An historical review of the theoretical development of rigid body displacements from Rodrigues parameters to the finite twist

The development of the finite twist or the finite screw displacement has attracted much attention in the field of theoretical kinematics and the proposed q-pitch with the tangent of half the rotation angle has demonstrated an elegant use in the study of rigid body displacements. This development can be dated back to Rodrigues’ formulae derived in 1840 with Rodrigues parameters resulting from the tangent of half the rotation angle being integrated with the components of the rotation axis.This paper traces the work back to the time when Rodrigues parameters were discovered and follows the theoretical development of rigid body displacements from the early 19th century to the late 20th century. The paper reviews the work from Chasles motion to Cayley’s formula and then to Hamilton’s quaternions and Rodrigues parameterization and relates the work to Clifford biquaternions and to Study’s dual angle proposed in the late 19th century. The review of the work from these mathematicians concentrates on the description and the representation of the displacement and transformation of a rigid body, and on the mathematical formulation and its progress.The paper further relates this historic development to the contemporary development of the finite screw displacement and the finite twist representation in the late 20th century.