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Machine Theory
Mechanism and Machine Theory 43 (2008) 335–346
ate/mechmt
Architecturally singular non-planar parallel manipulators
Adolf Karger
Faculty of Mathematics and Physics, Charles University Praha, Sokolovska´ 83, 186 00 Praha, Czech Republic
Received 14 April 2006; accepted 27 March 2007
Available online 7 June 2007
Abstract
In this contribution we present plete list of all non-planar architecture singular parallel manipulators (Stewart-
Gough type manipulators). The main result is that they are in a certain sense trivial, because any such parallel manipulator
has four collinear points in the platform or in the base. This follows that they are not useful for practical purposes.
We see that all of them must satisfy certain assembly conditions (the space of joint parameters is degenerated, we cannot
prescribe freely the length of legs). All non-planar architecture singular parallel manipulators have self-motions at every
possible configuration. To every non-planar manipulator we can add infinitely many legs without disturbing its geometrical
properties and self-motions, for planar platform and base it is not true. Inspection of the list of non-planar architecture
singular parallel manipulators shows that they are invariant with respect to projective transformations in both fixed
and moving spaces.
Ó 2007 Elsevier Ltd. All rights reserved.
Keywords: Stewart-Gough platform; Architecturally singular parallel manipulators
1. Introduction
Architecture singular parallel manipulators have been introduced by Ma and Angeles as such Stewart-
Gough type manipulators (SGP) which are singular at every possible configuration (see [11]). We consider sin-
gular configurations of SGP as such configurations that screws of their legs are linearly dependent. This means
that the determinant of coordinates of screws of all six legs of the manipulator is equal to zero [12]. The set of
all singular positions is an algeb