Quaternions
Definition
There are two quaternion conventions commonly in use, Hamilton quaternions and JPL quaternions. The set of Hamilton quaternions, \(\mathbb{H}\), and the set of JPL quaternions, \(\mathbb{S}\) are defined as:
\[
\begin{alignat}{3}
\mathbb{H} &= \mathbb{C} + \mathbb{C}j, &&\quad k \triangleq ij = -ji, \quad &&&i^2 = j^2 = k^2 = ijk = -1 \\
\mathbb{S} &= \mathbb{C} - \mathbb{C}j, &&\quad k \triangleq ji = -ij, \quad &&&i^2 = j^2 = k^2 = ijk = -1.
\end{alignat}
\]
Elements of \(\mathbb{H}\) and \(\mathbb{S}\) are of the form:
\[
\begin{align}
\mathbf{q}_{\mathbb{H}} &= q_w + q_x \mathbf{i} + q_y \mathbf{j} + q_z \mathbf{k} = q_w + \mathbf{q}_v \in \mathbb{H} \\
\mathbf{q}_{\mathbb{S}} &= q_x \mathbf{i} + q_y \mathbf{j} + q_z \mathbf{k} + q_w = \mathbf{q}_v + q_w \in \mathbb{S}.
\end{align}
\]
Note that we can conclude \(\mathbf{q}_{\mathbb{S}} = \mathbf{q}_{\mathbb{H}}\), but the equality is only present in the quaternion values and the two quaternions, when employed in formulas, mean and represent different things. The Hamilton notation (right-handed) and JPL notation (left-handed) are related by:
\[
\mathbf{q}_{\mathbb{S}} = \mathbf{q}^*_{\mathbb{H}}.
\]
| Quaternion Type | Hamilton | JPL |
|---|---|---|
| Definition | \(\mathbf{q} = \left[ \begin{array}{cc} q_w & \mathbf{q}_v \end{array} \right]^T\) | \(\mathbf{q} = \left[ \begin{array}{cc} \mathbf{q}_v & q_w \end{array} \right]^T\) |
| Conjugate | \(\mathbf{q}^{*} = \left[ \begin{array}{cc} q_w & - \mathbf{q}_v \end{array} \right]^T\) | \(\mathbf{q}^{*} = \left[ \begin{array}{cc} -\mathbf{q}_v & q_w \end{array} \right]^T\) |
| Inverse | \(\mathbf{q}^{-1} = \mathbf{q}^* / \left\|\mathbf{q}\right\|^2\) | |
| Attitude Representation Constraint | \(\mathbf{q}^{-1} = \mathbf{q}^*\) with \(\left\|\mathbf{q} \right\| = \sqrt{\mathbf{q} \otimes \mathbf{q}^*} = 1\) | |
| Multiplication | \(\mathbf{q}\otimes \mathbf{p} = \left[ \mathbf{q} \right]_L \mathbf{p} = \left[ \mathbf{p}\right]_R \mathbf{q}\) | \(\mathbf{q} \otimes \mathbf{p} = \left[ \mathbf{q} \right]_L \mathbf{p} = \left[ \mathbf{p} \right]_R \mathbf{q}\) |
| Left Product Matrix | \(\left[ \mathbf{q} \right]_L = \left[ \begin{array}{cccc} q_w & -q_x & -q_y & -q_z \\ q_x & q_w & - q_z & q_y \\ q_y & q_z & q_w & -q_x \\ q_z & -q_y & q_x & q_w \end{array}\right]\) | \(\left[ \mathbf{q} \right]_L = \left[ \begin{array}{cccc} q_w & q_z & -q_y & q_x \\ -q_z & q_w & q_x & q_y \\ q_y & -q_x & q_w & q_z \\ -q_x & -q_y & -q_z & q_w \end{array}\right]\) |
| Right Product Matrix | \(\left[\mathbf{q} \right]_R= \left[ \begin{array}{cccc} q_w & -q_x & -q_y & -q_z \\ q_x & q_w & q_z & -q_y \\ q_y & -q_z & q_w & q_x \\ q_z & q_y & -q_x & q_w \end{array}\right]\) | \(\left[\mathbf{q} \right]_R= \left[ \begin{array}{cccc} q_w & -q_z & q_y & q_x \\ q_z & q_w & -q_x & q_y \\ -q_y & q_x & q_w & q_z \\ -q_x & -q_y & -q_z & q_w \end{array}\right]\) |
| Rotation Type | Passive | Passive |
| Rotation Vector | \(\mathbf{q}^g_l = \cos(\phi / 2) + \mathbf{u} \sin(\phi / 2)\) | \(\begin{align*} \mathbf{q}^l_g &= \mathbf{u} \sin (\phi / 2) + \cos (\phi / 2) \\ &= \left[ \begin{array}{1} \mathbf{u} \sin(\phi / 2) \\ \cos (\phi / 2) \end{array} \right] \end{align*}\), where \(\mathbf{u}\) is expressed in \(F_l\) |
| Rotation Representation | Local-to-Global, i.e., \(\mathbf{q} \triangleq \mathbf{q}^g_l\) | Global-to-Local, i.e., \(\mathbf{q} \triangleq \mathbf{q}^l_g\) |
| Default Operation | \(\mathbf{x}^g_{a b} = \mathbf{q} \otimes \mathbf{x}^{l}_{a b} \otimes \mathbf{q}^*\) | \(\begin{align*}\left[ \begin{array}{c}\mathbf{x}^l_{a b} \\ 0 \end{array} \right]= \mathbf{q} \otimes \left[ \begin{array}{c} \mathbf{x}^{g}_{a b} \\ 0 \end{array} \right] \otimes \mathbf{q}^*\end{align*}\) |
| Corresponding Rotation Matrix | \(\begin{align*}\mathbf{R}^g_l &= \left[ \begin{array}{ccc} q^2_w + q^2_x - q^2_y - q^2_z & 2(q_x q_y - q_w q_z) & 2(q_x q_z + q_w q_y) \\ 2(q_x q_y + q_w q_z) & q^2_w - q^2_x + q^2_y - q^2_z & 2(q_y q_z - q_w q_x) \\ 2(q_x q_z - q_w q_y) & 2(q_y q_z + q_w q_x) & q^2_w - q^2_x - q^2_y + q^2_z \end{array}\right] \\ &= \left(q^2_w - \mathbf{q}^T_v \mathbf{q}_v \right)\mathbf{I} + 2 \mathbf{q}_v \mathbf{q}^T_v + 2 q_w \left[ \mathbf{q}_v \right]_\times \end{align*}\) | \(\begin{align*} \mathbf{R}^l_g &= \left[ \begin{array}{ccc} q^2_x - q^2_y - q^2_z + q^2_w & 2(q_x q_y + q_z q_w) & 2(q_x q_z - q_y q_w) \\ 2(q_x q_y - q_z q_w) & -q^2_x + q^2_y - q^2_z + q^2_w & 2(q_y q_z + q_x q_w) \\ 2(q_x q_z + q_y q_w) & 2(q_y q_z - q_x q_w) & -q^2_x - q^2_y + q^2_z + q^2_w \end{array} \right] \\ &= \left(2q^2_w - 1 \right) \mathbf{I} - 2q_w \left[ \mathbf{q}_v \right]_\times + 2 \mathbf{q}_v \mathbf{q}^T_v \\ &= \exp \left( -\left[ \mathbf{u} \right]_\times \phi \right) \end{align*}\) |
| Rotation Matrix Properties | \(\begin{align*} \mathbf{R}\left\{ -\mathbf{q} \right\} &= \mathbf{R}\left\{\mathbf{q} \right\} \\ \mathbf{R}\left\{ \mathbf{q}^* \right\} &= \mathbf{R}\left\{ \mathbf{q} \right\}^T \\ \mathbf{R}\left\{ \mathbf{q} \otimes \mathbf{p} \right\} &= \mathbf{R}\left\{ \mathbf{q} \right\} \mathbf{R} \left\{ \mathbf{p} \right\}\end{align*}\) | |
| Perturbation | Perturbation is in the local vector space \(\delta \boldsymbol{\phi}\): \(\begin{align*} \delta \mathbf{q}^g_l &= \left[ \begin{array}{c} \delta q_w \\ \delta \mathbf{q}_v \end{array} \right] \\ &= \left[ \begin{array}{c} \cos \left( \delta \phi / 2 \right) \\ \mathbf{u} \sin \left( \delta \phi / 2 \right) \end{array} \right] \\ &\approx \left[ \begin{array}{c} 1 \\ \frac{1}{2} \delta \phi \end{array} \right] \\ \mathbf{R}^g_l \left\{ \delta \mathbf{q}^g_l \right\} &= \mathbf{I}_3 + \left[\delta \boldsymbol{\phi} \right]_\times \end{align*}\) |
Perturbation is in the global vector space \(\delta \boldsymbol{\phi}\): \(\begin{align*} \delta \mathbf{q}^l_g &= \left[ \begin{array}{c} \delta \mathbf{q}_v \\ \delta q_w \end{array} \right] \\ &= \left[ \begin{array}{c} \mathbf{u} \sin \left( \delta \phi / 2 \right) \\ \cos \left( \delta \phi / 2 \right) \end{array} \right] \\ &\approx \left[ \begin{array}{c} \frac{1}{2} \delta \boldsymbol{\phi} \\ 1 \end{array} \right] \\ \mathbf{R}^l_g\left\{ \delta \mathbf{q}^l_g \right\} &\approx \mathbf{I}_3 - \left[ \delta \boldsymbol{\phi} \right]_\times \end{align*}\) |
Cross-Relations to Rotation Matrix
Let \(\boldsymbol{\phi} = \phi \mathbf{u}\) be a rotation vector representing a rotation of \(\phi\) radians around the axis of \(\mathbf{u}\) which is expressed in the local frame.
| Rotation Matrix, \(\mathbf{R}^g_l\) | Hamilton, \(\mathbf{q}^g_l\) | JPL, \(\mathbf{q}^l_g\) | |
|---|---|---|---|
| ODE | \(\begin{align*}\dot{\mathbf{R}}^g_l &= \mathbf{R}^g_l \left[ \boldsymbol{\omega}^l_{g l} \right]_\times \\ &= \left[ \boldsymbol{\omega}^g_{g l} \right]_\times \mathbf{R}^g_l \end{align*}\) | \(\begin{align*} \dot{\mathbf{q}}^g_l &= \frac{1}{2} \mathbf{q}^g_l \otimes \boldsymbol{\omega}^l_{g l} \\ &= \frac{1}{2} \boldsymbol{\Omega}^l_{g l} \mathbf{q}^g_l \\ &= \frac{1}{2} \left[ \begin{array}{cccc} 0 & -w_x & -w_y & -w_z \\ w_x & 0 & w_z & -w_y \\ w_y & -w_z & 0 & w_x \\ w_z & w_y & -w_x & 0 \end{array}\right] \mathbf{q}^g_l \end{align*}\) | \(\begin{align*} \dot{\mathbf{q}}^l_g &= \frac{1}{2} \boldsymbol{\omega}^l_{g l} \otimes \mathbf{q}^l_g \\ &= \frac{1}{2} \boldsymbol{\Omega}^l_{g l} \mathbf{q}^l_g \\ &= \frac{1}{2} \left[ \begin{array}{cccc} 0 & w_z & -w_y & w_x \\ -w_z & 0 & w_x & w_y \\ w_y & - w_x & 0 & w_z \\ -w_x & -w_y & -w_z & 0 \end{array} \right] \mathbf{q}^l_g \end{align*}\) |
| Exponential Map | \(\mathbf{R}^g_l = \exp \left( \left[ \phi \mathbf{u} \right]_\times \right)\) | \(\mathbf{q}^g_l = \exp \left( \left[ \phi \mathbf{u} / 2 \right]_\times \right)\) | \(\mathbf{q}^l_g = \exp \left( \left[ -\phi \mathbf{u} / 2 \right]_\times \right)\) |
| Logarithmic Map | \(\log \left( \mathbf{R}^g_l \right) = \left[ \phi \mathbf{u} \right]_\times\) | \(\log \left( \mathbf{q}^g_l \right) = \phi \mathbf{u} / 2\) | \(\log (\mathbf{q}^l_g) = -\phi \mathbf{u} / 2\) |
| Rotation Operator | \(\mathbf{R}^g_l = \mathbf{I} + \sin \phi \left[ \mathbf{u} \right]_\times + (1 - \cos \phi) \left[ \mathbf{u} \right]^2_\times\) | \(\mathbf{q}^g_l = \cos \phi / 2 + \mathbf{u} \sin \phi / 2\) | \(\mathbf{q}^l_g = -\mathbf{u} \sin \phi / 2 + \cos \phi / 2\) |
| Rotation Action | \(\mathbf{x}^g_{a b} = \mathbf{R}^g_l \mathbf{x}^l_{a b}\) | \(\mathbf{x}^g_{a b} = \mathbf{q}^g_l \otimes \mathbf{x}^l_{a b} \otimes \left( \mathbf{q}^g_l \right)^*\) | \(\mathbf{x}^l_{a b} = \mathbf{q}^l_g \otimes \mathbf{x}^g_{a b} \otimes \left( \mathbf{q}^l_g \right)^*\) |
| Rotation Composition | \(\mathbf{R}^a_c = \mathbf{R}^a_b \mathbf{R}^b_c\) | \(\mathbf{q}^a_{c} = \mathbf{q}^a_b \otimes \mathbf{q}^b_c\) | |
| Interpolation | \(\begin{align*} \mathbf{R}(t) &= \mathbf{R}_0 (\mathbf{I} + \sin (t \Delta \phi) \left[ \mathbf{u} \right]_\times + (1 - \cos( t \Delta \phi)) \left[ \mathbf{u} \right]^2_\times ) \\ &= \mathbf{R}_0 (\mathbf{R}^T_0 \mathbf{R}_1)^t \end{align*}\) | \(\begin{align*}\mathbf{q}(t) &= \mathbf{q}_0 \otimes \left[ \begin{array}{c} \cos(t \Delta \phi / 2) \\ \mathbf{u} \sin (t \Delta \phi / 2) \end{array} \right] \\ &= \mathbf{q}_0 \otimes \left( \mathbf{q}^*_0 \otimes \mathbf{q}_1 \right)^t \end{align*}\) |
References
- Sola, J., Quaternion Kinematics for the Error-State Kalman Filter
- Trawny, N., et al. Indirect Kalman Filter for 3D Attitude Estimation, 2005