Worked examples — Modified Rodrigues parameters — singularity-free, compact

Before symbols fly:

• $\hat{e}$ (read "e-hat") is a length-1 arrow pointing along the rotation axis — the pole you spin around.
• $\Phi$ (capital phi) is how far you spin. Convention on this page: $\Phi$ is in degrees, unless a formula explicitly says "in radians" (only the small-angle approximation in Example 6 does).
• $\tan$ is "opposite over adjacent" of a right triangle; here it turns the angle $\Phi /4$ into a number, and that number is the whole magnitude of $\mathbf{\sigma }$.

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The scenario matrix

Every case this topic can throw at you falls into one of these cells. Each worked example below names the cell it covers.

Cell	What makes it special	Covered by
A. Small angle ($\Phi <180^{\circ }$)	$\Vert \mathbf{\sigma }\Vert <1$, no switch needed	Ex 1
B. Exactly $180^{\circ }$	$\Vert \mathbf{\sigma }\Vert =1$ exactly, the switch boundary	Ex 2
C. Large angle ($180^{\circ }<\Phi <360^{\circ }$)	$\Vert \mathbf{\sigma }\Vert >1$, must switch to shadow set	Ex 3
D. Zero / identity	$\Phi =0$, $\mathbf{\sigma }=0$ — degenerate axis	Ex 4
E. Tilted axis, signed components	axis not along a coordinate line; negative entries	Ex 5
F. Limiting behaviour	$\Phi \to 360^{\circ }$, blow-up; and $\Phi \to 0^{\circ }$	Ex 6
G. Propagation (kinematics)	use $\dot{\mathbf{\sigma }}=\frac{1}{4}B\mathbf{\omega }$ off-identity	Ex 7
H. Real-world word problem	detumble → convert → decide switch	Ex 8
I. Exam twist	recover $\Phi ,\hat{e}$ back from $\mathbf{\sigma }$	Ex 9

Read the figure carefully — it is the map for every example:

• The horizontal axis is the rotation angle $\Phi$, in degrees, from $0^{\circ }$ to about $375^{\circ }$.
• The vertical axis is the MRP magnitude $\Vert \mathbf{\sigma }\Vert =\tan (\Phi /4)$.
• The chalk-blue curve is that tangent. It starts flat near the origin (small angles give small $\mathbf{\sigma }$), then curves upward ever faster.
• The pale-yellow dashed line at height $1$ is the switch threshold; the yellow arrow marks where the curve crosses it, exactly at $\Phi =180^{\circ }$.
• The pink dotted vertical line at $\Phi =360^{\circ }$ is the singularity, and the pink arrow shows the curve racing to infinity as it approaches.
• The shaded blue band under the curve, left of the yellow line, is the safe region where $\Vert \mathbf{\sigma }\Vert \le 1$.

The whole game is: stay in the shaded region, and if you ever drift out, jump to the shadow set.

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The examples

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Recall

Recall One-line rules for every cell

Safe vs switch threshold? $\Vert \mathbf{\sigma }\Vert =1$ at $\Phi =180^{\circ }$.

Undo the MRP map for the angle? $\Phi =4\arctan \Vert \mathbf{\sigma }\Vert$.

Undo the MRP map for the axis? $\hat{e}=\mathbf{\sigma }/\Vert \mathbf{\sigma }\Vert$.

Shadow set magnitude relation? $\Vert {\mathbf{\sigma }}^S\Vert =\tan (((360^{\circ }-\Phi )/4))$.

Kinematic coefficient? $\frac{1}{4}$ (quarter-angle), never $\frac{1}{2}$.

Related maps to compare against: Quaternions (Euler symmetric parameters), Classical Rodrigues parameters (Gibbs vector), Euler angles and Gimbal Lock, Direction Cosine Matrix (DCM), and the propagation background in Attitude kinematics and $\boldsymbol\omega$.