Quaternions¶

Quaternions are the tool we use to represent 3D orientation. Here, we build the algebra from scratch — definition, product, conjugate, norm, inverse, unit quaternions.

Definition¶

A quaternion $\mathbf q$ extends the complex numbers with three imaginary units $i,j,k$. Following the Cayley–Dickson construction, taking two complex numbers $A=a+bi$ and $C=c+di$ and forming $\mathbf q = A + Cj$ with $k\triangleq ij$ gives a number with one real and three imaginary parts:

$$\mathbf q = q_0 + q_1 i + q_2 j + q_3 k \in \mathbb H ,\qquad \{q_0,q_1,q_2,q_3\}\in\mathbb R,$$

where $\mathbb H$ is the space of quaternions. We split it into a scalar part $q_w$ and a vector part $\mathbf q_v=(q_x,q_y,q_z)$, and write it as a 4-vector in scalar-first order:

$$\mathbf q = q_w + \mathbf q_v = \begin{bmatrix} q_w \\ \mathbf q_v \end{bmatrix} = \begin{bmatrix} q_w \\ q_x \\ q_y \\ q_z \end{bmatrix}.$$

Convention. We use the Hamilton convention throughout: scalar part first, and $ij=+k$ (right-handed). This matches Eigen/ROS/Ceres and Joan Solà's reference.

The imaginary units¶

The units $i,j,k$ are defined by Hamilton's multiplication rules:

$$i^2 = j^2 = k^2 = ijk = -1,$$

from which all pairwise products follow:

$$ij = k,\quad jk = i,\quad ki = j,\qquad ji = -k,\quad kj = -i,\quad ik = -j.$$

Because $ij=k$ but $ji=-k$, the units anti-commute, and quaternion multiplication is therefore non-commutative: in general $\mathbf q_a \otimes \mathbf q_b \neq \mathbf q_b \otimes \mathbf q_a$ (the symbol $\otimes$ is defined below). This is the algebraic root of the fact that 3D rotations do not commute.

Special quaternions¶

• Real quaternion: zero vector part, $\mathbf q = \begin{bmatrix} q_w \\ \mathbf 0\end{bmatrix}$ — behaves like an ordinary real number.
• Pure quaternion: zero scalar part, $\mathbf q = \begin{bmatrix} 0 \\ \mathbf v\end{bmatrix}$ — this is how a 3-vector $\mathbf v\in\mathbb R^3$ is embedded as a quaternion, the key step when rotating vectors.
• Identity: $\mathbf q = \begin{bmatrix} 1 \\ \mathbf 0\end{bmatrix}$ — the neutral element of the product, $\mathbf q\otimes\mathbf 1 = \mathbf 1\otimes\mathbf q = \mathbf q$.

The Hamilton product $\otimes$¶

Multiplying out $(q_{a,w}+\mathbf q_{a,v})(q_{b,w}+\mathbf q_{b,v})$ with the unit rules above and collecting scalar and vector parts gives the compact form

$$\mathbf q_a \otimes \mathbf q_b = \begin{bmatrix} q_{a,w}\,q_{b,w} - \mathbf q_{a,v}\!\cdot\!\mathbf q_{b,v} \\[2pt] q_{a,w}\,\mathbf q_{b,v} + q_{b,w}\,\mathbf q_{a,v} + \mathbf q_{a,v}\times\mathbf q_{b,v}\end{bmatrix}.$$

The cross-product term is exactly what makes the product non-commutative (swapping $a,b$ flips its sign). The product is also bilinear, so it can be written as a matrix–vector product in two ways — with the left and right product matrices:

$$\mathbf q_a \otimes \mathbf q_b = [\mathbf q_a]_L\,\mathbf q_b = [\mathbf q_b]_R\,\mathbf q_a ,\qquad [\mathbf q]_{L,R} = q_w\mathbf I + \begin{bmatrix} 0 & -\mathbf q_v^\top \\ \mathbf q_v & \pm[\mathbf q_v]_\times\end{bmatrix}$$

($+$ for $[\mathbf q]_L$, $-$ for $[\mathbf q]_R$; see the digest §B). These matrix forms are handy for the Kalman-filter Jacobians later.

Conjugate, norm and inverse¶

• Conjugate — negate the vector part: $\;\mathbf q^* = \begin{bmatrix} q_w \\ -\mathbf q_v\end{bmatrix}$. It reverses product order: $(\mathbf q_a\otimes\mathbf q_b)^* = \mathbf q_b^*\otimes\mathbf q_a^*$.
• Norm — $\;\lVert\mathbf q\rVert = \sqrt{\mathbf q^\top\mathbf q} = \sqrt{q_w^2+q_x^2+q_y^2+q_z^2} = \sqrt{\mathbf q^*\otimes\mathbf q}$, and it is multiplicative: $\lVert\mathbf q_a\otimes\mathbf q_b\rVert = \lVert\mathbf q_a\rVert\,\lVert\mathbf q_b\rVert$.
• Inverse — $\;\mathbf q^{-1} = \dfrac{\mathbf q^*}{\lVert\mathbf q\rVert^2}$, so that $\mathbf q\otimes\mathbf q^{-1} = \mathbf q^{-1}\otimes\mathbf q = \begin{bmatrix}1\\\mathbf 0\end{bmatrix}$.

Unit quaternions — the rotations¶

A unit quaternion has $\lVert\mathbf q\rVert = 1$. For these the inverse is simply the conjugate, $\mathbf q^{-1}=\mathbf q^*$, and — because the norm is multiplicative — the product of two unit quaternions is again a unit quaternion. The unit quaternions thus form a group, and it is this group that represents 3D rotations:

$$\mathbf q = \begin{bmatrix}\cos(\phi/2)\\ \mathbf u\,\sin(\phi/2)\end{bmatrix}\quad\text{encodes a rotation of angle }\phi\text{ about axis }\mathbf u.$$

How a unit quaternion rotates a vector (the sandwich product) we be covered in the next chapter.

Numerical check¶

A few lines confirm the algebra: the product is non-commutative, the norm is multiplicative, the conjugate reverses product order, and $\mathbf q\otimes\mathbf q^{-1}$ is the identity.

Literature¶

• J. B. Kuipers. Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace and Virtual Reality. Princeton University Press, 1999.
• Joan Solà, Quaternion kinematics for the error-state Kalman filter, arXiv:1711.02508, 2017