Direction cosine matrix (DCM) — construction from Euler angles: Question bank

3.5.3 · D5 · Physics › Guidance, Navigation & Control (GNC) › Direction cosine matrix (DCM) — construction from Euler angl

Questions se pehle, ek self-contained picture hai un sab cheezein ki jo neeche ke traps use karte hain.

Definition Teen Euler angles aur unke elementary matrices

Ek $3$ - $2$ - $1$ attitude teen turns se banta hai, har ek ka naam us axis ke naam par rakha gaya hai jiske around wo ghoomta hai. Figure s03 teen rotation planes dikhata hai taaki tum dekh sako ki har turn kaun se do axes ko stir karta hai. Poora likhne par ( $c = cos, s = sin$ ): $R_{3} (ψ) = c ψ - s ψ 0 s ψ c ψ 0 001 (yaw, about 3-axis, first)$ $R_{2} (θ) = c θ 0 s θ 010 - s θ 0 c θ (pitch, about 2-axis, second)$ $R_{1} (ϕ) = 100 0 c ϕ - s ϕ 0 s ϕ c ϕ (roll, about 1-axis, last)$ Poora DCM inhe is tarah stack karta hai: $C_{B N} = R_{1} (ϕ) R_{2} (θ) R_{3} (ψ)$ — aakhri physical turn sabse baayein multiply hota hai (Figure s04 dikhata hai kyun). Is mapping ko dhyan mein rakho ( $ϕ \to 1$ , $θ \to 2$ , $ψ \to 3$ ); neeche kai traps isi par hinge karte hain.

True or false — justify

TF1. "Ek DCM vector ko store karta hai, isliye usse vector par apply karne se wo vector space mein rotate ho jaata hai."

False. Vehicle ghoomta hai, lekin

C_{B N}

sirf usi physical vector ke components ko naye frame mein re-express karta hai — space mein arrow kabhi nahi hilta. Axes ko

+ θ

se rotate karna numerically vector ko

- θ

se rotate karne jaisa lagta hai, isliye signs ulte lagte hain.

TF2. "DCM ki har entry

- 1

aur

+ 1

ke beech hoti hai."

True. Har cell

\hat{b}_{i} \cdot \overset{n}{^}_{j} = cos (θ_{ij})

hai (Figure s02 ki grid), aur cosine kabhi

[- 1, 1]

se bahar nahi jaata. Agar tumne kabhi magnitude

> 1

wali cell compute ki, to tumhari matrix corrupt hai (orthonormal nahi hai).

TF3. "

C_{B N}

aur

C_{N B}

alag-alag matrices hain, lekin

C_{N B} = C_{B N}^{T}

True. Translation ki direction ko reverse karna matrix ko invert karta hai, aur rotation ke liye inverse hi transpose hota hai:

C_{N B} = C_{B N}^{- 1} = C_{B N}^{T}

. Grid ko diagonal ke across flip karna turn ko undo karta hai.

TF4. "Kyunki

C_{B N}^{- 1} = C_{B N}^{T}

hai, isliye har symmetric

3 \times 3

matrix ek DCM hai."

False.

C_{B N}^{- 1} = C_{B N}^{T}

ka matlab orthogonal hai, symmetric nahi. Zyaadatar DCMs symmetric nahi hote (ek

9 0^{\circ}

yaw nahi hota). Reflections ko exclude karne ke liye tumhe

det C_{B N} = + 1

bhi chahiye.

TF5. "DCM ke columns unit vectors hote hain, aur rows bhi."

True. Orthonormality (

C_{B N} C_{B N}^{T} = I

aur

C_{B N}^{T} C_{B N} = I

) dono rows aur columns ko mutually perpendicular unit vectors hone par majboor karti hai — ek statement same fact ka row version hai, doosri column version hai.

TF6. "Do valid DCMs ko multiply karne se hamesha ek aur valid DCM milta hai."

True. Do rotations ka product ek rotation hai: orthogonality aur

det = + 1

dono multiplication mein survive karte hain (

det (A B) = det A det B = 1 \cdot 1

). Ye closure exactly wahi hai jo SO(3) ko ek group banata hai.

TF7. "Ek

3

2

1

sequence aur ek

3

1

2

sequence same angle values ke saath same DCM dete hain."

False. Matrix multiplication commute nahi karta, isliye rotation order result badal deta hai. Same numbers, alag sequence = generally alag attitude.

TF8. "Bahut chhote angles ke liye teen rotations ki order matter karna band ho jaati hai."

True, lekin sirf first order tak. Har tiny turn

R_{k} \approx I - ϵ [\overset{e}{^}_{k} \times]

hai; do ko multiply karne se milta hai

I - ϵ ([\overset{e}{^}_{a} \times] + [\overset{e}{^}_{b} \times]) + ϵ^{2} (\dots)

. Linear part skew matrices ka plain sum hai, aur addition commute karta hai, isliye order swap karne se sirf discarded

ϵ^{2}

block change hota hai. Wo surviving sum exactly

I - [α \times]

hai.

TF9. "Gimbal lock ka matlab hai DCM singular (non-invertible) ho jaata hai."

False. DCM ek perfectly good rotation raha hai

det = + 1

ke saath; ye Euler-angle parameterization singular hoti hai, kyunki pitch

θ = \pm 9 0^{\circ}

par teen angles mein se do ek hi physical axis ke around ghoomte hain.

Spot the error

SE1. "

R_{2} (θ)

mein bottom-left mein

+ sin θ

hona chahiye,

R_{1}

aur

R_{3}

se match karte hue."

Error. Mini-derivation: pitch

3

–

1

plane mein ghoomta hai, isliye

(x_{3}, x_{1})

par ek point

x_{1}^{'} = x_{1} c θ + x_{3} s θ

x_{3}^{'} = - x_{1} s θ + x_{3} c θ

par jaata hai (standard 2D turn

1, 3

order mein). Kyunki ordered pair yahan

(3, 1)

hai na ki

(1, 2)

, jab tum matrix ko normal

1, 2, 3

row order mein likhte ho to

- sin

top-right (

C_{13}

) mein aaता hai aur

+ sin

bottom-left (

C_{31}

) mein —

R_{1}, R_{3}

ke opposite. Figure s03 ka middle panel ye plane dikhata hai; hamesha re-derive karo ye poochh kar ki kaun se do axes rotation plane ko span karte hain aur kis cyclic order mein.

SE2. "

N

B

jaane ke liye hum pehle yaw apply karte hain, isliye yaw ka matrix

R_{3}

sabse baayein hona chahiye:

C_{B N} = R_{3} R_{2} R_{1}

Error. Har matrix pichle wale ke output par act karta hai (Figure s04), isliye aakhri physical rotation pehle (sabse baayein) multiply hota hai. Sahi order hai

C_{B N} = R_{1} (ϕ) R_{2} (θ) R_{3} (ψ)

— yaw (

R_{3}

), jo pehle apply hota hai, sabse daayein baithta hai; roll (

R_{1}

), jo aakhir mein apply hota hai, sabse baayein baithta hai.

SE3. "Yaw recover karne ke liye main

ψ = arctan (C_{12} / C_{11})

use karta hoon."

Error. Plain

arctan

un do quadrants ko collapse kar deta hai jo ek tangent value share karte hain, isliye

C_{11}, C_{12}

ke signs kho jaate hain.

ψ = atan2 (C_{12}, C_{11})

use karo, jo dono signs alag-alag padhta hai aur poore

36 0^{\circ}

mein correct quadrant return karta hai.

SE4. "Pitch

θ = arcsin (C_{13})

se aata hai."

Error — ek sign slip hai. 3-2-1 DCM mein entry

C_{13} = - sin θ

hai (yahan

θ

pitch angle hai), isliye

θ = - arcsin (C_{13})

. Minus bhoolne se tumhara pitch ulta ho jaata hai.

SE5. "Kyunki

C C^{T} = I

hai, main koi bhi teen perpendicular vectors rows ke roop mein lekar ek DCM bana sakta hoon."

Half-error. Perpendicular unit rows orthogonality dete hain lekin phir bhi reflection (

det = - 1

, left-handed frame) produce kar sakte hain. Tumhe right-handedness bhi ensure karni hogi:

\hat{b}_{1} \times \hat{b}_{2} = \hat{b}_{3}

, jo

det = + 1

deta hai.

SE6. "Small-angle DCM

C_{B N} \approx I - [α \times]

mein, vector

α = (ψ, θ, ϕ)

hai yaw-pitch-roll order mein."

Error. Skew vector components ko axis number ke hisaab se order karta hai, rotation sequence ke hisaab se nahi:

α = (ϕ, θ, ψ)

axes

1, 2, 3

ke liye (yaad karo

ϕ \to 1

θ \to 2

ψ \to 3

). Ye ordering hi ise directly ek linearized error state mein plug karne deti hai.

Why questions

WHY1. DCM ki har entry cosine kyun hoti hai, na ki koi distance ya raw coordinate?

Kyunki ye do unit vectors ka dot product hai, aur

\overset{u}{^} \cdot \overset{v}{^} = cos (angle)

jab dono ki length

1

ho. Cosine cleanly measure karta hai "ek axis kitni doosri ki taraf point karti hai" ek fixed

[- 1, 1]

scale par.

WHY2. Hum single-axis rotations ko angles add karke nahin balki matrices multiply karke kyun chain karte hain?

Alag-alag axes ke around rotations numbers ki tarah add nahi hote — ye interact karte hain. Matrix multiplication exactly wahi operation hai jo "ye transform karo, phir wo karo" compose karta hai, ye respect karte hue ki doosra pehle ka result par act karta hai (Figure s04).

WHY3.

C_{B N}^{- 1} = C_{B N}^{T}

"free mein" milna flight software mein itna kyun matter karta hai?

Ek general

3 \times 3

matrix ko invert karna arithmetic khaata hai aur round-off amplify kar sakta hai; transpose karna sirf entries ko relabel karna hai — exact aur instant. Ek power- aur cycle-limited flight computer par wo saving, jo har second mein hazaron baar hoti hai, bahut badi hai.

WHY4. Entry

C_{13} = - sin θ

sirf pitch par kyun depend karti hai, yaw ya roll par nahi?

C_{13}

inertial

3

-axis ko body

1

-axis par map karta hai.

3

2

1

build mein, sirf pitch rotation (

R_{2} (θ)

) body

1

-axis ko inertial horizontal plane se bahar tilt karta hai; yaw us plane ke andar ghoomta hai aur roll baad mein hota hai, isliye dono is particular projection ko nahi chhoote. Ye ek handy sanity anchor hai.

WHY5. GNC attitude ko quaternion ke roop mein kyun store karta hai aur sirf display ke liye Euler angles mein convert karta hai?

Euler angles gimbal-lock singularity hit karte hain aur unke rate equations pitch

θ = \pm 9 0^{\circ}

ke paas blow up kar jaate hain. Quaternions same rotations ko bina singularity ke parameterize karte hain, isliye internal state well-behaved rehti hai; insaan bas angles padhna pasand karte hain.

WHY6.

det C_{B N} = + 1

specifically kyun hona chahiye,

- 1

kyun nahi?

det = - 1

ek reflection encode karta — right-handed frame ko left-handed mirror image mein baadal deta, jo koi physical rotation nahi kar sakta. Ek proper rotation handedness preserve karta hai, isliye

det = + 1

compulsory hai.

WHY7. Small-angle DCM

I

minus ek skew-symmetric matrix kyun nikalta hai?

Ek vector ka infinitesimal rotation ek tiny cross product hai,

δ v = - α \times v

, aur cross product exactly wahi hai jo skew-symmetric operator

[α \times]

perform karta hai. Isliye

C_{B N} \approx I - [α \times]

bas "identity plus ek tiny cross-product nudge" hai.

Edge cases

EC1. Zero rotation ke liye DCM kaisa dikhta hai, aur kyun?

Ye identity matrix

I

hai. Sab angles zero hone par, har body axis apne inertial counterpart se coincide karta hai, isliye

\hat{b}_{i} \cdot \overset{n}{^}_{j} = 1

jab

i = j

aur

0

otherwise — exactly

I

EC2. Dono pitch

θ = + 9 0^{\circ}

aur

θ = - 9 0^{\circ}

par, recovered yaw aur roll ka kya hota hai — aur dono cases symmetric kyun hain?

Dono cases mein

cos θ = 0

hai, isliye yaw (

R_{3}

) aur roll (

R_{1}

) ek hi physical axis ke around ghoomte hain aur sirf unka combination recover ho sakta hai, har ek alag-alag nahi.

θ = + 9 0^{\circ}

par DCM yaw aur roll par sirf difference

ϕ - ψ

ke through depend karta hai;

θ = - 9 0^{\circ}

par sirf sum

ϕ + ψ

ke through — usi singularity ke mirror-image versions, kyunki pitch ka sign flip karne se do collapsed axes ka relative sense flip ho jaata hai.

EC3. Kya poora

36 0^{\circ}

yaw same DCM hai jaisa

0^{\circ}

yaw?

Haan —

cos 36 0^{\circ} = cos 0^{\circ} = 1

aur

sin 36 0^{\circ} = sin 0^{\circ} = 0

, identical matrices dete hain. DCM har angle mein

36 0^{\circ}

period ke saath periodic hai; alag angle triples same attitude naam de sakte hain.

EC4. Kya yaw

ψ = + 18 0^{\circ}

aur roll+pitch jo vehicle ko doosri taraf flip karte hain kabhi same DCM ke roop mein milte hain?

Haan — ye Euler-angle non-uniqueness hai. Alag

(ψ, θ, ϕ)

triples ek identical DCM produce kar sakte hain (ek

18 0^{\circ}

flip kai tarahon se reachable hai), isliye extraction formulas ek canonical representative pick karte hain.

EC5. Agar koi tumhe orthonormal rows wali matrix de par

det = - 1

ho, to wo kya hai?

Ek improper rotation (rotation combined with a reflection). Ye ek valid orthogonal matrix hai lekin DCM nahi, kyunki ye handedness flip karta hai — koi bhi rigid body physically ise achieve nahi kar sakta.

EC6. Roll

ϕ \to 0

hote time jab yaw aur pitch arbitrary hon, DCM ka kya behaviour hai?

ϕ

terms collapse ho jaate hain (

cos ϕ \to 1, sin ϕ \to 0

) aur

C_{B N} \to R_{2} (θ) R_{3} (ψ)

— ek clean two-rotation matrix. Yahan koi singularity nahi; sirf pitch

θ = \pm 9 0^{\circ}

gimbal lock trigger karta hai, roll nahi karta.

Recall Ek-line self-test

Sab kuch cover karo aur zor se jawab do: General case mein multiplication ka order kyun crucial hai lekin chhote angles ke liye first order tak irrelevant kyun ho jaata hai? ::: Rotations commute nahi karte, isliye full products order par depend karte hain; lekin small-angle DCM sirf linear terms $I - [α \times]$ rakhta hai, aur tiny skew matrices ko add karna commute karta hai — order entirely discarded second-order terms mein rehta hai.

Connections

Direction cosine matrix (DCM) — construction from Euler angles (index 3.5.3)
Quaternions — avoiding gimbal lock
Euler angles — kinematic differential equations
Rotation group SO(3) and orthogonal matrices
Attitude determination — TRIAD & QUEST
Kalman filter — linearized attitude error state
Angular velocity and skew-symmetric matrices