants II aur III (angles 9 0 ∘ 27 0 ∘
Axis-rotation cos ( ϕ − θ ) Axes ko + θ ϕ ϕ − θ
C v i C i v
Transpose C T C ij Ise row j i
arctan ( y / x ) atan2 ( y , x ) Yeh y x
Identity matrix I Diagonal par ones, baaki jagah zeros; isse multiply karne par kuch nahi badlata.
Teen Euler rotations ka order kyun matter karta hai? Matrix multiplication commutative nahi hai: A B = B A
