Reference frames — body frame, inertial frame; rotation between them

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

WHAT are the frames?

WHAT is a rotation between frames?

Ek vector $\vec{v}$ ek physical arrow hai — yeh change nahi hota. Sirf uske components change hote hain jab aap use alag frames mein describe karte hain. Agar ${\vec{v}}^{\mathcal{I}}$ uske components inertial frame mein hain aur ${\vec{v}}^{\mathcal{B}}$ body frame mein, toh woh ek matrix se related hain:

${\vec{v}}^{\mathcal{B}}=R_{\mathcal{B}\mathcal{I}}{\vec{v}}^{\mathcal{I}}$

$R_{\mathcal{B}\mathcal{I}}$ (padho "R body-from-inertial") Direction Cosine Matrix (DCM) / rotation matrix hai.

HOW to derive the rotation matrix from first principles

Maano $\{{\hat{x}}_{\mathcal{B}},{\hat{y}}_{\mathcal{B}},{\hat{z}}_{\mathcal{B}}\}$ body axes hain jo inertial coordinates mein express hain. Koi bhi vector: $\vec{v}=v_x^{\mathcal{B}}{\hat{x}}_{\mathcal{B}}+v_y^{\mathcal{B}}{\hat{y}}_{\mathcal{B}}+v_z^{\mathcal{B}}{\hat{z}}_{\mathcal{B}}$

Yeh step kyun? Kyunki ek vector apne components ka sum hota hai ek orthonormal basis ke along.

${\hat{x}}_{\mathcal{B}}$ pe project karo (dono sides pe dot karo, ${\hat{x}}_{\mathcal{B}}\cdot {\hat{y}}_{\mathcal{B}}=0$ etc. use karke): $v_x^{\mathcal{B}}=\vec{v}\cdot {\hat{x}}_{\mathcal{B}}$ $\vec{v}=v_x^{\mathcal{I}}\hat{X}+v_y^{\mathcal{I}}\hat{Y}+v_z^{\mathcal{I}}\hat{Z}$ likhte hain: $v_x^{\mathcal{B}}=({\hat{x}}_{\mathcal{B}}\cdot \hat{X})v_x^{\mathcal{I}}+({\hat{x}}_{\mathcal{B}}\cdot \hat{Y})v_y^{\mathcal{I}}+({\hat{x}}_{\mathcal{B}}\cdot \hat{Z})v_z^{\mathcal{I}}$

Yeh step kyun? Dot product linear hai, isliye $\vec{v}$ ki projection inertial basis vectors ki projections ka weighted sum hai.

Teeno rows stack karke: $\boxed{R_{\mathcal{B}\mathcal{I}}=\left[\begin{array}{ccc}{\hat{x}}_{\mathcal{B}}\cdot \hat{X}& {\hat{x}}_{\mathcal{B}}\cdot \hat{Y}& {\hat{x}}_{\mathcal{B}}\cdot \hat{Z}\\ {\hat{y}}_{\mathcal{B}}\cdot \hat{X}& {\hat{y}}_{\mathcal{B}}\cdot \hat{Y}& {\hat{y}}_{\mathcal{B}}\cdot \hat{Z}\\ {\hat{z}}_{\mathcal{B}}\cdot \hat{X}& {\hat{z}}_{\mathcal{B}}\cdot \hat{Y}& {\hat{z}}_{\mathcal{B}}\cdot \hat{Z}\end{array}\right]}$

Har entry ek direction cosine hai. Rows body axes hain jo inertial components mein likhe hain.

Why $R^{\mathsf{T}}R=I$ (derive it)

$R$ ki row $i$ body axis ${\hat{e}}_i^{\mathcal{B}}$ hai inertial coordinates mein. $R{R}^{\mathsf{T}}$ ki entry $(i,j)$ hai ${\hat{e}}_i^{\mathcal{B}}\cdot {\hat{e}}_j^{\mathcal{B}}$. Kyunki body axes orthonormal hain, yeh ${\delta }_{ij}$ hai → $R{R}^{\mathsf{T}}=I$. Yeh kyun matter karta hai: inverse free hai — bas transpose karo, onboard koi matrix inversion nahi chahiye.

Elementary rotations (building blocks)

$z$-axis ke baare mein angle $\theta$ ka rotation (rotated axes ko project karke derive kiya): $R_z(\theta )=\left[\begin{array}{ccc}\cos \theta & \sin \theta & 0\\ -\sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right]$

Signs kyun aisi hain? Yeh inertial components ko body components mein map karta hai jab body $+\theta$ se rotated ho. $-\sin$ wali row isliye aati hai kyunki naya $y_{\mathcal{B}}$ axis $-X$ ki taraf jhuk jaata hai. (Kuch texts transpose use karte hain — hamesha check karo ki matrix "frame-to-frame" hai ya "vector-rotation".)

Ek general attitude = teen elementary rotations ka product (jaise yaw-pitch-roll 3-2-1): $R_{\mathcal{B}\mathcal{I}}=R_x(\varphi )R_y(\theta )R_z(\psi )$

Worked examples

Common mistakes

Flashcards

Connections

• Euler angles and gimbal lock
• Quaternions for attitude representation
• [[Angular velocity and the kinematic equation $\dot R = -[\omega\times]R$]]
• IMU and gyroscope sensing
• Coriolis and fictitious forces in rotating frames
• ECI, ECEF and orbit reference frames
• Attitude determination — star trackers

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