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

3.5.2 · Physics › Guidance, Navigation & Control (GNC)

HUM KYA rotate kar rahe hain?

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

KAISE: Scratch se har elementary rotation derive karna

Passive rotation convention (hum frame rotate karte hain, ek fixed vector ke coordinates transform hote hain). Frame ko $+ α$ se rotate karna, coordinates ko vector-rotation ke transpose se transform karta hai — isse $- sin$ lower-left mein aa jaata hai. Toh:

R_{x} (ϕ) = 100 0 cos ϕ - sin ϕ 0 sin ϕ cos ϕ

R_{y} (θ) = cos θ 0 sin θ 010 - sin θ 0 cos θ

R_{z} (ψ) = cos ψ - sin ψ 0 sin ψ cos ψ 0 001

Multiply karke (maano $s \equiv sin, c \equiv cos$ ):

R_{n}^{b} = c θ c ψ s ϕ s θ c ψ - c ϕ s ψ c ϕ s θ c ψ + s ϕ s ψ c θ s ψ s ϕ s θ s ψ + c ϕ c ψ c ϕ s θ s ψ - s ϕ c ψ - s θ s ϕ c θ c ϕ c θ

Worked Example 1 — Pure yaw of 90°

Vehicle $ϕ = 0, θ = 0, ψ = 9 0^{\circ}$ par hai. Nav coordinates mein nose ( $x_{b}$ ) kahan point karta hai?

Why start here? Sirf yaw ke saath, $R_{n}^{b} = R_{z} (9 0^{\circ})$ , sabse simple case jo confidence build karta hai. $R_{z} (9 0^{\circ}) = 0 - 1 0 100001$
Body frame mein nose direction $[1, 0, 0]^{T}$ hai. Ise nav mein express karne ke liye, $R_{b}^{n} = (R_{n}^{b})^{T}$ use karo: nose $_{n} = (R_{n}^{b})^{T} [1, 0, 0]^{T} = [0, 1, 0]^{T}$ = East.
Why this step? $R_{n}^{b}$ nav→body map karta hai; inverse (transpose, kyunki rotations orthogonal hoti hain) body→nav map karta hai. Nose North se East ki taraf swing kiya — exactly ek $+ 9 0^{\circ}$ yaw. ✓

Worked Example 2 — Combined pitch + roll

$ϕ = 3 0^{\circ}, θ = 2 0^{\circ}, ψ = 0$ . $R_{n}^{b}$ ki third row find karo (nav coords mein body- $z$ axis).

Third row $= [c ϕ s θ c ψ + s ϕ s ψ, c ϕ s θ s ψ - s ϕ c ψ, c ϕ c θ]$ .
$ψ = 0$ ke saath: $= [c ϕ s θ, - s ϕ, c ϕ c θ]$ .
Plug in karo: $c 30 = 0.866, s 30 = 0.5, s 20 = 0.342, c 20 = 0.940$ : $= [0.866 (0.342), - 0.5, 0.866 (0.940)] = [0.296, - 0.5, 0.814]$
Why this step? $R_{n}^{b}$ ki third row batati hai ki body ka "Down" axis nav frame mein kahan point karta hai; uska downward tilt ( $z$ -comp $0.814 < 1$ ) reflect karta hai ki vehicle ab level nahi hai. ✓ (Check: magnitude $0.29 6^{2} + 0. 5^{2} + 0.81 4^{2} = 1.00$ , unit vector as required.)

Worked Example 3 — Recover angles

Diya gaya $R_{13} = - 0.5, R_{23} = 0.433, R_{33} = 0.75, R_{12} = 0, R_{11} = 0.866$ .

$θ = - arcsin (- 0.5) = 3 0^{\circ}$ . Why: $R_{13} = - sin θ$ .
$ϕ = atan2 (0.433, 0.75) = 3 0^{\circ}$ . Why: ratio roll ko isolate karta hai.
$ψ = atan2 (0, 0.866) = 0^{\circ}$ . ✓

Gimbal Lock

Active Recall

Recall Khud test karo (answers chupaao)

3-2-1 mein, pehle kis axis ke baare mein rotate karte hain? → 3rd axis ( $z$ ), yaani yaw.
$R_{n}^{b} = R_{x} R_{y} R_{z}$ kyun hai aur reverse product kyun nahi? → intrinsic 3-2-1 (moved axes ke baare mein) extrinsic $R_{x} R_{y} R_{z}$ ke barabar hai jo original vector par act karta hai.
$R$ se pitch kaise recover karte hain? → $θ = - arcsin (R_{13})$ .
Gimbal lock kab hota hai aur kyun? → $θ = \pm 9 0^{\circ}$ ; roll & yaw axes coincide ho jaate hain, ek DOF kho jaata hai.

Recall Feynman: ek 12-saal ke bachche ko explain karo

Apna toy airplane imagine karo. Use jis bhi direction mein point karna ho, teen spins order mein karo: pehle use flat spin karo jaise merry-go-round (yaw — nose ko left/right point karna). Phir nose ko upar ya neeche tilt karo (pitch). Finally use sideways roll karo jaise woh turn kar raha ho (roll). Yeh teen spin-amounts hi Euler angles hain. Ek tricky jagah hai: agar tum nose ko seedha upar point karo, toh "flat spin" aur "roll spin" ek hi kaam karne lagte hain, toh airplane confuse ho jaata hai ki kaun sa kaun sa hai — yahi gimbal lock hai.

Connections

Rotation matrices & orthogonality
Quaternions — avoiding gimbal lock
Direction Cosine Matrix (DCM)
Angular velocity & body rates p, q, r
NED navigation frame
Attitude estimation (Kalman / complementary filter)

3-2-1 Euler convention mein "3-2-1" kya refer karta hai?

Jis order mein axes ke baare mein rotate karte hain: 3rd (z, yaw), phir 2nd (y, pitch), phir 1st (x, roll).

3-2-1 ke liye

R_{n}^{b}

ko elementary rotations ke product ke roop mein likho.

R_{n}^{b} = R_{x} (ϕ) R_{y} (θ) R_{z} (ψ)

DCM se pitch

θ

kaise extract karte hain?

θ = - arcsin (R_{13})

DCM se roll aur yaw kaise extract karte hain?

ϕ = atan2 (R_{23}, R_{33})

ψ = atan2 (R_{12}, R_{11})

Gimbal lock kis pitch angle par hota hai aur kya kho jaata hai?

θ = \pm 9 0^{\circ}

par; roll aur yaw axes align ho jaate hain, rotational freedom ka ek degree kho jaata hai.

Passive rotation matrix ka off-diagonal sign active wale se flipped kyun hota hai?

Passive (frame) rotation = active (vector) rotation ka transpose, jo

- sin

ko lower-left mein le jaata hai.

Aircraft body frame mein

x_{b}, y_{b}, z_{b}

physically kaunse axes hain?

x_{b}

nose ke bahar,

y_{b}

right wing ke bahar,

z_{b}

belly ke neeche.