It is known that the twist space of a (plunging) constant-velocity (CV) coupling with intersecting shafts consists, in all configurations, of a planar field of zero-pitch screws, namely Hunt's fourth special three system (with principal pitches (1; 0; 0)), hereafter denoted S_(3,4). The input and output shafts form a mirror symmetry about the system plane, hereafter referred to as the bisecting plane. The persistence of the twist space, i.e. the invariance of its principal and reciprocal principal pitches, is achieved by a parallel mechanism with prescribed connecting chains, with each one of them having mirror symmetric joint axes about the bisecting plane. Recently, the first author reported an important discovery, namely that S_(3,4), though not being a Lie subalgebra of the Lie algebra se(3) of the special Euclidean group SE(3), presents similar characteristics, which underlie the operation of a CV coupling. First, S_(3,4) is closed under two consecutive Lie bracket operations (or a Lie triple product), thus being referred to as a Lie triple system; second, taking the exponential of all twists in S_(3,4) generates a submanifold of SE (3), which is exactly the continuous-motion manifold of the coupling. In this paper, we first give a geometric characterization of the Lie product and the Lie triple product of a screw system. Then, we present a systematic identification of all Lie triple screw systems of se (3), by an approach based on both algebraic Lie group theory and descriptive screw theory.
Identification and Geometric Characterization of Lie Triple Screw Systems / Wu, Yuanqing; Carricato, Marco. - STAMPA. - 4:(2015), pp. OS2-005.77-OS2-005.86. (Intervento presentato al convegno 14th IFToMM World Congress tenutosi a Taipei, Taiwan nel Oct. 25-30, 2015) [10.6567/IFToMM.14TH.WC.OS2.005].
Identification and Geometric Characterization of Lie Triple Screw Systems
WU, YUANQING;CARRICATO, MARCO
2015
Abstract
It is known that the twist space of a (plunging) constant-velocity (CV) coupling with intersecting shafts consists, in all configurations, of a planar field of zero-pitch screws, namely Hunt's fourth special three system (with principal pitches (1; 0; 0)), hereafter denoted S_(3,4). The input and output shafts form a mirror symmetry about the system plane, hereafter referred to as the bisecting plane. The persistence of the twist space, i.e. the invariance of its principal and reciprocal principal pitches, is achieved by a parallel mechanism with prescribed connecting chains, with each one of them having mirror symmetric joint axes about the bisecting plane. Recently, the first author reported an important discovery, namely that S_(3,4), though not being a Lie subalgebra of the Lie algebra se(3) of the special Euclidean group SE(3), presents similar characteristics, which underlie the operation of a CV coupling. First, S_(3,4) is closed under two consecutive Lie bracket operations (or a Lie triple product), thus being referred to as a Lie triple system; second, taking the exponential of all twists in S_(3,4) generates a submanifold of SE (3), which is exactly the continuous-motion manifold of the coupling. In this paper, we first give a geometric characterization of the Lie product and the Lie triple product of a screw system. Then, we present a systematic identification of all Lie triple screw systems of se (3), by an approach based on both algebraic Lie group theory and descriptive screw theory.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
