1.6. Quaternions

Quaternions

A quaternion is a 4D number system extension of complex numbers that provides a compact, numerically stable way to represent 3D rotations without the woes of gimbal lock. A quaternion is written
$q=w+x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\equiv (w,\mathbf{u}),\mathbf{u}=(x,y,z)\in {\mathbb{R}}^3,$
where $w$ is the scalar part and $\mathbf{u}$ is the vector part. For rotations, one uses unit quaternions satisfying $\Vert q\Vert =\sqrt{w^2+x^2+y^2+z^2}=1$. The quaternion basis elements $\mathbf{i},\mathbf{j},\mathbf{k}$ satisfy:

&\mathbf{i}^2=\mathbf{j}^2=\mathbf{k}^2=\mathbf{i}\mathbf{j}\mathbf{k}=-1, \\ &\mathbf{i}\mathbf{j}=\mathbf{k},\quad \mathbf{j}\mathbf{k}=\mathbf{i},\quad \mathbf{k}\mathbf{i}=\mathbf{j}, \\ &\mathbf{j}\mathbf{i}=-\mathbf{k},\quad \mathbf{k}\mathbf{j}=-\mathbf{i},\quad \mathbf{i}\mathbf{k}=-\mathbf{j}. \end{align*}$$ Note that pairwise multiplication is non-commutitative. A unit quaternion encodes a rotation by an angle $\theta$ about a unit axis $\hat{a}\in\mathbb{R}^3$ via $$q = \left(\cos\frac{\theta}{2},\; \hat{a}\,\sin\frac{\theta}{2}\right).$$ In this representation, $q$ and $-q$ correspond to the **same physical rotation**, so the mapping from unit quaternions to $SO(3)$ is two-to-one. Quaternions compose rotations through quaternion multiplication: if $q_1$ and $q_2$ represent rotations $R_1$ and $R_2$, then $q_2 q_1$ represents the composed rotation $R_2\circ R_1$ (order matters). The inverse rotation is given by the quaternion **conjugate**: $$q^* = (w,\; -\mathbf{u}),\qquad q^{-1}=q^*\ \text{for}\ \lVert q\rVert=1.$$

Quaternions avoid the singularities of Euler angles (no gimbal lock in the representation) and are widely used for orientation integration, interpolation, and filtering. Converting to a rotation matrix $R\in SO(3)$ is standard and yields an orthonormal matrix with $\det (R)=1$, while conversion from $R$ back to $q$ requires choosing a consistent sign convention because of the $q\sim -q$ ambiguity.

Intuition

Unit quaternions live on the 3-sphere $S^3\subset {\mathbb{R}}^4$, which double-covers $SO(3)$: each rotation corresponds to two antipodal quaternions $\pm q$. This 4-parameter representation trades one extra parameter for complete singularity freedom—unlike 3-parameter representations (Euler angles, exponential coordinates), there is no configuration where the parameterization degenerates.

The half-angle $\theta /2$ in $q=(\cos \frac{\theta }{2},\hat{a}\sin \frac{\theta }{2})$ is not arbitrary—it ensures that the quaternion product correctly composes rotations. Geometrically, $S^3$ wraps around $SO(3)$ twice: as $\theta$ goes from $0$ to $2\pi$, $q$ traverses a semicircle on $S^3$ from $(1,0,0,0)$ to $(-1,0,0,0)$, and the return trip corresponds to $\theta \in [2\pi ,4\pi ]$.

Properties

(Quaternion Multiplication) For $q_1=(w_1,{\mathbf{u}}_1)$ and $q_2=(w_2,{\mathbf{u}}_2)$:

$q_1{q}_2=(w_1{w}_2-{\mathbf{u}}_1\cdot {\mathbf{u}}_2,w_1{\mathbf{u}}_2+w_2{\mathbf{u}}_1+{\mathbf{u}}_1\times {\mathbf{u}}_2).$
The cross product term is what makes quaternion multiplication non-commutative.

(Rotation Action) To rotate a vector $p\in {\mathbb{R}}^3$ by quaternion $q$, embed $p$ as the pure quaternion $(0,p)$ and compute $p^{\prime }=qp{q}^{\ast }$ (quaternion sandwich product). This is equivalent to $Rp$ where $R$ is the corresponding rotation matrix.

(Quaternion to Matrix) For $q=(w,x,y,z)$:

$R=\left[\begin{array}{ccc}1-2(y^2+z^2)& 2(xy-wz)& 2(xz+wy)\\ 2(xy+wz)& 1-2(x^2+z^2)& 2(yz-wx)\\ 2(xz-wy)& 2(yz+wx)& 1-2(x^2+y^2)\end{array}\right].$

(SLERP) Spherical linear interpolation between $q_0$ and $q_1$ at parameter $t\in [0,1]$:

$\mathrm{SLERP}(q_0,q_1,t)=q_0\frac{\sin ((1-t)\Omega )}{\sin \Omega }+q_1\frac{\sin (t\Omega )}{\sin \Omega },$
where $\cos \Omega =q_0\cdot q_1$. This produces constant angular velocity interpolation—the optimal path on $SO(3)$.

(Angular Velocity) For a unit quaternion trajectory $q(t)$, the body angular velocity is ${\omega }^b=2\mathrm{Im}(q^{\ast }\dot{q})$ and the spatial angular velocity is ${\omega }^s=2\mathrm{Im}(\dot{q}{q}^{\ast })$, where $\mathrm{Im}$ extracts the vector part.

Examples

Composing Two 90° Rotations

$q_x=(\cos 45°,\sin 45°,0,0)=(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2},0,0)$ (90° about $x$).
$q_z=(\frac{\sqrt{2}}{2},0,0,\frac{\sqrt{2}}{2})$ (90° about $z$).

Quaternion product $q_z{q}_x$ Using the multiplication formula: $w=\frac{\sqrt{2}}{2}\cdot \frac{\sqrt{2}}{2}-(0,0,\frac{\sqrt{2}}{2})\cdot (\frac{\sqrt{2}}{2},0,0)=\frac{1}{2}$. $\mathbf{u}=\frac{\sqrt{2}}{2}(\frac{\sqrt{2}}{2},0,0)+\frac{\sqrt{2}}{2}(0,0,\frac{\sqrt{2}}{2})+(0,0,\frac{\sqrt{2}}{2})\times (\frac{\sqrt{2}}{2},0,0)=(\frac{1}{2},0,0)+(0,0,\frac{1}{2})+(0,\frac{1}{2},0)=(\frac{1}{2},\frac{1}{2},\frac{1}{2})$.

Result: $q_z{q}_x=(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2})$, which represents a 120° rotation about the axis $(1,1,1)/\sqrt{3}$ (since $\cos (\theta /2)=1/2\Rightarrow \theta =120°$).