Rotation Representations

In Module 3, we used Euler angles and rotation matrices to represent orientation. Both have significant drawbacks: Euler angles suffer from gimbal lock, and rotation matrices use 9 numbers with 6 redundant constraints. This lesson introduces representations that solve these problems — axis-angle, rotation vectors, and quaternions — the workhorses of modern robotics and 3D graphics.

Why Euler Angles Aren’t Enough

Quick recap of the problems (from Module 3):

1. Gimbal lock: When pitch = $\pm 90°$, roll and yaw become coupled — one degree of freedom is lost. The robot can still physically reach all orientations, but the mathematical representation becomes singular.

2. Non-unique representation: Multiple Euler angle triples can represent the same rotation (e.g., $(0°, 0°, 90°)$ and $(90°, 0°, 0°)$ with different conventions).

3. Interpolation artifacts: Linearly interpolating between two sets of Euler angles doesn’t produce a smooth, shortest-path rotation. The intermediate orientations can take bizarre detours.

4. 12 conventions: The lack of a single standard means convention mismatches are inevitable in multi-library systems.

We need representations that are singularity-free, unique (or nearly so), and interpolation-friendly.

Axis-Angle Representation

Euler’s rotation theorem states that any 3D rotation can be described as a single rotation by angle $\theta$ about a fixed axis $\hat{k}$:

This is the rotation’s axis-angle form. It uses only 4 numbers (3 for the unit axis + 1 for the angle), compared to 9 for a rotation matrix.

Axis-Angle to Rotation Matrix: Rodrigues’ Formula

Given axis $\hat{k}$ and angle $\theta$, the rotation matrix is:

where $[\hat{k}]_\times$ is the skew-symmetric matrix of $\hat{k}$:

This is Rodrigues’ rotation formula — one of the most important formulas in 3D geometry. A useful identity for the squared term: $[\hat{k}]_\times^2 = \hat{k}\hat{k}^T - I$, which follows from the vector triple product $\hat{k} \times (\hat{k} \times \vec{v}) = \hat{k}(\hat{k}\cdot\vec{v}) - \vec{v}$.

Rotation Matrix to Axis-Angle

Given a rotation matrix $R$:

1. Angle: $\theta = \arccos\left(\frac{\text{tr}(R) - 1}{2}\right)$

2. Axis (when $\theta \neq 0, \pi$): $\hat{k} = \frac{1}{2\sin\theta}\begin{bmatrix} R_{32} - R_{23} \\ R_{13} - R_{31} \\ R_{21} - R_{12} \end{bmatrix}$

Rotation Vectors

The rotation vector $\vec{\omega} = \theta \hat{k}$ packs the axis and angle into a single 3D vector:

• The direction of $\vec{\omega}$ is the rotation axis
• The magnitude $|\vec{\omega}|$ is the rotation angle

This is the most compact non-redundant representation — just 3 numbers for 3 DOF. It’s widely used in optimization and Lie group theory (where it parameterizes the tangent space of SO(3), called $\mathfrak{so}(3)$).

The mapping from rotation vectors to rotation matrices is the matrix exponential:

and Rodrigues’ formula is precisely the closed-form evaluation of this exponential.

The inverse mapping (rotation matrix to rotation vector) is the matrix logarithm:

Quaternions

Quaternions are the gold standard for rotation representation in robotics, aerospace, and computer graphics. They use 4 numbers, are singularity-free, and enable efficient composition and interpolation.

Definition

A quaternion extends complex numbers with three imaginary units $i, j, k$:

or equivalently, as a scalar-vector pair:

The imaginary units satisfy Hamilton’s rules: $i^2 = j^2 = k^2 = ijk = -1$.

Unit Quaternions and Rotations

A unit quaternion ($|q| = \sqrt{w^2 + x^2 + y^2 + z^2} = 1$) represents a rotation. Given axis $\hat{k}$ and angle $\theta$:

Note the half angle — this is key to how quaternions work. It means that a 360° rotation gives $q = (-1, \vec{0})$, not the identity $(1, \vec{0})$. In fact, $q$ and $-q$ represent the same rotation (double cover of SO(3)).

Rotating a Vector with a Quaternion

To rotate a vector $\vec{p}$ by the rotation represented by unit quaternion $q$:

1. Encode the vector as a pure quaternion: $p = (0, \vec{p})$
2. Compute: $p' = q \, p \, q^*$
3. Extract the vector part of $p'$

where $q^* = (w, -\vec{v})$ is the conjugate (which equals the inverse for unit quaternions).

Quaternion Multiplication

Two quaternions multiply as:

Composing rotations is just quaternion multiplication: if $q_1$ rotates and then $q_2$ rotates, the combined rotation is $q_2 \otimes q_1$.

This is much cheaper than multiplying two $3 \times 3$ matrices (16 multiplications + 12 additions for quaternions vs. 27 multiplications + 18 additions for matrices).

Quaternion to Rotation Matrix

Rotation Matrix to Quaternion

Given $R$, the quaternion can be extracted via the Shepperd method (numerically stable):

1. Compute $w = \frac{1}{2}\sqrt{1 + R_{11} + R_{22} + R_{33}}$
2. If $w > \epsilon$: $x = \frac{R_{32}-R_{23}}{4w}$, $y = \frac{R_{13}-R_{31}}{4w}$, $z = \frac{R_{21}-R_{12}}{4w}$
3. If $w \approx 0$: use alternative formulas based on the largest diagonal element

SLERP — Spherical Linear Interpolation

One of the most compelling reasons to use quaternions is SLERP (Spherical Linear intERPolation). Given two orientations $q_0$ and $q_1$, SLERP produces a smooth, constant-angular-velocity interpolation:

where $\Omega = \arccos(q_0 \cdot q_1)$ is the angle between the quaternions and $t \in [0, 1]$.

Properties of SLERP:

• Constant angular velocity: the rotation rate is uniform throughout the interpolation
• Shortest path: it follows the great circle on the unit quaternion sphere (after ensuring $q_0 \cdot q_1 > 0$)
• Torque-minimal: it minimizes the total angular acceleration

for t = 0 to 1 step 0.01:
    q_target = slerp(q_pick, q_place, t)
    send_orientation_command(q_target)

Comparison of Representations

Conversions Summary

Axis-Angle $\leftrightarrow$ Quaternion

Axis-angle → Quaternion: $q = \left(\cos\frac{\theta}{2}, \; \sin\frac{\theta}{2} \, \hat{k}\right)$

Quaternion → Axis-angle: $\theta = 2\arccos(w), \quad \hat{k} = \frac{\vec{v}}{|\vec{v}|}$

Quaternion $\leftrightarrow$ Rotation Matrix

Use the formulas in the “Quaternion to Rotation Matrix” and “Rotation Matrix to Quaternion” sections above.

Rotation Vector $\leftrightarrow$ Quaternion

Rotation vector → Quaternion: extract $\theta = |\vec{\omega}|$ and $\hat{k} = \vec{\omega}/|\vec{\omega}|$, then apply axis-angle → quaternion.

Quaternion → Rotation vector: extract axis-angle from quaternion, then $\vec{\omega} = \theta \hat{k}$.

Practice Problems

1. Find the axis-angle representation of $R_z(90°)$ using the trace formula and axis extraction.

2. Use Rodrigues’ formula to compute the rotation matrix for a 180° rotation about axis $\hat{k} = \frac{1}{\sqrt{2}}(1, 1, 0)$.

3. Convert the axis-angle pair $(\hat{k} = (0, 0, 1), \theta = 120°)$ to a unit quaternion.

4. Compute $q_1 \otimes q_2$ where $q_1 = (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}, 0, 0)$ and $q_2 = (\frac{\sqrt{2}}{2}, 0, \frac{\sqrt{2}}{2}, 0)$.

5. Two orientations are represented as quaternions: $q_0 = (1, 0, 0, 0)$ (identity) and $q_1 = (0, 0, 0, 1)$ (180° about z). What is $\text{slerp}(q_0, q_1, 0.5)$?