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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.

Rotations of 4nπ and the Kinematic Design of Parallel Manipulators