Joint position

Joint position task map computes the difference between the current joint configuration and a reference joint configuration:

$$\Phi_\text{Ref}(\boldsymbol{x}) = \boldsymbol{x}-\boldsymbol{x}_{\text{ref}},$$

where $\boldsymbol{x}$ is state vector of the joint configuration and $\boldsymbol{x}_{\text{ref}}$ is the reference configuration. The whole state vector $x$ may be used or a subset of joints may be selected. This feature is useful for constraining only some of the joints, e.g. constraining the back joints of a humanoid robot while performing a manipulation task. The Jacobian and Jacobian derivative are identity matrices.

We use notation $x$ for scalar values, $\boldsymbol{x}$ for vectors, $X$ for matrices, and $\boldsymbol{X}$ for vectorized matrices.