Key Identities

For a rotation matrix $\mathbf{C}$ defined in Rotation Representations by the Euler rotational theorem

$${\mathbf{C}}_{21}=\cos \varphi 1+(1-\cos \varphi )\mathbf{a}{\mathbf{a}}^T-\sin \varphi {\mathbf{a}}^{\times }$$

The partial derivative with respect to the amount of rotation is given by

$$\frac{\partial \mathbf{C}}{\partial \varphi }=-{\mathbf{a}}^{\times }\mathbf{C}$$

Euler Angles

Say we are doing an arbitrary Euler sequence (from Rotation Representations, like how we said 1-2-3). We can get a partial derivative as

$$\frac{\partial \mathbf{C}(\mathbf{\theta })\mathbf{v}}{\partial \mathbf{\theta }}=(\mathbf{C}(\mathbf{\theta })\mathbf{v}{)}^{\times }\underset{\mathbf{S}({\theta }_2,{\theta }_3)}{\underbrace{\left[\begin{array}{ccc}{\mathbf{C}}_{\gamma }({\theta }_3){\mathbf{C}}_{\beta }({\theta }_2)1_{\alpha }& {\mathbf{C}}_{\gamma }({\theta }_3)1_{\beta }& 1_{\gamma }\end{array}\right]}}$$