2.4 Jacobians

Consider $\mathbf{R\in}\mathrm{SO(3)}$ and $\mathbf{x}\in\mathcal{R}^{3}$ . The rotation of vector $\mathbf{x}$ by matrix $\mathbf{R}$ is given by multiplication:

$\displaystyle \mathbf{y}=f(\mathbf{R},\mathbf{x})=\mathbf{R}\cdot\mathbf{x}$

(26)

Then differentiation by the vector is straightforward, as

is linear in $\mathbf{x}$ :

$\displaystyle \dfrac{\partial\mathbf{y}}{\partial\mathbf{x}}=\mathbf{R}$

(27)

Differentiation by the rotation parameters is performed by implicitly left multiplying the rotation by the exponential of a tangent vector and differentiating the resulting expression around the origin. This is equivalent to left multiplying the product by the generators.

$\displaystyle \dfrac{\partial\mathbf{y}}{\partial\mathbf{R}}$	$\displaystyle =$	$\displaystyle \dfrac{\partial}{\partial\mathbf{\boldsymbol{\omega}}}\vert{}_{\m... ...\left(\exp\left(\boldsymbol{\omega}\right)\cdot\mathbf{R}\right)\cdot\mathbf{x}$	(28)
	$\displaystyle =$	$\displaystyle \dfrac{\partial}{\partial\mathbf{\boldsymbol{\omega}}}\vert{}_{\m... ...\exp\left(\boldsymbol{\omega}\right)\cdot\left(\mathbf{R}\cdot\mathbf{x}\right)$	(29)
	$\displaystyle =$	$\displaystyle \dfrac{\partial}{\partial\mathbf{\boldsymbol{\omega}}}\vert{}_{\m... ...dsymbol{\omega}}=\mathbf{0}}\exp\left(\boldsymbol{\omega}\right)\cdot\mathbf{y}$	(30)
	$\displaystyle =$	$\displaystyle \left(\begin{array}{c\vert c\vert c} G_{1}\mathbf{y} & G_{2}\mathbf{y} & G_{3}\mathbf{y}\end{array}\right)$	(31)
	$\displaystyle =$	$\displaystyle -\mathbf{y}_{\times}$	(32)

Differentiation of a product of rotations is performed by the same method. The differentation by the tangent space element $\mathbf{\boldsymbol{\omega}}$ is always performed around $\mathbf{\boldsymbol{\omega}}=\mathbf{0}$ , and the adjoint is employed to shift the tangent vector left. The result is simple:

$\displaystyle \mathbf{R}$	$\displaystyle \equiv$	$\displaystyle \mathbf{R}_{1}\cdot\mathbf{R}_{0}$	(33)
$\displaystyle \dfrac{\partial\mathbf{R}}{\partial\mathbf{R}_{0}}$	$\displaystyle =$	$\displaystyle \dfrac{\partial}{\partial\mathbf{\boldsymbol{\omega}}}\left[R_{1}\cdot\exp\left(\mathbf{\boldsymbol{\omega}}\right)\cdot R_{0}\right]$	(34)
	$\displaystyle =$	$\displaystyle \dfrac{\partial}{\partial\mathbf{\boldsymbol{\omega}}}\left[\exp\... ...bf{R}_{1}}\cdot\mathbf{\boldsymbol{\omega}}\right)\cdot R_{1}\cdot R_{0}\right]$	(35)
	$\displaystyle =$	$\displaystyle \mathrm{Adj}_{\mathbf{R}_{1}}$	(36)
	$\displaystyle =$	$\displaystyle \mathbf{R}_{1}$	(37)