Applications of the 4D geometric algebra to dimensional mobility criteria of delassus-parallelogram and bennett paradoxical linkages

Abstract

Geometric algebra is also termed Clifford-Grassmann algebra or hypercomplex number. It allows studying space geometric problems in an easy and compact way. Transforming threedimensional (3D) Euclidean geometric entities to actual elements of four-dimensional (4D) geometric algebra (abbreviated to G4) through a methodical approach of geometric algebra, one can describe motion displacements as even elements of G4. This article relies on the combined rotation and translation in G4 to establish the dimensional constraints of two non-exceptional overconstrained paradoxical linkages. Firstly, fundamentals of geometric algebra are recalled. Then, the single finite rotation and the composition of two successive finite rotations are introduced. After that, a general rigid-body motion in G4 is revealed for a possible application in exploring paradoxical chains using the geometric algebra. Finally, the metric or dimensional mobility criteria of Delassusparallelogram four-screw and Bennett four-revolute paradoxical linkages are algebraically verified.