Mobility analysis of overconstrained parallel mechanism using Grassmann-Cayley algebra

In this work we apply Grassmann-Cayley algebra (GCA) to the mobility analysis of overconstrained parallel mechanisms (PMs). Limb twist spaces are constructed by using join operator and extensor. The platform twist space or output twist space is the intersection of all limb twist spaces. A key step is to handle redundant constraints in overconstrained PMs, which has been a major obstacle in using GCA in mobility analysis of overconstrained PMs. To bridge the gap, new independent extensors are added to limb twist spaces to guarantee that the join of two limb twist space is a 6-dimensional vector space, thereby eliminating redundant constraints. Then the correct intersection of limb twist spaces can be obtained by using meet operator. The result is an analytic expression of output twist space of the target overconstrained PM in a coordinate-free manner, providing very helpful insights into motion property of an overconstrained PM. Three typical overconstrained PMs are chosen as examples to testify the effectiveness of the improved method.