Synthesis, Mobility, and Multifurcation of Deployable Polyhedral Mechanisms With Radially Reciprocating Motion

Extending the method coined virtual-center-based (VCB) for synthesizing a group of deployable platonic mechanisms with radially reciprocating motion by implanting dual-plane-symmetric 8-bar linkages into the platonic polyhedron bases, this paper proposes for the first time a more general single-plane-symmetric 8-bar linkage and applies it together with the dual-plane-symmetric 8-bar linkage to the synthesis of a family of one-degree of freedom (DOF) highly overconstrained deployable polyhedral mechanisms (DPMs) with radially reciprocating motion. The two 8-bar linkages are compared, and geometry and kinematics of the single-plane-symmetric 8-bar linkage are investigated providing geometric constraints for synthesizing the DPMs. Based on synthesis of the regular DPMs, synthesis of semiregular and Johnson DPMs is implemented, which is illustrated by the synthesis and construction of a deployable rectangular prismatic mechanism and a truncated icosahedral (C60) mechanism. Geometric parameters and number synthesis of typical semiregular and Johnson DPMs based on the Archimedean polyhedrons, prisms and Johnson polyhedrons are presented. Further, movability of the mechanisms is evaluated using symmetry-extended rule, and mobility of the mechanisms is verified with screw-loop equation method; in addition, degree of overconstraint of the mechanisms is investigated by combining the Euler's formula for polyhedrons and the Grubler–Kutzbach formula for mobility analysis of linkages. Ultimately, singular configurations of the mechanisms are revealed and multifurcation of the DPMs is identified. The paper hence presents an intuitive and efficient approach for synthesizing PDMs that have great potential applications in the fields of architecture, manufacturing, robotics, space exploration, and molecule research.

[1]  Jian S. Dai,et al.  A Spatial Eight-Bar Linkage and Its Association With the Deployable Platonic Mechanisms , 2014 .

[2]  Jian S. Dai,et al.  Constraint-Based Limb Synthesis and Mobility-Change-Aimed Mechanism Construction , 2011 .

[3]  H. Lipkin,et al.  Mobility of Overconstrained Parallel Mechanisms , 2006 .

[4]  Jian S. Dai,et al.  Mobility analysis of a complex structured ball based on mechanism decomposition and equivalent screw system analysis , 2004 .

[5]  Simon D. Guest,et al.  Deployable structures: concepts and analysis , 1994 .

[6]  Otto Roeschel.  OVERCONSTRAINED MECHANISMS BASED ON TRAPEZOHEDRA , 2012 .

[7]  S. Pellegrino,et al.  Matrix analysis of statically and kinematically indeterminate frameworks , 1986 .

[8]  Jian S. Dai,et al.  Synthesis of a Family of Regular Deployable Polyhedral Mechanisms (DPMs) , 2012, ARK.

[9]  Jian S. Dai,et al.  Screw System Analysis of Parallel Mechanisms and Applications to Constraint and Mobility Study , 2004 .

[10]  Jing-Shan Zhao,et al.  A theory of degrees of freedom for mechanisms , 2004 .

[11]  Jian S. Dai,et al.  Synthesis and Construction of a Family of One-DOF Highly Overconstrained Deployable Polyhedral Mechanisms (DPMs) , 2012 .

[12]  Jian S. Dai,et al.  Mobility and Geometric Analysis of the Hoberman Switch-Pitch Ball and Its Variant , 2010 .

[13]  R. Bricard.  Mémoire sur la théorie de l'octaèdre articulé , 1897 .

[14]  Harvey Lipkin,et al.  A Note on Denavit-Hartenberg Notation in Robotics , 2005 .

[15]  Guowu Wei,et al.  Overconstrained Mechanisms with Radially Reciprocating Motion , 2010 .

[16]  Gökhan Kiper.  Fulleroid-Like Linkages , 2009 .

[17]  Thierry Lalibert,et al.  Polyhedra with Articulated Faces , 2007 .

[18]  Patrick W. Fowler,et al.  A symmetry extension of Maxwell's rule for rigidity of frames , 2000 .

[19]  Karl Wohlhart,et al.  Kinematics and Dynamics of the Fulleroid , 1997 .

[20]  Patrick W. Fowler,et al.  TWO-ORBIT SWITCH-PITCH STRUCTURES , 2012 .

[21]  Clément Gosselin,et al.  Expandable Polyhedral Mechanisms Based on Polygonal One-Degree-of-Freedom Faces , 2006 .

[22]  Jian Feng,et al.  Generalized Eigenvalue Analysis of Symmetric Prestressed Structures Using Group Theory , 2012, J. Comput. Civ. Eng..

[23]  Sunil K. Agrawal,et al.  Polyhedral Single Degree-of-freedom Expanding Structures: Design and Prototypes , 2002 .

[24]  Jian S. Dai,et al.  Axis Constraint Analysis and Its Resultant 6R Double-Centered Overconstrained Mechanisms , 2011 .

[25]  Patrick W. Fowler,et al.  A class of expandable polyhedral structures , 2004 .

[26]  S. C. O'brien,et al.  C60: Buckminsterfullerene , 1985, Nature.

[27]  J. Michael McCarthy,et al.  Introduction to theoretical kinematics , 1990 .

[28]  Patrick W. Fowler,et al.  A symmetry-extended mobility rule , 2005 .

[29]  Jian Feng,et al.  Efficient Method for Moore-Penrose Inverse Problems Involving Symmetric Structures Based on Group Theory , 2014 .

[30]  D. R. Kerr,et al.  Finite Twist Mapping and its Application to Planar Serial Manipulators with Revolute Joints , 1995 .

[31]  X-Z Wei,et al.  A new method of creating expandable structure for spatial objects , 2006 .

[32]  Jian S. Dai,et al.  Finite Displacement Screw Operators With Embedded Chasles' Motion , 2012 .

[33]  Jian S. Dai,et al.  Origami-inspired integrated planar-spherical overconstrained mechanisms , 2014 .

[34]  Karl Wohlhart,et al.  Double-Ring Polyhedral Linkages , 2012 .

[35]  H. F. Verheyen.  THE COMPLETE SET OF JITTERBUG TRANSFORMERS AND THE ANALYSIS OF THEIR MOTION , 1989 .