The classical method to find possible solutions for path synthesis problem of planar mechanisms is continuation method. However, this method has some disadvantages such as producing unwanted extraneous and degenerate solutions and also producing mechanisms having defects. Moreover, many of the solutions are cognate of each other which can be obtained geometrically. Thus, finding the most feasible solution among all solutions is a cumbersome and time-consuming task. The main purpose of this paper is to explore applicability of heuristic algorithms to find multiple cognate- and defect-free solutions for path synthesis problems of planar four-bar and slider-crank mechanisms. To this aim, a new modified error function and an optimization-based algorithm is presented. The gravitational search algorithm (GSA) is utilized to minimize the modified error function. Efficiency of the method is proved through five case studies for path synthesis of four-bar and slider-crank mechanisms with and without prescribed timing.
HronesJANelsonGL.Analysis of the four bar linkage. New York: MIT Press and Wiley, 1951.
4.
ShigleyJUickerJ.Theory of machines and mechanisms., McGraw-Hill, New York, 1980.
5.
FreudensteinF.An analytical approach to the design of four-link mechanisms. Trans ASME1954; 76: 483–492.
6.
WamplerCWMorganAPSommeseAJ.Complete solution of the nine-point path synthesis problem for four-bar linkages. ASME J Mech Des1992; 114: 153–159.
7.
DhingraAKChengJCKohliD.Synthesis of six-link, slider-crank and four-link mechanisms for function, path and motion generation using homotopy with m-homogenization. ASME J Mech Des1994; 116: 1122–1131.
8.
TariHSuH-J.Complete solution to the eight-point path generation of slider-crank four-bar linkages. ASME J Mech Des2010; 132: 081003–081001.
9.
AngelesJAlivizatosAAkhrasR.An unconstrained nonlinear least-square method of optimization of RRRR planar path generators. Mech Mach Theory1988; 23: 343–353.
10.
KrishnamurtySTurcicDA.Optimal synthesis of mechanisms using nonlinear goal programming techniques. Mech Mach Theory1992; 27: 599–612.
11.
SancibrianRViaderoFGarcı́aP, et al. Gradient based optimization of path synthesis problems in planar mechanisms. Mech Mach Theory2004; 39: 839–856.
12.
ParadisMJWillmertKD.Optimal mechanism design using the gauss constrained method. J Mech Trans Autom1983; 105: 187–196.
HansenJM.Synthesis of mechanisms using time-varying dimensions. Multibody Syst Dyn2002; 7: 127–144.
16.
SancibrianRSarabiaEGSedanoA, et al. A general method for the optimal synthesis of mechanisms using prescribed instant center positions. Appl Math Model2016; 40: 2206–2222.
17.
CabreraJASimonAPradoM.Optimal synthesis of mechanisms with genetic algorithms. Mech Mach Theory2002; 37: 1165–1177.
18.
CabreraJAOrtizANadalF, et al. An evolutionary algorithm for path synthesis of mechanisms. Mech Mach Theory2011; 46: 127–141.
19.
Martínez-AlfaroH. Four-bar mechanism synthesis for n desired path points using simulated annealing. In: Siarry P and Michalewicz Z (eds) Advances in meta-heuristics for hard optimization. Berlin: Springer, 2008, pp.23–37.
20.
AcharyyaSKMandalM.Performance of EAS for four-bar linkage synthesis. Mech Mach Theory2009; 44: 784–1794.
21.
LinWYHsiaoKM.A new differential evolution algorithm with a combined mutation strategy for optimum synthesis of path-generating four-bar mechanisms. Proc IMechE, Part C: J Mechanical Engineering Science2017; 231: 2690–2705.
22.
BulatovićRRĐorđevićSRĐorđevićVS.Cuckoo search algorithm: a metaheuristic approach to solving the problem of optimum synthesis of a six-bar double dwell linkage. Mech Mach Theory2013; 61: 1–13.
23.
EbrahimiSPayvandyP.Efficient constrained synthesis of path generating four-bar mechanisms based on the heuristic optimization algorithms. Mech Mach Theory2015; 85: 189–204.
24.
LaribiMAMlikaARomdhaneL, et al. A combined genetic algorithm–fuzzy logic method (GA–FL) in mechanisms synthesis. Mech Mach Theory2004; 39: 717–735.
25.
LinWY.A GA–DE hybrid evolutionary algorithm for path synthesis of four-bar linkage. Mech Mach Theory2010; 45: 1096–1107.
26.
SmailiAADiabNAAtallahNA.Optimum synthesis of mechanisms using tabu-gradient search algorithm. J Mech Des2005; 127: 917–923.
27.
SmailiADiabN.Optimum synthesis of hybrid-task mechanisms using ant-gradient search method. Mech Mach Theory2007; 42: 115–130.
28.
KhorshidiMSoheilypourMPeyroM, et al. Optimal design of four-bar mechanisms using a hybrid multi-objective GA with adaptive local search. Mech Mach Theory2011; 46: 1453–1465.
29.
BulatovićRRMiodragovićGBoškovićMS.Modified krill herd (MKH) algorithm and its application in dimensional synthesis of a four-bar linkage. Mech Mach Theory2016; 95: 1–21.
30.
ZangwillWIGarciaCB.Pathways to solutions,fixed points, and equilibria. Englewood Cliffs, NJ: Prentice Hall, 1981.
31.
RheinboldtWC.Numerical analysis of parametrized nonlinear equations. New York: John Wiley, 1986.
32.
MorganAP.Solving polynomial systems using continuation for scientific and engineering problems. Englewood Cliffs, NJ: Prentice Hall.
33.
MorganAPSommeseAJ.Computing all solutions to polynomial systems using homotopy continuation. Appl Math Comput1987; 24: 115–138.
34.
TsaiLWMorganAP.Solving the kinematics of the most general six- and five-degree of freedom manipulators by continuation methods. ASME J Mech Transm Autom Des1985; 107: 189–200.
35.
WamplerCWMorganAPSommeseAJ.Numerical continuation methods for solving polynomial systems arising in kinematics. ASME J Mech Des1990; 112: 59–68.
36.
RaghavanM.The Stewart platform of general geometry has 40 configurations. ASME J Mech Des1993; 115: 277–282.
37.
VarediSMDanialiHMGanjiDD.Kinematics of an offset 3-UPU translational parallel manipulator by the homotopy continuation method. Nonlinear Anal: Real World Appl2009; 10: 1767–1774.
38.
Tari H, Su JJ and Li TY.A constrained homotopy technique for excluding unwanted solutions from polynomial equations arising in kinematics problems. Mech Mach Theory2010; 45: 898–910.
39.
VerstratenE. Cognate linkages the Roberts–Chebyshev theorem. In: Explorations in the history of machines and mechanisms. Netherlands: Springer, 2012, pp.505–519.
40.
RashediENezamabadi-PourHSaryazdiS.GSA: a gravitational search algorithm. Inform Sci2009; 179: 2232–2248.
41.
SuZWangH.A novel robust hybrid gravitational search algorithm for reusable launch vehicle approach and landing trajectory optimization. Neurocomputing2015; 162: 116–127.
42.
Derafshi BeigvandSAbdiHScalaML.Combined heat and power economic dispatch problem using gravitational search algorithm. Electric Power Syst Res2016; 133: 160–172.
43.
SinghADeepK.Reconstruction of 3D curves and surfaces using new variants of gravitational search algorithm. J Inform Optim Sci2018; 39: 1199–1221.
44.
BansalJCJoshiSKNagarAK.Fitness varying gravitational constant in GSA. Appl Intell2018; 48: 3446–3461.
45.
FiglioliniGConteMReaP.Algebraic algorithm for the kinematic analysis of slider-crank/rocker mechanisms. J Mech Robot2012; 4: 011003–011001.
46.
NadalFCabreraJABatallerA, et al. Turning functions in optimal synthesis of mechanisms. J Mech Des2015; 137: 062302.
47.
BuśkiewiczJ.A method for optimal path synthesis of four-link planar mechanisms. Inverse Prob Sci Eng2015; 23: 818–850.
