Foundations — Modified Rodrigues parameters — singularity-free, compact

This page builds every symbol the parent note Modified Rodrigues parameters leans on, from the ground up. Read top to bottom: each idea is used only after it is defined and drawn.

0. What does "orientation" even mean?

Before any symbol, picture the object.

Everything on this page answers one question: how do we write down "how turned am I" as numbers?

1. The axis $\hat{e}$ — the line you spin around

Why length 1? We only want $\hat{e}$ to carry direction, never size. If we let its length vary, we could not tell "how much to spin" (the angle) apart from "how long the arrow is." Fixing the length at 1 keeps those two jobs separate — the axis says where, the angle says how much.

2. The angle $\Phi$ — how far you spin

Together, the pair $(\hat{e},\Phi )$ is called the axis–angle description. It is the honest, physical picture. Every other representation — including MRPs — is just a repackaging of these two things into numbers a computer likes better.

3. Radians vs degrees — and why halving matters

You will repeatedly see fractions of the angle: $\Phi /2$ (half-angle), $\Phi /4$ (quarter-angle). Keep this table in your head — the whole MRP story is a game of how many times we halve the angle:

Each extra halving pushes the trouble-point further away. That is the single mechanical fact behind why MRPs are "better."

4. Sine, cosine, tangent — reading a right triangle

The parent note is built on $\sin$, $\cos$, $\tan$. Here is what they mean, on a picture, from zero.

Why does the topic reach for $\tan$ specifically? Because $\tan$ encodes the steepness of the angle in a single number, and — crucially — it grows without bound as the angle climbs toward its limit. Look at s03: as $\theta \to 90^{\circ }$ the "run" (adjacent, red) shrinks to zero, so $\tan \theta \to \infty$. That blow-up is deliberately used: MRPs put the blow-up of $\tan (\Phi /4)$ at $\Phi /4=90^{\circ }$, i.e. $\Phi =360^{\circ }$ — as far from normal operation as possible.

5. Vectors and their notation $\mathbf{v}$, $\mathbf{\sigma }$, $\mathbf{\omega }$

The players you will meet:

• $\mathbf{\sigma }$ ("sigma") — the MRP vector, our three-number attitude. Its direction is the axis $\hat{e}$; its length is $\tan (\Phi /4)$.
• $\mathbf{q}$ and $q_0$ — the quaternion: a vector part $\mathbf{q}$ (3 numbers) plus a scalar part $q_0$ (1 number). See Quaternions (Euler symmetric parameters).
• $\mathbf{\omega }$ ("omega") — the angular velocity: an arrow whose direction is the instantaneous spin axis and whose length is the spin rate (rad/s). See Attitude kinematics and $\boldsymbol\omega$.

6. The matrix objects ${\mathbb{I}}_3$, $[{\mathbf{\sigma }}^{\times }]$, $B(\mathbf{\sigma })$

• ${\mathbb{I}}_3$ — the identity matrix ($3\times 3$): the "do nothing" grid. ${\mathbb{I}}_3\mathbf{v}=\mathbf{v}$. Picture: it leaves every arrow untouched.
• $[{\mathbf{\sigma }}^{\times }]$ — the cross-product (skew) matrix. It is the grid that turns "$\mathbf{\sigma }$ crossed with $\mathbf{v}$" into a plain matrix multiply. A cross product produces an arrow perpendicular to both inputs — i.e. it encodes rotation-like sideways pushes.
• $B(\mathbf{\sigma })$ — the assembled grid $(1-{\sigma }^2){\mathbb{I}}_3+2[{\mathbf{\sigma }}^{\times }]+2\mathbf{\sigma }{\mathbf{\sigma }}^{\top }$ that maps the spin $\mathbf{\omega }$ into how fast the MRPs change.

7. The dot on top: $\dot{\mathbf{\sigma }}$ means "rate of change"

Why a derivative and not just algebra? Because attitude evolves continuously as the body spins. We measure $\mathbf{\omega }$ (rate), and integration turns rate into the new orientation.

8. The one edge case: the shadow set

$\tan (\Phi /4)$ blows up at $\Phi =360^{\circ }$, so $\mathbf{\sigma }$ has a singularity there. Spacecraft software never actually hits it, thanks to a switch every GNC engineer keeps in their pocket.

Equipment checklist

Test yourself — cover the right side and answer aloud.