3.3 Quaternions

Quaternions are extended complex numbers.

A quaternion q has a real part and three imaginary parts. $$\pmb{q} = q_0 + q_1 i + q_2 j + q_3 k$$ where i, j, k are three imaginary parts of the quaternion. These three imaginary parts satisfy the following relationship: $$\left\{ \begin{array}{l} {i^2} = {j^2} = {k^2} = - 1\\ ij = k, ji = - k \\ jk = i,kj = - i\\ ki = j, ik = - j \end{array} \right.$$ We can also use a scalar and a vector to express quaternions: $$\pmb{q} = \left[ s, \pmb{v} \right]^T, \quad s=q_0 \in \mathbb{R},\quad \pmb{v} = [q_1, q_2, q_3]^T \in \mathbb{R}^3$$ Here, $s$ is the real part of the quaternion, and $\pmb {v}$ is its imaginary part. If the imaginary part of a quaternion is $\pmb {0}$, it is called real quaternion. Conversely, if its real part is $0$, it is called imaginary quaternion.

3.3.1 Quaternion Operations

$$\pmb{q} _a = s_a + x_ai + y_aj + z_ak, \quad \pmb {q} _b = s_b + x_bi + y_bj + z_bk$$

1. Addition and subtraction $$\pmb{q}_a \pm \pmb{q}_b = \left[ s_a \pm s_b, \pmb{v}_a \pm \pmb{v}_b \right]^T$$

2. Multiplication $$\pmb{q}_a \pmb{q}_b = \left[ s_a s_b - \pmb{v}_a^T \pmb{v}_b, s_a\pmb{v}_b + s_b\pmb{v}_a + \pmb{v}_a \times \pmb{v}_b \right]^T$$

3. Length $$\| \pmb{q}_a \| = \sqrt{ s_a^2 + x_a^2 + y_a^2 + z_a^2 }.$$ It can be verified that the length of the product is the product of the lengths. This makes the unit quaternion keep unit-length when multiplied by another unit quaternion: $$\| \pmb{q}_a \pmb{q}_b \| = \|\pmb{q}_a \| \| \pmb{q}_b \|.$$

4. Conjugate

   The conjugate of a quaternion is to take the imaginary part as the opposite: $$\pmb{q}_a ^ * = s_a - x_ai - y_aj - z_ak = [s_a, - \pmb{v}_a] ^T.$$ We get a real quaternion if the quaternion is multiplied by its conjugate. The real part is the square of its length: $$\pmb{q}^* \pmb{q} = \pmb{q} \pmb{q}^* = [s^2+\pmb{v}^T \pmb{v}, \pmb{0} ]^T$$

5. Inverse $$\pmb{q} ^ { - 1 } = \pmb{q} ^ * / \| \pmb{q} \| ^ 2.$$ According to this definition, the product of the quaternion and its inverse is the real quaternion $\pmb {1}$: $$\pmb{q} \pmb{q}^{-1} = \pmb{q}^{-1} \pmb{q} = \pmb{1}.$$ If $\pmb{q}$ is a unit quaternion, its inverse and conjugate are the same. So the inverse of the product has properties similar to matrices: $$\left( \pmb{q}_a \pmb{q}_b \right)^{-1} = \pmb{q}_b^{-1} \pmb{q}_a^{-1}.$$

6. Scalar Multiplication

$$k \pmb{q} = \left[ ks, k\pmb{v} \right]^T.$$

3.3.2 Use Quaternion to Represent a Rotation

Suppose a spatial 3D point $\pmb{p} = [x,y,z]^T \in \mathbb {R}^3$, and a rotation is specified by a unit quaternion $\pmb{q}$.

1. Extend the 3D point to an imaginary quaternion: $$\pmb{p} = [0, x, y, z]^T = [0, \pmb{v}]^T.$$

2. The rotated point $p'$ can be expressed as such a product: $$\pmb{p}' = \pmb{q} \pmb{p} \pmb{q}^{-1}.$$

3.3.3 Conversion of Quaternions to Other Rotation Representations

The relationship between quaternions and rotation vectors / matrices.

Let $\pmb{q}=[s,\pmb{v}]^T$, then define the following symbols $^{+}$ and $^{\oplus}$ $$\pmb{q}^{+}=\left[\begin{array}{cc} s&-\pmb{v}^T \\ \pmb{v}&s\pmb{I}+\pmb{v}^{\wedge} \end{array}\right],\quad \pmb{q}^{\oplus}= \left[\begin{array}{cc} s & -\pmb{v}^T \\ \pmb{v} & s\pmb{I}-\pmb{v}^{\wedge} \end{array}\right]$$ These two symbols map the quaternion to a 4x4 matrix. $$\pmb{q}_1^ + {\pmb{q}_2} = \left[ {\begin{array}{*{20}{c}} s_1&-\pmb{v}_1^T\\ \pmb{v}_1 & s_1 \pmb{I} + \pmb{v}_1^\wedge \end{array}} \right]\left[ {\begin{array}{*{20}{c}} { {s _2}} \\ { {\pmb{v} _2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - \pmb{v} _1^T{\pmb{v} _2} + {s _1}{s _2}} \\ { {s _1}{\pmb{v} _2} + {s _2}{\pmb{v} _1} + \pmb{v} _1^ \wedge {\pmb{v} _2}} \end{array}} \right] = \pmb{q}_1 \pmb{q}_2$$

$$\pmb{q}_1 \pmb{q}_2 = \pmb{q}_1^{+} \pmb{q}_2 = \pmb{q}_2^{\oplus} \pmb{q}_1.$$

Then, consider the problem of using a quaternion to rotate a spatial point. $$\begin{split} \pmb{p}' &= \pmb{q} \pmb{p} \pmb{q}^{-1} = \pmb{q}^+ \pmb{p}^+ \pmb{q}^{ -1} \\ &= \pmb{q}^+ \pmb{q}^{ {-1}^{\oplus}} \pmb{p}. \end{split}$$ In summary, the conversion formula from quaternion to rotation vector can be written as: $$\begin{cases} \theta = 2\arccos {q_0}\\ {\left[ { {n_x},{n_y},{n_z}} \right]^T} = { { {\left[ { {q_1},{q_2},{q_3}} \right] }^T}}/{\sin \frac{\theta }{2}} \end{cases}$$ And for converting from other representations to quaternions, we only need to reversely follow the above steps.