**4. Study cases**

50 Bio-Inspired Computational Algorithms and Their Applications

determination of control parameters, experimenting with different values and selecting those that gave better results. (De Jong, 1975) recommended, after experimenting, values for the probability of the interbreeding of simple point and the movement of a bit in the mutation. In this work, the following parameters are defined: a population-based measure of 50 individuals, probability of crossing 0.6, probability of mutation of 0.001 and elitist selection; however, it presents the disadvantage that these parameters only worked for a

(De Jong, 1975) described that the operation on-line is based on the monitoring of the best solution in every generation, while the operation off-line takes into account all the solutions in the population to obtain the optimum value. (Grefenstette, 1986) used the metaalgorithms as a method of optimization, in order to obtain values with similar parameters

In order to have a good performance on-line of a search algorithm, it must quickly decide where the most promising search region is and concentrate their efforts there. The performance off-line does not always penalize the search algorithm to explore poor regions of the search space, since this will contribute to achieving the best possible solutions (in terms of fitness). The best sets of parameters analyzed on and off- line were population of 30 and 80 individuals, probability of crossing 0.95 and 0.45, probability of mutation 0.01 for both, either using a strategy of elitist selection for the on-line case or not elitist for the off-line case .

(Smith, 1993) proposes a genetic algorithm which adjusts the extent of the population taking into account the likelihood of error. This is linked with the number of generations, if under the conditions of little use is determined a small value (20 to 50) to the number of

(Endre Eiben et al., 1999) expose technical drawbacks of the analysis of parameters on the

• Parameters are not independent, but trying all possible combinations of these

• The process of tuning parameters is time-consuming, but if the parameters are

• For a given problem, the values for the selected parameters are not necessarily the best, but if they are used to analyze uniformly, more meaningful values will be obtained. In general, here are listed some important observations made by authors such as (Holland, 1975),with respect to the genetic algorithms that must be considered for the use of this tool,

• A high generational interval and the use of an elitist strategy also improve the

• The use of large populations *(> 200)* with a high percentage of mutation *(> 0.05)* does

• The use of small populations *(< 20)* with low percentage of mutation *(< 0.002)* does not

• The mutation seems to have greater importance in the performance of a GA. • If the size of the population is increased, the effect of crosses seems to be diluted.

evaluations, the convergence will be quick, but it is not ensured an optimum result.

optimized one by one, it is possible to handle their interactions.

particular problem with very specific restrictions.

for the operation on-line and off-line of the algorithm.

basis of experimentation, observing the following points:

systematically it is almost impossible.

not improve the performance of a GA.

improve the performance of a GA.

performance of the GA.

such as:

The case of study is based on mechanisms synthesis, for that reason the basic concepts are presented.

A mechanism is a set of rigid members that are jointed together in order to develop a specific function. The mechanisms design, which is described by (Varbanov et al., 2006), consists of two parts: the analysis and synthesis. The first one consists of techniques to determine position, velocities and accelerations of points onto the members of mechanisms and the angular position, velocities and accelerations of those members. The second type explains the determination of the optimal length of the bars and the spatial disposition that best reproduces the desired movement of the coupler link. The optimal dimensional synthesis problem of mechanisms can be seen as a minimization process, since it is required that the structural error being as small as possible. The point of the coupler link will have to be able to generate a trajectory defined through separate points, with a minimum error. The

Performance of Simple Genetic Algorithm

**4.1 Optimal design in the mechanisms synthesis** 

presented original version in the works of (Goldberg, 1989), which is:

*N ni*

<sup>1</sup> <sup>1</sup> \*

*F*

of Grashof.

defined as:

reaching a solution.

**four- bar mechanism** 

=

*i*

=

Inserting Forced Inheritance Mechanism and Parameters Relaxation 53

The formulation of this problem demands the definition of several aspects like the space of design, the objective function, the algorithm of optimization and the restrictions (Lugo-Gonzalez et al., 2010). In the case corresponding to the synthesis of mechanisms, it is desired to diminish the error between the desired and generated trajectories besides analyzing the changes in the response of the algorithm when modifying parameters like the probability of mutation and crossing, the number of individuals and the maximum of generations, that will be evaluated by the proposed equation (3) that has characteristics applied to the approximated evaluation of the function, that involves the addition of the penalty to the

( ) () () ( ) () () 2 2

(3)

− +−

*CvCv CvCv*

*<sup>N</sup> ii ii xd xg yd yg*

Applying a division of the number of individuals *ni* in addition to a factor of division by the reciprocal of *N,* that is the number of precision points, it adds a penalty whose objective is to recover the individuals that do not fulfill the initial restrictions known as the conditions

In order to finalize, the optimization algorithm uses four criteria of convergence that are

*reng***=** Is the first restriction, this one is the first evaluation in which it is verified if the population fulfills the restrictions of Grashof (specific condition of mechanism synthesis). *maximogen =* Defines the maximum number of times that the algorithm can evaluate the objective function. An additional call to this implies the conclusion of the search without

*minimerror =* Defines the minimum value of error allowed in the objective function to being compared with the generated function. A change of value in the parameter of minimum

*condrep =* Defines the number of times that the same value can be repeated into the

Being fulfilled these last conditions to the evaluation; the algorithm will stop its search having presented the optimal values that better satisfy the restrictions and conditions.

**4.2 Elliptical trajectory with parameters optimization of GA, 18 precision points, and a** 

The obtained research results in (Cabrera et al., 2002, Laribi et al., 2004, Starosta, 2006) are taken as a basis for describing an elliptical path with a four-link mechanism. The study case was proposed for the first time by (Kunjur and Krishnamurty, 1997). The synthesis was carried out using some variants of application using genetic algorithms or combining these with tools such as fuzzy logic. In the table 1 is shown the desired precision points to be

followed by the mechanism. In the figure 6 is showed the corresponding graphic.

error implies the conclusion of the search without reaching a solution.

evaluation before proceeding to the following operation.

generation of a desired trajectory consists controlling a point of the coupler link, figure 5 (case I four-bar mechanism and case II six-bar mechanism), so that its described trajectory drives the coupler through a discreet set of giving points, known as precision points (Norton, 1995). In order to determine this point it is necessary to obtain the open and close chain mechanism.

In the last century, have been developing a variety of mechanisms synthesis methods. These are usually based on graphical procedures originally developed by (Freudenstein, 1954); or on analytical methods of research of (Denavit and Hartenberg, 1964). Other techniques include the application of least squares in the finite synthesis of four-bar spatial synthesis proposed by (Levitski and Shakvazian, 1960), or on the mathematical model and simulation for the exact mechanisms synthesis as is described in (A. K. Mallik and A. Ghosh, 1994) and (Tzong-Mou and Cha'o-Kuang, 2005). However while these works have represented major contributions in the area, the principal restriction are the number of points of precision that can be taken into account to define the desired path. The foregoing refers to the fact that each point of precision defined for the desired path represents a new set of equations to be solved. For example, the synthesis of a four- bar mechanism involves a set of 7 holonomics restrictions that describe the kinematic relationship of the links that make it up; if the designer consider 4 points of accuracy, the problem to be solved is a set of 28 non-linear equations with 29 unknowns, which represents a non-linear indeterminate problem with an infinite number of possible solutions.

With all these arguments in mind and taking into account that exist a wide variety of applications that require a large number of precision points to define more accurately the trajectory to be reproduced by the mechanism, the synthesis of these can be seen as an optimization multi-objective problem. For this purpose, researchers have developed different methodologies that include non-linear optimization (Levitski and Shakvazian, 1960), genetic algorithms (Quintero-R et al., 2004, Laribi et al., 2004, Cabrera et al., 2002, Michalewicz, 1999, Roston and Sturges, 1996), neuronal networks (Vasiliu and Yannou, 2001), (Starosta, 2006), (Walczak, 2006)), Monte Carlo optimization (Kalnas and Kota, 2001), or the controlled method (Bulatovic and Djordjevic, 2004). All the above methods have been used for four-bar mechanisms synthesis and have helped to identify the constraints of space that lead to the synthesis of mechanisms and programs developed for applications.

a) Coupler point on the coupler link of a four bar linkage

b) Type 6-bar Watt mechanism.

#### **4.1 Optimal design in the mechanisms synthesis**

52 Bio-Inspired Computational Algorithms and Their Applications

generation of a desired trajectory consists controlling a point of the coupler link, figure 5 (case I four-bar mechanism and case II six-bar mechanism), so that its described trajectory drives the coupler through a discreet set of giving points, known as precision points (Norton, 1995). In order to determine this point it is necessary to obtain the open and close chain mechanism.

In the last century, have been developing a variety of mechanisms synthesis methods. These are usually based on graphical procedures originally developed by (Freudenstein, 1954); or on analytical methods of research of (Denavit and Hartenberg, 1964). Other techniques include the application of least squares in the finite synthesis of four-bar spatial synthesis proposed by (Levitski and Shakvazian, 1960), or on the mathematical model and simulation for the exact mechanisms synthesis as is described in (A. K. Mallik and A. Ghosh, 1994) and (Tzong-Mou and Cha'o-Kuang, 2005). However while these works have represented major contributions in the area, the principal restriction are the number of points of precision that can be taken into account to define the desired path. The foregoing refers to the fact that each point of precision defined for the desired path represents a new set of equations to be solved. For example, the synthesis of a four- bar mechanism involves a set of 7 holonomics restrictions that describe the kinematic relationship of the links that make it up; if the designer consider 4 points of accuracy, the problem to be solved is a set of 28 non-linear equations with 29 unknowns, which represents a non-linear indeterminate problem with an

With all these arguments in mind and taking into account that exist a wide variety of applications that require a large number of precision points to define more accurately the trajectory to be reproduced by the mechanism, the synthesis of these can be seen as an optimization multi-objective problem. For this purpose, researchers have developed different methodologies that include non-linear optimization (Levitski and Shakvazian, 1960), genetic algorithms (Quintero-R et al., 2004, Laribi et al., 2004, Cabrera et al., 2002, Michalewicz, 1999, Roston and Sturges, 1996), neuronal networks (Vasiliu and Yannou, 2001), (Starosta, 2006), (Walczak, 2006)), Monte Carlo optimization (Kalnas and Kota, 2001), or the controlled method (Bulatovic and Djordjevic, 2004). All the above methods have been used for four-bar mechanisms synthesis and have helped to identify the constraints of space

that lead to the synthesis of mechanisms and programs developed for applications.

b) Type 6-bar Watt mechanism.

infinite number of possible solutions.

a) Coupler point on the coupler link of a four bar linkage

Fig. 5. Diferent mechanisms configuration.

The formulation of this problem demands the definition of several aspects like the space of design, the objective function, the algorithm of optimization and the restrictions (Lugo-Gonzalez et al., 2010). In the case corresponding to the synthesis of mechanisms, it is desired to diminish the error between the desired and generated trajectories besides analyzing the changes in the response of the algorithm when modifying parameters like the probability of mutation and crossing, the number of individuals and the maximum of generations, that will be evaluated by the proposed equation (3) that has characteristics applied to the approximated evaluation of the function, that involves the addition of the penalty to the presented original version in the works of (Goldberg, 1989), which is:

$$F = \frac{1}{N} \ast \sqrt{\frac{\sum\_{i=1}^{N} \left(\mathcal{C}\_{\text{xd}}^{i}\left(\upsilon\right) - \mathcal{C}\_{\text{xg}}^{i}\left(\upsilon\right)\right)^{2} + \left(\mathcal{C}\_{yd}^{i}\left(\upsilon\right) - \mathcal{C}\_{\text{yg}}^{i}\left(\upsilon\right)\right)^{2}}{m}}{m} \tag{3}$$

Applying a division of the number of individuals *ni* in addition to a factor of division by the reciprocal of *N,* that is the number of precision points, it adds a penalty whose objective is to recover the individuals that do not fulfill the initial restrictions known as the conditions of Grashof.

In order to finalize, the optimization algorithm uses four criteria of convergence that are defined as:

*reng***=** Is the first restriction, this one is the first evaluation in which it is verified if the population fulfills the restrictions of Grashof (specific condition of mechanism synthesis).

*maximogen =* Defines the maximum number of times that the algorithm can evaluate the objective function. An additional call to this implies the conclusion of the search without reaching a solution.

*minimerror =* Defines the minimum value of error allowed in the objective function to being compared with the generated function. A change of value in the parameter of minimum error implies the conclusion of the search without reaching a solution.

*condrep =* Defines the number of times that the same value can be repeated into the evaluation before proceeding to the following operation.

Being fulfilled these last conditions to the evaluation; the algorithm will stop its search having presented the optimal values that better satisfy the restrictions and conditions.

#### **4.2 Elliptical trajectory with parameters optimization of GA, 18 precision points, and a four- bar mechanism**

The obtained research results in (Cabrera et al., 2002, Laribi et al., 2004, Starosta, 2006) are taken as a basis for describing an elliptical path with a four-link mechanism. The study case was proposed for the first time by (Kunjur and Krishnamurty, 1997). The synthesis was carried out using some variants of application using genetic algorithms or combining these with tools such as fuzzy logic. In the table 1 is shown the desired precision points to be followed by the mechanism. In the figure 6 is showed the corresponding graphic.

Performance of Simple Genetic Algorithm

Inserting Forced Inheritance Mechanism and Parameters Relaxation 55

give the best results, as it is observed in table 2. For this study case, the best result appears in interjection *k* with the value of minimum error. This value is affected directly by the

The dominion is a determining value to obtain the optimal result, since all the variables are related to each other by the calculations required for the synthesis. For example if the restriction of some angles are modified, it changes the value of lengths of the links and by consequence the value of the error, since perhaps the bars must increase or decrease them length to cover the specified trajectory. Although the parameters are designed well, if this

a) b) c)

d) e) f)

g) h) i)

Error:0.0019202 generación:1000 tiemp

j) k) l)

Fig. 7. Different parameters in elliptical trajectory.

dominion of the variables and in addition to the number of individuals.

definition of variables are incorrect, it does not fulfilled the objective.


Table 1. Precision points of desired elliptical trajectory by(Kunjur and Krishnamurty, 1997).

Fig. 6. Elliptical trajectory, by(Kunjur and Krishnamurty, 1997).

The realized parameters change is presented in table 2 and the results obtained by (Kunjur and Krishnamurty, 1997, Cabrera et al., 2002, Laribi et al., 2004, Starosta, 2006) and our results are shown in table 3. The analysis procedure is shown in figures 7 and 8. But this indicates that is necessary to make a change of value in the parameters of crossover and mutation. The changes are the number of individuals, crossover and mutation, affecting with this time and the number of generations for the convergence. It has a maximum number of 1500 generations and a precision of 6 digits.

Of this series of tests one concludes that:

The individual number is an important factor for the convergence, since although a response time with a small number of individuals is obtained, it does not make sure that the result is the optimal one. With a greater number of individuals the response time increases but the possibility of obtaining a better result also increases. As it has been mentioned previously, the program will have an optimal rank of individuals to operate satisfactorily, but this must be verified by trial and error, being a program that has as a basis the random generation of the population. However, the performance of the algorithm when the FIM is implemented only registers a minor reduction compared with the one obtained for the previously considered systems.

Do not exist a rule to determine the optimal value for the crossing and the mutation probability. Not always the maximum values, that produce a total change in the individual, 54 Bio-Inspired Computational Algorithms and Their Applications

**Point 1 2 3 4 5 6 7 8 9 X** 0.5 0.4 0.3 0.2 0.1 0.005 0.02 0.0 0.0 **Y** 1.1 1.1 1.1 1.0 0.9 0.75 0.6 0.5 0.4 **Point 10 11 12 13 14 15 16 17 18 X** 0.03 0.1 0.15 0.2 0.3 0.4 0.5 0.6 0.6 **y** 0.3 0.25 0.2 0.3 0.4 0.5 0.7 0.9 1.0 Table 1. Precision points of desired elliptical trajectory by(Kunjur and Krishnamurty, 1997).

Fig. 6. Elliptical trajectory, by(Kunjur and Krishnamurty, 1997).

number of 1500 generations and a precision of 6 digits.

Of this series of tests one concludes that:

considered systems.

The realized parameters change is presented in table 2 and the results obtained by (Kunjur and Krishnamurty, 1997, Cabrera et al., 2002, Laribi et al., 2004, Starosta, 2006) and our results are shown in table 3. The analysis procedure is shown in figures 7 and 8. But this indicates that is necessary to make a change of value in the parameters of crossover and mutation. The changes are the number of individuals, crossover and mutation, affecting with this time and the number of generations for the convergence. It has a maximum

The individual number is an important factor for the convergence, since although a response time with a small number of individuals is obtained, it does not make sure that the result is the optimal one. With a greater number of individuals the response time increases but the possibility of obtaining a better result also increases. As it has been mentioned previously, the program will have an optimal rank of individuals to operate satisfactorily, but this must be verified by trial and error, being a program that has as a basis the random generation of the population. However, the performance of the algorithm when the FIM is implemented only registers a minor reduction compared with the one obtained for the previously

Do not exist a rule to determine the optimal value for the crossing and the mutation probability. Not always the maximum values, that produce a total change in the individual, give the best results, as it is observed in table 2. For this study case, the best result appears in interjection *k* with the value of minimum error. This value is affected directly by the dominion of the variables and in addition to the number of individuals.

The dominion is a determining value to obtain the optimal result, since all the variables are related to each other by the calculations required for the synthesis. For example if the restriction of some angles are modified, it changes the value of lengths of the links and by consequence the value of the error, since perhaps the bars must increase or decrease them length to cover the specified trajectory. Although the parameters are designed well, if this definition of variables are incorrect, it does not fulfilled the objective.

Fig. 7. Different parameters in elliptical trajectory.

Performance of Simple Genetic Algorithm

randomly chosen to display the specified path.

2.3

2.4

2.5

Error

2.6 x 10-3

precision.

Inserting Forced Inheritance Mechanism and Parameters Relaxation 57

In spite of applying more generations that in the last researcher work, satisfactory results are obtained. Due to the high amount of generations, computation time is more demanding, but this offers less error among the generated and desired path, and therefore, greater

Figure 8 shows how the error behavior decreases at the beginning of the path and at the end of the evaluation in each generation (*a* and *b)*. Figures *c* and *d* illustrates the four-bar mechanism along the path, covering the first and sixth precision point, which were

> Generation Generation a) b)

0

1

2

Error

3 x 10-3

0 500 1000

c) d)

There are two main six-link configuration mechanisms Watt and Stephenson type, whose features make them suitable for the manufacture of polycentric prostheses such as

The first example illustrates a six-bar mechanism for covering 21 precision points. To evaluate the effectiveness of the analyzed mechanism a Watt-I type will follow a path with arbitrarily proposed restrictions on the initial 18 points, being the conditions reported in

Figure 9 presents the proposed path to be followed by the Watt-I type mechanism. As in the previous cases, settings in population, crossing and mutation probability, time and number of generation analysis, will be varied in order to demonstrate that these adjustments are not

Fig. 8. Four-bar mechanism evolution to cover 18 precision points.

0 2 4 6 8

**4.3 Six bar mechanisms optimization** 

(Radcliffe, 1977, Dewen et al., 2003).

independent and that they are affected each other.

table 4.


Table 2. The parameters modification for a generated elliptical figure by a four bar mechanism.

In table 3 is presented the comparison of the researchers mentioned above with the proposed algorithm. With these results it can be seen that there is a correspondence of values in the bars length, angles and among desired and generated trajectory. Another variable not found in the mentioned investigation is time, a factor that is critical for the optimization. This will depend on the crossover probability, mutation parameters, individuals and generation number. Varying a small value to these parameters can mean a short time in convergence but not always the optimal value is guaranteed. With the specific parameters, the time elapsed by the GA optimization analysis is 280.508318 seconds.


Table 3. Dimensions and angles definition of an elliptical path obtained by some authors.

56 Bio-Inspired Computational Algorithms and Their Applications

*a)* 500 0.6 0.01 0.12607 974 124.70116 *b)* 1000 0.6 0.01 0.116874 992 200.83390 *c)* 1000 0.8 0.8 0.146653 994 281.61255 *d)* 1000 0.8 0.7 0.140036 992 318.54909 *e)* 1500 0.8 0.7 0.128140 992 499.81671 *f)* 1500 0.85 0.85 0.113155 979 384.667514 *g)* 2000 0.3 0.1 0.253548 992 416.934751 *h)* 2000 0.6 0.2 0.2020683 988 374.043950 *i)* 2000 0.6 0.4 0.1852588 991 431.125409 *j)* 2000 0.7 0.2 0.0986130 988 335.516387 *k)* 2000 0.7 0.4 0.0854537 995 402.078071 *l)* 2000 0.85 0.85 0.09922667 989 776.17100

Table 2. The parameters modification for a generated elliptical figure by a four bar

parameters, the time elapsed by the GA optimization analysis is 280.508318 seconds.

*Autor Xo Yo R2 R1 R4 R3 R5*  **Kunjur** 1.132062 0.663433 0.274853 1.180253 2.138209 1.879660 0.91 **Cabrera** 1.776808 -0.641991 0.237803 4.828954 2.056456 3.057878 2 **Laribi** -3.06 -1.3 0.42 2.32 3.36 4.07 3.90 **Starosta** 0.074 0.191 0.28 0.36 0.98 1.01 0.36

Table 3. Dimensions and angles definition of an elliptical path obtained by some authors.

3.88548 0.907087 0.286753 4.52611 3.59121 4.29125 3.613847

In table 3 is presented the comparison of the researchers mentioned above with the proposed algorithm. With these results it can be seen that there is a correspondence of values in the bars length, angles and among desired and generated trajectory. Another variable not found in the mentioned investigation is time, a factor that is critical for the optimization. This will depend on the crossover probability, mutation parameters, individuals and generation number. Varying a small value to these parameters can mean a short time in convergence but not always the optimal value is guaranteed. With the specific

*Pc Pm Error Generation Time (S)* 

*Population number* 

mechanism.

**A-G prop.** 

**A-G prop.** 

**Laribi** 0.20

*Autor Error No. Eval.* **Kunjur** 0.62 5000 **Cabrera** 0.029 5000

0.0152 2000

**Starosta** 0.0377 200

In spite of applying more generations that in the last researcher work, satisfactory results are obtained. Due to the high amount of generations, computation time is more demanding, but this offers less error among the generated and desired path, and therefore, greater precision.

Figure 8 shows how the error behavior decreases at the beginning of the path and at the end of the evaluation in each generation (*a* and *b)*. Figures *c* and *d* illustrates the four-bar mechanism along the path, covering the first and sixth precision point, which were randomly chosen to display the specified path.

Fig. 8. Four-bar mechanism evolution to cover 18 precision points.

#### **4.3 Six bar mechanisms optimization**

There are two main six-link configuration mechanisms Watt and Stephenson type, whose features make them suitable for the manufacture of polycentric prostheses such as (Radcliffe, 1977, Dewen et al., 2003).

The first example illustrates a six-bar mechanism for covering 21 precision points. To evaluate the effectiveness of the analyzed mechanism a Watt-I type will follow a path with arbitrarily proposed restrictions on the initial 18 points, being the conditions reported in table 4.

Figure 9 presents the proposed path to be followed by the Watt-I type mechanism. As in the previous cases, settings in population, crossing and mutation probability, time and number of generation analysis, will be varied in order to demonstrate that these adjustments are not independent and that they are affected each other.

Performance of Simple Genetic Algorithm

of convergence is achieved.

of crossing at least greater than 0.6.

results.

mechanism.

g pg

<sup>50</sup> <sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>15</sup> -50


Error:0.11947 generación:1000 tiempo:458.0269seg


Error:0.19621 generación:1000 tiempo:632.0186seg


Inserting Forced Inheritance Mechanism and Parameters Relaxation 59

• A small number of individuals decreases the search and does not offer satisfactory

• In order to obtain the optimal values is necessary to increase the value of the probability

• The rate of mutation can vary up to a maximum of 0.9, because if it increases to 1, it would be completely changing the individual without having a real meaning of the best for the evaluation, which was obtained with the elitism and the forced inheritance

• High values of probability of crossover and mutation do not ensure that the best value

*a) b) c) d)* 


Error:0.059796 generación:1000 tiempo:468.7599seg




Error:0.027759 generación:1200 tiempo:1295.9432seg



Error:0.053299 generación:1000 tiempo:480.8841seg


Error:0.031187 generación:1000 tiempo:550.6851seg


g pg


Error:0.14441 generación:1000 tiempo:250.5145seg

g pg

<sup>50</sup> <sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>1</sup> -50

Error:0.055823 generación:1000 tiempo:290.6977seg


Error:0.032607 generación:1000 tiempo:524.1338seg


g pg


Error:0.20885 generación:1000 tiempo:164.1856seg

*e) f) g) h)* 

*i) j) k) l)* 

*m) n) o) p)* 

 *q) r)* 

g pg


Fig. 10. Adjustment of parameters for a specific path.


Table 4. Six-bar mechanism restriction.

The path is obtained as a result of the evolution of the synthesis of the genetic mechanism (figure 10). In the subsequent figures and in table 5 it can be seen how decrease the error, while passing generations and changing some parameters to obtain the best adjustment.

Fig. 9. Trajectory of 20 points for a six-bar Watt-I type mechanism.

This path was proposed with the objective of demonstrating that a six-link mechanism can follow paths that would be difficult to follow by a four-bar mechanism.

From this analysis it is concluded that:

• An increase in precision points is directly proportional to the number of individuals in the population, since to obtain a minor error, it is necessary to have a greater field of search.

58 Bio-Inspired Computational Algorithms and Their Applications

xd=[ 25 10 5 10 20 10 5 10 15 25 40 43 50 55 50 40 50 55 50 40 25] yd=[[ 130 120 100 80 65 55 35 20 15 10 10 15 20 40 55 65 80 100 120 130 130]

1000 generations

*x0,y0*∈ [−60,60] ∈ [−60,60]

*in mm*

*inmm* 

**Polycentric Mechanisms Description** 

**Restriction for each links** *r1,r2,r3,r4,r5,r6,r7,r8,r9,r10*

Fig. 9. Trajectory of 20 points for a six-bar Watt-I type mechanism.

follow paths that would be difficult to follow by a four-bar mechanism.

**Movements range** *0º to 360*º degrees

**Population numbers** niindividuals 200 **Crossover probability** Proportional type varied **Mutation probability** Only one point varied **Precision** Digits after point 6

The path is obtained as a result of the evolution of the synthesis of the genetic mechanism (figure 10). In the subsequent figures and in table 5 it can be seen how decrease the error, while passing generations and changing some parameters to obtain the best adjustment.

This path was proposed with the objective of demonstrating that a six-link mechanism can

• An increase in precision points is directly proportional to the number of individuals in the population, since to obtain a minor error, it is necessary to have a greater field of

**Characteristic Desired points Variables limits** 

**Maximum number of generations** 

Table 4. Six-bar mechanism restriction.

From this analysis it is concluded that:

search.


Fig. 10. Adjustment of parameters for a specific path.

Performance of Simple Genetic Algorithm

in the solution space of the problem.

**6. Conclusions** 

Inserting Forced Inheritance Mechanism and Parameters Relaxation 61

Besides, the excessive population causes that the algorithm requires of a greater time of calculation to process and to obtain a new generation. In fact, there is not a limit wherein it is inefficient to increase the size of the population since it neither obtains a faster speed in the resolution of the problem, nor the convergence makes sure. For the referred study cases in this chapter, when increasing the population to 3500 individuals no acceptable results are presented and the program became extremely slow. If the population remains so large, like for example 1000 individuals, this means that it can improve the performance of the algorithm, although this is affected by slower initial responses. It is important to do emphasis on the relation that exists among the population size and the probabilistic relation

The study cases of this work are over determined and nonlinear type, which implies by necessity a space of multidimensional, nonlinear and non-homogenous solution, therefore, large initial values cover different regions of the solution space wherein the algorithm could converge prematurely to a solution that implies optimal premises costs, but when maintaining a low probability of mutation is not possible to assure that the population, although extensive in the number of individuals, continues being probabilistic representative of the problem solution. With this in mind and considering that the computation time to evaluate and to generate a new population of individuals from the present initial or, directly is the bound to the number of individuals of this one, requires a

When operating with a population reduced in number of individuals, a sufficient representative quantity of the different regions of the solution space is not achieved, but the necessary computation time to create a new generation of possible solutions diminishes dramatically. When considering a high percentage of the probability of mutation in the algorithm, one assures a heuristic search made in different regions of the solution space, this combined with the forced inheritance mechanism has demonstrated that for the problem treated in this work, it is a strategy that power the heuristic capacities of the GA, for nonlinear multidimensional problems, non-homogenous, becoming the algorithm metaheuristic; it is demonstrated then that an important improvement in the diminution of the error is obtained, around 20% with respect to the reported works previously. Also it was observed that the increase in the percentage of mutation improves the off-line performance, since all the solutions in the population are taken into account to obtain the optimal value. The off-line performance does not penalize the algorithm to explore poor regions of the search

space, as long as it contributes to reach the best possible solutions in terms of aptitude.

the elitism and the forced inheritance mechanism.

It was verified that for the crossover the rule is fulfilled of which applying values smaller to 0.6, the performance is not optimal and it does not change the expected result for a specific problem. In the case of mutation, one demonstrated that this one can change no mattering the number of times and increasing its value to obtain optimal results, reaching almost at the unit, but avoiding to muter totally all the chromosomes eliminating the benefits created by

By means of the trial and error, also one concludes that the parameters are not independent, and searching systematically to obtain all the possible combinations of these, is almost

greater number of operations to obtain a new generation of possible solutions.


Table 5. Adjustment of six-bar mechanism parameters for a specific path.
