Rotations of 4nπ and the Kinematic Design of Parallel Manipulators

For parallel manipulator systems a fundamental distinction is drawn between displacement and motion. The former is a spatial relation adequately modelled by standard vector and matrix algebra. The latter is a spatial trajectory whose ‘history’ also requires modelling. Quaternion, spinor and Clifford algebra representations are utilised for this purpose specifically for rigid body finite rotation in 3D space. Each involves rotation half-angles and hence exhibits apparently counter-intuitive features, notably that rotations of 0 and 2π are not equivalent, whereas rotations of 0 and 4π are equivalent. In general, rotations of 4nπ are not equivalent to rotations of (4n+2)π, where n is any integer. These representations have real physical manifestations, demonstrated here for parallel manipulator designs, adapted from a mechanical model devised by Dirac.