Euler–Rodrigues formula variations, quaternion conjugation and intrinsic connections

Abstract

This paper reviews the Euler-Rodrigues formula in the axis-angle representation of rotations, studies its variations and derivations in different mathematical forms as vectors, quaternions and Lie groups and investigates their intrinsic connections. The Euler-Rodrigues formula in the Taylor series expansion is presented and its use as an exponential map of Lie algebras is discussed particularly with a non-normalized vector. The connection between Euler-Rodrigues parameters and the Euler-Rodrigues formula is then demonstrated through quaternion conjugation and the equivalence between quaternion conjugation and an adjoint action of the Lie group is subsequently presented. The paper provides a rich reference for the Euler-Rodrigues formula, the variations and their connections and for their use in rigid body kinematics, dynamics and computer graphics.

A way of relating instantaneous and finite screws based on the screw triangle product

“…The dual quaternion based method can be traced back to description of rotations of a rigid body by means of Euler's four-square identity, Euler-Rodrigues parameters [23,35] and Hamilton quaternions [36]. Perez and McCarthy [37] seem to be the first to use the dual quaternions to do analyses for finite and instantaneous motions of serial kinematic chains.…”

“…With the aid of group theory, Selig [39][40] and Dai [26,35] investigated the algebraic properties of the exponential and Cayley maps between unit dual quaternions and unit pure dual quaternions. By introducing the notation of high-dimensional Clifford algebra, Selig [40] and Featherstone [41] extended the dual quaternions representation to deal with rigid body dynamics.…”

“…Considering the algebraic characteristic of finite screws, Dai [22] demonstrated the relationship between finite displacement screw operation and the different matrix representations of SE(3) elements as well as quaternions [25,35]. By solving the eigenscrew and derivative of finite displacement screw matrix, Dai [25] formulated the eigen and differential mappings between finite screws and instantaneous screws.…”

A finite screw approach to type synthesis of three-DOF translational parallel mechanisms

“…This is because the composition of a number of successive finite screws can be explicitly represented and algebraically operated by screw triangle product which is the specific representation of Baker-Campbell-Hausdorff formula [14]. The idea of finite screw was first proposed by Dimentberg [15] and developed by many others over the last few decades [16][17][18][19][20][21][22][23][24][25]. For example, Parkin [16][17] proposed a specific finite screw in a quasi-vector form with elaborately designed magnitude that is particularly suitable for finite motion composition.…”

“…Huang [18][19] developed the screw triangle product of two finite screws as a linear combination of five meaningful terms. Recently, intensive efforts have been made by Dai [20] to investigate the interrelationships among finite displacement screws, the point/line transformation matrix representations of SE(3) as well as dual quaternions [21][22][23]. Having firstly developed the eigen/differential map of finite/instantaneous screws [22] and rigorously proven its consistency with the exponential map of Lie group/algebra or quaternions and Euler-Rodrigues formula [22][23][24], Dai [25] established the interrelationship theory that enables algebraic properties of various mathematic descriptions of rigid body motions to be closely connected.…”