Representational singularities in the torque optimization problem of an active closed loop mechanism

Resolution of actuation redundancy in active mechanisms that include a closed kinematic loop is addressed here. The minimization of the active effort in these mechanisms is explored here instead of the more frequently studied problem of optimization of the contact force conditions in the presence of multiple frictional contacts. This minimal effort problem is relevant if these mechanisms are performing tasks where slipping at the frictional contacts is extremely unlikely, or if the mechanisms do not include frictional contacts. An analytical formulation is presented using a spatial notation that is an extension of the screw representation of motion and force systems. Representational singularities (singularities that have no physical significance) that arise during the algorithmic development of the problem are presented. It is also shown that different representational singularities arise if the solution for the active effort is arrived at with different manipulation of the equations governing the active closed loop mechanism.