End-effector orientation

End-effector orientation captures the rotation of the relative frame transformation:

$$\Phi_\text{EffRot}(\boldsymbol{x}) = \boldsymbol{R}_A^B,$$

where $\boldsymbol{R}_A^B\in SO(3)$ is rotational part of $\boldsymbol{M}_A^B$. Similarly to the end-effector frame task map, the storage and the operations on the resulting $SO(3)$ space are implemented within the task space vector. The Jacobian of this task consists of the rows of the geometric Jacobian corresponding to the rotation of the frame.