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:

$$\mathbf{y}=f(\mathbf{R},\mathbf{x})=\mathbf{R}\cdot\mathbf{x}$$ (26)

Then differentiation by the vector is straightforward, as $f$ is linear in $\mathbf{x}$:

$$\dfrac{\partial\mathbf{y}}{\partial\mathbf{x}}=\mathbf{R}$$ (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.

$$\dfrac{\partial\mathbf{y}}{\partial\mathbf{R}} = \dfrac{\partial}{\partial\mathbf{\boldsymbol{\omega}}}\vert{}_{\m... ...\left(\exp\left(\boldsymbol{\omega}\right)\cdot\mathbf{R}\right)\cdot\mathbf{x}$$ (28)

$$= \dfrac{\partial}{\partial\mathbf{\boldsymbol{\omega}}}\vert{}_{\m... ...\exp\left(\boldsymbol{\omega}\right)\cdot\left(\mathbf{R}\cdot\mathbf{x}\right)$$ (29)

$$= \dfrac{\partial}{\partial\mathbf{\boldsymbol{\omega}}}\vert{}_{\m... ...dsymbol{\omega}}=\mathbf{0}}\exp\left(\boldsymbol{\omega}\right)\cdot\mathbf{y}$$ (30)

$$= \left(\begin{array}{c\vert c\vert c} G_{1}\mathbf{y} & G_{2}\mathbf{y} & G_{3}\mathbf{y}\end{array}\right)$$ (31)

$$= -\mathbf{y}_{\times}$$ (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:

$$\mathbf{R} \equiv \mathbf{R}_{1}\cdot\mathbf{R}_{0}$$ (33)

$$\dfrac{\partial\mathbf{R}}{\partial\mathbf{R}_{0}} = \dfrac{\partial}{\partial\mathbf{\boldsymbol{\omega}}}\left[R_{1}\cdot\exp\left(\mathbf{\boldsymbol{\omega}}\right)\cdot R_{0}\right]$$ (34)

$$= \dfrac{\partial}{\partial\mathbf{\boldsymbol{\omega}}}\left[\exp\... ...bf{R}_{1}}\cdot\mathbf{\boldsymbol{\omega}}\right)\cdot R_{1}\cdot R_{0}\right]$$ (35)

$$= \mathrm{Adj}_{\mathbf{R}_{1}}$$ (36)

$$= \mathbf{R}_{1}$$ (37)