Mathematical Model Design & Validation of Planer Fourbar Mechanism for Path Point Synthesis Problem using Genetic Optimization Technique

Mathematical Model Design & Validation of Planer Fourbar Mechanism for Path Point Synthesis Problem using Genetic Optimization Technique

Hitesh R. Patel1

Mechanical Engineering Department,

Gujarat Power Engineering and Research Institute, Mevad, Mehsana, Gujarat, India

Dr. V. K. Bhatt2

Mechanical Engineering Department, Indus University, Rancharda, Ahmedabad, Gujarat, India

AbstractAn analytical approach for path point generation of planer fourbar mechanism is illustrated in this work, Mathematical model of planer fourbar mechanism is designed based on loop closer and Freudensteins equation, here genetic optimization technique is used to find global optimum solution and validated with similar work carried out by different authors using different optimization technique, also illustrated how better solution can be obtained by modifying design space for the objective function. Here modification is carried out with the starting position of crank, it can start from any angular position with respect to fixed link of mechanism and also angular position for each trace point on desired path is left on algorithm to decide so that will further increase the search space for the algorithm.

KeywordsSynthesis of Mechanism; Path synthesis; four bar Mechanism; Genetic Algorithm.

1. INTRODUCTION

   In the synthesis of planer four bar mechanism. Three categories are there, namely function generation, path generation and body Guidance. In function generation, considered point on the coupler will generate definite function as an output when motion provided to the input link. In path Generation, Considered point on the coupler link will pass through predefines guide point or trace points as an output motion. And lastly in body guidance coupler link will pass through predefined point and orientation of the body.

   Firstly. Cabrera and et al [1] had used genetic optimization technique to find the optimum solution for synthesis problem of the four-bar mechanism. Here Grashofs criteria and sequence of the input angles taken as constraint. Also Kinzel and et al [2] had proposed geometric constrained programming approach for synthesis problem of planar fourbar mechanism and later on Acharaya & Mandal [3] employed particle swarm optimization (PSO) & differential evolution (DE) approach for synthesis of 4-bar linkages. Zadah and et al [4] used multi-objective genetic algorithms (GA) for pareto optimal synthesis of 4-bar linkages. Two objectives namely tracking error and transmission angle deviation from 90 degree is accounted. Erkaya and Uzmay

   [5] presented a joint clearance influence on path generation and transmission angle by adapting GA. Todorov [6] described a new dimensional synthesis method. The position function of the four-bar mechanism is presented by the Freudensteins equation and it is minimized by the

   Chebyshevs best approximation theory. Khare & Dave[7] developed a closed form equations for the synthesis of the 4-bar crank rocker mechanism in which the angle between dead-centre positions of the rocker and the corresponding angle turned by the crank are prescribed. Ahmed and Waldron

   1. outlined synthesis techniques for 4-bar linkages, having adjustable driven crank pivots, for different motion generation problems. The method of solution is analytical in nature, and, therefore well suited for use on a digital computer. Levitskii et al [9] considered the general problem of determining five parameters 17 specifying a for 4-bar linkages which synthesizes a given function and at the same time satisfies some limiting conditions. Hobson & Torfason [10] presented the design of mechanism, which approximate desired centrodes and the applied to two prosthetic knee mechanisms. Geem & Kim [11] presented a new structural optimization method based on the harmony search algorithm. Mahdavi, Fesanghary & Damangir [12] presented an improvised HS algorithm for solving optimization problem.

2. MATHEMATICAL MODEL

   Fig. 1. Configuration of fourbar mechanism

   In a four bar linkage synthesis problem need to trace number of desired points on a coupler curve generated by a mechanism. For this purpose configuration of four bar coupler mechanism shown in figure 1 is considered in which a, b, c, d, Lx, and Ly are the basic linkage dimensions used in mechanism, here XY is the global co-ordinate system and XrYr is the reference co-ordinate system on mechanism, and point P (Px, Py) is the coupler point on the mechanism.

   Here for tracing desired points by coupler point P need to evaluate the values of X0, Y0, 0, a, b, c, d, Lx, Ly and as many input angles (2) as Number of desired points given to be trace. For formulating mathematical model loop closure method and Freudensteins equation is used.

   The vector position of trace point P is defined as

   = + + (1)

   So the position of coupler point or trace point P with respect to reference coordinate system XrYr as

   = 0 + 0 0 (17)

3. GENETIC OPTIMIZATION ALGORITHM There is several global optimization techniques are available

   likewise stochastic optimization, genetic algorithm, neural networks, particle swarm optimization algorithm, Tabu search optimization, ant colony method etc. Here out of these algorithms genetic optimization algorithm is selected because of following reasons. It takes the encoding of decision variables as the operational objects. It takes the objective function directly as the searching information. It uses searching information of several searching points at the same time. It uses probability searching technology.

   In genetic search algorithm number of solution generated randomly is known as population and which is repetitively applied on objective function is known as iteration. In every iteration reproduction, crossover and mutation operation carried out for making new generation of population and that

   = 2 + 3 3 (2)

   newly generated population again applied on objective function, when objective criteria satisfied then this iterative

   = 2 + 3 + 3 (3) For evaluation of angle 3 vectors loop considered as

   + = 0 (4)

   So its X and Y components about reference axis XrYr will become zero

   acos 2 + 3 4 = 0 (5)

   asin 2 + 3 4 = 0 (6)

   According to Freudensteins equation with the use of above two equation eliminating angle 4 so we can obtain

   13 + 42 + 5 = cos(2 3) (7)

   procedure will stop and algorithm will gives a solution. Here in this work size of population is taken as 100, crossover fraction considered as 0.8 and mutation fraction considered as 0.1. At this values algorithm gives a better performance checked by author Hitesh Patel and J. R. Mevada [13] in their work based on genetic optimization technique.

4. OBJECTIVE FUNCTION AND CONSTRAINTS FOR OPTIMIZATION

   Here the main objective of this work is to minimize the error between the position of desired guide points (Pxdi,Pydi) and position of coupler or trace point (Pxi,Pyi) on the mechanism. The error function is considered as

   Where,

   1 =

   (8)

   () = [(

   =1

   2

   ) (

   )2]

   (18)

   4

   = (9)

   Where n is the number of desired points considered for synthesis problem.

   5 =

   2 2 2 2 2

   (10)

   Here mechanism needs to satisfy some constraint like Grashofs law for full rotation of input lin, also Input link a

   From above equation by known value of input angle 2 value of angle 3 can be evaluate as following.

   3

   1 = 2 tan1 ( + 2 4) (11)

   2

   2 = 2 tan1 ( 2 4) (12)

   need to make small so it can be work as a crank and continuous

   increment require in input angle In of crank for continuous one direction rotation so it can be possible to maintaining sequences of desired points to be trace by the mechanism.

   In mechanism requirement is that having one of the link will act as a crank and become an input link, so mechanism need to

   3

   Where,

   2

   satisfy Grashofs criterion. By Grashofs law summation of shortest link length (Ls) and longest link length (Ll) is less than the summation of remaining two links from fig. link O2A is

   = 2 1 + 42 + 5 (13)

   = 22 (14)

   = 1 + (4 1)2 + 5 (15)

   Now position of coupler point P with respect to global coordinate system as

   = 0 + 0 0 (16)

   considered as a shortest link so

   ( + ) < ( ) 2 ( + )

   < ( + ) + ( ) 2 ( + ) < ( + + + )

   For maintaining sequences of desired points to be trace by the mechanism constraint considered as following.

   2 2

   If values of i i1 < 0 then add penalty to the objective function.

   So the total number of Design variables and its boundary limits with this optimization problem are as following.

   Crossover

   TABLE I. BOUNDARY LIMITS OF DESIGN VARIABLE.

   START

   t = t + 1

   Initialize population for Error function input

   Error Function

   t = 0

   Mutation

   Sr.

   Description Symbol and

   No. of Design

   No. boundary limits Variable.

   1. Links Length 0.05 a, b, c, d 5 4

      Reproduction

      Coordinates of reference coordinate system with

   2. respect to global coordinate system,

      Coupler Point position on

      -5 X0, Y0 5 2

      Termination

      Condition NO

   3. connecting rod. -5 LX, LY 5 2

      Rotation of reference coordinate system with

   4. respect to global -180 0 180 1

      coordinate system

      Equal to

      YES

      End

5. RESULTS AND DISCUSSION

   1. Input crank angles -360 2 540

   number of

   Here, firstly 18 points considered to trace by coupler point

   desired points

   In below figure shown a Flow chart for error function which is used to calculate error and this error function is utilized in genetic algorithm for find out optimum solution.

   firstly proposed by Kunjur and Krishnamurthy (KK) [14] to compare the performance of genetic algorithm with central difference and exact gradient method also Cabrera et al. [1] had modified operators and tested same problem with genetic algorithm, also same problem is discussed by A. Smaili, N. Diab [15] had discussed same problem using ACO algorithm, all the authors have taken fixed values of input angle, here in this solution input angle values were to decide by algorithm itself, also constraints discussed above are applied to the algorithm and then obtained results are compared with existing work.

   Error function Start

   Defining Design Variable for input of the function

   If

   (E2-4DF) < 0

   Error function end

   Error Function

   TABLE II. DESIRED 18 POINTS [14]

   Desired

   1 2 3 4 5 6 7 8 9

   Evaluate Variable K1, K4, K5, D, E, F

   point i Xdi 0.5 0.4 0.3 0.2 0.1 0.05 0.02 0 0

   Ydi 1.1 1.1 1.1 1 0.9 0.75 0.6 0.5 0.4

   	10	11	12 13 14	15	16	17 18
   Xdi	0.03	0.1	0.15 0.2 0.3	0.4	0.5	0.6 0.6

   Take Error = 1E10

   Using Mathematical Model Evaluate Coupler coordinates PX, PY

   Ydi 0.3 0.25 0.2 0.3 0.4 0.5 0.7 0.9 1

   Calculate Error between Desired path point and Coupler points using equation 18.

   Fig. 2. Coupler curve generated by ACO Optimal Mechanism

   [15]

   Fig. 3. Coupler curve obtained by GA in current work

   after applying Genetic algorithm obtained results are compared with the results of A. Smaili, N. Diab [15] comparative results

   Desired

   TABLE V. DESIRED 25 POINTS [16]

   graph was shown in figure 2 & 3, also the configuration for obtained mechanism is shown in below figure 4,

   Fig. 4. Obtained Mechansim configuration for 18 desired points

   For above configuration of mechanism the linkage parameters are tabulated in below table III,

   TABLE III. OBTAINED PARAMETERS OF 18 POINTS PATH

   point i 1 2 3 4 5 6 7 8 9

   Xdi 7.03 6.95 6.77 6.4 5.91 5.43 4.93 4.93 4.67

   Ydi 5.99 5.45 5.03 4.6 4.03 3.56 2.94 2.6 2.2

   	10	11	12 13 14 15 16 17	18
   Xdi	4.39	4.04	3.76 3.76 3.76 3.76 3.76 3.76	3.8
   Ydi 1.67 1.22 1.97 2.78 3.56 4.34 4.91 5.47 5.98
   	19	20	21 22 23 24 25
   Xdi	4.07	4.53	5.07 5.05 5.89 6.41 6.92

   Ydi 6.4 6.75 6.85 6.84 6.83 6.8 6.58

   After applying genetic algorithm, results obtained results are compared with the same work carried out by author A. Smaili,

   N. Diab [15] as shown in figure 5 & 6. And the configuration for the obtained mechanism is shown in below figure 7. And the dimensional parameters for the linkage of mechanism is tabulated in below table VI.

   Variable	Value	Variable	Value	Variable	Value
   X0	1.201835	2 – 1	39.10128	2 10	211.0341
   Y0	-1.35544	2 2	57.2049	2 11	237.6793
   Lx	0.904845	2 3	73.09223	2 12	254.0793
   Ly	2.060674	2 4	94.02072	2 13	273.3342
   a	0.34529	2 5	114.6813	2 14	292.9728
   b	3.057001	2 6	137.6361	2 15	307.8849
   c	0.497724	2 7	159.6725	2 16	329.3006
   d	2.983992	2 8	174.8861	2 17	353.1682
   0	40.71874	2 – 9	191.0274	2 18	368.4727

   Fig. 5. Coupler curve generated by ACO Optimal

   Mechanism [15]

   Fig. 6. Coupler curve obtained by GA in current work

   Here, error obtained in this problem is compared with same work carried out by different authors using different optimization techniques are tabulated in below table IV, error generated in this work is comparatively very lower as compared to the work of other authors.

   TABLE IV. COMPARISON OF OBTAINED RESULTS

   Variable	Value	Variable	Value	Variable	Value
   X0	-10.0758	2 4	71.68049	2 16	259.7496
   Y0	-1.51267	2 5	88.31685	2 17	270.6838
   Lx	16.38188	2 6	102.1246	2 18	281.0881
   Ly	-1.46501	2 7	117.8924	2 19	292.1885
   a	2.273927	2 8	126.6046	2 20	305.7724
   b	9.578672	2 – 9	137.0034	2 21	319.7508
   c	4.516432	2 10	152.2369	2 22	330.7251
   d	11.38357	2 11	180.744	2 23	340.1309
   0	5.786575	2 12	205.8736	2 24	352.1487
   2 – 1	21.32654	2 13	221.4302	2 25	365.061
   2 2	42.6952	2 14	235.3666
   2 3	56.78272	2 15	249.0915

   Optimization algorithm	No. of Evaluation	Final Error
   Exact Gradient [14]	240	0.0168
   GA KK [14]	5000	0.043	Fig. 7. Obtained Mechansim configuration for 18 desired points
   GA CSP [1]	5000	0.0245	TABLE VI. OBTAINED PARAMETERS OF 25 POINTS PATH
   Tabu-gradient [15]	550	0.0137
   Ant [15]	50000	0.0526
   Ant colony [15]	479	0.0109
   GA (Current work)	5000	0.0035
   One more example is considered here to trace 25 points by 25 desired points are tabulated in below table 5, same example is considered by Mc Garva[16] which is used for a packaging machinery. And later on it is utilized by Laribi and et al [17] by

   coupler point of four bar planer mechanism, the position of all

   using genetic algorithm.

   TABLE VII. COMPARISON OF OBTAINED RESULTS

   Optimization Algorithm Error

   GA FL [17] 0.902

   Ant [15] 3.56

   Ant-gradient [15] 0.682

   GA (current work) 0.1656

   Here comparative study being carried out based on mathematical modeling and optimization algorithm, Here using mathematical modeling in the program, change is considered like crank of four bar mechanism can start from any angular position with respect to fixed link so for tracing a path, restriction on the crank to start always from 0o initial position with respect to crank is removed and also the values of angular position of crank for every desired tracing points is to be left on the algorithm to decide so the solution space for the algorithm is increased and the algorithm can find more better solution. Due to this reasons in the obtained results you can see within a same number of generation, algorithm can find better solutions as compared to the results obtained for the same examples by different authors, comparative results are shown in above two examples. In example for 18 nos of desired points obtained results are shown in above figure 2 & 3 , also error is reduced to much more level as compared to the work being carried out by other authors. And the configuration of mechanism for the same is shown in figure – 4. Same as also in another example of 25 desired points work being carried and the obtained results are plotted in figure 5 & 6, also comparative results in the terms of error is shown in table VII, the configuration for obtained mechanism is shown in figure – 7.

6. CONCLUSION

   Genetic algorithm is a global optimum search technique and can find a solution from the available search field, by considering constraints to the design variable participated in formulation of objective functions, as search space increased there is a more chances to find better solution. Here in above discussed examples restriction to the crank is to start angular displacements always with respect to fixed link and also fixed angular position for each desired trace point is removed so the design space for search algorithm is increased and also chances to obtain better and better results are increased, in above discussed examples same fundamental is applied to synthesis a mechanism for a given path which can pass through given number of desired points and obtained a better results as compared with work being carried out by different authors.

7. REFERENCES

1. J.A .Carbrea. A. Simon & M.Prado, Optimal synthesis of mechanism with GA, Mechanism & Machine Theory, Vol.37, pp 1165-1177, 2002.

2. E.C.Kinzel,J.P.Schniedeler & G.R.Pennock, Kinetic synthesis for finitely separated positive any geometric coordinate programming, J.

   Machine design, Trans.ASME, Vol.128,pp.1070-1079,2006

3. S.K.Acharyya and M.Mandal, Performance of EAs for four-bar linkage synthesis, Mechanism & Machine Theory, vol. 44, pp.1784-1794,2009.

4. N.N.Zadah, M.Fklezi,A.Jamalize M.Garj, Pareto optimal synthesis of four-bar mechanism for path generation, Mechanism & Machine Theory, Vol.44, pp.180-191, 2009.

5. S.Erkaya & I.Uzmay, Determining parameters using GA in mechanism with joint clearance, Mechanism & Machine Theory, Vol.44, pp.222-234, 2009.

6. T.S.Todorov, Synthesis of four-bar mechanisms by Freudenstein Chebyshev theory, Mechanism and Machine Theory, Volume 37,Pages 1505-1512,2002.

7. A.K. Khare and R. K. Dave, Optimizing 4-bar crank-rocker mechanism, Mechanism and Machine Theory, Volume 14, Pages 319-325,1979.

8. A.Ahmad and K.J.Waldron, Synthesis of adjustable planar 4bar mechanisms, Mechanism and Machine Theory, Volume 14, pp.405- 411,1979.

9. N.I.Levitskii, Y.L.Sarkissyan and G.S.Gekchian, Optimum synthesis of for 4-bar function generating, Mechanism and Machine Theory, Volume 7, Pages 387-398,1972.

10. D.A. Hobson and L.E. Torfason, optimization of four-bar knee mechanisms A computerized approach, Journal of Biomechanics, Volume 7, Pages 371-376,1974.

11. Z.W.Geem & J.H.kim, new heurisrc optimization algorithm H , imulation, vol.76, pp.60-68, 2001.

12. M.Mahdavi, M.Fesanghary & E.Damangir, An improved HS algorithm for solving optimization problem, Applied Mathematical computation, Vol.188, pp-1567-1579, 2007.

13. Hitesh R. Patel, J. R. Mevada, Shape Control and Optimization using Cantilever Beam, International Journal of Engineering Research and Application, Vol. 3, Issue 3, May-June 2013, pp 155-161.

14. S. Kunjur, S. Krishnamurty, Genetic algorithms in mechanism synthesis,

    J. App. Mech. Rob. 4 (2) (1997) 1824

15. A. Smaili, N. Diab, Optimum synthesis of hybrid-task mechanisms using ant-gradient search method, Mechanism and Machine Theory 42 (2007) 115130.

16. J.R. McGarva, Rapid search and selection of path generating mechanism from a library, Mech. Mach.Theory 29 (2) (1994) 223235.

17. M.A. Laribi, A. Mlika, L. Romdhane, S. Zeghloul, A combined genetic algorithm-fuzzy logic method (GA-FL) in mechanism synthesis, Mech. Mach. Theory 39 (2004) 717735.