Abstract
This study aims to make approximate motion generation that has multiple solutions to facilitate the selection of the best mechanism parameters. Therefore, an approximate synthesis method for an epicyclic gear train (EGT) with noncircular gears based on optimization–homotopy algorithm is proposed in this paper. First, the multipose synthesis objective function of a simplified model of EGT (planar RR dyad) is established. The approximate motion generation is transformed into the extremum problem of multivariate function, and a variable is introduced to expand the solution space. A homotopy algorithm is used to solve the gradient equation of the objective function to obtain the finite range solution domain. Second, the feasible solution domain of EGT is obtained by identifying the total rotation of revolute joints of the RR dyad, and a mechanism error evaluation index is established. In accordance with actual constraints, the most suitable solution is selected in the feasible solution domain, and the relative angular displacement relationship of the selected RR dyad is used to determine the transmission ratio and pitch curve of noncircular gears to realize the design of an EGT. Lastly, a design example is provided to illustrate the feasibility of the method.
Issue Section:
Research Papers
References
1.
Erdman
, A. G. S. G. N.
, 1997
, Mechanism Design: Analysis and Synthesis (Vol. 2)
, Prentice-Hall, Inc.
, Hoboken, NJ
.2.
Suh
, C. H. R. C. W.
, 1978
, Kinematics and Mechanisms Design
, John Wiley
, New York
.3.
Mccarthy
, J. M.
, and Soh
, G. S.
, 2011
, Geometric Design of Linkages
, Springer
, New York
.4.
Burmester
, L.
, 1888
, Lehrbuch Der Kinematik
, Verlag Von Arthur Felix
, Leipzig, Germany
.5.
Freudenstein
, F.
, 1954
, “An Analytical Approach to the Design of Four-Link Mechanisms
,” Trans. ASME
, 76
(3
), pp. 483
–492
.6.
Ge
, Q. J.
, Zhao
, P.
, and Purwar
, A.
, 2013
, Advances in Mechanisms, Robotics and Design
, V.
Kumar
, J.
Schmiedeler
, S.
Sreenivasan
, and H. J.
Su
, eds., Springer
, Heidelberg
.7.
Bai
, S.
, Wang
, D.
, and Dong
, H.
, 2016
, “A Unified Formulation for Dimensional Synthesis of Stephenson Linkages
,” ASME J. Mech. Rob.
, 8
(4
), p. 041009
. 8.
Mirth
, J. A.
, and Chase
, T. R.
, 1995
, “Circuit Rectification for Four Precision Position Synthesis of Four-Bar and Watt Six-Bar Linkages
,” ASME J. Mech. Des.
, 117
(4
), pp. 612
–619
. 9.
Han
, J.
, and Qian
, W.
, 2009
, “On the Solution of Region-Based Planar Four-Bar Motion Generation
,” Mech. Mach. Theory
, 44
(2
), pp. 457
–465
. 10.
Han
, J.
, and Liu
, W.
, 2019
, “On the Solution of Eight-Precision-Point Path Synthesis of Planar Four-Bar Mechanisms Based on the Solution Region Methodology
,” ASME J. Mech. Rob.
, 11
(6
), p. 064504
. 11.
Liu
, H.
, and Han
, J.
, 2021
, “Solution Region Synthesis Methodology of Spatial 1cs-4ss Linkages for Six Given Positions
,” Mech. Mach. Theory
, 162
(9
), p. 104369
. 12.
Han
, J.
, and Cao
, Y.
, 2019
, “Analytical Synthesis Methodology of Rccc Linkages for the Specified Four Poses
,” Mech. Mach. Theory
, 133
(3
), pp. 531
–544
. 13.
Alizade
, R.
, Can
, F. C.
, and Kilit
, Ö
, 2013
, “Least Square Approximate Motion Generation Synthesis of Spherical Linkages by Using Chebyshev and Equal Spacing
,” Mech. Mach. Theory
, 61
(61
), pp. 123
–135
. 14.
Nariman-Zadeh
, N.
, Felezi
, M.
, Jamali
, A.
, and Ganji
, M.
, 2009
, “Pareto Optimal Synthesis of Four-Bar Mechanisms for Path Generation
,” Mech. Mach. Theory
, 44
(1
), pp. 180
–191
. 15.
Kim
, J. W.
, Seo
, T. W.
, and Kim
, J.
, 2016
, “A New Design Methodology for Four-Bar Linkage Mechanisms Based on Derivations of Coupler Curve
,” Mech. Mach. Theory
, 100
(6
), pp. 138
–154
. 16.
Cabrera
, J. A.
, Simon
, A.
, and Prado
, M.
, 2002
, “Optimal Synthesis of Mechanisms With Genetic Algorithms
,” Mech. Mach. Theory
, 37
(10
), pp. 1165
–1177
. 17.
Russell
, K. S. Q.
, and Sodhi
, R. S.
, 2013
, Mechanism Design: Visual and Programmable Approaches
, CRC Press
, New York
.18.
Ravani
, B.
, and Roth
, B.
, 1983
, “Motion Synthesis Using Kinematic Mappings
,” ASME. J. Mech. Trans. Autom.
, 105
(3
), pp. 460
–467
. 19.
Zhao
, P.
, Ge
, X.
, Zi
, B.
, and Ge
, Q. J.
, 2016
, “Planar Linkage Synthesis for Mixed Exact and Approximated Motion Realization Via Kinematic Mapping
,” ASME J. Mech. Rob.
, 8
(5
), p. 051004
. 20.
Ge
, Q. J.
, Purwar
, A.
, Zhao
, P.
, and Deshpande
, S.
, 2017
, “A Task Driven Approach to Unified Synthesis of Planar Four-Bar Linkages Using Algebraic Fitting of a Pencil of G-Manifolds
,” ASME J. Comput. Inf. Sci. Eng.
, 17
(3
), p. 031011
. 21.
Deshpande
, S.
, and Purwar
, A.
, 2017
, “A Task-Driven Approach to Optimal Synthesis of Planar Four-Bar Linkages for Extended Burmester Problem
,” ASME J. Mech. Rob.
, 9
(6
), p. 061005
. 22.
Guan
, Y.
, and Verschelde
, J.
, 2008
, “Phclab: A Matlab/Octave Interface to Phcpack
,” IMA Volumes in Mathematics And Its Applications
, 148
(1
), pp. 15
–32
.23.
Lee
, E.
, and Mavroidis
, C.
, 2004
, “Geometric Design of 3r Robot Manipulators for Reaching Four End-Effector Spatial Poses
,” Int. J. Rob. Research
, 23
(3
), pp. 247
–254
. 24.
Wampler
, C. W.
, and Sommese
, A. J.
, 2011
, “Numerical Algebraic Geometry and Algebraic Kinematics
,” Acta Numer.
, 20
(1
), pp. 469
–567
. 25.
Liu
, A. X.
, and Yang
, T. L.
, 1999
, “Finding All Solutions to Unconstrained Nonlinear Optimization for Approximate Synthesis of Planar Linkages Using Continuation Method
,” ASME J. Mech. Des.
, 121
(3
), pp. 368
–374
. 26.
Yu
, G.
, Tong
, Z.
, Sun
, L.
, Tong
, J.
, and Zhao
, X.
, 2019
, “Novel Gear Transmission Mechanism With Twice Unequal Amplitude Transmission Ratio
,” ASME J. Mech. Des.
, 141
(9
), p. 092304
. 27.
Alexandru
, P.
, Macaveiu
, D.
, and Alexandru
, C.
, 2012
, “A Gear With Translational Wheel for a Variable Transmission Ratio and Applications to Steering Box
,” Mech. Mach. Theory
, 52
(6
), pp. 267
–276
. 28.
Xu
, G.
, Chen
, J.
, and Zhao
, H.
, 2018
, “Numerical Calculation and Experiment of Coupled Dynamics of the Differential Velocity Vane Pump Driven by the Hybrid Higher-Order Fourier Non-Circular Gears
,” J. Thermal Sci.
, 27
(3
), pp. 285
–293
. 29.
Mundo
, D.
, 2006
, “Geometric Design of a Planetary Gear Train With Non-Circular Gears
,” Mech. Mach. Theory
, 41
(4
), pp. 456
–472
. 30.
Guo
, L. S.
, and Zhang
, W. J.
, 2001
, “Kinematic Analysis of a Rice Transplanting Mechanism With Eccentric Planetary Gear Trains
,” Mech. Mach. Theory
, 36
(11
), pp. 1175
–1188
. 31.
Wang
, L.
, Liang
, S.
, Huang
, H.
, Yu
, Y.
, and Yu
, G.
, 2021
, “Design of Clamping-Pot-Type Planetary Gear Train Transplanting Mechanism for Rice Wide–Narrow-Row Planting
,” Int. J. Agric. Biol. Eng.
, 14
(2
), pp. 62
–71
. 32.
Zhao
, X.
, Chu
, M.
, Ma
, X.
, Dai
, L.
, and Chen
, J.
, 2018
, “Research on Design Method of Non-Circular Planetary Gear Train Transplanting Mechanism Based on Precise Poses and Trajectory Optimization
,” Adv. Mech. Eng.
, 10
(11–12
), pp. 1
–12
.33.
Sun
, L.
, Wang
, Z.
, Wu
, C.
, and Zhang
, G.
, 2020
, “Novel Approach for Planetary Gear Train Dimensional Synthesis Through Kinematic Mapping
,” Proc. Inst. Mech. Eng., Part C
, 234
(1
), pp. 273
–288
. 34.
Ye
, J.
, Zhao
, X.
, Wang
, Y.
, Sun
, X. C.
, and Xia
, X. D.
, 2019
, “A Novel Planar Motion Generation Method Based on the Synthesis of Planetary Gear Train With Noncircular Gears
,” J. Mech. Sci. Technol.
, 33
(10
), pp. 4939
–4949
. 35.
Tsai
, L. W.
, 2000
, “Mechanism Design: Enumeration of Kinematic Structures According to Function
,” ASME Appl. Mech. Rev.
, 122
(4
), pp. B85
–B86
. 36.
Uyar
, K.
, and Olker
, E.
, 2017
, “B-Spline Curve Fitting With Invasive Weed Optimization
,” Appl. Math. Model.
, 52
(12
), pp. 320
–340
. 37.
Zhou
, M.
, Sun
, L.
, Du
, X.
, Zhao
, Y.
, and Xin
, L.
, 2016
, “Optimal Design and Experiment of Rice Pot Seedling Transplanting Mechanism With Planetary Bezier Gears
,” Trans. ASABE
, 57
(6
), pp. 1537
–1548
. 38.
Bai
, S.
, Li
, Z.
, and Li
, R.
, 2020
, “Exact Synthesis and Input–Output Analysis of 1-Dof Planar Linkages for Visiting 10 Poses
,” Mech. Mach. Theory
, 143
(1
), p. 103625
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