Model-based control of redundantly actuated parallel manipulators in redundant coordinates

Model-based control of parallel kinematics machines (PKM) relies on computationally efficient formulations in terms of a set of independent joint coordinates. Since PKM models are commonly expressed in terms of actuator or end-effector coordinates the models are not valid at input- or output-singularities, respectively. Moreover input-singularities limit the motion range of PKM. Actuation redundancy is a means to increase the singularity-free range of motion. However, due to the redundancy only a subset of the actuator coordinates constitute independent coordinates. This subset corresponds to the actuator coordinates of the non-redundant PKM, which does generally not constitute proper minimal coordinates for the entire workspace. Hence a redundantly actuated PKM (RA-PKM) controlled by a model-based controller in terms of minimal coordinates would exhibit the same limitations as the non-redundant PKM. One approach to tackle this problem is to switch between different minimal coordinates, i.e., different motion equations are used within the controller.In this contribution a computed torque and augmented PD control scheme in redundant coordinates is proposed, as an alternative to coordinate switching, and applied to the control of redundantly actuated PKM. That is, no minimal coordinates are selected. This novel formulation is numerically robust and does not suffer from input- or output-singularities. Even more the formulation is always valid except at configuration space singularities. For the redundancy resolution within the inverse dynamics the pseudoinverse of a rank deficient matrix is required, for which an explicit formulation is presented. For both controllers exponential trajectory tracking is shown. Experimental results are reported for a planar 2 DOF RA-PKM.