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

3.5.2 · D2 · Physics › Guidance, Navigation & Control (GNC) › Euler angles — roll φ, pitch θ, yaw ψ; rotation sequence (3-

Pehli line se pehle, teen plain-word promises:

Ek vector bas ek arrow hai jiske paas ek length aur ek direction hai — socho ek toy plane ki naak.
Ek frame teen arrows ka ek set hai jo right angles par hote hain, jo hum apni "measuring sticks" ki tarah use karte hain: ek aage ki taraf, ek daayein, ek neeche. Kisi arrow ke coordinates bas itne hain ki woh har measuring stick tak kitna pahuncha.
Frame rotate karna matlab hai measuring sticks ko uthao aur ghumaao, jabki physical arrow wahi rehta hai. Arrow ke numbers isliye change hote hain kyunki hum ab use ghumi hui sticks ke against measure kar rahe hain.

Step 1 — Ek single 2D turn measuring sticks ke saath kya karta hai

KYA. Abhi ke liye 3D bhool jao. Ek flat sheet lo jisme do measuring sticks hain: $x$ (daayein) aur $y$ (upar). Puri sheet ko ek angle $α$ se counter-clockwise ghuma do. Pucho: unit sticks ke tips kahan gaye?

KYUN. Har 3D rotation jo humein chahiye woh secretly yahi ek 2D turn hai jo ek flat plane mein ho raha hai, aur teesra axis bas khada rehta hai. Agar hum 2D case ek baar pakad lein, to hum teeno 3D matrices free mein le lete hain.

PICTURE. Figure mein, purana $x$ -stick (pale yellow) ek naye spot par jhulta hai; purana $y$ -stick (chalk blue) bhi jhulta hai. (Yaad raho hat matlab "length-1 stick"; prime matlab "turn ke baad.")

Rotated $x$ -stick ka tip horizontal amount $cos α$ aur vertical amount $sin α$ par aata hai: $\overset{x}{^}^{'} = (kitna abhi bhi daayein point karta hai cos α, kitna ab upar point karta hai sin α)$

Rotated $y$ -stick aata hai: $\overset{y}{^}^{'} = (peeche (baayein) jhuk jaata hai - sin α, abhi bhi mostly upar cos α)$

Step 2 — "Arrow ghumaane" se "frame ghumaane" tak (transpose)

KYA. Step 1 mein humne sticks ko counter-clockwise ghummaya. Lekin aerospace mein hum physical arrow ko fixed rakhte hain aur uske neeche frame ko ghummayein. Frame ko $+ α$ se ghummana, arrow ke coordinates ke liye, bilkul waisa hi dikhta hai jaise arrow ko $- α$ se ghummao.

KYUN. Yahi ek passive rotation ka poora matlab hai: hum airplane ki naak ko duniya ke relative kabhi nahi hilate; hum sirf change karte hain ki hum use kin measuring sticks ke against padhein. Ghumi hui sticks ke against padhne se angle ka sign flip ho jaata hai.

PICTURE. Wahi picture, ulta spin. Jahan pehle off-diagonal minus top-right mein tha, $α \to - α$ flip karne se ( $sin (- α) = - sin α$ ) minus lower-left mein aa jaata hai.

Coordinate rules ko rows ki tarah stack karne se (har row kehta hai ek naya coordinate kaise banana hai) passive 2D turn milta hai:

R_{2 D} (α) = [cos α - sin α sin α cos α]

Step 3 — 2D turn ko 3D mein embed karo: yaw matrix $R_{z} (ψ)$

KYA. Yaw vertical $z$ -axis (NED Down) ke baare mein ek flat spin hai. $z$ -stick nahi hilta; $x$ aur $y$ sticks Step 2 ka 2D turn karte hain.

KYUN. Yaw 3-2-1 mein pehla rotation hai — pilot pehle naak ko left/right karta hai baaki sab se pehle. Ise pehle banana physical order se match karta hai. Right-hand rule se, positive $ψ$ ko $+ z_{n}$ ke along dekhte hue (yaani neeche ki taraf) dekha jaata hai, isliye NED mein naak-East-ki-taraf-jhulna ek positive yaw hai.

PICTURE. $z$ -axis ke seedha neeche dekhte hue, hum bilkul Step 2 ki picture dekhte hain. $z$ -row hai $[0, 0, 1]$ : "vertical coordinate touch nahi hota."

R_{z} (ψ) = cos ψ - sin ψ 0 sin ψ cos ψ 0 00 z unmoved 1

Top-left $2 \times 2$ block = $x$ – $y$ plane mein passive 2D turn.
Akela $1$ kehta hai spin axis ( $z$ ) har $z$ -coordinate ko rakhta hai.

Step 4 — Pitch $R_{y} (θ)$ : wahi turn, alag plane, sign dekho

KYA. Pitch naak ko upar/neeche jhukkaata hai: $y$ -axis ke baare mein $x$ – $z$ plane mein ek turn. $y$ -stick wahi rehta hai.

KYUN. Pitch doosra rotation hai. Yeh us $y$ -axis ke baare mein hota hai jise yaw ne abhi reposition kiya. Matrix ke roop mein yeh phir se hamaara 2D turn hai, lekin $x$ – $z$ plane mein. Right-hand rule se ( $+ y$ ki taraf angutha), positive $θ$ NED mein naak uthata hai.

PICTURE. Yahaan ek sign trap hai. Agar tum axes ko $x$ phir $z$ ke order mein label karo aur counter-clockwise jao, to plane ki "handedness" Steps 3 aur 5 ke comparison mein flip ho jaati hai, isliye minus top-right mein aa jaata hai lower-left ki jagah.

R_{y} (θ) = cos θ 0 sin θ 0 y unmoved 1 0 - sin θ 0 cos θ

Step 5 — Roll $R_{x} (ϕ)$ : aakhri spin, naak ke baare mein

KYA. Roll wings ko bank karta hai: $x$ -axis (naak) ke baare mein $y$ – $z$ plane mein ek turn. $x$ -stick wahi rehta hai.

KYUN. Roll 3-2-1 mein teesra aur aakhri rotation hai, us nose axis ke baare mein jise yaw aur pitch ne pehle se aim kiya. Right-hand rule se (naak $+ x$ se angutha bahar), positive $ϕ$ daaya wing neeche jhukkaata hai.

PICTURE. Naak ke neeche se dekhte hue, hum phir Step 2 ka flat turn dekhte hain, ab $y$ – $z$ plane mein; minus lower-left par wapas aata hai.

R_{x} (ϕ) = x unmoved 1 00 0 cos ϕ - sin ϕ 0 sin ϕ cos ϕ

Step 6 — Teeno spins ko chain karna: kyun product right-to-left padhta hai

KYA. Ab hum teeno ko stack karte hain. Ek vector ke nav coordinates ( ${n}$ ) andar jaate hain; yaw, phir pitch, phir roll ke baad, uske body coordinates ( ${b}$ ) bahar aate hain.

KYUN. Matrices jo unke seedhe daayein hota hai uske upar act karti hain. Hum yaw pehle apply karte hain, isliye vector pehle $R_{z}$ se milta hai — matlab $R_{z}$ sabse daayein (vector ke sabse karib) baithna chahiye. Pitch us result par act karta hai, isliye $R_{y}$ uske baayein baithta hai; roll aakhri, isliye $R_{x}$ sabse baayein hai.

PICTURE. Operations ka arrow right→left padhein: nav vector daayein se andar aata hai, har machine se guzarta hai, aur baayein body vector ke roop mein nikalta hai.

Step 7 — Ise term by term multiply karo

KYA. Pehle $R_{y} R_{z}$ ke do matrix multiplications karo, phir $R_{x} (\dots)$ , aur collect karo. $s \equiv sin, c \equiv cos$ use karte hue:

KYUN. Attitude software yeh ek $3 \times 3$ Direction Cosine Matrix store karta hai bajaay har step par teen matrices ko re-multiply karne ke — yeh faster hai aur yahi hai jo Direction Cosine Matrix (DCM) entry propagate karta hai.

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

Sabse saaf entries aur unka matlab notice karo:

Top-right $R_{13} = - s θ$ : sirf pitch ise set karta hai — koi roll, koi yaw ise contaminate nahi karta. Isliye baad mein $θ = - arcsin (R_{13})$ aata hai.
Bottom-right column $[s ϕ c θ, c ϕ c θ]$ (entries $R_{23}, R_{33}$ ): woh $c θ$ share karte hain, isliye unka ratio $tan ϕ$ hai, aur $c θ$ cancel ho jaata hai — isliye $ϕ = atan2 (R_{23}, R_{33})$ aata hai.
Pehla column-pair $R_{11}, R_{12} = c θ c ψ, c θ s ψ$ : $c θ$ share karte hain, ratio $= tan ψ$ .

Yeh poora rows-and-orthogonality structure Rotation matrices & orthogonality mein unpack hota hai.

Step 8 — Angles recover karna: $atan2$ kya hai aur hume iska kyun zarurat hai

KYA. Numbers $R_{ij}$ diye jaane par, hum unhe ulta chalate hain $ϕ, θ, ψ$ paane ke liye: $θ = - arcsin (R_{13}), ϕ = atan2 (R_{23}, R_{33}), ψ = atan2 (R_{12}, R_{11})$

KYUN $atan2$ aur plain $arctan$ nahi? Ratio $R_{23} / R_{33} = tan ϕ$ akela roll ka quadrant nahi bata sakta: $tan$ har $18 0^{\circ}$ par repeat karta hai, isliye ek angle aur woh angle plus $18 0^{\circ}$ same ratio dete hain. Do-argument $atan2 (y, x)$ define hota hai "woh angle jiska sine $y$ ka sign le aur cosine $x$ ka sign le" — yeh dono numbers alag rakhta hai, isliye yeh char mein se sahi quadrant mein land karta hai, $(- 18 0^{\circ}, 18 0^{\circ}]$ mein value return karta hai. Isliye hum ise $R_{23}$ aur $R_{33}$ (sirf unka ratio nahi) feed karte hain: $R_{23} = s ϕ c θ$ aur $R_{33} = c ϕ c θ$ $sin ϕ$ aur $cos ϕ$ ke signs carry karte hain.

PICTURE. Figure ek point har quadrant mein dikhata hai jo same $tan$ ratio share karta hai; $arctan$ unhe collapse karta hai, $atan2$ unhe alag karta hai.

Pitch ka branch. $arcsin$ sirf $[- 9 0^{\circ}, 9 0^{\circ}]$ return karta hai, isliye $θ$ hamesha us band mein liya jaata hai — standard convention jo pitch bounded rakhta hai aur roll/yaw free.

Step 9 — Degenerate cases: pitch seedha upar AUR seedha neeche

KYA. Do orientations recovery tod dete hain: $θ = + 9 0^{\circ}$ (naak seedha upar) aur $θ = - 9 0^{\circ}$ (naak seedha neeche). Dono mein, $c θ = 0$ , isliye $c θ$ carry karne wali har entry mar jaati hai: $R_{11} = R_{12} = R_{23} = R_{33} = 0$ , aur $atan2$ ratios $0/0$ ban jaate hain.

KYUN. Naak vertical hone ke saath, pehla spin (yaw, world-Down ke baare mein) aur aakhri spin (roll, ab-vertical naak ke baare mein) ek hi line ke baare mein ghumte hain. Hamare teen knobs mein se do identical kaam karte hain — ek degree of freedom gayab ho jaata hai. Yeh gimbal lock hai.

PICTURE. Yaw circle aur roll circle ek shared circle par collapse ho jaate hain; sirf $ϕ$ aur $ψ$ ka ek combination observable hai, har ek alag nahi.

Case $θ = + 9 0^{\circ}$ ( $s θ = + 1$ ). Surviving entries ek single combined angle $ϕ - ψ$ par collapse ho jaati hain:

R_{n}^{b}_{θ = + 9 0^{\circ}} = 0 sin (ϕ - ψ) cos (ϕ - ψ) 0 cos (ϕ - ψ) - sin (ϕ - ψ) - 1 00

Sirf $ϕ - ψ$ dikhta hai — tum ise kabhi ek unique $ϕ$ aur $ψ$ mein wapas nahi tod sakte.

Case $θ = - 9 0^{\circ}$ ( $s θ = - 1$ ). Ab har $s θ$ term ke aage sign flip ho jaata hai, aur surviving combination sum $ϕ + ψ$ ban jaata hai:

R_{n}^{b}_{θ = - 9 0^{\circ}} = 0 - sin (ϕ + ψ) - cos (ϕ + ψ) 0 cos (ϕ + ψ) - sin (ϕ + ψ) + 1 00

Yahaan sirf $ϕ + ψ$ observable hai. Toh dono poles lock karte hain, lekin woh alag combinations par lock karte hain (upar $ϕ - ψ$ , neeche $ϕ + ψ$ ) — ek subtlety jise ek baar dekhna zaroori hai.

Ek-picture summary

Poora page ek single frame mein: nav ( ${n}$ , NED) coordinates daayein se andar aate hain, yaw → pitch → roll (machines mein right-to-left) se spin kiye jaate hain, aur body ( ${b}$ ) coordinates ke roop mein nikalta hai — dono vertical-nose poles pink mein flagged hain.

Recall Feynman retelling — poora walkthrough plain words mein

Socho ek flat sheet ghumana: jo bhi ghumaao, ek stick abhi bhi "along" point karti hai (woh cosine hai) aur kuch "sideways" spill hota hai (woh sine hai). Yeh ek fact har rotation matrix ka beej hai. Kyunki hum plane ko still rakhte hain aur bajaye apni measuring sticks ko ghummaate hain, hum ulti taraf spin karte hain — jo ek minus sign lower-left corner mein park kar deta hai. Ab wahi choti flat-turn teen baar karo, 3D space ke har alag flat slice mein, aur har ek right-hand-rule sign ke saath: ek baar neeche dekhte hue (woh yaw hai, aur upar-neeche stick nahi hilti), ek baar side se (pitch — lekin yeh wala apna minus top corner mein flip karta hai kyunki axes wrap karne ke tarike ki wajah se), aur ek baar naak ke neeche dekhte hue (roll). Hamare world sticks hain North, East, Down (NED). Ek airplane ki attitude describe karne ke liye hum pilot ke order mein karte hain: naak aim karo (yaw), uthao (pitch), wings bank karo (roll). Kyunki har matrix jo apne daayein hai usse grab karti hai, aur yaw pehle jaata hai, yaw sabse daayein baithta hai, roll sabse baayein: $R_{x} R_{y} R_{z}$ , jo nav numbers ko body numbers mein turn karta hai. Unhe multiply karo aur tumhein cosines aur sines ki ek tidy 3×3 table milti hai — aur woh tidy purpose se hai: pure-pitch entry ek corner mein akeli baithti hai, aur entries ke pairs ek $cos θ$ share karte hain isliye unke ratios tumhe roll aur yaw cleanly wapas dete hain (hum $atan2$ use karte hain, jo dono numbers rakhta hai isliye quadrant pata chal jaata hai). Ek jagah yeh toot jaata hai woh hai naak-seedha-upar ya seedha-neeche: yaw aur roll tab bilkul ek hi line ke baare mein spin karte hain, do knobs ek kaam karte hain, aur tum unhe phir kabhi nahi suljha sakte. Woh gimbal lock hai, aur isliye pros apni back pocket mein quaternions rakhte hain.

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)

R_{n}^{b}

mein subscript

n

aur superscript

b

ka kya matlab hai?

n

= navigation frame (NED: North-East-Down) input hai;

b

= body frame (nose-right wing-belly) output hai.

Yahaan hat (jaise

\overset{x}{^}

mein) kya denote karta hai?

Ek unit vector — ek arrow bilkul length 1 ka us axis ke along.

Har positive Euler rotation ki direction kis sign convention se fix hoti hai?

Right-hand rule: positive spin axis ke along angutha, curling fingers positive rotation (CCW +axis se dekhte hue) deti hain.

Roll ke liye

arctan (R_{23} / R_{33})

ki jagah

atan2 (R_{23}, R_{33})

kyun use karte hain?

tan

har 180° par repeat karta hai;

atan2

dono signs rakhta hai isliye

(- 180°, 180°]

mein sahi quadrant choose karta hai.

θ = + 90°

aur

θ = - 90°

par kaun se angle combinations survive karte hain?

θ = + 90°

sirf

ϕ - ψ

chodta hai;

θ = - 90°

sirf

ϕ + ψ

— dono poles par gimbal lock.