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.