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

3.5.2 · D1 · Physics › Guidance, Navigation & Control (GNC) › Euler angles — roll φ, pitch θ, yaw ψ; rotation sequence (3-

Yeh page kuch bhi assume nahi karti. Agar parent note (the topic note) mein koi symbol use hua hai, toh hum use yahan pehle ek picture se build karte hain.

0. "Vector" aur "frame" actually hain kya

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

Parent do frames use karta hai:

Curly braces ${n}$ aur ${b}$ sirf labels hain — " ${n}$ " padho "the nav frame" aur " ${b}$ " padho "the body frame."

1. Angles, right-hand rule, aur hum "turning" ko unse kyun measure karte hain

Definition Right-hand rule — POSITIVE turn kis taraf hota hai

Turn ko ek sign chahiye: kya $+ 3 0^{\circ}$ clockwise hai ya counter-clockwise? Hum ise ek baar right-hand rule se fix karte hain. Apna right thumb axis arrow ke saath point karo (e.g. $+ z$ ke saath). Tumhari curling fingers positive rotation ki direction dikhati hain. Toh positive turn counter-clockwise hota hai jab tum axis ke neeche origin ki taraf dekh rahe ho (matlab axis tumhari aankhon ki taraf point kar raha ho).

Teen vehicle turns par apply karke (har ek apni axis ke baare mein):

Positive ==yaw $ψ$ == ( $+ z$ ke baare mein, jo Down point karta hai) → nose right/East ki taraf swing karta hai.
Positive ==pitch $θ$ == ( $+ y$ ke baare mein, right wing) → nose upar jata hai.
Positive ==roll $ϕ$ == ( $+ x$ ke baare mein, nose) → right wing neeche jhukti hai.

Yeh batana kyun zaroori hai? Bina fixed positive direction ke, " $ψ = 3 0^{\circ}$ " ambiguous hai aur matrix mein $sin$ / $cos$ ke signs kisi bhi taraf ja sakte hain. Neeche har formula yeh right-hand-rule convention assume karta hai.

Parent ke teen named angles sab sirf "ek axis ke baare mein turn ki signed matra" hain:

Symbol	Naam	Picture (nose kya karta hai, positive sense)	Jis axis ke baare mein turn hai
$ψ$ (psi)	yaw	right (East) ki taraf swing karta hai	$z$ (down)
$θ$ (theta)	pitch	nose upar uthta hai	$y$ (right wing)
$ϕ$ (phi)	roll	right wing neeche jhukti hai	$x$ (nose)

$ψ, θ, ϕ$ Greek letters hain — inhe purely un teen signed turn-amounts ke names samjho. Inhe "yaw number, pitch number, roll number" kehne se zyada mysterious kuch nahi.

2. Sine aur cosine — woh do numbers jo ek turn produce karta hai

3. tan aur arcsin — ratios ko wapas angles mein convert karna

4. Rotation matrix — ek machine jo arrow ko re-label karti hai

Formula Concrete

$R_{n}^{b}$ jo parent actually use karta hai (3-2-1) Har single-axis turn ek 2D shadow-rotation hai jo 3D mein embed hui hai spin axis par ek $1$ ke saath (woh axis move nahi karti). Right-hand-rule/passive convention (Section 1) use karke, teen elementary matrices hain:

R_y(\theta)=\begin{bmatrix}\cos\theta&0&-\sin\theta\\0&1&0\\\sin\theta&0&\cos\theta\end{bmatrix},\quad R_x(\phi)=\begin{bmatrix}1&0&0\\0&\cos\phi&\sin\phi\\0&-\sin\phi&\cos\phi\end{bmatrix}$$ Pehle yaw, phir pitch, phir roll apply karna (3-2-1) inhe right-to-left multiply karta hai: $$R^b_n(\phi,\theta,\psi)=R_x(\phi)\,R_y(\theta)\,R_z(\psi)$$ Multiply out karne par (jahan $s\equiv\sin,\ c\equiv\cos$): $$R^b_n=\begin{bmatrix} c\theta\,c\psi & c\theta\,s\psi & -s\theta\\ s\phi\,s\theta\,c\psi - c\phi\,s\psi & s\phi\,s\theta\,s\psi + c\phi\,c\psi & s\phi\,c\theta\\ c\phi\,s\theta\,c\psi + s\phi\,s\psi & c\phi\,s\theta\,s\psi - s\phi\,c\psi & c\phi\,c\theta \end{bmatrix}$$ Yeh **woh ek symbol hai jis par parent lean karta hai**; ab tumhare paas uska concrete grid hai. Parent ke angle-recovery formulas simply entries wapas padhte hain: $\theta=-\arcsin(R_{13})$, $\phi=\operatorname{atan2}(R_{23},R_{33})$, $\psi=\operatorname{atan2}(R_{12},R_{11})$.

Definition Transpose, orthogonality, aur

$det = + 1$ — free reverse gear Transpose $R^{T}$ grid ko uske diagonal ke across flip karta hai (rows columns ban jaati hain). Rotation matrices ke liye ek khoobsoorat gift hai: reverse rotation sirf transpose hai, $R_{b}^{n} = (R_{n}^{b})^{T} = (R_{n}^{b})^{- 1} .$ Ek matrix jisme $R^{T} R = I$ (identity, diagonal par $1$ 's, aur $0$ 's baaki jagah) hai use orthogonal kehte hain. Lekin sirf orthogonal kaafi nahi: yeh reflections (mirror-flips) bhi allow karta hai, jo real rotations nahi hain. Ek proper rotation additionally satisfy karta hai $det R = + 1,$ jahan determinant $det R$ ek akela number hai jo matrix volume ko kaise scale karta hai yeh measure karta hai; $+ 1$ matlab "koi stretching nahi, koi mirror-flip nahi." Saath mein $R^{T} R = I$ aur $det R = + 1$ exactly rotations ko pin down karte hain (is set ko $S O (3)$ kehte hain). Parent ko kyun parwah hai: kisi arrow ko body → nav bhejna ho toh kuch naya compute nahi karna — bas transpose karo (Worked Example 1 exactly yahi use karta hai), aur $det = + 1$ ki guarantee woh hai jo "rotation" ko secretly tumhara aircraft mirror karne se rokti hai.

5. Rotations compose karna — order kyun matter karta hai

Figure same do turns ko dono orders mein alag attitudes par end hote hua dikhata hai — proof ki order decoration nahi hai.

6. atan2 — sign-aware angle finder

7. Ek cheez aage ki taraf: angular rates aur gimbal lock

Prerequisite map

Equipment checklist

Right side chhupao aur parent note kholne se pehle khud test karo.

Vector kya hai, plain words mein?

Ek arrow jisme length aur direction hoti hai; teen coordinates se represent hota hai jab ek frame choose kar lo.

Frame kya hai?

Teen mutually perpendicular axes (

x, y, z

) jo origin par milte hain, vector ko uske coordinates dene ke liye rulers ki tarah use hote hain.

{n}

aur

{b}

ka kya matlab hai?

Fixed navigation (ground) frame aur moving body (vehicle) frame.

v_{n}

v_{b}

mein subscript kya batata hai?

Kis frame ke axes ke against USE SAME arrow ke coordinates measure hue hain — nav vs body.

Positive rotation kya define karta hai?

Right-hand rule: thumb axis ke saath, curling fingers positive turn direction deti hain (CCW axis ke neeche wapas dekhte hue).

Unit circle par,

cos α

aur

sin α

kya hain?

Turned unit arrow ki

x

-axis par (

cos

) aur

y

-axis par (

sin

) shadows; tip

(cos α, sin α)

par baithi hai.

sin 3 0^{\circ}

aur

cos 3 0^{\circ}

kya hain?

0.5

aur

\approx 0.866

tan α

kya measure karta hai?

Arrow ki slope (rise over run),

sin α / cos α

ke barabar.

arcsin (v)

kaun sa sawaal answer karta hai, aur kis range mein?

"Kaun se angle ka sine

v

ke barabar hai?"; yeh

[- 9 0^{\circ}, + 9 0^{\circ}]

mein principal value return karta hai.

3-2-1 ke liye

R_{n}^{b}

3 \times 3

form kya hai?

R_{x} (ϕ) R_{y} (θ) R_{z} (ψ)

, jo woh matrix deta hai jiska top row

[c θ c ψ, c θ s ψ, - s θ]

hai.

R_{n}^{b}

vector ke saath kya karta hai?

Uske nav-frame coordinates

v_{n}

leta hai aur uske body-frame coordinates

v_{b}

return karta hai.

Ek matrix ko proper rotation banane wali DO conditions kaun si hain?

R^{T} R = I

(orthogonal) AUR

det R = + 1

(koi reflection nahi).

(R_{n}^{b})^{T}

reverse rotation kyun hai?

Rotation matrices orthogonal hoti hain, toh unka transpose unke inverse ke barabar hota hai.

Product

R_{x} R_{y} R_{z}

mein, kaun sa rotation pehle hota hai?

Sabse right wala,

R_{z}

(products time mein right-to-left padhti hain).

ϕ

aur

ψ

recover karne ke liye plain

arctan

use kyun nahi kar sakte?

tan

har

18 0^{\circ}

mein repeat karta hai, toh plain arctan quadrant khota hai; atan2 dono inputs ke signs use karke full-circle answer deta hai.

atan2 result ko

0^{\circ}

–

36 0^{\circ}

heading mein kaise convert karte hain?

Kisi bhi negative value mein

36 0^{\circ}

add karo (ise

36 0^{\circ}

mod le lo).

p, q, r

kya hain?

Body angular rates — instantaneous roll, pitch aur yaw turn-speeds jo body frame mein measure ki gayi hain.