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

Abstract

Copies of full items can be used for personal research or study, educational, or not-for-profit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP URL' above for details on accessing the published version and note that access may require a subscription. Abstract: It has been a desire to unify the models for structural and parametric analyses and design in the field of robotic mechanisms.This requires a mathematical tool that enables analytical description, formulation and operation possible for both finite and instantaneous motions. This paper presents a method to investigate the algebraic structures of finite screws represented in a quasivector form and instantaneous screws represented in a vector form. By revisiting algebraic operations of screw compositions, this paper examines associativity and derivative properties of the screw triangle product of finite screws and produces a vigorous proof that a derivative of a screw triangle product can be expressed as a linear combination of instantaneous screws. It is proved that the entire set of finite screws forms an algebraic structure as Lie group under the screw triangle product and its time derivative at the initial pose forms the corresponding Lie algebra under the screw cross product, allowing the algebraic structures of finite screws in quasi-vector form and instantaneous screws in vector form to be revealed.

Type Synthesis of Lower Mobility Parallel Mechanisms: A Review

“…Derived from the expression of Chasles' motion with a unit dual quaternion [53], Sun et al [54] presented another approach to describe motion using a finite screw as where s f is a unit vector that represents the axis of the finite screw, r f is a position vector from the origin of a reference frame to an arbitrary point on the screw axis, and θ and t represent rotational and translational displacements, respectively.…”

“…The investigation on systematic type synthesis approaches of lower mobility PMs dates back to the last century and has experienced a gold period with great progress at the beginning of this century. Based on various mathematical tools, a variety of approaches have been presented in the literature, including the Lie group theory-based method [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29], the method based on G F (generalized function) sets [30][31][32][33][34][35], linear transformation method [36][37][38][39][40][41][42][43][44][45][46], position and orientation characteristic (POC) set method [47][48][49][50][51][52], finite screw method [53][54][55][56][57][58][59], screw theory-based method …”

Elastodynamic modeling and parameter sensitivity analysis of a parallel manipulator with articulated traveling plate

“…The parameterized topological models were further analyzed [21][22][23]. By filling in the gap between finite and instantaneous screw theory, Sun [24,25] succeeded in connecting topology analysis to the following performance analysis and even optimal design, which is a milestone in the topology synthesis of parallel manipulators.…”

Static and dynamic optimization of a pose adjusting mechanism considering parameter changes during construction

“…To find out solutions for pose adjusting mechanism with parallel architecture, the first problem is topology synthesis that invents PMs to achieve large orientation workspace. In this regard, Sun [14,15] proposed a topology synthesis method that is generic and recognized as a milestone in the topology synthesis of PMs. In his proposed method, derivative mapping between the finite and instantaneous screws are built, from which the full cycle finite motion pattern of limbs specified to moving platform can be computed in an analytical manner.…”