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.