Three Novel Symmetric Waldron–Bricard Metamorphic and Reconfigurable Mechanisms and Their Isomerization

This paper explores a class of metamorphic and reconfigurable linkages belonging to both Waldron's double-Bennett hybrid linkage and Bricard linkages, which include three novel symmetric Waldron–Bricard metamorphic and reconfigurable mechanisms, and further presents their three extended isomeric metamorphic linkages. The three novel Waldron–Bricard metamorphic and reconfigurable linkages are distinguished by line-symmetric, plane-symmetric, and line-plane-symmetric characteristics. The novel line-symmetric Waldron–Bricard metamorphic linkage with one Waldron motion branch and two general and three special line-symmetric Bricard motion branches is obtained by integrating two identical general Bennett loops. The novel plane-symmetric Waldron–Bricard reconfigurable linkage with two plane-symmetric motion branches is obtained by coalescing two equilateral Bennett loops. The novel line-plane-symmetric Waldron–Bricard metamorphic linkage with six motion branches is obtained by blending two identical equilateral Bennett loops, including the plane-symmetric Waldron motion branch, the line-plane-symmetric Bricard motion branch, the spherical 4R motion branch, and three special line-symmetric Bricard motion branches. With the isomerization that changes a mechanism structure but keeps all links and joints, each of the three novel Waldron–Bricard linkages results in an extended isomeric metamorphic linkage. This further evolves into the study of the three isomeric mechanisms. The study of these three novel metamorphic and reconfigurable mechanisms and their isomerization are carried out to demonstrate the characteristics of bifurcation and to reveal motion-branch transformation. Furthermore, by exploring the intersection of given motion branches and using the method of isomerization, more metamorphic and reconfigurable linkages can be discovered to usefully deal with transitions among possible submotions.

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The equilibrium and stability of sessile drops

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1977.0060 ◽

1977 ◽

Vol 354 (1676) ◽

pp. 127-136 ◽

Cited By ~ 8

Keyword(s):

Internal Pressure ◽

Three Dimensional ◽

Two Dimensional ◽

Axisymmetric Mode ◽

Plane Symmetric ◽

Symmetric Plane ◽

Fixed Base ◽

Pressure Maximum ◽

Axisymmetric Instability ◽

Sessile Drops

The paper gives a mathematical analysis of the equilibrium and stability of two dimensional and axisymmetric sessile drops, when controlled by their volume or by their internal pressure. Sessile drops of either form when standing on a horizontal plane with a prescribed contact angle are shown to be stable when controlled by their volume or pressure, except for the neutrally stable sideways translation, without change of shape, which can occur formally. For two dimensional drops grown on a fixed base, and controlled by their volume, plane symmetric, plane asymmetric, and three dimensional forms of instability can occur, the last named being the first one to occur, at the position of the internal pressure maximum. For the same drop when controlled by its pressure plane symmetric and three dimensional instabilities arise together at the pressure maximum. For axisymmetric drops grown on a fixed base and controlled by their volume, axisymmetric and asymmetric instabilities can arise. Asymmetric instability with azimuthal wavenumber m = 1 arises when the drop profile becomes horizontal at its base, and this instability always precedes the axisymmetric instability arising at the volume maximum. However, the same drop when controlled by its pressure becomes unstable first in the axisymmetric mode arising at the pressure maximum.

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Kinematic Analysis of Conventional and Multi-Mode Spatial Mechanisms Using Dual Quaternions

Volume 5B: 40th Mechanisms and Robotics Conference ◽

10.1115/detc2016-59194 ◽

2016 ◽

Cited By ~ 3

Author(s):

Xianwen Kong

Keyword(s):

Algebraic Geometry ◽

Kinematic Analysis ◽

Fundamental Issue ◽

Input Output ◽

Plane Symmetric ◽

Spatial Mechanism ◽

Dual Quaternions ◽

Reconfigurable Mechanisms ◽

Kinematic Loop ◽

Multi Mode

Although kinematic analysis of conventional mechanisms is a well-documented fundamental issue in mechanisms and robotics, the emerging reconfigurable mechanisms and robots (or mechanisms and robots with multiple operation modes) require re-examining this fundamental issue. Recent advances in mathematics, especially algebraic geometry and numerical algebraic geometry, make it possible to develop an efficient method for the kinematic analysis of not only conventional mechanisms and robots but also reconfigurable mechanisms and robots. This paper first presents a method for setting up a set of kinematic loop equations for mechanisms using dual quaternions. Using this approach, a set of kinematic loop equations of a spatial mechanism is composed of six equations. The effectiveness of the proposed kinematic loop equations is then demonstrated by deriving the explicit input-output equations of a line symmetric 1-DOF (degree-of-freedom) 7R single-loop spatial mechanism, the re-configuration analysis of a novel multi-mode 1-DOF 7R spatial mechanism. In the former case, an explicit input-output equation of degree 8 is derived. In the latter case, it is found that the 7R multi-mode mechanism has three motion modes, including a planar 4R mode, an orthogonal Bricard 6R mode, and a plane symmetric 6R mode. Unlike the 7R multi-mode mechanisms in the literature, the 7R multi-mode mechanism presented in this paper does not have a 7R mode in which all the seven R joints can move simultaneously.

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Reconfiguration Analysis of Multimode Single-Loop Spatial Mechanisms Using Dual Quaternions

Journal of Mechanisms and Robotics ◽

10.1115/1.4037111 ◽

2017 ◽

Vol 9 (5) ◽

Cited By ~ 10

Author(s):

Xianwen Kong

Keyword(s):

Algebraic Geometry ◽

Single Degree Of Freedom ◽

Plane Symmetric ◽

Dual Quaternions ◽

Revolute Joints ◽

Spatial Mechanisms ◽

Reconfigurable Mechanisms ◽

Motion Modes ◽

Kinematic Loop ◽

And Robotics

Although kinematic analysis of conventional mechanisms is a well-documented fundamental issue in mechanisms and robotics, the emerging reconfigurable mechanisms and robots pose new challenges in kinematics. One of the challenges is the reconfiguration analysis of multimode mechanisms, which refers to finding all the motion modes and the transition configurations of the multimode mechanisms. Recent advances in mathematics, especially algebraic geometry and numerical algebraic geometry, make it possible to develop an efficient method for the reconfiguration analysis of reconfigurable mechanisms and robots. This paper first presents a method for formulating a set of kinematic loop equations for mechanisms using dual quaternions. Using this approach, a set of kinematic loop equations of spatial mechanisms is composed of six polynomial equations. Then the reconfiguration analysis of a novel multimode single-degree-of-freedom (1DOF) 7R spatial mechanism is dealt with by solving the set of loop equations using tools from algebraic geometry. It is found that the 7R multimode mechanism has three motion modes, including a planar 4R mode, an orthogonal Bricard 6R mode, and a plane symmetric 6R mode. Three (or one) R (revolute) joints of the 7R multimode mechanism lose their DOF in its 4R (or 6R) motion modes. Unlike the 7R multimode mechanisms in the literature, the 7R multimode mechanism presented in this paper does not have a 7R mode in which all the seven R joints can move simultaneously.

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