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Journal ArticleDOI

Oblique circular torus, Villarceau circles and four types of Bennett linkages

TL;DR: In this paper, the oblique circular torus (OCT) and its main geometric properties are introduced and the coordinate-free approach leads to the algebraic equation of an OCT in a privileged Cartesian reference frame.
Abstract: The oblique circular torus (OCT) and its main geometric properties are introduced. Intrinsic vector calculation is utilized to mathematically describe the OCT. The coordinate-free approach leads to the algebraic equation of an OCT in a privileged Cartesian reference frame. The OCT equation is used to confirm a theorem of Euclidean geometry. In a broad category of OCT, through any point five circles can be drawn on the surface, namely the parallel of latitude and four circular generatrices whose planes pass through the OCT center of symmetry. In the special case of a right circular torus, the Villarceau theorem is verified. Next, consider the four RRS open chains whose S spherical-joint centers move on the same OCT and their possible in-parallel assemblies in single-loop RRRS chains. From a category of the foregoing RRRS chains, a new derivation of the amazing Bennett 4R linkage is proposed. Two kinds of Bennett linkages are further verified and each kind contains two enantiomorphic or symmetric linkages. ...
Citations
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Journal ArticleDOI
TL;DR: In this article, the singularities of the generated surfaces can be used to obtain mechanisms which can change their number of degrees-of-freedom depending on its configuration, which can be obtained by means of two methods presented in this contribution.
Abstract: The method of intersection of surfaces generated by kinematic dyads is applied to obtain mechanisms that are able to shift from one mode of motion to another. Then a mobility analysis shows that the singularities of the generated surfaces can be used to obtain mechanisms which can change their number of degrees-of-freedom depending on its configuration. The generator dyads are connected as usually done by a spherical pair. However, in the cases shown in this contribution the three-degrees-of-freedom of the spherical pair are not all necessary to keep the kinematic chain closed and movable, and the spherical pair can be substituted by either a pair of intersecting revolute joints or a single revolute joint. This substitution can be obtained by means of two methods presented in this contribution.

24 citations

Journal ArticleDOI
TL;DR: In this article, the authors investigated a line-symmetric Bricard mechanism by means of two generated toroids and revealed their intersection that leads to a set of special Bricards with various branches of reconfiguration.
Abstract: This paper for the first time investigates a family of line-symmetric Bricard mechanisms by means of two generated toroids and reveals their intersection that leads to a set of special Bricard mechanisms with various branches of reconfiguration. The discovery is made in the concentric toroid-toroid intersection. By manipulating the construction parameters of the toroids any possible bifurcation point is explored. This leads to the common bi-tangent planes that present singularities in the intersection set. The study reveals the presence of Villarceau and secondary circles in the toroids intersection. Therefore, a way to reconfigure the Bricard linkage to two different types of Bennett mechanism is uncovered. Further a linkage with two Bricard and two Bennett motion branches is explored. In addition, the paper reveals the Altmann linkage as a member of the family of special line-symmetric Bricard linkages studied in this paper.

22 citations

Journal ArticleDOI
TL;DR: In this article, a set of special plane-symmetric Bricard linkages with various branches of reconfiguration by means of intersection of two generating toroids is revealed.
Abstract: This paper for the first time reveals a set of special plane-symmetric Bricard linkages with various branches of reconfiguration by means of intersection of two generating toroids, and presents a complete theory of the branch reconfiguration of the Bricard plane-symmetric linkages. An analysis of the intersection of these two toroids reveals the presence of coincident conical singularities, which lead to design of the plane-symmetric linkages that evolve to spherical 4R linkages. By examining the tangents to the curves of intersection at the conical singularities, it is found that the linkage can be reconfigured between the two possible branches of spherical 4R motion without disassembling it and without requiring the usual special configuration connecting the branches. The study of tangent intersections between concentric singular toroids also reveals the presence of isolated points in the intersection, which suggests that some linkages satisfying the Bricard plane-symmetry conditions are actually structures with zero finite degrees-of-freedom (DOF) but with higher instantaneous mobility. This paper is the second part of a paper published in parallel by the authors in which the method is applied to the line-symmetric case.

20 citations


Cites background or methods from "Oblique circular torus, Villarceau ..."

  • ...Using the toroid as a generated surface the existence of several overconstrained linkages can be explained: The Bennett 4R linkage [52, 53, 54, 55], the Myard 5R linkage [23] and the family of overconstrained 5R loops with two consecutive parallel axes presented by...

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  • ...It is well known [52, 53, 54] that a Bennett linkage can be explained as a generator of a toroid where E is confined to move in a Villarceau circle by means of link between E and a revolute joint that generates the circle....

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01 Jan 2015
TL;DR: Based on the four distinct types of Bennett mechanisms, this paper proposed a novel family of discontinuously movable mechanisms with two postures of trifurcation by adding two coaxial revolute pairs along their opposite bars.
Abstract: Based on the four distinct types of Bennett mechanisms, this paper proposes a novel family of discontinuously movable mechanisms with two postures of trifurcation by adding two coaxial revolute pairs along their opposite bars. These mechanisms are called metamorphic Bennett linkages (MetaMBLs) for brevity. They are all special six-revolute linkages with the discontinuous mobility characterized by three working modes: the 1-DoF coaxial revolute rotations, Bennett motion with the same handedness of the four links and other Bennett motion with the alternate link handedness. Two special cases, rectangular and equilateral MetaMBLs, are studied, too. The proposed MetaMBLs are further animated for the verification of the metamorphic postures.

4 citations


Cites background from "Oblique circular torus, Villarceau ..."

  • ...Recently, a new geometric derivation of four types of Bennett 4R linkages [13] was done by considering RRS open chains whose spherical S joint center moves on the same oblique circular torus (OCT)....

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  • ...properties of symbolic notations :(b,;a), (a,;b), (b,-;a) and (a,-; b) respectively can produce the same given oblique circular torus [13]....

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  • ...An equilateral isogram is an achiral object and only one type of equilateral Bennett mechanism exist [13] when one ignores the partitioned mobility with the bifurcation toward two modes of rotation about aligned axes [14]....

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  • ...The four types of Bennett mechanisms are degenerated into only two types [13]....

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Journal ArticleDOI
TL;DR: A method for designing asymmetric multi-mode parallel mechanism that can realize multiple motion modes by using fewer driving pairs and does not need to reassemble the mechanism when the motion mode is changed.
Abstract: A method for designing asymmetric multi-mode parallel mechanism is proposed. Parallel mechanisms with 2R1T and 2T1R motion modes are synthesized by using displacement manifold theory. Based on the parallel mechanism, a kind of mechanism with hybrid variable DOF kinematic chain is proposed. The DOF characteristics of the mechanism in the process of motion mode transformation are analyzed by using screw theory, and the rationality of the selection of driving pairs in different motion modes is verified. The results show that the mechanism with hybrid variable DOF kinematic chain has 3R, 2T1R, and 2R1T motion modes. When the mechanism is in the general configuration of the four motion modes mentioned above, three driving pairs can control the mechanism. When the mechanism is in the transformation configuration having 2R2T or 3R1T instantaneous DOF, an additional auxiliary driving pair is needed to control the mechanism. The mechanism can realize multiple motion modes by using fewer driving pairs. It does not need to reassemble the mechanism when the motion mode is changed.

3 citations

References
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Journal ArticleDOI
TL;DR: In this article, the authors describe two premieres qui correspondent aux mecanismes les plus importants du point de vue des possibilites d'applications.

340 citations

Journal Article

236 citations

Journal ArticleDOI

125 citations


"Oblique circular torus, Villarceau ..." refers background in this paper

  • ...Substituting the relations of equation (9) into equation (8) leads to the algebraic equation of OCT with respect to the x-y-z system, which is...

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  • ...Equation (8) expresses the Cartesian equation of OCT in the fixed X-Y-Z coordinate system that is the frame (O, i, j, k)....

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  • ...The surface modeled by equation (8) is an OCT....

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BookDOI
01 Jun 2000
TL;DR: The Selig design of new mechanisms via the displacement subgroups, J.M. Selig subgroups and representations, and variational problems associated with kinematic chains.
Abstract: Groups, J.M. Selig subgroups and representations, J.M. Selig design of new mechanisms via the displacement subgroups, J.M. Herve kinematics from the singular viewpoint, G.G. Gibson singularity analysis of serial robot-manipulators, A. Karger variational problems associated with kinematic chains, R. Brockett computational differential algebra, B. Mishra.

51 citations

Journal ArticleDOI
01 Jun 2005
TL;DR: In this paper, the authors present a mobility analysis based on algebraic concepts of set theory and show that the Lie group algebraic properties of the displacement set can be used to explain the discontinuous mobility of planar and spherical 4R chains.
Abstract: Combining planar and spherical 4R chains generates three kinds of new discontinuously movable (DM) seven-link mechanisms. The mobility analysis is based on algebraic concepts of set theory. These mechanisms are called hybrid planar-spherical 7R, hybrid spherical-spherical 7R, and hybrid planar-planar 6RIP DM mechanisms. Their discontinuous mobility is explained employing the Lie group algebraic properties of the displacement set. Moreover, the same given spatial arrangement of joints can be linked in two ways constituting two distinct chains, which have a quite different mobility. One chain has two global degrees of freedom (d.o.f.), which disobey the general Grubler-Kutzbach mobility criterion. The other chain exhibits a singular pose, which is a bifurcation towards two distinct working modes of mobility with one d.o.f. Then the set of relative motions between any two specific links is not manifold but consists of the union of displacement one-dimensional manifolds. In general, when two or more s...

34 citations