Finite Displacement Screw Operators With Embedded Chasles’ Motion

Abstract

Rigid body displacement can be presented with Chasles’ motion by rotating about an axis and translating along the axis. This motion can be implemented by a finite displacement screw operator in the form of either a 3 × 3 dual-number matrix or a 6 × 6 matrix that is executed with rotation and translation as an adjoint action of the Lie group. This paper investigates characteristics of this finite displacement screw matrix and decomposes the secondary part that is the off diagonal part of the matrix into the part of an equivalent translation due to the effect of off-setting the rotation axis and the part of an axial translation. The paper hence presents for the first time the axial translation matrix and reveals its property, leading to discovery of new results and new formulae. The analysis further reveals two new traces of the matrix and presents the relationship between the finite displacement screw matrix and the instantaneous screw, leading to the understanding of Chasles’ motion embedded in a rigid body displacement. An algebraic and geometrical interpretation of the finite displacementscrew matrix is thus given, presenting an intrinsic property of the matrix in relation to the finite displacement screw. The paper ends with a case study to verify the theory and illustrate the principle.

Geometric methods and formulations in computational multibody system dynamics

“…It actually took some time until their relation was fully understood since Ball [8][9][10] introduced the concept of screws. An excellent review on this topic can be found in [47,48]. With a chosen reference frame, a general screw is represented by a screw coordinate vector X = (ξ , η), as in (5), where ξ ∈ R 3 is the angular and η ∈ R 3 is the translational component.…”

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

“…For example, the finite screw of RRPRR dose not equal to Eq. (22) In this way, five 5-DOF TPM limb structures can be generated and denoted by PRRRR, RPRRR, RRPRR (Eq. (21), RRRPR and RRRRP, respectively.…”

“…Fig. 1, general finite motion of a rigid body moving from pose 1 to pose 2 can be represented by a finite screw f S in terms of a rotation about an axis followed by a translation along the same axis [18,22,25]. The axis is referred to as the Chasles' axis or the finite screw axis [21,[24][25][26][27], and hereafter we refer the axis as screw axis for simplicity.…”

“…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.…”