Quaternion Rotation

Quaternions are a number system that extends complex numbers. A quaternion provides a convenient mathematical notation for representing orientations and rotations of an object in three dimensions. This section discusses some of the useful properties and operations which are used in quaternion rotation.

Conventions used here: quaternions follow the Hamilton convention (\(ijk = -1\)) and are written scalar-first, i.e. \(q = [q_1, q_2, q_3, q_4]\) where \(q_1\) is the real (scalar) part. Frames are denoted with leading super/sub-scripts following Craig’s notation, as used by Madgwick [1]. Be careful: many libraries (e.g. SciPy) store quaternions scalar-last, and some literature uses the JPL convention instead [2].

Let there be two coordinate frames in a 3-dimensional space, \(A\) and \(B\). Let \(\hat{r}^{A}\) be the unit axis of rotation, described in frame \(A\), as shown in the figure below.

\[\hat{r}^{A} = (r^{A}_{x}, r^{A}_{y}, r^{A}_{z}), \qquad \lVert \hat{r}^{A} \rVert_2 = 1\]

The axis must be normalized; otherwise the quaternion built from it below will not be of unit length and will not represent a rotation.

Using Quaternions we can rotate from frame \(A\) to \(B\) by angle \(\theta\) around the axis \(\hat{r}^{A}\).

Quaternion describing this orientation is \(q^{A}_{B}\):

\[q^{A}_{B} = [q_1, q_2, q_3, q_4] = [cos(\frac{\theta}{2}), -r^{A}_{x} sin(\frac{\theta}{2}), -r^{A}_{y} sin(\frac{\theta}{2}), -r^{A}_{z} sin(\frac{\theta}{2})]\]

\(q^{A}_{B}\) describes the orientation of frame \(B\) relative to frame \(A\).

Note the minus signs. They are not a typo: they are what makes \(q^{A}_{B}\) the operator that takes the coordinates of a vector in frame \(A\) to its coordinates in frame \(B\). Concretely, if \({}^{A}v\) and \({}^{B}v\) are the same vector expressed in the two frames (each written as a 4-element quaternion with a leading \(0\)), then

\[{}^{B}v = q^{A}_{B} \bigotimes {}^{A}v \bigotimes \left( q^{A}_{B} \right)^{*}\]

Every sign and ordering convention below follows from this one equation, so it is worth fixing it in mind before reading on.

A rotation quaternion must be of unit length.
Quaternion arithmetic itself works for any quaternion, but only unit quaternions represent rotations. It is therefore conventional to normalize any quaternion describing an orientation.

\[\lVert q \rVert_2 = 1\]

Conjugate quaternion:

\[\left( q^{A}_{B} \right)^{*} = q^{B}_{A} = [q_1, -q_2, -q_3, -q_4]\]

The conjugate swaps the two frames described by an orientation. For a unit quaternion the conjugate is also the inverse, \(q^{-1} = q^{*}\); this is not true in general (in general \(q^{-1} = q^{*} / \lVert q \rVert_2^2\)).

Compound orientations using quaternions: Let \(q^{A}_{B}\) and \(q^{B}_{C}\) be two quaternions. \(q^{A}_{B}\) - orientation of \(B\) w.r.t. \(A\).
\(q^{B}_{C}\) - orientation of \(C\) w.r.t. \(B\).

The compound orientation \(q^{A}_{C}\) is defined as:

\[q^{A}_{C} = q^{B}_{C} \bigotimes q^{A}_{B}\]

\(q^{A}_{C}\) - orientation of \(C\) w.r.t. \(A\).
\(\bigotimes\) represents quaternion product.

Why the order looks reversed: applying the rotation operation twice gives

\[{}^{C}v = q^{B}_{C} \bigotimes \left( q^{A}_{B} \bigotimes {}^{A}v \bigotimes \left( q^{A}_{B} \right)^{*} \right) \bigotimes \left( q^{B}_{C} \right)^{*}\] \[{}^{C}v = \left( q^{B}_{C} \bigotimes q^{A}_{B} \right) \bigotimes {}^{A}v \bigotimes \left( q^{B}_{C} \bigotimes q^{A}_{B} \right)^{*}\]

hence \(q^{A}_{C} = q^{B}_{C} \bigotimes q^{A}_{B}\). If you instead adopt the (nowadays more common) convention without the minus signs, where \(q^{A}_{B}\) maps coordinates from \(B\) into \(A\), the composition reverses to \(q^{A}_{C} = q^{A}_{B} \bigotimes q^{B}_{C}\). Mixing the two is the single most common source of errors.

Quaternion Product (Hamilton rule):
Let \(p\) and \(q\) be two quaternions. Then the product of these two quaternions is:

\[p \bigotimes q = [p_1, p_2, p_3, p_4] \bigotimes [q_1, q_2, q_3, q_4]\] \[p \bigotimes q = \begin{pmatrix} p_1 q_1 - p_2 q_2 - p_3 q_3 - p_4 q_4 \\ p_1 q_2 + p_2 q_1 + p_3 q_4 - p_4 q_3 \\ p_1 q_3 - p_2 q_4 + p_3 q_1 + p_4 q_2 \\ p_1 q_4 + p_2 q_3 - p_3 q_2 + p_4 q_1 \end{pmatrix}^T\]

Quaternion product is not commutative, i.e., \(p \bigotimes q \ne q \bigotimes p\).

NOTE: I know there may be a lot of confusion in Quaternion mathematics. The blogpost by Fan Zheng provides an excellent overview of conventions used in Quaternion representations. [2]

References

1. Madgwick, S. (2010). An efficient orientation filter for inertial and inertial/magnetic sensor arrays. Report x-io and University of Bristol (UK), 25, 113-118. [link]
2. Zheng, F. (2017). Quaternion Conventions: Hamilton and JPL. [link]