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

This page assumes nothing. If the parent note (the topic note) used a symbol, we build it here from a picture first.

0. What a "vector" and a "frame" even are

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

The parent uses two frames:

The curly braces ${n}$ and ${b}$ are just labels — read " ${n}$ " as "the nav frame" and " ${b}$ " as "the body frame."

1. Angles, the right-hand rule, and why we measure "turning" with them

Definition The right-hand rule — which way is a POSITIVE turn

A turn needs a sign: is $+ 3 0^{\circ}$ clockwise or counter-clockwise? We fix this once with the right-hand rule. Point your right thumb along the axis arrow (e.g. along $+ z$ ). Your curling fingers show the direction of a positive rotation. So a positive turn is counter-clockwise when you look back down the axis toward the origin (i.e. with the axis pointing at your eye).

Applied to the three vehicle turns (each about its own axis):

Positive ==yaw $ψ$ == (about $+ z$ , which points Down) → nose swings toward the right/East.
Positive ==pitch $θ$ == (about $+ y$ , the right wing) → nose goes up.
Positive ==roll $ϕ$ == (about $+ x$ , the nose) → right wing dips down.

Why state this at all? Without a fixed positive direction, " $ψ = 3 0^{\circ}$ " is ambiguous and the $sin$ / $cos$ signs in the matrix could go either way. Every formula below assumes this right-hand-rule convention.

The parent's three named angles are all just "signed amount of turn about one axis":

Symbol	Name	Picture (what the nose does, positive sense)	Axis it turns about
$ψ$ (psi)	yaw	swings toward the right (East)	$z$ (down)
$θ$ (theta)	pitch	nose lifts up	$y$ (right wing)
$ϕ$ (phi)	roll	right wing dips	$x$ (nose)

$ψ, θ, ϕ$ are Greek letters — treat them purely as names for these three signed turn-amounts. Nothing more mysterious than calling them "the yaw number, the pitch number, the roll number."

2. Sine and cosine — the two numbers that a turn produces

3. tan and arcsin — turning ratios back into angles

4. The rotation matrix — a machine that re-labels an arrow

Formula The concrete

$R_{n}^{b}$ the parent actually uses (3-2-1) Each single-axis turn is a 2D shadow-rotation embedded in 3D with a $1$ on the spin axis (that axis doesn't move). Using the right-hand-rule/passive convention (Section 1), the three elementary matrices are:

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}$$ Applying yaw first, then pitch, then roll (3-2-1) multiplies them right-to-left: $$R^b_n(\phi,\theta,\psi)=R_x(\phi)\,R_y(\theta)\,R_z(\psi)$$ Multiplied out (with $s\equiv\sin,\ c\equiv\cos$): $$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}$$ This is the **one symbol the parent leans on**; now you have its concrete grid. The parent's angle-recovery formulas simply read entries back out: $\theta=-\arcsin(R_{13})$, $\phi=\operatorname{atan2}(R_{23},R_{33})$, $\psi=\operatorname{atan2}(R_{12},R_{11})$.

Definition Transpose, orthogonality, and

$det = + 1$ — the free reverse gear The transpose $R^{T}$ flips the grid across its diagonal (rows become columns). For rotation matrices there is a beautiful gift: the reverse rotation is just the transpose, $R_{b}^{n} = (R_{n}^{b})^{T} = (R_{n}^{b})^{- 1} .$ A matrix with $R^{T} R = I$ (the identity, $1$ 's on the diagonal, $0$ 's elsewhere) is called orthogonal. But orthogonal alone is not enough: it also allows reflections (mirror-flips), which are not real rotations. A proper rotation additionally satisfies $det R = + 1,$ where the determinant $det R$ is a single number measuring how the matrix scales volume; $+ 1$ means "no stretching, no mirror-flip." Together $R^{T} R = I$ and $det R = + 1$ pin down exactly the rotations (this set is called $S O (3)$ ). Why the parent cares: to send an arrow body → nav you don't compute anything new — you just transpose (Worked Example 1 uses exactly this), and the $det = + 1$ guarantee is what stops a "rotation" from secretly mirroring your aircraft.

5. Composing rotations — why order matters

The figure shows the same two turns in the two orders ending at different attitudes — proof that order is not decoration.

6. atan2 — the sign-aware angle finder

7. One thing pointed forward: angular rates & gimbal lock

Prerequisite map

Equipment checklist

Hide the right side and test yourself before opening the parent note.

What is a vector, in plain words?

An arrow with a length and a direction; represented by three coordinates once you pick a frame.

What is a frame?

Three mutually perpendicular axes (

x, y, z

) meeting at an origin, used as rulers to give a vector its coordinates.

What do

{n}

and

{b}

mean?

The fixed navigation (ground) frame and the moving body (vehicle) frame.

What does the subscript in

v_{n}

v_{b}

tell you?

Which frame's axes the SAME arrow's coordinates are measured against — nav vs body.

What defines a POSITIVE rotation?

The right-hand rule: thumb along the axis, curling fingers give the positive turn direction (CCW looking back down the axis).

On the unit circle, what are

cos α

and

sin α

The shadows of a turned unit arrow on the

x

-axis (

cos

) and the

y

-axis (

sin

); the tip sits at

(cos α, sin α)

What are

sin 3 0^{\circ}

and

cos 3 0^{\circ}

0.5

and

\approx 0.866

What does

tan α

measure?

The slope (rise over run) of the arrow, equal to

sin α / cos α

What question does

arcsin (v)

answer, and over what range?

"Which angle has sine equal to

v

?"; it returns a principal value in

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

What is the 3×3 form of

R_{n}^{b}

for 3-2-1?

R_{x} (ϕ) R_{y} (θ) R_{z} (ψ)

, giving the matrix whose top row is

[c θ c ψ, c θ s ψ, - s θ]

What does

R_{n}^{b}

do to a vector?

Takes its nav-frame coordinates

v_{n}

and returns its body-frame coordinates

v_{b}

Which TWO conditions make a matrix a proper rotation?

R^{T} R = I

(orthogonal) AND

det R = + 1

(no reflection).

Why is

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

the reverse rotation?

Rotation matrices are orthogonal, so their transpose equals their inverse.

In the product

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

, which rotation happens first?

The rightmost one,

R_{z}

(products read right-to-left in time).

Why can't we just use plain

arctan

to recover

ϕ

and

ψ

tan

repeats every

18 0^{\circ}

, so plain arctan loses the quadrant; atan2 uses the signs of both inputs to get the full-circle answer.

How do you turn an atan2 result into a

0^{\circ}

–

36 0^{\circ}

heading?

Add

36 0^{\circ}

to any negative value (take it mod

36 0^{\circ}

What are

p, q, r

Body angular rates — the instantaneous roll, pitch and yaw turn-speeds measured in the body frame.