Quaternion kinematics — q̇ = ½ Ξ(q) ω — Hinglish notes

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

Quaternion KYA hai (fast recap)

Quaternion multiplication (Hamilton product): $p = (p_{0}, p_{v})$ , $q = (q_{0}, q_{v})$ ke liye, $p \otimes q = [p_{0} q_{0} - p_{v} \cdot q_{v} p_{0} q_{v} + q_{0} p_{v} + p_{v} \times q_{v}]$

Cross product term ka WHY: quaternion multiplication non-commutative hai, bilkul waisi hi jaise rotations compose karna. $p_{v} \times q_{v}$ term usi non-commutativity ko encode karta hai.

First principles se kinematics derive karna

Step 1 — Tiny rotation quaternion

Body-frame axis $\overset{ω}{^} = ω /∥ ω ∥$ ke baare mein small angle $Δ θ = ∥ ω ∥ Δ t$ ka rotation hai: $δ q = [cos \frac{Δ θ}{2} \overset{ω}{^} sin \frac{Δ θ}{2}]$

Yeh step kyun? Yeh sirf quaternion ki definition hai jo infinitesimal rotation par apply ki gayi hai jo $Δ t$ ke dauran hoti hai.

Small $Δ t$ ke liye: $cos \frac{Δ θ}{2} \approx 1$ aur $sin \frac{Δ θ}{2} \approx \frac{Δ θ}{2} = \frac{∥ ω ∥Δ t}{2}$ . Toh $δ q \approx [1 \frac{1}{2} ω Δ t] = 1000 + \frac{Δ t}{2} [0 ω] .$

$\frac{1}{2}$ yahan kyun aata hai: $sin \frac{θ}{2}$ mein half-angle quaternion definition mein baked in hai. Yahi famous $\frac{1}{2}$ ki origin hai.

Step 2 — Difference quotient → derivative

$q (t + Δ t) - q (t) = q \otimes δ q - q = q \otimes (δ q - q_{identity}) = q \otimes \frac{Δ t}{2} [0 ω]$

jahan $q_{identity} = (1, 0)$ hai. $Δ t$ se divide karo aur $Δ t \to 0$ hone do:

$\overset{q}{˙} = \frac{1}{2} q \otimes [0 ω]$

Yeh step kyun? Humne "ek tiny rotation compose karna" ko ek genuine time-derivative mein badla. Pure quaternion $(0, ω)$ ko angular-velocity quaternion kehte hain.

Step 3 — Product ko $ω$ par acting matrix ke roop mein rewrite karo

Quaternion multiplication linear hai, isliye $q \otimes (0, ω)$ ko ek $4 \times 3$ matrix times $ω$ likha ja sakta hai. Hamilton product se $q \otimes (0, ω)$ workout karne par:

$q \otimes [0 ω] = [- q_{v} \cdot ω q_{0} ω + q_{v} \times ω] = Ξ (q) ω$

ke saath

Yeh step kyun? GNC filters (EKF, etc.) ko ek linear relation $\overset{q}{˙} = A ω$ chahiye. Hamilton product ko $Ξ (q)$ ke roop mein package karna hume bilkul yahi deta hai — matrix code ke liye clean.

Norm 1 kyun rehta hai (sanity check)

$∥ q ∥^{2}$ ki rate of change zero honi chahiye — warna $q$ unit sphere se drift kar jaata hai aur valid rotation represent nahi karta.

$\frac{d}{d t} ∥ q ∥^{2} = 2 q^{⊤} \overset{q}{˙} = 2 q^{⊤} (\frac{1}{2} Ξ (q) ω) = (q^{⊤} Ξ (q)) ω .$

Key algebraic fact: $q^{⊤} Ξ (q) = 0^{⊤}$ (ek zero $1 \times 3$ row). Block form use karke row-by-row check karo: $q^{⊤} Ξ (q) = q_{0} (- q_{v}^{⊤}) + q_{v}^{⊤} (q_{0} I + [q_{v} \times]) = - q_{0} q_{v}^{⊤} + q_{0} q_{v}^{⊤} + q_{v}^{⊤} [q_{v} \times] = 0^{⊤}$ kyunki $q_{v}^{⊤} [q_{v} \times] = - (q_{v} \times q_{v})^{⊤} = 0$ . Isliye $\frac{d}{d t} ∥ q ∥^{2} = 0$ . ✔

Body frame vs inertial frame (ek real GNC subtlety)

Agar $ω$ body frame mein express kiya gaya hai (usual case, gyros body rates measure karte hain): right multiplication use karo $\overset{q}{˙} = \frac{1}{2} q \otimes (0, ω)$ → upar wala $Ξ (q)$ .
Agar $ω$ inertial frame mein hai: left multiplication use karo $\overset{q}{˙} = \frac{1}{2} (0, ω) \otimes q$ , jo ek alag matrix deta hai (often $Ω (ω)$ kehte hain).

Worked examples

Common mistakes (Steel-manned)

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

Imagine karo tum ek office chair par spin kar rahe ho. Tumhara munh kis taraf hai yeh describe karne ke liye, tum ek special 4-number "compass" use karte ho jise quaternion kehte hain. Ab, agar main tumhe batata hun tum kitni fast spin kar rahe ho, tum jaanna chahoge ki tumhare compass numbers kitni fast change ho rahe hain. Turns out compass numbers spin speed ki half se change hote hain — kyunki compass secretly half-turns measure karta hai (ek quirky quirk jo math ko khoobsurat banata hai). Ek chhota bookkeeping table bhi hai ( $Ξ$ matrix) jo tumhari current facing ko tumhari spin se mix karta hai sahi change dene ke liye. Aur poori cheez aise built hai ki tumhara compass hamesha ek valid "unit" compass rehta hai — kabhi toot nahi sakta.

Active recall

\overset{q}{˙} = \frac{1}{2} Ξ (q) ω

mein ½ kyun aata hai?

Kyunki quaternions half-angle encode karte hain (

sin \frac{θ}{2}, cos \frac{θ}{2}

); half-angle differentiate karne par chain rule se ½ milta hai.

Continuous-time quaternion kinematic equation (body rates) likho.

\overset{q}{˙} = \frac{1}{2} q \otimes (0, ω) = \frac{1}{2} Ξ (q) ω

Kinematics mein

∥ q ∥

1 rehna kya guarantee karta hai?

q^{⊤} Ξ (q) = 0^{⊤}

, isliye

\frac{d}{d t} ∥ q ∥^{2} = q^{⊤} \overset{q}{˙} \cdot 2 = 0

;

\overset{q}{˙} ⊥ q

Ξ (q)

ki structure?

4 \times 3

: top row

- q_{v}^{⊤}

; neeche

3 \times 3

block

q_{0} I + [q_{v} \times]

Body-frame

ω

: pre- ya post-multiply?

Post-multiply,

q \otimes (0, ω)

. Inertial-frame

ω

pre-multiply karta hai.

Equivalent

Ω (ω)

form kya hai?

\overset{q}{˙} = \frac{1}{2} Ω (ω) q

ω

ki jagah

q

factor out karke;

Ω (ω)

4 \times 4

skew hai.

Agar norm provably preserved hai toh code mein renormalize kyun karo?

Preservation sirf continuous time mein exact hai; discrete integrators norm leak karte hain, isliye har step mein renormalize karo.

Pure spin

ω_{z}

identity se,

q_{3} (t)

kya hoga?

q_{3} (t) = sin \frac{ω _{z} t}{2}

(half-angle),

q_{0} = cos \frac{ω _{z} t}{2}

ke saath.

Connections

Rotation matrices and SO(3) — quaternions $S O (3)$ ka double cover hain.
Euler angles and gimbal lock — quaternions is equation se singularity avoid karte hain.
Angular velocity and the body frame — $ω$ kahan se aata hai (gyros).
Extended Kalman Filter for attitude estimation — $\overset{q}{˙} = \frac{1}{2} Ξ (q) ω$ ko process model ke roop mein use karta hai.
Numerical integration RK4 — is ODE ko discretize karna + renormalization.
Rodrigues rotation formula — is kinematics ka exponential-map cousin.