Mobility of overconstrained parallel mechanisms with reconfigurable end-effectors

Reconfigurable parallel mechanisms have legs or end-effectors with additional degrees of freedom or with the ability to change their mobility. This allows them to reconfigure into a variety of conventional parallel mechanisms. One type of reconfigurable parallel mechanism can be obtained by replacing the end-effector with a closed-loop linkage. The parts of this linkage that are connected to the legs may be considered as the end-effectors, having their own mobility relative to the base and a coupled mobility with each other. Classical methods for mobility analysis cannot be directly applied to this type of mechanism. This article presents a general method to calculate and represent the coupled mobility of the multiple end-effectors using nullspace operations on a matrix of mobility. The nullspace approach provides an efficient way to calculate the mobility in a closed-loop chain while the matrix of mobility allows for calculating the multiple end-effectors mobility interaction. The general method can be performed analytically to obtain insights on the mechanism in any configuration, or algorithmically to obtain numerical values of the mobility in a particular configuration. Two illustrative examples showcasing the method are presented.