Forward Kinematics of the 3RPR planar Parallel

Download PDF
0 downloads 0 Views 105KB Size Report
Comment
The conversion of an equation system solving problem into an optimization one is ... Firstly, Primerose and Freudenstein have counted 8 as- sembly modes ...
Forward Kinematics of the 3RPR planar Parallel Manipulators Using Real Coded Genetic Algorithms Luc Rolland

Rohitash Chandra

Mechanical Engineering dept. Middle East Technical University, North Cyprus Campus Kalkanli, Turkish Republic of North Cyprus [email protected] or [email protected]

Computer Science dept. Middle East Technical University, North Cyprus Campus Kalkanli, Turkish Republic of North Cyprus Rohitash [email protected]

Abstract—This article examines Genetic Algorithms to solve the forward kinematics problem applied to planar parallel manipulators. Their majority can be modeled by the tripod 3-RPR. The conversion of an equation system solving problem into an optimization one is investigated. Parallel manipulator kinematics are formulated using the explicit Inverse Kinematics Model (IKM). From the displacement based equation systems, the objective function is formulated specifically for the FKP using one squared error performance criteria. The proposed approach implements an elitist selection process where a new mutation operator for Real-Coded GA is analyzed. Experiments on a typical manipulator example show the fast convergence of the proposed method. Index Terms—Parallel robot, planar manipulator, forward kinematics, displacement based model, genetic algorithms, elitist selection, real root isolation.

I. I NTRODUCTION

L3

A3

B3

Yc B2

C Yo

L2 A2

Xc

B1 L1 3 RPR

O A1 Figure 1.

Xo

The general planar manipulator and the typical 3-RPR tripod

Planar parallel manipulators are characterized by one base, one mobile platform and 3 kinematics chains which lie in one plane, namely the 3-RPR, [2]. The manipulator end-effector displacements are restricted to that same plane, fig. (1). The design hypotheses state that the bodies are infinitely rigid and the joints do not yield friction nor play.

c 2009 IEEE 978-1-4244-2794-9/09/$25.00

Firstly, Primerose and Freudenstein have counted 8 assembly modes (manipulator postures) for the 3-RPR and 3RRR, [3]. Later, Hunt geometrically demonstrated that the 3RPR yields 6 assembly modes, [4]. For the general 3-RPR, six complex solutions were found using an algorithm based on variable elimination, [2]. Fast numerical methods usually implement Newtons but it is sometimes plagued by Jacobian inversion problems and numerical instabilities. Moreover, it can only find one solution. Resultant or dialytic elimination methods have been applied but they might add spurious solutions, [5]. Moreover, homothopy methods might also be advocated but they are prone to miss some solutions, [5]. Hence, this justified the implementation of another method which could find solutions numerically. Genetic algorithms were introduced by Holland, [6] and they have evolved significantly in order to suit real-world optimization challenges faced by engineers. In robotics, evolutionary algorithms have been applied for solving the FKP of parallel manipulators. Boudreau and Turkkan proposed a realcoded genetic algorithm (RCGA), also called floating point genetic algorithm since real numbers cannot be represented in a digital computer, [11]. It integrates crossover and mutation operators inspired by operators used in binary GA. However, it was reported that the GA method is more time consuming than Newton-Raphsons method. However, it was shown that the domain in which the GA will converge to a solution is larger allowing to initiate the process with a more distant initial guess. More recently, the FKP solving of the spherical and spatial manipulators has also been approached with genetic algorithms, the 3RRR by [12] and the the Gough platform by [13]. In this work, RCGA is implemented with Wright’s heuristic crossover operator, [15]. The Pivot Mutation operator is introduced to simply add a small real random number to selected genes in the chromosome. The RCGA uses roulette wheel selection combined with the elitist strategy in order to avoid oscillation during the search. We will also introduce an elitist strategy for the selection process and study its impact of on resolution performances in terms of response times. Using the exact results coming from computer algebra, [16], it will be possible to verify the results. The article is divided into three topics. The first section

ISCIS09

reviews GA theory and details the selected elitist strategy. The second part addresses the issue of the kinematics modeling and its conversion as an optimization problem. The third part presents the resolution results obtained on one typical plan.

B 1

C

II. T HE F ORWARD K INEMATICS P ROBLEM F ORMULATION L

A. The kinematics of parallel manipulators

1

Manipulator configuration OA , CB Rf

Rm

A 1

O

Joint coordinates Generalized coordinates

rho1

Figure 3.

The vectorial formulation for one kinematics chain

MODEL

...

F=0

rho3

Figure 2.

X

Kinematics model

Any manipulator is characterized by its mechanical configuration parameters and the posture variables. The configuration parameters are thus OA|Rf , the base attachment point coordinates in Rf (the base reference frame), and CB|Rm , the mobile platform attachment point coordinates in Rm (the mobile platform reference frame). The kinematics model variables are the joint coordinates and end-effector generalized coordinates. The joint variables are described as li , the prismatic joint or linear actuator positions. The generalized coordinates are −→ expressed as X , the end-effector position and orientation. The kinematics model is an implicit relation between the configuration parameters and the posture variables, −→ F ( X , ρ, OA|Rf , CB|Rm ) = 0 where L = {l1 , l2 , l3 }. This article shall only concentrate on the forward kinematics problem (FKP ), fig. 2. Usually the inverse kinematics problem is required to model the FKP and is defined as: given the generalized coordinates of the manipulator end-effector, find the joint positions. Accordingly, the forward kinematics problem is defined as: given the joint positions, find the generalized coordinates of the manipulator end-effector. In the majority of parallel manipulator cases, the FKP is a difficult problem, [17]. B. Vectorial formulation of the basic kinematics model Containing as many equations as variables, vectorial formulation constructs an equation system [18], fig. (3), as a closed vector cycle between the following points: Ai and Bi , kinematics chain attachment points, O the fixed base reference frame and C the mobile platform reference frame. For each −→ kinematics chain, an implicit function Ai Bi = U1 (X) can be written between joint positions Ai and Bi . Each vector

−→

−→

Ai Bi is expressed knowing the joint coordinates Li and X −→ giving function U2 (X, L ). The following equality has to be −→ solved: U1 (X) = U2 (X, L ). The distance between Ai and Bi is set to Li . Thus, the end-effector position X or C can −→ be derived by one platform displacement OC and then one platform general rotation expressed by the rotation matrix R. Vectorial formulation 2 evolves as a displacement based equation system using the following relation : −→

−→

−→

−→

Ai Bi = OC + RCBi − OAi

(1)

−→ OBi |Rf

with i = 1, ..., 3, For each distinct platform point each kinematics chain can be expressed using the distance norm constraint, [19]: L2i = ||Ai Bi ||2

(2)

C. The inverse kinematics problem A review of planar parallel manipulators shows us that the majority of proposals fall into the four following classes: 3RPR, 3-RRP, 3-PRR and 3-RRR, [16]. We shall only study the 3-RPR. For each kinematics chain, the RPR manipulator is constituted by a prismatic actuator located between two ball joints fixed on the base and the platform. Lets have Li = li . Taking the 1 equation and substituting it in equation 2, one obtains the following equation : −→

−→

−→

li2 = ||OC + RCBi − OAi ||2 , i = 1 . . . 3

(3)

The rotation matrix R is expressed in terms of one rotation parameter. D. The Forward Kinematics Problem The IKP expression gives an algebraic system comprising the first three equations in terms of the classical variables : xc , yc , θ, equation (3). This system contains trigonometric functions which can be handled by the numerical solvers implemented in GA.

ISCIS 2009

E. The conversion to an optimization problem GA implements a selection process which takes populations of solutions guesses and retains according to a certain set of rules. Being only applicable to optimization problems, the GA process handles a performance or cost function respectively to be maximized or minimized. Therefore, we need to effectively convert the problem which solves a system of equations into an optimization problem. Hence, only the inverse kinematic model is required from which we can easily derived a cost function, also called fitness function. This function will be calculated on each solution estimate for the FKP and it will represent the total error on each leg lengths. Let lgi be the leg length of kinematics chain i which is given as input of the problem. Then, the objective or fitness function is set to : 3 X

(li − lgi )2

(4)

i=1

li2

If we set Hi = coming from equation (3), then the fitness function becomes : 3 X

(sqrt(Hi ) − lgi )2

(5)

i=1

This function is to then be minimized by our selected optimization technique to be covered in the next section. III. S OLVING

WITH

G ENETIC A LGORITHMS

A. Real Coded Genetic Algorithms The basic idea used in GA optimization is as follows: Initially, a number of possible solutions (chromosomes) make up a population. Over time, the GA evaluates each chromosome according to its fitness and employs genetic operators like selection, crossover and mutation for producing new solutions. These new solutions are called offsprings which are added into the new population. The process is repeated until the algorithm obtains the best solution. The selection of parent chromosomes from the population is done using the selection operator. The optimization procedure of a standard RCGA is shown in Algorithm 1. The major advantages of real-coded GA over standard binary-coded GA are 1) their maintainability of precision which is usually lost in binary representation of a real number and 2) the reduction in the size of the chromosome which directly reduces the computation time. Furthermore, realvalued encoding gives a close conceptual approximation of real weight values for real world application problems. The computational and optimization power of RCGA has also been demonstrated in several theoretical studies [7], [22], [23]. The choice of the appropriate genetic operator is important as it directly influences the convergence of the GA. However, different forms of the main genetic operators i.e. selection, crossover and mutation are needed according to the type of the GA and the nature of the optimization problem. An overview of each genetic operator is discussed below:

Algorithm 1 Real Coded Genetic Algorithm Initialize Population Evaluate fitness while !termination do while i < P opulationSize do 1) selection 2) crossover 3) mutation i++ end while Update population end while Get the best solution

Selection: The selection operator plays an important role in the GA as it ensures that qualities from the fittest chromosomes always survive in future generations. Some common selection strategies are rank selection [25], roulette wheel selection [26] and the elitist strategy [24]. In rank selection, the selection is done according to the fitness rank of the individual; the mostfit individuals have priority over lesser ones . In roulette wheel selection, selection is done according to the wheel which gives priority to those individuals with greater fitness. However, in this way, lesser fit chromosomes are also chosen as they may contain useful genetic material for the future. In the elitist strategy, some of the fittest individuals are always retained in the new population to make sure that the best performing chromosomes always survive. Crossover: The crossover operator exchanges genetic material from selected parents and forms either a single or multiple offspring’s. Some common crossover operators for real-coded GA’s are flat crossover [22], simple crossover [7], [27], arithmetic crossover [27], blend crossover [29], linear breeder crossover [30] and Wright’s heuristic crossover. Wright’s heuristic crossover has also shown superior performance compared to binary GAs for a set of optimization problems [15]. Mutation: In the GA optimization process, the mutation operator provides random diversity in the population. This is important when the GA gets trapped in a local minima. This usually happens the whole population is made up of similar solutions. The effect of mutation rates, the strength of mutation and its impact in building better solutions has been studied in [31]–[33]. Some common mutation operators are random uniform mutation and Michalewicz non-uniform mutation [27], [28]. In uniform mutation, a random number in the range of [a, b] is added to a selected gene where a and b are the highest and lowest values in the chromosome, respectively. In nonuniform mutation, the strength of mutation is decreased as the number of generations increases. Other mutation operators include Muhlenbeins BGA mutation [34] which creates new chromosomes in the neighborhood restricted by some probability, and Voigt discrete and continuous modal mutation [35] which works in a similar

ISCIS 2009

way. Furthermore, Guaussian mutation works by acquiring the mean (set to zero initially) and the standard deviation which determines the strength of mutation [36], [37]. The pointed direction mutation (PoD) [38] and momentum notion mutation [42] employ a set of co-evolving points which set directions that govern the resulting chromosome. The PCA-mutation [39] is based on the principal component analysis while the wavelet mutation [40] is based on wavelet theory. The polynomial mutation has demonstrated effectiveness in a set of constrained optimization problems [41]. Power mutation [44] has also been proposed which has demonstrated effiteveneess when compared to non-uniform mutation and Makinen, Periaux and Toivanen mutation [43]. In the past, a number of techniques have been proposed to improve the performance of GAs. Fan et.al. introduced three improved genetic algorithms which use 1) a dual fitness function to adapt the mutation probability, 2) a new evolutionary directional operator, and 3) probabilistic binary search to reveal a new offspring, respectively. All these approaches aim in building better offsprings in relation to traditional methods [46]. The crossover and mutation operators are improved from the introduction of the local gradient based operators such as gradient optimizer [45] and evolutionary gradient operators [47] . Other modifications include a new stopping rule based on asymptotic considerations and mutation scheme based on Particle Swarm Optimization [48]. B. Pivot Mutation The Pivot Mutation operator selects a pivot point in the chromosome and all the genes after the selected pivot are mutated by adding a small real-random numbers, respectively. For instance, given a chromosome x = (x1 , x2 , x3 , ...xn ), the resulting pivoted chromosome becomes y = (x1 , x2 , y3 , ...yn ), where ,yi = xi + r given that r is a small real random number in the interval [a, b], where a and b are small negative and positive real numbers chosen by the user, respectively. C. Other genetic operators used in this study Wrights heuristic crossover operator has shown superior performance in comparison with other crossover operators for a set of optimization problems [15] , therefore, it is used in this study. The Wrights heuristic operator produces a single offspring given two parents. For a pair of parents x1 = (x11 , x12 , x13 , ...x1n ) and x2 = (x21 , x22 , x23 , ...x2n ) , an offspring is produced as follows: yi = r(x1i − x2i ) + x1i where r is a real random number belonging to [0,1] and x1 is the parent with the best fitness. A new ofspring is produced with a new r until a chromosome with a a better fitness is created. The uniform mutation operator developed by Michalewicz adds a small random number [a, b] to a selected gene, where a and b are the greatest and the least values found in the chromosomes, respectively. The non-uniform mutation operator [28] is one of the widely used operators in real-coded

GAs. In this method, from a point xt = (xt1 , xt2 , xt3 , ...xtn ), the t+1 t+1 t+1 muted point xt+1 = (xt+1 1 , x2 , x3 , ...xn ) is created as follows: t+1

x

=



xti + ∆(t, xui − xti ), xti − ∆(t, xui − xti ),

if r ≤ 0.5 otherwise

where t is the current generation number and r is a uniformly distributed random number in the interval [0, 1]. xli and xui are the lower and upper bounds of the selected chromosome, respectively. The function ∆(t, y) given below takes value in the interval [0, y].   t b ∆(t, y) = y 1 − u(1− T ) where u is a uniformly distributed random number in the interval [0, 1], T is the maximum number of generations and b is a number which determines the strength of the mutation operator. This mutation operator performs global search during the initial search and local search in the later generations. IV. R ESULTS

AND

A NALYSIS

A. Modeling the planar 3RPR In this section, we shall examine one example of the FKP resolution on a typical 3RPR manipulator configurations. The manipulator configuration is written in a text file, noted 3RPR8. The file includes the manipulator base coordinates of the joint center positions OA|Rf in the base reference frame Rf and the mobile platform coordinates of the joint center positions CB|Rm in the platform reference frame Rm , the minimum bar lengths if applicable. The corresponding configuration examples are shown in table I. Config 3RPR8 Id 3RPR8

A1 (x) 0 B1 (x) 0

A1 (y) 0 B1 (y) 0

A2 (x) 200 B2 (x) 50

A2 (y) 0 B2 (y) 0

A3 (x) 0 B3 (x) 40

A3 (y) 200 B3 (y) 40

Table I P LANAR PARALLEL MANIPULATOR CONFIGURATION TABLE

The active joint variables are then specified. For the 3RPR8, the joint variables are the kinematics chain lengths. To complete the FKP input data in the aforementioned examples, the square of the joint variables are set respectively to L := [100, 120, 150]. B. Algebraic system exact solution In order to verify the method implementing GA, we revert to an exact algebraic method which can compute the certified exact results, [16]. For the aforementionned planar parallel manipulator, the exact results are shown with four exact digits after the point: sol1 = (97.9911, 19.9193, 85.0154) sol2 = (52.8654, 84.9536, -33.4487)

ISCIS 2009

C. Genetic Algorithm results The computations were achieved on a 3 GHz Pentium IV personal computer equipped with 512k bytes of random access memory. The operating system was Mandrake LINUX version 9.2. In this section, the performance of 3 different combinations of genetic operators in RCGA’s is evaluated. In all experiments, the roulette wheel selection is used in conjunction with the elitist strategy method. This ensures that the fittest chromosome is retained in future generations. This prevents the GA from oscilating or becoming trapped in a local mimimum while searcing for the global optimum solution. Note that if the population size is P , then P selections are done in order to make P offsprings for the new population. The following combinations were used in order to get the best combination of genetic operators in order to evaluate the performnace of the prposed Pivot Mutation operator. Note that the respective mutation and crossover rates were determined in trial experimental runs. 1. Hx Non-uniform: The wrights heuristic crossover combined with Michaelwicz nonuniform mutation. The crossover and mutation rate are 0.8 and 0.05, respectively. 2. Hx Crossover: In this strategy, the Wright’s heuristic crossover operator dominates the GA. There is no mutation. The crossover rate is 0.8. 3. Hx Pivot: The wrights heuristic crossover operator is combined with Pivot Mutation. The crossover and mutation rates are 0.8 and 0.1, respectively. The fitness function given by the forward kinamatic of 3 leg parallel manupilators is given in equation (n) is used. All experiments initialise the population with real numbers in the range of [−40, 40]. The population size of 40 and chromosome size of 3, representing the poition’s x, y and θ were used. The constant leg lengths for leg1, leg2 and leg3 were 100, 120 and 150, respectively. Figure 1 shows the convergence of a typical experimental run for each GA optimization technique. For each method, a total of 50 experimental runs were done. The results are summarised in Table 1. The optimization results for cordinates x, y, and orientation θ with its respective fitness is given in Table 2. The given forward kinematics problem has 2 unique solutions. Note that the experiments were done on a 2 Giga Hertz Linux dual core machine and the CPU time is given in milliseconds. The CPU time was calculated using the number of clock ticks taken for the GA in converging to a solution. The CPU executes 29 clock ticks in 1 second (2 GigaHz CPU). Therefore, the time in milliseconds was calculated by T (ms) = (ClockT icks ∗ (1/29 )) ∗ 1000.

Hx Non-Uniform Hx Crossover Hx Pivot

No. Generations 29±4 40±7 17±6

Fitness 0.75±0.06 0.77±0.06 0.61±0.08

CPU Time 1.49±0.24 2.11±0.39 0.86±0.10

Table II T HE MEAN AND 95 PERCENT CONFIDENCE INTERVAL FOR 50 EXPERIMENTAL RUNS

Figure 4. methods

A typical experimental run showing the convergence of the three

Hx Non-Uniform Hx Crossover Hx Pivot

Solution 1 2 1 2 1 2

x 53.2047 98.7197 52.4156 98.2234 52.4071 97.7789

y 85.6626 20.2729 85.0839 19.3598 85.4213 20.0296

θ(deg) -34.0811 85.5912 -33.3571 85.0078 -34.1143 84.4414

Fitness 0.765958 0.627518 0.527414 0.640394 0.57497 0.190192

Table III O PTIMAL x, y

AND

θ

VALUES FROM THE BEST EXPERIMENTAL RUN FOR EACH METHOD .

Note that the given optimization problem has 2 real solutions. The precision of the GA solutions depends on the number of generations allowed, and are thus proportional to time. D. Discussion on model solving V. C ONCLUSION In this paper, one complete method to solve the parallel manipulator FKP has been established and explained based on genetic algorithms. The method was applied to the planar 3RPR parallel manipulator. At first, the manipulator kinematics formulation was studied. The inverse kinematics equation system was written applying the classical displacement-based model. Then, the objective function was written as an error sum on all kinematics chain lengths. Real-coded genetic algorithms provide multiple solutions to optimization problem given multiple experimental runs with different initial search space. The two solutions reported in this work were found in different populations given by multiple experimental runs, since the entire population is made up of similar solutions towards convergence. This observation means that the best solutions survive over time. The proposed ’Pivot Mutation’ operator has demonstrated its strength when compared to standard methods. The two real solutions contained two position parameters found within 0.5 mm accuracy and angles within 0.6 degrees. For the first time, the GA method results were verified against the exact solutions. In future work, the implementation of the RCGA procedure into a real-time robotic system can be foreseen due to its fast convergence, accuracy and robustness the search procedure.

ISCIS 2009

R EFERENCES [1] D. A Stewart. Platform with 6 degrees of freedom. Proceedings of IMechE, vol. 180, pp. 371-386. 1965. [2] C. Gosselin and J.-P. Merlet. The direct kinematics of planar parallel manipulators : special architectures and umber of solutions. Mechanism and Machine Theory, vol. 29, pp. 1083-1097. 1994. [3] E. J. F. Primrose and F. Freudenstein, Spatial motions in mechanisms with four or fewer links, ASME J. Eng. Industry 91B, pp. 103-114. 1969. [4] K. H. Hunt, Structural kinematics of in-parallel-actuated robot-arms, J. Mech. Transmiss. Automat. Des. 105, pp. 705-712, 1983. [5] L. Rolland. Certified Solving and Synthesis on Modeling of the Kinematics Problems of Gough-Type Parallel Manipulators with an Exact Algebraic Method. In Parallel Manipulators, towards new Applications, edited by H. Wu, I-Tech Education and Publishing, 2008. [6] J. H. Holland. Adaptation in natural and artificial systems. The University of Michigan Press, 1975. [7] D. E. Goldberg. Real-coded genetic algorithms, virtual alphabets, and blocking. Complex Systems, vol. 5, pp. 139-167, 1991. [8] Yu, H., and Liang, W. Neural Network and genetic algorithm based hybrid approach of extended job-scheduling. Computers and Industrial Engineering, vol. 39, pp. 337-356, 2001. [9] H. Niska et al. Evolving the neural network model for forecasting air pollution time series. Engineering Applications of Artificial Intelligence, vol. 17, pp. 159-167, 2004. [10] G. Capi and K. Doya. Evolution of recurrent neural controllers using an extended parallel genetic algorithm. Robotic and Autonomous Systems, vol. 52, pp. 148-159, 2005. [11] R. Boudreau and N. Turkkan. Solving the Forward Kinematics of Parallel Manupilators with a Genetic Algorithmn. Journal of Robotics Systems, vol. 13, no. 2, 111-125, 1995. [12] R. Boudreau, S. Darenfeld S. and N. Turkkan. Etude comparative de trois nouvelles approches pour la solution du probl`eme g´eom´etrique direct des manipulateurs parall`eles. Mechanism and Machine Theory, vol. 33, no. 5, pp. 463-477, 1998. [13] A. Omran, G. El-Bayiumi, M. Bayoumi, and A. Kassem. Genetic Algorithm Based Optimal Control for a 6-DOF Non Redundant Stewart Manipulator. International Journal of Mechanical Systems Science and Engineering, vol. 2, no. 2, pp. 73-79, 2008. [14] X. Wang, M. Hao and Y. Cheng. On the use of differential evolution for forward kinematics of parallel manipulators. Applied Mathematics and Computation, vol. 205, no. 2, pp. 760-769, 2008. [15] A.H. Wright. (1991). Genetic algorithms for real parameter optimization. Foundations of Genetic Algorithms. Morgan Kaufman ed., 1991. [16] L. Rolland. Synthesis on the forward kinematics problem algebraic modeling for the planar parallel manipulator. Advanced Robotics. vol.20, no. 9, pp. 1035-1065, 2006. [17] M. Raghavan and B. Roth, Solving polynomial systems for the kinematic analysis and synthesis of mechanisms and robot manipulators, Trans. ASME 117, pp. 71-79, 1995. [18] R. Parrish, E. Dieudonn and R. Bardusch, An actuator extension transormation for a motion simulator and an inverse transformation applying newton-raphson’s method, Technical report D-7067, NASA, Washington, DC, 1972. [19] J. P. Merlet, Les Robots Parallles, Herm`es, Paris, 1997. [20] H. Sadjadian, H.D. Taghirad and A. Fatehi. Neural Networks Approaches for Computing the Forward Kinematics of a Redundant Parallel Manipulator. International Journal of Computational Intelligence 2(1),40-47, 2005. [21] Y. Xia and J. Wang. A Dual Neural Network for Kinematics Control of Redundant Robot Manupilators. IEEE Transactions on Systems, Man, and Cybernatics, Part B: Cybernatics, vol. 31, no. 1, pp. 147-154, 2001. [22] N. J. Radcliffe. Equivalence class analysis of genetic algorithms. Complex Syst. vol. 2, no. 5, pp. 183-205, 1991. [23] J. Schaffer, R. Caruana, L. Eshelman and R. Das. A study of control parameters affecting online performance of genetic algorithms for function optimization. In J. Schaffer ed. 3rd International Conference on Genetic Algorithms, pp. 51-60, Morgan Kaufmann, San Mateo, 1989. [24] K. A. DeJong. An analysis of the behavior of a class of genetic adaptive systems. Ph.D. dissertation. University of Michigan, 1975. [25] D. Whitley, D. The GENITOR algorithm and selection pressure: Why rank-based allocation of reproductive trials is best. Proceedings of the Third International Conference on Genetic Algorithms, pp. 116-121, 1989.

[26] J. E. Baker. Reducing Bias and Inefficiency in the Selection Algorithm. Proceedings of the Second International Conference on Genetic Algorithms and their Application, Hillsdale, New Jersey, Lawrence Erlbaum Associates, pp. 14-21, 1987. [27] Z. Michalewicz, Z. Genetic Algorithms + Data Structures = Evolution Programs. Springer, New York, 1992. [28] Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, vol. 4, no. 1, pp. 1-32, 1996. [29] L. J. Eshelman and J. D. Scahffer, J. D. Real-coded genetic algorithms and interval-schemata. Foundations of Genetic Algorithms 2, pp. 187-202, Morgan Kaufmann ed., San Mateo, 1993. [30] D. Schlierkamp-Voosen. Strategy adaptation by competition. Proceedings of the Second European Congress on Intelligent Techniques and Soft Computing, pp. 1270-1274, 1994. [31] Smith, J.E. (2001). Modelling GAs with self adapting mutation rates. Proceedings of the Genetic and Evolutionary Computation Conference. [32] Smith, J.E., and Fogarty, T.C. (1997). Operators and parameter adaptation in genetic algorithms. Soft Computing, 1(2), 81-87. [33] Krasnogor, N., and Smith, J.E. (2001).Emergence of profitable search strategies based on a simple inheritance mechanism. Proceedings of the International Conference on Genetic and Evolutionary Computations, (pp. 432-439), Morgan Kaufmann, San Mateo, CA. [34] Muhelenbein, H., and Schlierkamp-Voosen, D. (1993). Predictive models for the breeder genetic algorithm 1, continuous parameter optimization. Evolutionary Computations, 1, 25-49. [35] Voigt, H.M., and Anheyer, T. (1994). Modal mutations in evolutionary algorithms. Proceedings of the First IEEE Conference on Evolutionary Computation, (pp. 88-92). [36] Hinterding, R. (1995). Gaussian mutation and self-adaption in numeric genetic algorithms. Proceedings of Second IEEE Conference Evolutionary Computation, (pp. 384-389), IEEE Press, Piscataway, NJ. [37] Hinterding, R., Michalewicz, Z., and Peachey, T.C. (1996). Self-adaptive genetic algorithm for numeric functions. In: H.-M. Voigt, W. Ebeling, I. Rechenberg and H.-P. Schwefel, Editors, Proceedings of the Fourth Conference on Parallel Problem Solving from Nature, Lecture Notes in Computer Science, vol. 1141, (pp. 420-429). Springer, Berlin, Germany. [38] Berry, A., and Vamplew, P. (2004). PoD can mutate: a simple dynamic directed mutation approach for genetic algorithms. Proceedings of AISAT2004: International Conference on Artificial Intelligence in Science and Technology., Hobart, Tasmania. [39] Munteanu, C., and Lazarescu, C. (1999). Improving mutation capabilities in a real-coded GA. Proceedings of GoIASP99, Lecture Notes in Computer Science, 1596, (pp. 138-148), Springer-Verlag. [40] Ling, S.H., and Leung, F.H.F. (2007). An improved genetic algorithm with average-bound crossover and wavelet mutation operations, Soft Computing, 11, 7-31. [41] Deb, K. (2001). Muiti-objective Optimization Using Evolutionary Algorithms. Wiley, Chichester. [42] Temby, L., Vamplew, P., and Berry, A. (2005). Accelerating Real-Valued Genetic Algorithms Using Mutation-with-Momentum, Lecture Notes in Computer Science, 3809, (pp. 1108-1111), Springer-Verlag. [43] Makinen, R.A.E., Periaux, J., and Toivanen, J. (1999). Multidisciplinary shape optimization in aerodynamics and electromagnetic using genetic algorithms. International Journal for Numerical Methods in Fluids, 30(2), 149-159. [44] Deep, K., and Thakur, M. (2007). A new mutation operator for real coded genetic algorithms. Applied Mathematics and Computation, 193 (1), 211-230. [45] Davis, L. (1990). Handbook of genetic algorithms. Van Nostrand Reinhold, New York. [46] Fan, H. Y., Lu. J. W. Z. and Xu, Z. B. (2000). An empirical comparison of three novel genetic algorithms. Engineering Computation, 17 (8), 9811001. [47] Yamamoto, K., and Inoue, O. (1995). New evolutionary direction operator for genetic algorithms. AIAA J., 33(10), 1990-1993. [48] Tsoulos, I. G. (2008). Modifications of real code genetic algorithm for global optimization. Applied Mathematics and Computation, 203(2), 598607.

ISCIS 2009