A More General Mobility Criterion for Parallel Platforms

Abstract

This contribution presents a more general mobility criterion applicable to parallel platforms, unlike previously employed mobility criteria based on the well-known Kutzbach-Gru¨bler criterion that often fails to provide the correct number of degrees of freedom of parallel manipulators, the mobility criterion introduced in this contribution provides the correct number of degrees of freedom for a wider class of parallel manipulators. Furthermore, the analysis provides insight into why the criteria based on the Kutzbach-Gru¨bler criterion often fails. Moreover, the newly developed criterion computes the passive degrees of freedom in parallel platforms and the mobility of some classes of kinematically defficient parallel platforms. Finally, it is important to note that this criterion is based on an analysis of the subalgebras of the Lie algebra, se(3), also known as screw algebra, of the Euclidean group, SE(3). As is common in many areas of kinematics, the criterion presented here can be also obtained by an analysis of the subgroups of the Euclidean group.

Fault Tolerance of Parallel Manipulators with Passive Joints

“…The approach in Dai et al [2006] is based on the constraint space formulation. It is also shown that the mobility can be found by the motion space as in Rico et al [2003] and Rico et al [2006]. By this approach the chains are also classified and it is straight forward to determine the effect a passive joint has on the mechanism.…”

“…In the following, we will show that an alternative to the Modified Grübler formula given in (30) can be found from the motion space and we will show how this approach naturally leads to the classification of different types of overconstrained joints. This is the same classification of overconstrained chains as in Rico et al [2003] and Rico et al [2006]. Further, in the next section, we will use this to set up a set of simple rules on where the active joints need to be placed, i.e.…”

“…Paradoxical linkages are not treated here. In Rico et al [2006] this is generalised to the case of arbitrarily many chains. Here we will look at it from a different view in order to get a deeper understanding of the mobility criteria.…”

“…The total mobility of the mechanism is then given by adding (32) to (31). We will write this as a proposition and provide a different proof than that of Rico et al [2006].…”

“…The mobility can also be found by the motion space as in Rico et al [2003] and Rico et al [2006]. The degree of freedom of the motion of the end effector is first found.…”