Visual walkthrough — Gimbal lock — problem with Euler angles at θ = ±90°

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Step 1 — Three axes, three spins (the vocabulary)

WHAT. An orientation is "which way is the object tilted." We build it from three simple turns, each about one axis of a right-handed frame. A right-handed frame is three arrows at $90°$ to each other, labelled $x$ (forward), $y$ (left), $z$ (up), arranged so that curling the right hand from $x$ toward $y$ points the thumb along $z$.

WHY these three. Any tilt in 3D can be reached by stacking exactly three independent turns — no more, no less. That number three is the "degrees of freedom" (DOF): three independent dials. Losing one is the whole story of this page.

PICTURE. Figure s01 shows the three named turns:

• $\psi$ (yaw) — spin about the up-axis $z$. Think "turn left/right."
• $\theta$ (pitch) — spin about the left-axis $y$. Think "nose up/down."
• $\varphi$ (roll) — spin about the forward-axis $x$. Think "barrel roll."

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Step 2 — Turning a spin into a grid (the rotation matrix)

WHAT. To compute with these turns we replace each one by a $3\times 3$ box of numbers called a rotation matrix. A matrix here is just a bookkeeping grid: give it the arrow "before," it hands you the arrow "after."

WHY a matrix and not the angle alone. Because we must stack turns, and stacked turns multiply as grids in a fixed order. Angles alone can't be chained; grids can. See Rotation matrices SO(3) for the group they live in.

PICTURE. Figure s02 takes the pitch turn $R_y(\theta )$ and shows what each entry does to a test arrow.

$$R_y(\theta )=\left(\begin{array}{ccc}\underset{\text{keep-forward}}{\underbrace{\cos \theta }}& 0& \underset{\text{tip up}}{\underbrace{\sin \theta }}\\ 0& \underset{\text{left axis frozen}}{\underbrace{1}}& 0\\ \underset{\text{tip forward-from-up}}{\underbrace{-\sin \theta }}& 0& \underset{\text{keep-up}}{\underbrace{\cos \theta }}\end{array}\right)$$

Each symbol, right where it sits:

• $\cos \theta$ on the diagonal $=$ "how much of the old direction survives."
• $\sin \theta$ off-diagonal $=$ "how much leaks from forward into up (or back)."
• the lone $1$ in the middle row $=$ the pitch axis $y$ is untouched — it is the spin axis.

The other two dials, written out the same way. So no grid is left to guess, here are the yaw and roll matrices in full. Each keeps its own spin axis frozen (the lone $1$ on that axis) and mixes the other two components through $\cos$ (keep) and $\sin$ (leak):

$$R_z(\psi )=\left(\begin{array}{ccc}\underset{\text{keep}}{\underbrace{\cos \psi }}& -\sin \psi & 0\\ \underset{\text{leak}}{\underbrace{\sin \psi }}& \cos \psi & 0\\ 0& 0& \underset{z\text{frozen}}{\underbrace{1}}\end{array}\right),R_x(\varphi )=\left(\begin{array}{ccc}\underset{x\text{frozen}}{\underbrace{1}}& 0& 0\\ 0& \underset{\text{keep}}{\underbrace{\cos \varphi }}& -\sin \varphi \\ 0& \underset{\text{leak}}{\underbrace{\sin \varphi }}& \cos \varphi \end{array}\right)$$

• $R_z(\psi )$ freezes $z$ (the yaw axis) and mixes $x,y$.
• $R_x(\varphi )$ freezes $x$ (the roll axis) and mixes $y,z$.

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Step 3 — Freeze the middle dial at the danger value

WHAT. We set pitch to its suspected singular value $\theta =+90°$ and see what the pitch grid becomes. At $90°$: $\cos 90°=0$ and $\sin 90°=1$.

WHY here. The middle turn is the only one that tilts the last axis relative to the first. Push it to $90°$ and we suspect the last axis ($x$, roll) will lie exactly on the first axis ($z$, yaw). We test that suspicion directly rather than guess.

PICTURE. Figure s03: watch the forward-arrow $x$ swing up until it points along $z$. The two "leak" terms $\sin \theta$ hit their maximum $1$; the two "keep" terms $\cos \theta$ collapse to $0$.

$$R_y(90°)=\left(\begin{array}{ccc}\underset{\text{no forward kept}}{\underbrace{0}}& 0& \underset{\text{fully tipped up}}{\underbrace{1}}\\ 0& 1& 0\\ \underset{\text{up}\to \text{back}}{\underbrace{-1}}& 0& \underset{\text{no up kept}}{\underbrace{0}}\end{array}\right)$$

The key visual: the old forward direction now stands straight up. That is the alignment that will merge yaw and roll.

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Step 4 — Multiply it out and read the collapse

WHAT. We compute $R=R_z(\psi )R_y(90°)R_x(\varphi )$ using the three grids from Steps 2–3, and stare at the result:

$$R=\left(\begin{array}{ccc}0& \sin (\varphi -\psi )& \cos (\varphi -\psi )\\ 0& \cos (\varphi -\psi )& -\sin (\varphi -\psi )\\ -1& 0& 0\end{array}\right)$$

WHY this is the punchline. Look at every non-trivial entry: they contain $\psi$ and $\varphi$ only through the single combination $(\varphi -\psi )$. So if you turn the yaw dial up by $\delta$ and the roll dial up by the same $\delta$, the difference $\varphi -\psi$ is unchanged — and therefore $R$ is unchanged. Two dials, one net effect: a degree of freedom has vanished.

PICTURE. Figure s04 plots the entries as $\psi$ and $\varphi$ each vary. The surface is flat along the diagonal $\varphi =\psi +\text{const}$ — a whole valley of $(\psi ,\varphi )$ pairs give the same orientation.

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Step 5 — The other pole, and everything in between

WHAT. We must cover all cases, not just $+90°$. Three regimes:

• $\theta =-90°$: same collapse, opposite sign. Set $\cos \theta =0$, $\sin \theta =-1$, so the pitch grid becomes $R_y(-90°)=\left(\begin{array}{ccc}0& 0& -1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)$. Multiplying $R_z(\psi )R_y(-90°)R_x(\varphi )$ out gives the collapse below.
• $-90°<\theta <+90°$ (the safe interior): $\cos \theta \ne 0$, the two axes are genuinely tilted apart, all three dials act independently. Full 3 DOF.
• $\theta$ near $\pm 90°$ but not exactly there: not safe, as Step 6 shows.

WHY the sign flips. At $+90°$ the forward axis pointed straight up along $+z$, so roll and yaw ran the same way and their effects subtracted ($\varphi -\psi$). At $-90°$ the forward axis points straight down along $-z$ — roll now runs opposite to yaw, so their effects add ($\varphi +\psi$). The lone sign change $\sin \theta :+1\to -1$ is exactly what carries $-\psi \to +\psi$ through the multiplication. Explicitly:

$$R{|}_{\theta =-90°}=\left(\begin{array}{ccc}0& -\sin (\varphi +\psi )& -\cos (\varphi +\psi )\\ 0& \cos (\varphi +\psi )& -\sin (\varphi +\psi )\\ 1& 0& 0\end{array}\right)$$

— every non-trivial entry now depends only on $(\varphi +\psi )$.

PICTURE. Figure s05 lines up the three regimes side by side: yaw axis and roll axis fanned apart (interior), merged pointing up ($+90°$), merged pointing down ($-90°$).

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Step 6 — Why the rates explode (a neighborhood, not a point)

WHAT. Orientation is not static; a GNC computer measures body spin rates $(p,q,r)$ — where $p$ is the spin rate about the roll axis $x$, $q$ about the pitch axis $y$, and $r$ about the yaw axis $z$ (the same three axes from Step 1) — and integrates them into Euler angles. The bridge (from Angular velocity kinematics) is:

$$\left(\begin{array}{c}\dot{\varphi }\\ \dot{\theta }\\ \dot{\psi }\end{array}\right)=\left(\begin{array}{ccc}1& \sin \varphi \tan \theta & \cos \varphi \tan \theta \\ 0& \cos \varphi & -\sin \varphi \\ 0& \frac{\sin \varphi }{\cos \theta }& \frac{\cos \varphi }{\cos \theta }\end{array}\right)\left(\begin{array}{c}p\\ q\\ r\end{array}\right)$$

Two symbols are the culprits, defined here:

• $\tan \theta =\frac{\sin \theta }{\cos \theta }$ — "steepness"; it answers how fast the tilt grows per unit turn. As $\theta \to 90°$, denominator $\cos \theta \to 0$, so $\tan \theta \to \infty$.
• $\frac{1}{\cos \theta }$ — same vanishing denominator, same blow-up.

WHY it's a region not a point. Because $1/\cos \theta$ is huge for a band of angles around $90°$, not only at the exact value. At $\theta =89.9°$, $\cos \theta \approx 0.001745$, so $1/\cos \theta \approx 573$. A whisper of sensor noise in $r$ becomes a scream in $\dot{\psi }$.

PICTURE. Figure s06 plots $1/\cos \theta$ across $\theta$: a smooth curve for most of the range that rockets toward a vertical asymptote as $\theta \to \pm 90°$. The unstable band is shaded.

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Step 7 — The physics never broke; only the chart did

WHAT. The rigid body still rotates perfectly freely in every direction. What failed is the coordinate chart — the particular way $(\varphi ,\theta ,\psi )$ labels orientations. This is exactly what Singularities of coordinate charts describes: the map is fine in the interior, singular at the poles, like the meridian lines all crashing together at the North Pole of a globe.

WHY the fix works. Quaternions wrap all of $SO(3)$ smoothly with no singular orientation, so they never divide by $\cos \theta$. Their only quirk ($q\equiv -q$) is not a lost DOF.

PICTURE. Figure s07: a globe with meridians crowding at the pole (the Euler singularity) beside a smooth quaternion cover with no pinch point.

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The one-picture summary

WHAT. Figure s08 compresses the whole walkthrough: interior pitch keeps yaw and roll fanned apart (3 DOF); as pitch swings to $\pm 90°$ the roll axis climbs onto the yaw axis, the matrix depends only on $(\varphi \mp \psi )$, and the rate curve $1/\cos \theta$ blows through the roof. One glance tells the entire story.

Recall Feynman retelling — the whole walkthrough in plain words

Picture a toy plane with three dials: turn (yaw), nose up/down (pitch), barrel-roll (roll). We wrote each dial as a little grid of numbers so we could stack them (Steps 1–2). Then we cranked the middle dial — pitch — until the nose pointed dead at the ceiling (Step 3). When we multiplied the three grids together, something spooky happened: the answer only cared about the difference between the turn dial and the roll dial (Step 4). Turn both dials by the same amount and the plane doesn't move at all — two dials had fused into one. The same trap sits at nose-straight-down, except there the dials add instead of subtract (Step 5). Worse, the equation that turns real spin into dial-motion has a $1/\cos \theta$ in it, which becomes a division by almost-zero near the poles, so a tiny wobble in the sensor makes the computer command an insane spin (Step 6). But the plane itself was never stuck — only our three dials got confused. Swap the dials for quaternions, whose map never pinches, and the confusion disappears (Step 7).

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Connections

• Euler angles — the three-dial representation we dissected
• Rotation matrices SO(3) — the grids we multiplied in Steps 2–4
• Angular velocity kinematics — source of the $1/\cos \theta$ rate blow-up (Step 6)
• Quaternions — the pinch-free fix (Step 7)
• Singularities of coordinate charts — the general reason a chart fails while the physics is fine
• Attitude determination and control — where this bites in real GNC
• Apollo Guidance Computer — the historic "gimbal lock" alarm