On Crossley's contribution to the development of graph based algorithms for the analysis of mechanisms and gear trains

Abstract This paper celebrates a particular branch of Crossley's early work dedicated to Mechanism Science, which deals with a rigorous introduction of Graph Theory to the study of some fundamental and intrinsic properties of kinematic chains and mechanisms. Although such idea gave its main outcome in Type and Number Synthesis (which has been much better and extensively described in another paper of the present special issue) some other intriguing side effects appeared, later in Mechanism Science, which yielded several results, and are still in the center of research and industrial world interest, such as, to name but a few, the automatic generation of the equations governing kinematic, static force and dynamic analysis of mechanisms and geared trains, the power flow analysis, the computation of the efficiency and, finally, the never fully explored structure-to-function mapping, which the present contribution points out to be still a challenge in the field.

[1]  Lung-Wen Tsai,et al.  The kinematics of spatial robotic bevel-gear trains , 1988, IEEE J. Robotics Autom..

[2]  F.R.E. Crossley,et al.  On the response of the singly free linear oscillator , 1960 .

[3]  Zhen Huang,et al.  A unique representation of the kinematic chain and the atlas database , 2007 .

[4]  Pier Paolo Valentini,et al.  Kinematic Analysis of the Antikythera Gear Mechanism by Means of Graph Theory , 2002 .

[5]  Madhusudan Raghavan.  The Analysis of Planetary Gear Trains , 2006 .

[6]  Pier Paolo Valentini,et al.  A Review of Formulas for the Mechanical Efficiency Analysis of Two Degrees-of-Freedom Epicyclic Gear Trains , 2003 .

[7]  J. J. Uicker,et al.  IMP (Integrated Mechanisms Program), A Computer-Aided Design Analysis System for Mechanisms and Linkage , 1972 .

[8]  Horace M. Trent,et al.  Isomorphisms between Oriented Linear Graphs and Lumped Physical Systems , 1955 .

[9]  F.R.E. Crossley.  Über die Kräfte in Kombinationen von Gelenk-und Rädertrieben (On the forces which occur in Gear-Linkage-Combination Mechanisms): F. Heinrich Lehn: Dr. -Ingenieur Dissertation, Technische Hochschule Aachen, West Germany, July 1967. , 1970 .

[10]  Seizo Uematsu.  An application of graph theory to the kinematic analysis of planetary gear trains , 1996 .

[11]  Chao Chen.  Power Flow Analysis of Compound Epicyclic Gear Transmission: Simpson Gear Train , 2011 .

[12]  Ettore Pennestrì,et al.  A Systematic Approach to Power-Flow and Static-Force Analysis in Epicyclic Spur-Gear Trains , 1993 .

[13]  Nicola Pio Belfiore,et al.  A brief note on the concept of planarity for kinematic chains , 2000 .

[14]  Li Shujun,et al.  Topology of Kinematic Chains with Loops and Orientation of Joints Axes , 2009 .

[15]  Zhen Huang,et al.  Computer-aided structure decomposition theory of kinematic chains and its applications , 2008 .

[16]  Nicola Pio Belfiore,et al.  Distributed Databases for the development of Mechanisms Topology , 2000 .

[17]  B. Harrison.  Las Vegas, Nevada , 2002 .

[18]  W. T. Tutte.  A Census of Hamiltonian Polygons , 1962, Canadian Journal of Mathematics.

[19]  F.R.E. Crossley.  Nominations for new officers of IFToMM , 1979 .

[20]  D. R. Salgado,et al.  Analysis of the transmission ratio and efficiency ranges of the four-, five-, and six-link planetary gear trains , 2014 .

[21]  Hong-Sen Yan,et al.  Creative design of a wheel damping mechanism , 1985 .

[22]  J. R. Gomà Ayats,et al.  Hypergraphs for the analysis of complex mechanisms comprising planetary gear trains and other variable or fixed transmissions , 2012 .

[23]  Cheng-Ho Hsu.  A graph notation for the kinematic analysis of differential gear trains , 1992 .

[24]  Jorge Angeles,et al.  Virtual-Power Flow and Mechanical Gear-Mesh Power Losses of Epicyclic Gear Trains , 2007 .

[25]  Moez Mayourian.  The creation of mechanisms according to the separation of kinematic structure and function and its partial automation , 1985 .

[26]  Lung-Wen Tsai,et al.  Kinematic Analysis of Epicyclic-Type Transmission Mechanisms Using the Concept of Fundamental Geared Entities , 1996 .

[27]  Nicola Pio Belfiore,et al.  A method for the identification of the connectivity in multi-loop kinematic chains: Analysis of chains with total and partial mobility , 2006 .

[28]  Martín A. Pucheta,et al.  Automated sketching of non-fractionated kinematic chains , 2013 .

[29]  Wan-Chun Zhou,et al.  Topological analysis based on the kinematic chain of knitting machine needle guide agency creative design , 2010, 2010 International Conference on Mechanical and Electrical Technology.

[30]  Józef Wojnarowski,et al.  The application of signal flow graphs—the kinematic analysis of planetary gear trains , 1975 .

[31]  Ilie Talpasanu.  A General Method for Kinematic Analysis of Epicyclic Bevel Gear Trains Applied to Robotic Wrist Mechanisms , 2013 .

[32]  Bo Wu,et al.  The Design of Ternary Joint Robot Fingers without Drive Based on the Topological Graph of Kinematic Chain , 2010 .

[33]  Xian Zhi Li,et al.  Research on Kinematics for Planetary Bevel-Gear Trains , 2012 .

[34]  Marc J. Richard,et al.  The vector-network method for the modelling of mechanical systems , 1990 .

[35]  Chao Chen,et al.  Complete Efficiency Analysis of Epicyclic Gear Train With Two Degrees of Freedom , 2012 .

[36]  K. H. Hunt,et al.  Coefficient of Restitution Interpreted as Damping in Vibroimpact , 1975 .

[37]  N. I. Manolescu,et al.  For a united point of view in the study of the structural analysis of kinematic chains and mechanisms , 1968 .

[38]  Yong Liu.  NEW APPROACH TO AUTOMATIC SKETCHING OF PLANAR KINEMATIC CHAINS BASED ON LOOP CONFIGURATION AND IT'S COMPUTERIZATION , 2002 .

[39]  Ferdinand Freudenstein,et al.  The Mechanical Efficiency of Epicyclic Gear Trains , 1993 .

[40]  F. G. Umholtz,et al.  Design for a three-fingered hand , 1977 .

[41]  Donald R. Riley,et al.  An Algorithm for Automatic Sketching of Planar Kinematic Chains , 1985 .

[42]  B. Paul.  A Unified Criterion for the Degree of Constraint of Plane Kinematic Chains , 1960 .

[43]  Chao Chen,et al.  Theoretic study of efficiency of two-DOFs of epicyclic gear transmission via virtual power , 2011 .

[44]  Sundar Krishnamurty,et al.  A loop configuration approach to automatic sketching of mechanisms , 1996 .

[45]  Lung-Wen Tsai,et al.  Computer-Aided Sketching of Epicyclic-Type Automatic Transmission Gear Trains , 1996 .

[46]  Ettore Pennestrì.  On the kinematic analysis of geared robotic wrists , 1991 .

[47]  K. C. Gupta,et al.  Signal flow graphs for spatial gear trains , 1994 .

[48]  Donald R. Riley,et al.  Topological Analysis of Single-Degree-of-Freedom Planetary Gear Trains , 1991 .

[49]  J. Kopeć,et al.  Gears and graphs , 2006 .

[50]  D. Frederick,et al.  Developments in theoretical and applied mechanics , 1963 .

[51]  Hongzhao Liu,et al.  A Maximal Loop Approach to Automatic Sketching of Mechanisms , 2007 .

[52]  Norman K. Bucknor.  Automatic Determination of Transmission Powerflow Mechanizability Using Graph Theory , 2007 .

[53]  J. W. Polder.  A network theory for variable epicyclic gear trains , 1969 .

[54]  J. Drewniak,et al.  Comparison of graph-based methods of kinematical analysis of planetary gears , 2010 .

[55]  Ettore Pennestrì,et al.  Kinematic and Power-Flow Analysis of Bevel Gears Planetary Gear Trains with Gyroscopic Complexity , 2013 .

[56]  Arthur G. Erdman,et al.  Modern kinematics : developments in the last forty years , 1993 .

[57]  W. T. Tutte,et al.  A Census of Planar Triangulations , 1962, Canadian Journal of Mathematics.

[58]  Lin Woo.  An algorithm for straight-line representation of simple planar graphs , 1969 .

[59]  Daniel Dopico,et al.  Computational structural analysis of planar multibody systems with lower and higher kinematic pairs , 2014 .

[60]  W. T. Tutte.  A Census of Planar Maps , 1963, Canadian Journal of Mathematics.

[61]  Józef Wojnarowski.  The graph method of determining the loads in complex gear trains , 1976 .

[62]  Cheng-Ho Hsu.  Displacement isomorphism of planetary gear trains , 1994 .

[63]  A. Hedman.  Transmission Analysis—Automatic Derivation of Relationships , 1993 .

[64]  E. R. Maki,et al.  The Creation of Mechanisms According to Kinematic Structure and Function , 1979 .

[65]  Ettore Pennestrì,et al.  Efficiency Evaluation of Gearboxes for Parallel Hybrid Vehicles: Theory and Applications , 2012 .

[66]  Offer Shai,et al.  Extension of Graph Theory to the Duality Between Static Systems and Mechanisms , 2006 .

[67]  Zhen Huang,et al.  A new theory for the topological structure analysis of kinematic chains and its applications , 2007 .

[68]  Marc J. Richard,et al.  A general vector-network formulation for dynamic systems with kinematic constraints , 1988 .

[69]  G. C. Andrews,et al.  The vector-network model: A new approach to vector dynamics , 1975 .

[70]  E. Pennestrì,et al.  Characterization of Kinematic and Static Performances of Robotic Geared Wrists , 1994 .

[71]  John McPhee,et al.  Wittenburg's formulation of multibody dynamics equations from a graph-theoretic perspective , 1996 .

[72]  Goutam Chatterjee.  Enumeration and Automatic Sketching of Epicyclic-Type Automatic Transmission Gear Trains , 1993 .

[73]  Lung-Wen Tsai,et al.  Dynamic Analysis of Geared Robotic Mechanisms Using Graph Theory , 1998 .

[74]  J. Wittenburg,et al.  Dynamics of systems of rigid bodies , 1977 .

[75]  Stefan Staicu.  Inverse dynamics of a planetary gear train for robotics , 2008 .

[76]  F. R. E. Crossley,et al.  Multiple Impacts of a Ball Between Two Plates—Part 2: Mathematical Modelling , 1975 .

[77]  Ettore Pennestrì,et al.  Kinematic and static torque analysis of remotely actuated geared wrists , 1994 .

[78]  Nicola Pio Belfiore.  An Atlas of Remote Actuated Bevel Gear Wrist Mechanisms of up to Nine Links , 1993, Int. J. Robotics Res..

[79]  Nie Songhui.  Maximal Loop Method of Automatic Sketching of Planar Closed Kinematic Chains , 2009 .

[80]  F. R. E. Crossley.  A Contribution to Gruebler’s Theory in the Number Synthesis of Plane Mechanisms , 1964 .

[81]  Iulian Popescu,et al.  Kinematic chains with independent loops and spatial system groups , 2006 .

[82]  Jing Zhao,et al.  The establishment of edge-based loop algebra theory of kinematic chains and its applications , 2009, Engineering with Computers.

[83]  J. S. Rao,et al.  The mobility properties of kinematic chains , 1987 .

[84]  J. Drewniak,et al.  Kinematical and dynamical analysis of closed kinematic chains using graphs and contour equations , 2009 .

[85]  W Li,et al.  A Study on Determining Loops of Planar Kinematic Chains , 1994 .

[86]  Zhou Wan-chun,et al.  Innovative new-typed mechanism indicating flow by use of topological and structural characteristics and the graph of kinematic chains , 2010, 2010 International Conference On Computer Design and Applications.

[87]  Ettore Pennestrì,et al.  Kinematic design and optimization of automatic drive transmissions , 1995 .

[88]  Ferdinand Freudenstein,et al.  Some Applications of Graph Theory to the Structural Analysis of Mechanisms , 1967 .

[89]  Nicola Pio Belfiore,et al.  Connectivity and Redundancy in Spatial Robots , 2000, Int. J. Robotics Res..

[90]  Wei-Hua Chieng,et al.  A Combinatorial Approach for the Automatic Sketching of Planar Kinematic Chains and Epicyclic Gear Trains , 1988 .

[91]  Martín A. Pucheta,et al.  Combined Graph Layout Algorithms for Automated Sketching of Kinematic Chains , 2012 .

[92]  Amir Khajepour,et al.  A new multi-terminal pulley model for use in graph theoretic modelling , 2000 .

[93]  Iulian Popescu,et al.  The structure of planar mechanisms with dyads , 2005, SPIE Optics East.

[94]  Carl A. Nelson,et al.  Simplified kinematic analysis of bevel epicyclic gear trains with application to power-flow and efficiency analyses , 2005 .

[95]  Sundar Krishnamurty,et al.  Development of a standard code for colored graphs and its application to kinematic chains , 1994 .

[96]  Ettore Pennestrì,et al.  Automatic sketching of planar kinematic chains , 1994 .

[97]  Daniel Martins,et al.  Variety and connectivity in kinematic chains , 2008 .

[98]  Hong-Sen Yan,et al.  An improved algorithm for the construction of generalized kinematic chains , 2014 .

[99]  A. Srinath,et al.  Mechanical Efficiency of Planetary Gear Trains: An Estimate , 2011 .

[100]  H. Demirel,et al.  Kinematic analysis of bevel-gear trains using graphs , 2005 .

[101]  F. Freudenstein,et al.  On a theory for the type synthesis of mechanisms , 1966 .

[102]  L. W. Tsai.  An Application of Graph Theory to the Detection of Fundamental Circuits in Epicyclic Gear Trains , 1995 .

[103]  A. E. Samuel,et al.  Kinematic chains for robot hands—II. Kinematic constraints, classification, connectivity, and actuation , 1995 .

[104]  Huafeng Ding,et al.  THE TOPOLOGICAL GRAPHS CREATION AUTOMATICALLY OF KINEMATIC CHAINS AND ATLAS DATABASE ESTABLISHMENT , 2006 .

[105]  Z. Huang,et al.  The Novel Characteristic Representations of Kinematic Chains and Their Applications , 2005 .

[106]  T. S. Mruthyunjaya,et al.  Kinematic structure of mechanisms revisited , 2003 .

[107]  J. W. Polder.  Lumped Parameter Model of Planetary Gear Systems , 1978 .

[108]  Henrique Simas,et al.  Efficiency of gear trains determined using graph and screw theories , 2012 .

[109]  Chih Ching Hung,et al.  A procedure to count the number of planar mechanisms subject to design constraints from kinematic chains , 2008 .

[110]  Lung-Wen Tsai,et al.  The Selection of a Most Efficient Clutching Sequence Associated With Automatic Transmission Mechanisms , 1998 .

[111]  Ferdinand Freudenstein,et al.  An Application of Boolean Algebra to the Motion of Epicyclic Drives , 1971 .

[112]  G. C. Andrews,et al.  Simulating planar systems using a simplified vector-network method , 1975 .

[113]  Zongquan Deng,et al.  Adding sub-chain method for structural synthesis of planar closed kinematic chains , 2012 .

[114]  Moshe Shoham,et al.  Connectivity in open and closed loop robotic mechanisms , 1997 .

[115]  Ferdinand Freudenstein,et al.  Synthesis of kinematic structure of geared kinematic chains and other mechanisms , 1970 .

[116]  Y. T. Chiu,et al.  An algorithm for the construction of generalized kinematic chains , 2013 .

[117]  F. R. E. Crossley,et al.  Multiple Impacts of a Ball Between Two Plates—Part 1: Some Experimental Observations , 1975 .

[118]  Robert E. Tarjan,et al.  Efficient Planarity Testing , 1974, JACM.

[119]  H. K. Kesavan,et al.  Graph theoretic modeling and analysis of multibody planar mechanical systems , 2001, IEEE Trans. Syst. Man Cybern. Part A.

[120]  Sherman Y.T. Lang,et al.  Graph-theoretic modelling of epicyclic gear systems , 2005 .

[121]  Daniel Alazard,et al.  Kinematic Analysis of Spatial Geared Mechanisms , 2012 .

[122]  Cheng-Ho Hsu,et al.  A New Graph Representation for the Automatic Kinematic Analysis of Planetary Spur-Gear Trains , 1992 .

[123]  M. Uyguroğlu,et al.  Kinematic analysis of tendon-driven robotic mechanisms using oriented graphs , 2006 .

[124]  Ferdinand Freudenstein,et al.  Mechanics of Epicyclic Bevel-Gear Trains , 1973 .

[125]  Iulian Popescu,et al.  Structural design of planar mechanisms with dyads , 2008 .

[126]  Lung-Wen Tsai,et al.  Kinematic Analysis of Tendon-Driven Robotic Mechanisms Using Graph Theory. , 1989 .

[127]  P. A. Simionescu,et al.  Kinematic Analysis of Epicyclic Bevel Gear Trains With Matroid Method , 2012 .

[128]  L. Auslander,et al.  Incidence Matrices and Linear Graphs , 1959 .

[129]  Y. Tokad,et al.  Kinematic Analysis of Robotic Bevel-Gear Trains: An Application of Network Model Approach , 1998 .

[130]  Ming Chen.  Efficiency Sensitivity of Two-Stage Planetary Gear Transmission System Based on Graph Theory , 2012 .

[131]  Essam L. Esmail,et al.  An Approach to Power-Flow and Static Force Analysis in Multi-Input Multi-Output Epicyclic-Type Transmission Trains , 2010 .

[132]  Andrew Kusiak,et al.  A graph-theoretic approach to conceptual design with functional perspectives , 2000, Comput. Aided Des..

[133]  Artde D.K.T. Lam.  A Study on the Mechanical Efficiency of Power Split Unit of the Toyota Hybrid System , 2011 .

[134]  Jing Zhao,et al.  Unified structural synthesis of planar simple and multiple joint kinematic chains , 2010 .

[135]  V. P. Agrawal,et al.  Freedom Analysis of Planar Kinematic Chains Using the Concept of Distance , 1996 .

[136]  E. R. Maki,et al.  The Categorization Of Planetary Gear Trains For Automatic Transmissions According To Kinematic Topology , 1988 .

[137]  Giacomo Mantriota,et al.  Theoretical and Experimental Efficiency Analysis of Multi-Degrees-of-Freedom Epicyclic Gear Trains , 2003 .

[138]  Ferdinand Freudenstein,et al.  Kinematics and statics of a coupled epicyclic spur-gear train , 1972 .

[139]  Yong Liu,et al.  An Improved Approach to Automatic Sketching of Mechanism Based on Loop Configuration and Its Implementation , 2002 .

[140]  T. Davies.  Kirchhoff's circulation law applied to multi-loop kinematic chains , 1981 .

[141]  F. R. E. Crossley,et al.  Digital Simulation of Impact Phenomenon in Spur Gear Systems , 1977 .