Jacobians
Definition
Consider a rotation matrix \(\mathbf{R} \in SO(3)\) with Lie algebra \(\left[ \boldsymbol{\rho} \right]_\times = \left[ \phi \mathbf{u} \right]_\times\) where \(\phi \in \mathbb{R}\) is the angle of rotation and \(\mathbf{u} \in \mathbb{R}^3\) is the unit axis of rotation vector. Multiple Jacobians can be derived in this setting:
| Jacobian | Supplementary | ||
|---|---|---|---|
| Jacobian with respect to the rotated vector, \(\mathbf{x}\) | \(\begin{align*} \frac{\partial \left(\mathbf{R} \mathbf{x} \right)}{\partial \mathbf{x}} = \frac{\partial \left( \mathbf{q} \otimes \mathbf{x} \otimes \mathbf{q} \right)}{\partial \mathbf{x}} = \mathbf{R} \in SO(3) \end{align*}\) | ||
| Jacobian with respect to the angle of rotation, \(\phi\) | Using Rodrigues' formula \(\mathbf{R} = \cos \phi \mathbf{I} + \left( 1- \cos \phi \right) \hat{\mathbf{e}} \hat{\mathbf{e}}^T - \sin \phi \left[ \hat{\mathbf{e}} \right]_\times\), we have: \(\begin{align*} \frac{ \partial \mathbf{R}}{\partial \phi} &= - \sin \phi \mathbf{I} + \sin \phi \hat{\mathbf{e}} \hat{\mathbf{e}}^T - \cos \phi \left[ \hat{\mathbf{e}} \right]_\times \\ &= \sin \phi \underbrace{\left( -\mathbf{I} + \hat{\mathbf{e}} \hat{\mathbf{e}}^T \right)}_{\left[ \hat{\mathbf{e}} \right]^2_\times} - \cos \phi \left[ \hat{\mathbf{e}} \right]_\times \\ &= -\cos \phi \left[ \hat{\mathbf{e}} \right]_\times - (1 - \cos \phi) \underbrace{\left[ \hat{\mathbf{e}} \right]_\times \hat{\mathbf{e}}}_{\mathbf{0}} \hat{\mathbf{e}}^T + \sin \phi \left[ \hat{\mathbf{e}} \right]^2_\times \\ &= -\left[ \hat{\mathbf{e}} \right]_\times \underbrace{\left(\cos \phi \mathbf{I} + (1 - \cos \phi) \hat{\mathbf{e}} \hat{\mathbf{e}}^T - \sin \phi \left[ \hat{\mathbf{e}} \right]_\times \right)}_{\mathbf{R}} \\ &= - \left[ \hat{\mathbf{e}} \right]_\times \mathbf{R} \end{align*}\) |
||
| Jacobian with respect to the rotation vector, \(\boldsymbol{\rho}\) | Consider a rotation of an arbitrary vector \(\mathbf{x} \in \mathbb{R}^3\). We have: \(\begin{align*} \frac{\partial \left( \mathbf{R} \mathbf{x} \right)}{\partial \boldsymbol {\rho}} &= \lim_{\delta \boldsymbol{\rho} \rightarrow 0} \frac{\exp \left( \left[ \boldsymbol{\rho} + \delta \boldsymbol{\rho} \right]_\times \right) \mathbf{x} - \exp \left(\left[ \boldsymbol{\rho} \right]_\times \right) \mathbf{x}}{\delta \boldsymbol{\rho}} \\ &= \lim_{\delta \boldsymbol{\rho} \rightarrow \mathbf{0}} \frac{\exp \left( \left[ \mathbf{J}_l \delta \boldsymbol{\rho} \right]_\times \right) \exp \left(\left[ \boldsymbol{\rho} \right]_\times\right) \mathbf{x} - \exp \left(\left[ \boldsymbol{\rho} \right]_\times\right) \mathbf{x}}{\delta \boldsymbol{\rho}} \\ &= \lim_{\delta \boldsymbol{\rho} \rightarrow \mathbf{0}} \frac{\left(\mathbf{I} + \left[ \mathbf{J}_l \delta \boldsymbol{\rho} \right]_\times \right) \exp \left( \left[ \boldsymbol{\rho} \right]_\times\right) \mathbf{x} - \exp \left(\left[ \boldsymbol{\rho} \right]_\times\right) \mathbf{x}}{\delta \boldsymbol{\rho}} \\ &= \lim_{\delta \boldsymbol{\rho} \rightarrow \mathbf{0}} \frac{\left[ \mathbf{J}_l \delta \boldsymbol{\rho} \right]_\times \exp \left(\left[ \boldsymbol{\rho} \right]_\times\right) \mathbf{x}}{\delta \boldsymbol{\rho}} \\ &= \lim_{\delta \boldsymbol{\rho} \rightarrow \mathbf{0}} \frac{-\left[ \exp \left(\left[ \boldsymbol{\rho} \right]_\times\right) \mathbf{x} \right]_\times \mathbf{J}_l \delta \boldsymbol{\rho}}{\delta \boldsymbol{\rho}} \\ &= - \left[ \mathbf{R} \mathbf{x} \right]_\times \mathbf{J}_l(\boldsymbol{\rho}) \\ &= -\mathbf{R} \left[ \mathbf{x} \right]_\times \mathbf{R}^T \mathbf{R} \mathbf{J}_r (\boldsymbol{\rho}) = -\mathbf{R} \left[ \mathbf{x} \right]_\times \mathbf{J}_r (\boldsymbol{\rho}) \end{align*}\) |
Per Barfoot, the directional derivatives (along the \(i\)'th axes \(\mathbf{I}\), denoted as \(\mathbf{I}_i\)) with respect to \(\rho_i\) is: \(\begin{align*} \frac{\partial \left( \mathbf{R} \mathbf{x} \right)}{\partial \rho_i} = \lim_{h \rightarrow 0} \frac{\exp \left( \left[ \boldsymbol{\rho} + h \mathbf{I}_i \right]_\times \right) \mathbf{x} - \exp \left(\left[ \boldsymbol{\rho} \right]_\times \right) \mathbf{x}}{h} \end{align*}\) Since we are interested in the limit of \(h\) infinitely small, using the BCH formula, we get: \(\begin{align*} \exp \left( \left[ \boldsymbol{\rho} + h \mathbf{I}_i \right]_\times \right) &\approx \exp \left( \left[ \mathbf{J}_l h \mathbf{I}_i \right]_\times \right) \exp \left( \left[ \boldsymbol{\rho} \right]_\times \right) \\ &\approx \left(1 + h \left[ \mathbf{J}_l \mathbf{I}_i \right]_\times \right) \exp \left( \left[ \boldsymbol{\rho} \right]_\times \right) \end{align*}\) Finally, we have: \(\begin{align*} \frac{\partial \left( \mathbf{R} \mathbf{x} \right)}{\partial \rho_i} = \left[ \mathbf{J}_l \mathbf{I}_i \right]_\times \mathbf{R} \mathbf{x} = - \left[\mathbf{R} \mathbf{x} \right]_\times \mathbf{J}_l \mathbf{I}_i \end{align*}\) Stacking the three directional derivatives alongside one another yields to the desired Jacobian. |
If \(\mathbf{R} \mathbf{x}\) appears inside another scalar function, \(u(\mathbf{y})\), with \(\mathbf{y} = \mathbf{R} \mathbf{x}\), we can use the chain rule: \(\begin{align*} \frac{\partial u}{ \partial \boldsymbol{\rho}} = \frac{\partial u}{ \partial \mathbf{y}} \frac{\partial \mathbf{y}}{ \partial \boldsymbol{\rho}}= \frac{\partial u}{\partial \mathbf{y}} \left[ \mathbf{R} \mathbf{x} \right]_\times \mathbf{J}_l \end{align*}\) |
| Jacobian with respect to the Euler Angles, \((\theta_1, \theta_2, \theta_3)\) | Consider the 1-2-3 Euler angles and denote it as \(\boldsymbol{\theta} = \left( \theta_1, \theta_2, \theta_3 \right)\) corresponding to the roll, pitch, and yaw angles. The rotation matrix can be represented as: \(\begin{align*} \mathbf{R}(\boldsymbol{\theta}) &= \mathbf{R}_3 (\theta_3) \mathbf{R}_2(\theta_2) \mathbf{R}_1(\theta_1) \end{align*}\) We have (see proofs): \(\begin{align*} \frac{\partial \left(\mathbf{R}(\boldsymbol{\theta}) \mathbf{v} \right)}{ \partial \boldsymbol{\theta}} &= \left[ \begin{array}{ccc} \frac{\partial \left(\mathbf{R}(\boldsymbol{\theta}) \mathbf{v} \right)}{\partial \theta_1} & \frac{\partial \left(\mathbf{R}(\boldsymbol{\theta}) \mathbf{v} \right)}{\partial \theta_2} & \frac{\partial \left(\mathbf{R}(\boldsymbol{\theta}) \mathbf{v} \right)}{\partial \theta_3} & \end{array} \right] \\ &= \left[ \mathbf{R} (\boldsymbol{\theta}) \mathbf{v} \right]_\times \underbrace{\left[ \begin{array}{ccc} \mathbf{R}_3(\theta_3) \mathbf{R}_2 (\theta_2) \mathbf{I}_1 & \mathbf{R}_3(\theta_3) \mathbf{I}_2 & \mathbf{I}_3 \end{array} \right]}_{\mathbf{S}(\theta_2, \theta_3)} \\ &= \left[ \mathbf{R}(\boldsymbol{\theta}) \mathbf{v} \right]_\times \mathbf{S}(\theta_2, \theta_3) \end{align*}\) |
Proofs
Jacobian with respect to Euler Angles
Consider the 1-2-3 Euler angles and denote it as \(\boldsymbol{\theta} = \left( \theta_1, \theta_2, \theta_3 \right)\) corresponding to the roll, pitch, and yaw angles. The rotation matrix can be represented as:
\[
\begin{align*}
\mathbf{R}(\boldsymbol{\theta}) &= \mathbf{R}_3 (\theta_3) \mathbf{R}_2(\theta_2) \mathbf{R}_1(\theta_1).
\end{align*}
\]
Taking the derivative of the rotation matrix with respect to the Euler angles yields to:
\[
\begin{alignat*}{2}
\frac{\partial \mathbf{R}(\boldsymbol{\theta})}{\partial \theta_3} &=
\frac{\partial \left( \mathbf{R}_3 (\theta_3) \mathbf{R}_2(\theta_2) \mathbf{R}_1(\theta_1) \right)}{\partial \theta_3} &&= -\left[ \mathbf{I}_3 \right]_\times \mathbf{R}(\boldsymbol{\theta}) \\
\frac{\partial \mathbf{R}(\boldsymbol{\theta})}{\partial \theta_2} &=
\frac{\partial \left( \mathbf{R}_3 (\theta_3) \mathbf{R}_2(\theta_2) \mathbf{R}_1(\theta_1) \right)}{\partial \theta_2} &&= -\mathbf{R}_3(\theta_3) \left[ \mathbf{I}_2 \right]_\times \mathbf{R}_2 (\theta_2) \mathbf{R}_1 (\theta_1) \\
\frac{\partial \mathbf{R}(\boldsymbol{\theta})}{\partial \theta_1} &=
\frac{\partial \left( \mathbf{R}_3 (\theta_3) \mathbf{R}_2(\theta_2) \mathbf{R}_1(\theta_1) \right)}{\partial \theta_1} &&= -\mathbf{R}_3(\theta_3) \mathbf{R}_2 (\theta_2) \left[ \mathbf{I}_1 \right]_\times \mathbf{R}_1. \\
\end{alignat*}
\]
If we take the derivative of the rotated vector with repect to the Euler angles:
\[
\begin{align*}
\frac{\partial \left( \mathbf{R}(\boldsymbol{\theta}) \mathbf{v} \right)}{\partial \theta_3} &= -\left[\mathbf{I}_3 \right]_\times \mathbf{R}_3(\theta_3) \mathbf{R}_2(\theta_2) \mathbf{R}_1(\theta_1) \mathbf{v} \\
&= -\left[ \mathbf{I}_3 \right]_\times \mathbf{R}(\boldsymbol{\theta}) \mathbf{v} \\
&= \left[ \mathbf{R}(\boldsymbol{\theta}) \mathbf{v} \right]_\times \mathbf{I}_3 \\
\frac{\partial \left( \mathbf{R}(\boldsymbol{\theta}) \mathbf{v} \right)}{\partial \theta_2} &= -\mathbf{R}_3(\theta_3) \left[\mathbf{I}_2 \right]_\times \mathbf{R}_2(\theta_2) \mathbf{R}_1 (\theta_1)\mathbf{v} \\
&= -\mathbf{R}_3 (\theta_3) \left[ \mathbf{I}_2 \right]_\times \mathbf{R}^T_3(\theta_3) \mathbf{R}_3(\theta_3) \mathbf{R}_2(\theta_2) \mathbf{R}_1(\theta_1) \mathbf{v} \\
&= -\left[ \mathbf{R}_3(\theta_3) \mathbf{I}_2 \right]_\times \mathbf{R}(\boldsymbol{\theta}) \mathbf{v} \\
&= \left[\mathbf{R}(\boldsymbol{\theta}) \mathbf{v} \right]_\times \mathbf{R}_3(\theta_3) \mathbf{I}_2 \\
\frac{\partial \left( \mathbf{R}(\boldsymbol{\theta}) \mathbf{v} \right)}{\partial \theta_1} &= -\mathbf{R}_3(\theta_3) \mathbf{R}_2(\theta_2) \left[ \mathbf{I}_1 \right]_\times \mathbf{R}_1(\theta_1) \mathbf{v} \\
&= -\mathbf{R}_3(\theta_3) \mathbf{R}_2(\theta_2) \left[ \mathbf{I}_1 \right]_\times \mathbf{R}^T_2(\theta_2) \mathbf{R}^T_3(\theta_3) \mathbf{R}_3(\theta_3) \mathbf{R}_2(\theta_2) \mathbf{R}_1(\theta_1) \mathbf{v} \\
&= -\left[ \mathbf{R}_3 (\theta_3) \mathbf{R}_2 (\theta_2) \mathbf{I}_1 \right]_\times \mathbf{R}(\boldsymbol{\theta}) \mathbf{v} \\
&= \left[ \mathbf{R}(\boldsymbol{\theta}) \mathbf{v} \right]_\times \mathbf{R}_3(\theta_3) \mathbf{R}_2(\theta_2) \mathbf{I}_1.
\end{align*}
\]
Summarizing the result, we have:
\[
\begin{align*}
\frac{\partial \left(\mathbf{R}(\boldsymbol{\theta}) \mathbf{v} \right)}{ \partial \boldsymbol{\theta}} &=
\left[
\begin{array}{ccc}
\frac{\partial \left(\mathbf{R}(\boldsymbol{\theta}) \mathbf{v} \right)}{\partial \theta_1} &
\frac{\partial \left(\mathbf{R}(\boldsymbol{\theta}) \mathbf{v} \right)}{\partial \theta_2} &
\frac{\partial \left(\mathbf{R}(\boldsymbol{\theta}) \mathbf{v} \right)}{\partial \theta_3} &
\end{array}
\right] \\
&= \left[ \mathbf{R} (\boldsymbol{\theta}) \mathbf{v} \right]_\times
\underbrace{\left[
\begin{array}{ccc}
\mathbf{R}_3(\theta_3) \mathbf{R}_2 (\theta_2) \mathbf{I}_1
& \mathbf{R}_3(\theta_3) \mathbf{I}_2 & \mathbf{I}_3
\end{array}
\right]}_{\mathbf{S}(\theta_2, \theta_3)} \\
&= \left[ \mathbf{R}(\boldsymbol{\theta}) \mathbf{v} \right]_\times \mathbf{S}(\theta_2, \theta_3),
\end{align*}
\]
which is true regardless of the choice of Euler set.
Principle Rotation Matrix Derivatives
The principle rotation matrices are:
\[
\begin{align*}
\mathbf{R}_1(\theta_1) &=
\left[
\begin{array}{ccc}
1 & 0 & 0 \\
0 & \cos \theta_1 & \sin \theta_1 \\
0 & -\sin \theta_1 & \cos \theta_1
\end{array}
\right] \\
\mathbf{R}_2(\theta_2) &=
\left[
\begin{array}{ccc}
\cos \theta_2 & 0 & -\sin \theta_2 \\
0 & 1 & 0 \\
\sin \theta_2 & 0 & \cos \theta_2
\end{array}
\right] \\
\mathbf{R}_3(\theta_3) &=
\left[
\begin{array}{ccc}
\cos \theta_3 & \sin \theta_3 & 0 \\
-\sin \theta_3 & \cos \theta_3 & 0 \\
0 & 0 & 1
\end{array}
\right].
\end{align*}
\]
The derivatives of the principle rotation matrices with respect to their angle of rotations are:
\[
\begin{alignat*}{2}
\frac{\partial \mathbf{R}_1(\theta_1)}{ \partial \theta_1} &=
\left[
\begin{array}{ccc}
0 & 0 & 0 \\
0 & -\sin \theta_1 & \cos \theta_1 \\
0 & -\cos \theta_1 & -\sin \theta_1
\end{array}
\right] &&=
-\left[\mathbf{I}_1 \right]_\times \mathbf{R}_1 (\theta_1) \\
\frac{\partial \mathbf{R}_2(\theta_2)}{ \partial \theta_2} &=
\left[
\begin{array}{ccc}
-\sin \theta_2 & 0 & -\cos \theta_2 \\
0 & 0 & 0\\
\cos \theta_2 & 0 & -\sin \theta_2
\end{array}
\right] &&=
-\left[\mathbf{I}_2 \right]_\times \mathbf{R}_2 (\theta_2) \\
\frac{\partial \mathbf{R}_3(\theta_3)}{ \partial \theta_3} &=
\left[
\begin{array}{ccc}
-\sin \theta_3 & \cos \theta_3 & 0 \\
-\cos \theta_3 & -\sin \theta_3 & 0 \\
0 & 0 & 0
\end{array}
\right] &&=
-\left[\mathbf{I}_3\right]_\times \mathbf{R}_3 (\theta_3).
\end{alignat*}
\]
Note that this property holds regardless if it is passive or active rotation.
General Derivative of a Rotation Matrix
In general, given a rotation vector \(\boldsymbol{\rho} = \phi \mathbf{u}\) and the Rodrigues' formula:
\[
\begin{align*}
\mathbf{R}(\boldsymbol{\rho}) = \mathbf{I} + \frac{\sin \theta}{ \theta} \left[ \boldsymbol{\rho} \right]_\times + \frac{1 - \cos \theta}{\theta^2} \left[ \boldsymbol{\rho} \right]^2_\times,
\end{align*}
\]
the derivative of the rotation matrix with respect to a variable of interest (e.g., time or the rotation vector component etc.) can be expressed as:
\[
\begin{align*}
\frac{\partial \left( \mathbf{R} \right)}{\partial x} = \left[ \mathbf{J}_r(\boldsymbol{\rho}) \frac{\partial \boldsymbol{\rho}}{\partial x} \right]_\times \mathbf{R}.
\end{align*}
\]
References
- Barfoot, T., State Estimation for Robotics
- Solà, J., Quaternion Kinematics for the Error-State Kalman Filter, 2017