**Meta-Heuristic Optimization Techniques and Its Applications in Robotics**

Alejandra Cruz-Bernal

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Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/54460

## **1. Introduction**

Robotics is the science of perceiving and manipulating the physical world. Perceive informa‐ tion on their envioronments through sensors, and manipulate through physical forces. To do diversity tasks, robots have tobe able to accomodate the ennormous uncertainty that exist in the physical world. The level of uncertainty depends on the application domain. In some robotics applications, such as assembly lines, humans can cleverly engineer the system so that uncertainty is only a marginal factor. In contrast, robots operating in residential homes, militar operates or on other planets will have to cope with substantial uncertainty. Managing uncertainty is possibly the most important step towards robust real-world robot system.

If considerate that, for reduce the uncertainty divide the problem in two problems, where is the first is to robot perception, and another, to planning and control. Likewise, path planning is an important issue in mobile robotics. It is to find a most reasonable collision-free path a mobile ro‐ bot to move from a start location to a destination in an envioroment with obstacles. This path is commonly optimal in some aspect, such as distance or time. How to find a path meeting the need of such criterion and escaping from obstacles is the key problem in path planning.

Optimization techniques are search methods, where the goal is to find a solution to an optimiza‐ tion problem, such that a given quantity is optimized, possibly subject to a set of constraints. Modern optimization techniques start to demonstrate their power in dealing with hard optimi‐ zation problems in robotics and automation such as manufacturing cells formation, robot mo‐ tion planning, worker scheduling, cell assignment, vehicle routing problem, assembly line balancing, shortest sequence planning, sensor placement, unmanned-aerial vehicles (UAV) communication relaying and multirobot coordination to name just a few. By example, in parti‐ cle, path planning it is a difficult task in robotics, as well as construct and control a robot. The main propose of path planning is find a specific route in order to reach the target destination.

© 2013 Cruz-Bernal; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Cruz-Bernal; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Given an environment, where a mobile robot must determine a route in order to reach a target destination, we found the shortest path that this robot can follow. This goal is reach using bioinspired techniques, as Ant Colony Optimization (ACO)and the Genetics Algorithms (GA).

A first path planning approach of a mobile robot trated as non-deterministic polynomial time hard (*NP-hard*) problem is [31]. Moreover, even more complicated are the environment dynam‐ ic, the classic approaches to be incompetent [32]. Hence, evolutionary approaches such as Tabu Search (TS), Artificial Neural Network (ANN), Genetic Algorithm (GA), Particle Swarm Opti‐ mization (PSO), Ant Colony Optimization (ACO) and Simulated Annealing (SA), name a few, are employed to solve the path planning problem efficiently although not always optimal.

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55

Genetic Algorithm (GA) based search and optimization techniques have recently found increasing use in machine learning, scheduling, pattern recognition, robot motion planning, image sensing and many other engineering applications. The first research of Robot Motion Planning (RMP), according to Masehian et al.[53] although GA, was first used in [42] and [43]. An approach for solution the problem of collision-free paths is presented in [44].GA was applied [45, 46, 48] in planning multi-path for 2D and 3D environment dimension and shortest path problem. A novel GA searching approach for dynamic constrained multicast routing is developed in [49]. Parallel GA [50], is used for search and constrained multi-objective optimi‐ zation. Differentials optimization used hybrid GA, for path planning and shared-path protections has been extended in [51, 52]. In [62, 63, 64, 65], has been a compared of differential algorithms optimization GA (basically for dynamic environment), subjected to penalty

By other side, thetechnique PSO have some any advantages [35], such as simple implementa‐ tionwith a few parameters to be adjusted. Binary PSO [37] withouta mutation operator[36]are used to optimize the shortest path. Planning in dynamic environment, that containing invalid paths (repair by a operator mutation), are subjected to penalty function evaluation [38]. Recent‐ ly, [39] proposedwith multi-objective PSO and mutate operator path planning in dangerous dy‐

Ant Colony Optimization (ACO) algorithms have been developed to mimic the behavior of real ants to provide heuristic solutions for optimization problems. It was first proposed by M. Dorigoin 1992 in his Ph. D. dissertation [54]. The first instance of the application of Ant Colony Optimization in Probabilistic Roadmap is the work [55, 56]. In [57] an optimal path planning for mobile robots based on intensified ACO algorithm is developed. Also in 2004, ACO was used to plan the best path [58]. ACRMP is presented in [60]. An articulated RMP using ACO is introduced in [59]. Also, a path planning based on ACO and distributed local navigation for multi robot systems is developed in [66]. Finally, an approach based on numerical Potential Fields (PF) and ACO is introduced in [61] to path planning in dynamic environment. In [66] a

The notion of using Simulated Annealing (SA) for roadmap was initiated in [67]. PFapproach was integrated with SA to escape from local minimaand evaluation [68,70]. Estimates using SA for a multi-path arrival and path planning for mobile robotic based on PF, is introduced in [69, 72]. A path planning based on PF approach with SA is developed in [72].Finally, in [71] was presented a multi operator based SA approach for navigation in uncertain environments.

A case particle are militar applications, with an uninhabited combat air vehicle (UCAV). The techniques employed, have been proposed to solve this complicated multi-constrained

namics environment.Finally, a newperspective global optimization is propossed [40].

fast two-stage ACO algorithm for robotic path planning is used.

function evaluation.

A principal of these techniques, is by example, with a colony can solve problems unthinkable for individual ants, such as finding the shortest path to the best food source, allocating workers to different tasks, or defending a territory from neighbors. As individuals, ants might be tiny dummies, but as colonies they respond quickly and effectively to their environment. They do it with something called Swarm Intelligence.

These novel techniques are nature-inspired stochastic optimization methods that iteratively use random elements to transfer one candidate solution into a new, hopefully better, solution with regard to a given measure of quality.

We cannot expect them to find the best solution all the time, but expect them to find the good enough solutions or even the optimal solution most of the time, and more importantly, in a reasonably and practically short time. Modern meta-heuristic algorithms are almost guaran‐ teed to work well for a wide rangeof tough optimization problems.

## **1.1. Previous work**

Path planning is an essential task navigation and motion control of autonomous robot. This problem in mobile robotic is not simple, and the same is attached by two distint approaches. In the *classical approaches* present by Raja et al.[49]cited according to [3]the C-space, where the representation of the robot is a simple point. The same approach is described by Latombe's book [4]. Under concept of C-space, are developedpath planning approaches with roadmap and visibility graph was introduced[5].Sparce envioroments considering to polygonal obstacles and their edges [6, 7]. The Voroni diagram was introduced [8]. Other approach for roadmap andrecient applications in [9,10]. Cell descomposition approach [11, 12, 13, 14, 15,16]. A efficente use of grids [17].

A related problem is when both, the map and the vehicle position are not know. This problem is usually nown as Simultaneous Localization and Map Building (SLAM), and was originally introduced [18]. Until recently,have been significative advances in the solution of the SLAM problem [19,20,21,22].

Kalman filter methods can also be extended to perform simultaneous localization and map building. There have been several applications in mobile robotic, such as indoors, underwater and outdoors. The potential problems with SLAM algorithm have been the computational requeriments. The complexy of original algortihm is of *O(N3 )* but, can be reduced to *O (N2 )*where, N will be the number of landmarks in the map [23].

In computational complexity theory, path planning is classified as an *NP* (nondeterministic polynomial time) complete problem [33]. *Evolutionary approaches* provide these solutions. Where, one of the high advantage of heuristic algorithms, is that it can produce an acceptable solution very quickly, which is especially used for solving *NP-complete problems*.

A first path planning approach of a mobile robot trated as non-deterministic polynomial time hard (*NP-hard*) problem is [31]. Moreover, even more complicated are the environment dynam‐ ic, the classic approaches to be incompetent [32]. Hence, evolutionary approaches such as Tabu Search (TS), Artificial Neural Network (ANN), Genetic Algorithm (GA), Particle Swarm Opti‐ mization (PSO), Ant Colony Optimization (ACO) and Simulated Annealing (SA), name a few, are employed to solve the path planning problem efficiently although not always optimal.

Given an environment, where a mobile robot must determine a route in order to reach a target destination, we found the shortest path that this robot can follow. This goal is reach using bioinspired techniques, as Ant Colony Optimization (ACO)and the Genetics Algorithms (GA).

A principal of these techniques, is by example, with a colony can solve problems unthinkable for individual ants, such as finding the shortest path to the best food source, allocating workers to different tasks, or defending a territory from neighbors. As individuals, ants might be tiny dummies, but as colonies they respond quickly and effectively to their environment. They do

These novel techniques are nature-inspired stochastic optimization methods that iteratively use random elements to transfer one candidate solution into a new, hopefully better, solution

We cannot expect them to find the best solution all the time, but expect them to find the good enough solutions or even the optimal solution most of the time, and more importantly, in a reasonably and practically short time. Modern meta-heuristic algorithms are almost guaran‐

Path planning is an essential task navigation and motion control of autonomous robot. This problem in mobile robotic is not simple, and the same is attached by two distint approaches. In the *classical approaches* present by Raja et al.[49]cited according to [3]the C-space, where the representation of the robot is a simple point. The same approach is described by Latombe's book [4]. Under concept of C-space, are developedpath planning approaches with roadmap and visibility graph was introduced[5].Sparce envioroments considering to polygonal obstacles and their edges [6, 7]. The Voroni diagram was introduced [8]. Other approach for roadmap andrecient applications in [9,10]. Cell descomposition approach [11, 12, 13, 14, 15,16].

A related problem is when both, the map and the vehicle position are not know. This problem is usually nown as Simultaneous Localization and Map Building (SLAM), and was originally introduced [18]. Until recently,have been significative advances in the solution of the SLAM

Kalman filter methods can also be extended to perform simultaneous localization and map building. There have been several applications in mobile robotic, such as indoors, underwater and outdoors. The potential problems with SLAM algorithm have been the computational

In computational complexity theory, path planning is classified as an *NP* (nondeterministic polynomial time) complete problem [33]. *Evolutionary approaches* provide these solutions. Where, one of the high advantage of heuristic algorithms, is that it can produce an acceptable

solution very quickly, which is especially used for solving *NP-complete problems*.

*)* but, can be reduced to *O*

teed to work well for a wide rangeof tough optimization problems.

requeriments. The complexy of original algortihm is of *O(N3*

*)*where, N will be the number of landmarks in the map [23].

it with something called Swarm Intelligence.

54 Recent Advances on Meta-Heuristics and Their Application to Real Scenarios

with regard to a given measure of quality.

**1.1. Previous work**

A efficente use of grids [17].

problem [19,20,21,22].

*(N2*

Genetic Algorithm (GA) based search and optimization techniques have recently found increasing use in machine learning, scheduling, pattern recognition, robot motion planning, image sensing and many other engineering applications. The first research of Robot Motion Planning (RMP), according to Masehian et al.[53] although GA, was first used in [42] and [43]. An approach for solution the problem of collision-free paths is presented in [44].GA was applied [45, 46, 48] in planning multi-path for 2D and 3D environment dimension and shortest path problem. A novel GA searching approach for dynamic constrained multicast routing is developed in [49]. Parallel GA [50], is used for search and constrained multi-objective optimi‐ zation. Differentials optimization used hybrid GA, for path planning and shared-path protections has been extended in [51, 52]. In [62, 63, 64, 65], has been a compared of differential algorithms optimization GA (basically for dynamic environment), subjected to penalty function evaluation.

By other side, thetechnique PSO have some any advantages [35], such as simple implementa‐ tionwith a few parameters to be adjusted. Binary PSO [37] withouta mutation operator[36]are used to optimize the shortest path. Planning in dynamic environment, that containing invalid paths (repair by a operator mutation), are subjected to penalty function evaluation [38]. Recent‐ ly, [39] proposedwith multi-objective PSO and mutate operator path planning in dangerous dy‐ namics environment.Finally, a newperspective global optimization is propossed [40].

Ant Colony Optimization (ACO) algorithms have been developed to mimic the behavior of real ants to provide heuristic solutions for optimization problems. It was first proposed by M. Dorigoin 1992 in his Ph. D. dissertation [54]. The first instance of the application of Ant Colony Optimization in Probabilistic Roadmap is the work [55, 56]. In [57] an optimal path planning for mobile robots based on intensified ACO algorithm is developed. Also in 2004, ACO was used to plan the best path [58]. ACRMP is presented in [60]. An articulated RMP using ACO is introduced in [59]. Also, a path planning based on ACO and distributed local navigation for multi robot systems is developed in [66]. Finally, an approach based on numerical Potential Fields (PF) and ACO is introduced in [61] to path planning in dynamic environment. In [66] a fast two-stage ACO algorithm for robotic path planning is used.

The notion of using Simulated Annealing (SA) for roadmap was initiated in [67]. PFapproach was integrated with SA to escape from local minimaand evaluation [68,70]. Estimates using SA for a multi-path arrival and path planning for mobile robotic based on PF, is introduced in [69, 72]. A path planning based on PF approach with SA is developed in [72].Finally, in [71] was presented a multi operator based SA approach for navigation in uncertain environments.

A case particle are militar applications, with an uninhabited combat air vehicle (UCAV). The techniques employed, have been proposed to solve this complicated multi-constrained optimization problem, solved contradiction between the global optimization and excessive information. Such techniques used to solution this problem are differential evolution [24], artificial bee colony [29], genetic algorithm [25], water drops optimization (chaotic and intelligent) [30] and ant colony optimization algorithm [26, 27, 28].

A proposal of spatial representation is to sample discretely the two- or three-dimensional environment. This isa sample space in *cells* of a *uniform grid*for two-dimensional or considering

Meta-Heuristic Optimization Techniques and Its Applications in Robotics

http://dx.doi.org/10.5772/54460

57

*Geometric maps* are composed of the union of simple geometric primitives. Such maps are characterized by two key properties: the set of basic primitives used for describing objects, and the set of operators used to manipulate objects. The fundamentals problems with this technique

Within the *geometric representations*, the *topological representation* can be used to solve abstract tasks that are not void reliance on error-prone metric, provided an explicit representation of connectivity between regions or objects. A topological representation is based on an abstraction of the environment in terms of discrete places (*landmarks*) with edges connecting them. In [76], present an example of this topological representation, where after of delimited region of

The basic path planning problem refers to determining a path in configuration space between an initial configuration (start pose) of the robot and a final configuration (finish pose). Therefore, several approaches for path planning exist of course a particular problem of application, as well as the kinematic constraints of the robot [77]. Although is neccesary make a serie of simplied with respect to real environment, the techniques for path planning can be

To efficiently compute of search the path through techniques as A\* [14] and Dynamic pro‐ gramming. The difference between them usually, resides in the simplicity to define or compute the evaluation function, which hardly depends on the nature of the environment and the specific problem.Other techniques for mapping a robot's environment inside a discrete

Path planning in a *continuum space* is consideret as the determination of an appropriate trajectory within this continuum. Two of the most known techniques for continuous state space are the potentialfields [61] and the vector field histogram methods. Alternatively,these algorithms, can be based on the *bug algorithm*[41], guaranteed to find a path from the beginning until target, if such path exists. Unfortunately, these methods can be generated arbitrarytra‐

Many results in the literature indicate that metaheuristics are the state-of-the-art techniques for problems for which there is non efficient algortihm. Thus, meta-heuristics approach approximationsallow solving complex optimization problems. Although these methods, cannot guarantee that the best solution found after termination criteria are satisfied or indeed

the volume of elementsthat are used to represent objets named *voxel*.

interest, used a GA for the landmark search through the image.

searchable space include visibility graphs and Voronoi diagrams [75].

are lack of stability, uniqueness, and potentiality.

group in *discrete state space* and *continuum space*.

jectories worse than the optimal path to the target.

its global optimal solution to the problem.

**3. Population-based meta-heuristic optimization**

**2.3. Path planning**

## **2. Robot navigation**

## **2.1. Introduction**

Mobile robots and manipulator robots are increasingly being employed in many automated envionments. Potential applications of mobile robots include a wide range such a service robots for elderly persons, automated guide vehicles for transferring goods in a factory, unmmaned bomb disposal robots and planet exploration robots. In all thes applications, the mobile robots perform their navegation task using the building blocks (see figure 1) [1], the same, is based on [2] known with the Deliberative Focus.

**Figure 1.** Deliberative Focus.

While perception of enviroment refers to understanding its sensory data, finding its pose or configuration in the surroundings is localization and map building. Planning the path in accordance with the task by using cognitive decision making is an essential phase before actually accomplishing the preferred trajectory by controlling the motion. As each of the building blocks is by itself a vast research field.

## **2.2. Map representation methods**

When, Rencken in1993 [73] defined the map building problem as sensing capacity of robot, can be split in two, where robot know a pre-existing map or it has to build this, through information of the environment. According to above is assumed that the robot begins a exploration without having any knowledge of the environment, but with a exploration strategy, and it depends strongly on the kind of sensors used, the robot builds its own perception of environment[74].

A proposal of spatial representation is to sample discretely the two- or three-dimensional environment. This isa sample space in *cells* of a *uniform grid*for two-dimensional or considering the volume of elementsthat are used to represent objets named *voxel*.

*Geometric maps* are composed of the union of simple geometric primitives. Such maps are characterized by two key properties: the set of basic primitives used for describing objects, and the set of operators used to manipulate objects. The fundamentals problems with this technique are lack of stability, uniqueness, and potentiality.

Within the *geometric representations*, the *topological representation* can be used to solve abstract tasks that are not void reliance on error-prone metric, provided an explicit representation of connectivity between regions or objects. A topological representation is based on an abstraction of the environment in terms of discrete places (*landmarks*) with edges connecting them. In [76], present an example of this topological representation, where after of delimited region of interest, used a GA for the landmark search through the image.

## **2.3. Path planning**

optimization problem, solved contradiction between the global optimization and excessive information. Such techniques used to solution this problem are differential evolution [24], artificial bee colony [29], genetic algorithm [25], water drops optimization (chaotic and

Mobile robots and manipulator robots are increasingly being employed in many automated envionments. Potential applications of mobile robots include a wide range such a service robots for elderly persons, automated guide vehicles for transferring goods in a factory, unmmaned bomb disposal robots and planet exploration robots. In all thes applications, the mobile robots perform their navegation task using the building blocks (see figure 1) [1], the same, is based

While perception of enviroment refers to understanding its sensory data, finding its pose or configuration in the surroundings is localization and map building. Planning the path in accordance with the task by using cognitive decision making is an essential phase before actually accomplishing the preferred trajectory by controlling the motion. As each of the

When, Rencken in1993 [73] defined the map building problem as sensing capacity of robot, can be split in two, where robot know a pre-existing map or it has to build this, through information of the environment. According to above is assumed that the robot begins a exploration without having any knowledge of the environment, but with a exploration strategy, and it depends strongly on the kind of sensors used, the robot builds its own

intelligent) [30] and ant colony optimization algorithm [26, 27, 28].

56 Recent Advances on Meta-Heuristics and Their Application to Real Scenarios

**2. Robot navigation**

**Figure 1.** Deliberative Focus.

on [2] known with the Deliberative Focus.

building blocks is by itself a vast research field.

**2.2. Map representation methods**

perception of environment[74].

**2.1. Introduction**

The basic path planning problem refers to determining a path in configuration space between an initial configuration (start pose) of the robot and a final configuration (finish pose). Therefore, several approaches for path planning exist of course a particular problem of application, as well as the kinematic constraints of the robot [77]. Although is neccesary make a serie of simplied with respect to real environment, the techniques for path planning can be group in *discrete state space* and *continuum space*.

To efficiently compute of search the path through techniques as A\* [14] and Dynamic pro‐ gramming. The difference between them usually, resides in the simplicity to define or compute the evaluation function, which hardly depends on the nature of the environment and the specific problem.Other techniques for mapping a robot's environment inside a discrete searchable space include visibility graphs and Voronoi diagrams [75].

Path planning in a *continuum space* is consideret as the determination of an appropriate trajectory within this continuum. Two of the most known techniques for continuous state space are the potentialfields [61] and the vector field histogram methods. Alternatively,these algorithms, can be based on the *bug algorithm*[41], guaranteed to find a path from the beginning until target, if such path exists. Unfortunately, these methods can be generated arbitrarytra‐ jectories worse than the optimal path to the target.

## **3. Population-based meta-heuristic optimization**

Many results in the literature indicate that metaheuristics are the state-of-the-art techniques for problems for which there is non efficient algortihm. Thus, meta-heuristics approach approximationsallow solving complex optimization problems. Although these methods, cannot guarantee that the best solution found after termination criteria are satisfied or indeed its global optimal solution to the problem.

## **3.1. Optimization**

The theory of optimization refers to the quantitative study of optima and the methods for finding them. Global Opitimization is the branch of applied mathematics and numerical analysis that focuses on well optimization. Of course T. Weise in [81], the goal of global optimization is to find to the best possible elements x\* form a set X according to a set criteria *F = {f1, f2,..., fn}*These criteria are expressed as mathematical function, that so-called objetive functions. In the Weise's book[81] the Objective Function is described as:

∀ *g* ∈G ∃ *x* ∈X: *gpm* (*g*)= *x* (1)

Meta-Heuristic Optimization Techniques and Its Applications in Robotics

9

59

http://dx.doi.org/10.5772/54460

∀ *g* ∈G ∃ *x* ∈X:*P*(*gpm* (*g*)= *x*)>0 (2)

a. Calculated through evaluation to aptitude of individual

b. Select two individuals, assign its aptitude probabilistic. c. Applied Genetic Operators. Only two gene with best

ii. Mutation to the equal rate probabilistic. d. Extinction (or null reproduction) by a poor aptitude

i. Crossover to the couples.

Swarm intelligence (SI) is based on the collective behavior. The collective behaviorthat emerges is a form of autocatalytic behaviour [34], self-organized systems. It's typically made up of a population of simple agents interacting locally with one another and with their environment. Natural examples of SI include ant colonies, bird flocking, animal herding, bacterial growth, and fish schooling. Nowadays swarm intelligence more generically, referes design complex

Particle Swarm optimization (PSO), is a form of swarm intelligence, in which the behavior of a biological social systems. A particullaritied is when, looks for food, its individuals will spreadin the environment and move around independently. According to [81] Particle Swarm Optimization, a swarm of particles (individuals) in a n-dimensional search space G is simu‐ lated, where each particle *q* has a position genotype *p.g* ∈ Q ⊆ R<sup>n</sup> in this case n-dimensional is two, likewise a velocity *p.v* ∈ Rn. Therefore, each particle p has a memory holding its best position (can be reference to minim distance Euclidian of target (*p*) respect to *q*) best (*q*) ∈ G. In order to realize the social component, the particle furthermore knows a set of topological

This set could be defined to contain adjacent particles within a specific perimeter, using the

*<sup>n</sup>* (**p** – *<sup>q</sup>*(*x*,*<sup>y</sup>*))<sup>2</sup> (3)

∥<sup>2</sup> = ∑*i*=1

**3.4. Swarm intelligence**

**Figure 3.** GA Modified.

adaptive systems.

*3.4.1. Particle Swarm Optimization*

1. Random Initial Population.

(gpm) of course (2).

aptitude probabilistic.

probabilistic individual.

e. Reproduction.

3. Until finish condition

neighbors N(*q*) which could well be strong landmarks in the Rn.

*deuc* = ∥ *p* - *qi*

Euclidian distance measure deuc specified by

2. Repeat

*DefinitionObjective Funcition.* An objective funtion *f* : X↦ Y with Y ⊆ R is a mathematical function which is subject to optimization.

Global optimization comprises all techniques that can be used to find the best elements x\* in X with respect to such criteria *f Є F*.

#### **3.2. Nature-inspied meta-heuristic optimization**

The high increase in the size of the search space and the need of proccesing in real-time has motivated recent researchs to solving scheduling problem using nature inspired heuristic techniques. The principal components of any metaheuristic algorithms are: intensification and diversification, or explotation and exploration. The optimal combination of these will usually ensure that a global optimization is achievable. 7

**Figure 2.** General Description for Nature Inspired Algorithm

## **3.3. Genetic algorithms**

In essence, a Genetic Algorithm (GA) is a search method based on the abstraction Darwinian evolution and natural selection of biological systems and respresenting them in the mathe‐ matical operator: croosover or recombination, mutation, fitness, and selection of the fittest.

The application of GA to path planning problem requires development of a chromosome

representation of the path, appropriate constraint definition to minimize path distance, such that is providing smooth paths. It is assumed that the environment is static and known.

Cited according to [81], the genotypes are used in the reproduction operations whereas the values of the objective funtions *f*(or fitness function), where*f Є F*are computed on basis of the phenotypes in the problme space X which are obtained via the genotype-phenotype mapping (gpm) [82, 83, 84, 85], where G is a space any binary chain.

Meta-Heuristic Optimization Techniques and Its Applications in Robotics http://dx.doi.org/10.5772/54460 59

$$\forall \, \mathfrak{g} \in \mathbf{G} \, \exists \, \mathbf{x} \in \mathbb{X} . \operatorname{gpm} \left( \mathfrak{g} \right) = \mathbf{x} \tag{1}$$

$$\forall \, \forall \, \mathcal{g} \in \mathsf{G} \; \exists \, \mathbf{x} \; \in \mathsf{X} \colon P(\operatorname{\mathbf{g}pm}(\mathbf{g}) = \mathbf{x}) > 0 \tag{2}$$

 **Figure 3.** GA Modified.

**3.1. Optimization**

**3.3. Genetic algorithms**

The theory of optimization refers to the quantitative study of optima and the methods for finding them. Global Opitimization is the branch of applied mathematics and numerical analysis that focuses on well optimization. Of course T. Weise in [81], the goal of global optimization is to find to the best possible elements x\* form a set X according to a set criteria *F = {f1, f2,..., fn}*These criteria are expressed as mathematical function, that so-called objetive

*DefinitionObjective Funcition.* An objective funtion *f* : X↦ Y with Y ⊆ R is a mathematical

Global optimization comprises all techniques that can be used to find the best elements x\* in

The high increase in the size of the search space and the need of proccesing in real-time has motivated recent researchs to solving scheduling problem using nature inspired heuristic techniques. The principal components of any metaheuristic algorithms are: intensification and diversification, or explotation and exploration. The optimal combination of these will usually

1. Initialize the solution vectors and all parameters.

nature o social behaviors. b. Evaluate the new candidate solutions.

In essence, a Genetic Algorithm (GA) is a search method based on the abstraction Darwinian evolution and natural selection of biological systems and respresenting them in the mathe‐ matical operator: croosover or recombination, mutation, fitness, and selection of the fittest. The application of GA to path planning problem requires development of a chromosome

representation of the path, appropriate constraint definition to minimize path distance, such that is providing smooth paths. It is assumed that the environment is static and known.

Cited according to [81], the genotypes are used in the reproduction operations whereas the values of the objective funtions *f*(or fitness function), where*f Є F*are computed on basis of the phenotypes in the problme space X which are obtained via the genotype-phenotype mapping

a. Generation a new candidate solutions via the

2. Evaluate the candidate solutions.

4. Until meet optimal criteria.

7

functions. In the Weise's book[81] the Objective Function is described as:

function which is subject to optimization.

**3.2. Nature-inspied meta-heuristic optimization**

58 Recent Advances on Meta-Heuristics and Their Application to Real Scenarios

ensure that a global optimization is achievable.

**Figure 2.** General Description for Nature Inspired Algorithm

3. Repeat

(gpm) [82, 83, 84, 85], where G is a space any binary chain.

X with respect to such criteria *f Є F*.

#### **3.4. Swarm intelligence**

Swarm intelligence (SI) is based on the collective behavior. The collective behaviorthat emerges is a form of autocatalytic behaviour [34], self-organized systems. It's typically made up of a population of simple agents interacting locally with one another and with their environment. Natural examples of SI include ant colonies, bird flocking, animal herding, bacterial growth, and fish schooling. Nowadays swarm intelligence more generically, referes design complex adaptive systems.

#### *3.4.1. Particle Swarm Optimization*

Particle Swarm optimization (PSO), is a form of swarm intelligence, in which the behavior of a biological social systems. A particullaritied is when, looks for food, its individuals will spreadin the environment and move around independently. According to [81] Particle Swarm Optimization, a swarm of particles (individuals) in a n-dimensional search space G is simu‐ lated, where each particle *q* has a position genotype *p.g* ∈ Q ⊆ R<sup>n</sup> in this case n-dimensional is two, likewise a velocity *p.v* ∈ Rn. Therefore, each particle p has a memory holding its best position (can be reference to minim distance Euclidian of target (*p*) respect to *q*) best (*q*) ∈ G. In order to realize the social component, the particle furthermore knows a set of topological neighbors N(*q*) which could well be strong landmarks in the Rn.

This set could be defined to contain adjacent particles within a specific perimeter, using the Euclidian distance measure deuc specified by

$$d\_{euc} = \parallel \ p - \ q\_i \parallel\_2 = \sqrt{\sum\_{i=1}^{n} (\mathbf{p} - q\_{(x,y)})^2} \tag{3}$$

$$d\_{p, \text{g}} = \min \left\{ d\_{euc}(p\_\prime, q\_i) \, \middle| \, q\_i \in \mathbf{Q} \right\} \tag{4}$$

$$p.v.\ \neg \omega \cdot v\_i + \alpha \cdot rnd() \cdot (p\_i \cdot p.\mathbf{g}) + \beta \cdot rnd() \cdot (\mathbf{g}\_{best} \cdot p.\mathbf{g})\tag{5}$$

$$p.g = d\_{p.g} + p.v.\tag{6}$$

This pheromone update rule leads to an increase of pheromone on solution components that

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Applied the performance and success of an evolutionary optimization approach given by a set

**•** Its basic parameter settings like the population size *≤ ms* or the crossover and mutation rates.

**•** The genotype-phenotype mapping connecting the search Space and the problem space.

The properties of their crossover and mutatrion operations are well known and an extensive

**•** Select four members to population (two best explorers and two best workers).

A string chromosome can either be a fixed-length tuple (9) or a variable-length list (10).

. This implies, that *j* is visited only once.

G=∀(*g* 1 , …, *g n* : *g i* ∈G*i*∀*<sup>i</sup>* ∈1…*n*) (9)

G={∀*lists*(*g* : *g i* ∈G*<sup>T</sup>* ∀0≤*i* ≤*len*(*g*)} (10)

have been found in high quality solutions.

**4.1. Representation and initial population**

**•** Initial population with a frequency *f*.

is evaluated with a cost of T+

**4.2. Genetic operators and parameters**

body of search on them is avaible [87, 88].

The natural selections is improved the next form:

**•** The probability the mutation is same to the percentage this.

**•** The transition rule is (9).

**•** Applied crossover.

**•** Performed null reproduction.

of objective functions *F* and a problem space X is defined by:

**•** Whether it uses an archive Arc of the best individuals found.

**•** The fitness assignment process and the selection algorithm.

**•** All ants begin in the same node. Applied (6) to node initiate.

**4. Behavior fusion learning**

Such that:

**•** L+

#### *3.4.2. Ant Colony Optimization*

Ant Colony Optimization (ACO) has been formalized into a meta-heuristic for combinatorial optimization problems by Dorigo and co-workers [78], [79].This behavior, described by Deneubourg [80] enables ants to find shortest paths between food sources and their nest. While walking, they decide about a direction to go, they choose with higher probability paths that are marked by stronger pheromone concentrations.

ACO has risen sharply. In fact, several successful applications of ACO to a wide range of different discrete optimization problems are now available. The large majorities of these applications are to *NP-hard* problems; that is, to problems for which the best known algorithms that guarantee to identify an optimal solution have exponential time worst case complexity. The use of such algorithms is often infeasible in practice, and ACO algorithms can be useful for quickly finding high-quality solutions.

ACO algorithms are based on a parameterized probabilistic model the *pheromone model*, is used to model the chemical pheromone trails. Artificial ants incrementally construct solutions by adding opportunely defined solution components of the partial solutions in consideration. The first ACO algorithm proposed in the literature is called Ant System (AS) [56].

In the construction phase, an ant incrementally builds a solution by adding solution compo‐ nents to the partial solution constructed so far. The probabilistic choice of the next solution component to be added is done by means of transition probabilities, which in AS are deter‐ mined by the following *state transition rule*:

$$\mathbf{P}(\mathbf{c}\_{r}\big|\mathbf{s}\_{a}\big|\mathbf{c}\_{l}\big] = \begin{cases} \mathbbm{I}\eta\_{r}\big|\mathbf{c}\_{r}\big|\mathbf{f}^{\alpha}\\ \sum\_{\substack{\mathbf{c}\_{u}\neq\{\mathbf{s}\_{a}\big|\mathbf{c}\_{l}\big]\big\}}} \mathbbm{I}\eta\_{u}\big|\mathbf{f}^{\alpha}\big|\mathbf{f}\_{u}\big|^{\beta} & \quad \text{if}\quad \mathbf{c}\_{r}\in\operatorname{J}(\mathbf{s}\_{a}[\mathbf{c}\_{l}])\\ \mathbf{0} & \quad \text{else} \end{cases} \tag{7}$$

otherwise

Once all ants have constructed a solution, the *online delayed pheromone update rule* is applied:

$$\begin{aligned} \pi\_j &\leftarrow (1-\rho)\cdot\tau\_j + \sum\_{a \in A} \Delta \tau^{s\_a}\_{\ \ r\_j} \\ \text{where} \\ \Delta \tau^{s\_a}\_{\ \ j} &= \begin{cases} F(s\_q) & \text{if } c\_j \text{ is a component of } s\_a \\ & \text{otherwise} \end{cases} \end{aligned} \tag{8}$$

This pheromone update rule leads to an increase of pheromone on solution components that have been found in high quality solutions.

## **4. Behavior fusion learning**

## **4.1. Representation and initial population**

Applied the performance and success of an evolutionary optimization approach given by a set of objective functions *F* and a problem space X is defined by:


## Such that:

*dp*.*<sup>g</sup>* =min {*deuc*(*p*, *qi*

*p*.*g* = *dp*.*<sup>g</sup>*

Ant Colony Optimization (ACO) has been formalized into a meta-heuristic for combinatorial optimization problems by Dorigo and co-workers [78], [79].This behavior, described by Deneubourg [80] enables ants to find shortest paths between food sources and their nest. While walking, they decide about a direction to go, they choose with higher probability paths that

ACO has risen sharply. In fact, several successful applications of ACO to a wide range of different discrete optimization problems are now available. The large majorities of these applications are to *NP-hard* problems; that is, to problems for which the best known algorithms that guarantee to identify an optimal solution have exponential time worst case complexity. The use of such algorithms is often infeasible in practice, and ACO algorithms can be useful

ACO algorithms are based on a parameterized probabilistic model the *pheromone model*, is used to model the chemical pheromone trails. Artificial ants incrementally construct solutions by adding opportunely defined solution components of the partial solutions in consideration. The

In the construction phase, an ant incrementally builds a solution by adding solution compo‐ nents to the partial solution constructed so far. The probabilistic choice of the next solution component to be added is done by means of transition probabilities, which in AS are deter‐

first ACO algorithm proposed in the literature is called Ant System (AS) [56].

( [ ])

*r r ral r al c js c u u csc Js c* a b

Î

*a A*

å

Î

*u al*

h t

[ ][ ] P( [ ]) if c ( [ ]) [ ][ ]

h t

a b

Once all ants have constructed a solution, the *online delayed pheromone update rule* is applied:

*a*

*s*

 t

( ) if isacomponentof

otherwise

*jq j <sup>a</sup> Fs c s*

<sup>ï</sup> <sup>=</sup> <sup>í</sup> <sup>Î</sup>

otherwise

å (7)

(8)

0

{ 0

(1 )

 rt

*j jj*

¬- ×+ D

where

t

*a*

D =

*s*

t

ì

ï î

*3.4.2. Ant Colony Optimization*

are marked by stronger pheromone concentrations.

60 Recent Advances on Meta-Heuristics and Their Application to Real Scenarios

for quickly finding high-quality solutions.

mined by the following *state transition rule*:

)|*qi* ∈ Q} (4)

+ *p*.*v*. (6)

*p*.*v*. =*ω* ⋅*vi* + *α* ⋅ *rnd*() ⋅ (*pi* - *p*.*g*) + *β* ⋅ *rnd*() ⋅ (*gbest* - *p*.*g*) (5)


#### **4.2. Genetic operators and parameters**

The properties of their crossover and mutatrion operations are well known and an extensive body of search on them is avaible [87, 88].

The natural selections is improved the next form:


A string chromosome can either be a fixed-length tuple (9) or a variable-length list (10).

$$\mathbf{G} = \forall \left( \emptyset \mathbf{1} \mathbf{1} \mathbf{1} \dots \mathbf{1}, \; \lg \mathbf{n} \mathbf{1}; \lg \mathbf{i} \mathbf{j} \in \mathbf{G}\_i \forall\_i \in \mathbf{1} \dots n \right) \tag{9}$$

$$\mathbf{G} = \{ \forall \; lits \& \{ \mathbf{g} : \mathbf{g} \| \mathbf{i} \} \in \mathbf{G}\_T \; \forall \; 0 \le i \le len(\mathbf{g}) \} \tag{10}$$

## *4.2.1. Selection*

Elitist selection, returns the *k*<*ms*(number of individuals to be placed into the mating)best elements from the list. In general evolutionary algorithms, it should combined with a fitness assignment processs that incorporates diversity information in order to prevent premature convergence.The elitist selection, force to the best individuals, according with to fitness function, to move to the next iteration.

*4.2.3. Mutation*

*4.2.4. Null reproduction or extinction*

initialize it with random values.

**4.3. Path planning through evolutive ACO**

*J* ={

is obtain the value more high of pheromone.

A new transiction rule. The rule of the equation(7) is modified by:

*argmaxu ji*

range *[0, 1]*, q*0* is a parameter adjusts in the range (0 ≤ q*0* ≤0) y J Є *Ji*

*pij* 

<sup>∆</sup>*τij*(*t*)= <sup>1</sup>

The update rule allows a global update, only for the path best.

*<sup>k</sup>* <sup>=</sup>*α*sin *<sup>β</sup>* <sup>⋅</sup>*τij*

optimized using a genetic algorithm.

The mutation operation applied, mutate N↦N is used to create a new genotype *gn* Є N by modifying an existing one. The way this modification is performed is application-dependent.

Therefore, the mutation performed only one of the three gene in the chromosome offspring. The selected gen must a change in his allele (inherited), by a new random value in the same.

After the crossover of the mutation (if is the case) is need performed a null reproduction or extinction of the ant workers or ant explorer. This allows the creation of fixed-length strings individuals means simple to create a new tuple of the structure defined by the genome and

The modifcation of the proposed ACO algorithm is applied. Due to in this modification have several parameters that determine the behavior of proposed algorithm, these parameters were

*<sup>k</sup>* { *τiu*(*t*) ⋅ *ηiu* } *if q* ≤*q*<sup>0</sup>

This rule allows the exploration. Where, an ant *k* is move through *i* and *j*, with a distribution *q* in a

The transition rule, therefore is different (of equation 7) when q≤q*0*, this meaning is a heuristic knowledge characterized in the pheromone. The amount of pheromone τij ∈Z, can be positive and negative, allowing a not-extinction, but rather to continue comparison between its these,

 *J if q* >*q*<sup>0</sup>

*gn* =*mutate*(*g*): *g* ∈N (11)

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*k*

, is a state select based in

*<sup>k</sup>* (*t*) + *γ* (13)

*<sup>L</sup>* <sup>+</sup> *whereL* <sup>+</sup> <sup>∈</sup>*<sup>T</sup>* <sup>+</sup> (14)

(12)

It may happen in a randomized or in a deterministic fashion.

#### *4.2.2. Crossover*

Amongst all evolutionary algortihms, genetic algorithms have the recombination operation which probably comes closest to the natural paragon.

Example of a fixed-length string, applied to the best explorer and the best workers obtain the two best gene.

14 Book Title

16 Book Title

**Figure 4.** Crossover Algorithm.

**Figure 5.** Example of Crossover, of course steps (b) and (c) crossover algorithm.

## *4.2.3. Mutation*

*4.2.1. Selection*

*4.2.2. Crossover*

two best gene.

**Figure 4.** Crossover Algorithm.

**α1 0 β1 1 γ1 0** 

 **α1 0 β1 1 γ2 1** 

function, to move to the next iteration.

which probably comes closest to the natural paragon.

62 Recent Advances on Meta-Heuristics and Their Application to Real Scenarios

2. Repeat

3. Until ms.

**α2 1 β2 0 γ2 1** 

**Figure 5.** Example of Crossover, of course steps (b) and (c) crossover algorithm.

Best ant-explorer Best ant-worker

Elitist selection, returns the *k*<*ms*(number of individuals to be placed into the mating)best elements from the list. In general evolutionary algorithms, it should combined with a fitness assignment processs that incorporates diversity information in order to prevent premature convergence.The elitist selection, force to the best individuals, according with to fitness

Amongst all evolutionary algortihms, genetic algorithms have the recombination operation

Example of a fixed-length string, applied to the best explorer and the best workers obtain the

14 Book Title

b. The value is one, pass to gen father. c. The value is cero, pass to gen mother.

16 Book Title

**<sup>α</sup> <sup>β</sup> <sup>γ</sup>** Chromosome.

> **α3 1 β3 0 γ3 1**

father mother father mother

 **α4** 

**α4 0 β4 0 γ4 1** 

**0 β3 0 γ4 1** 

1. Init random value α, β and γ.

a. Applied mask.

The mutation operation applied, mutate N↦N is used to create a new genotype *gn* Є N by modifying an existing one. The way this modification is performed is application-dependent. It may happen in a randomized or in a deterministic fashion.

$$\mathfrak{g}\_n = \text{mutate}(\mathfrak{g}) \colon \mathfrak{g} \in \mathbb{N} \tag{11}$$

Therefore, the mutation performed only one of the three gene in the chromosome offspring. The selected gen must a change in his allele (inherited), by a new random value in the same.

#### *4.2.4. Null reproduction or extinction*

After the crossover of the mutation (if is the case) is need performed a null reproduction or extinction of the ant workers or ant explorer. This allows the creation of fixed-length strings individuals means simple to create a new tuple of the structure defined by the genome and initialize it with random values.

#### **4.3. Path planning through evolutive ACO**

The modifcation of the proposed ACO algorithm is applied. Due to in this modification have several parameters that determine the behavior of proposed algorithm, these parameters were optimized using a genetic algorithm.

A new transiction rule. The rule of the equation(7) is modified by:

$$f = \begin{cases} \arg\max\_{u \in \hat{h}} \left[ \mathbb{T} \tau\_{iu}(t) \right] \cdot \left[ \mathbb{T} \eta\_{iu} \right] & \text{if } q \le q\_0 \\\\ f & \text{if } q > q\_0 \end{cases} \tag{12}$$

This rule allows the exploration. Where, an ant *k* is move through *i* and *j*, with a distribution *q* in a range *[0, 1]*, q*0* is a parameter adjusts in the range (0 ≤ q*0* ≤0) y J Є *Ji k* , is a state select based in

$$p\_{i\uparrow}^{k} = \alpha \sin \beta \cdot \tau\_{i\uparrow}^{k}(t) + \gamma \tag{13}$$

The transition rule, therefore is different (of equation 7) when q≤q*0*, this meaning is a heuristic knowledge characterized in the pheromone. The amount of pheromone τij ∈Z, can be positive and negative, allowing a not-extinction, but rather to continue comparison between its these, is obtain the value more high of pheromone.

$$
\Delta \tau\_{ij}(t) = \frac{1}{L^\*} \omega \\
where \ L^\* \in T^\* \tag{14}
$$

The update rule allows a global update, only for the path best.

18 Book Title

**Figure 6.** ACO Modified.

## **5. Experimental results**

In this section, the accuracy of the proposed algorithms described above. Present different start node, implying different grade of complexity.

Firtstly obtain the graph of each environment, the same applied of the landmarks of init and finish, also the centroid of the initial node. Finally, applied the algorithms over each graph, search the shortest path.

## **6. Conclusions and perspectives**

This work was implemented with call selection natural and evolution natural, through basict two types of ants: jobs and explorer. The offspring are considerated for the jobs, the best foragers and for the explorer, the offspring least have lost their path. Each ant have three gene, α, β and γ owned to transition function. The parameters was applied in the same algorithms, but, the best result is obtanained with the ACO-GA algorithm.

The implementation of the ACO-GA in a robot manipulator, is a job in prosess, but the first results prove the effectively of the same. See figures (9) and (10).

**Figure 7.** Images of Solution Process applied the ACO-GA Algorithm and parameters of table 1.

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18 Book Title

2. Put the *m*-ant's created with a frequency *f* in the node

i. Select to *J* with a transition rule

ሺݐሻ

a. Natural selection (GA Modified)

c. Calculate the *Lk* of the k-th ant. d. Update the short best path.

ii. Move to ant to *J* iii. Update the tour with (6) iv. If *J* was visited (Kill ant) v. Else (*J* is part of the list)

e. For any edge (i, j) applied (13) and (14).

In this section, the accuracy of the proposed algorithms described above. Present different start

Firtstly obtain the graph of each environment, the same applied of the landmarks of init and finish, also the centroid of the initial node. Finally, applied the algorithms over each graph,

This work was implemented with call selection natural and evolution natural, through basict two types of ants: jobs and explorer. The offspring are considerated for the jobs, the best foragers and for the explorer, the offspring least have lost their path. Each ant have three gene, α, β and γ owned to transition function. The parameters was applied in the same algorithms,

The implementation of the ACO-GA in a robot manipulator, is a job in prosess, but the first

1. For all edge (i, j) initialize τij = c and ∆τij(t) = 0.

b. For k:=1 to ms

5. If find a path finish else go to step 2.

top. 3. Repeat

64 Recent Advances on Meta-Heuristics and Their Application to Real Scenarios

4. Until Find a path.

**Figure 6.** ACO Modified.

**5. Experimental results**

search the shortest path.

node, implying different grade of complexity.

**6. Conclusions and perspectives**

but, the best result is obtanained with the ACO-GA algorithm.

results prove the effectively of the same. See figures (9) and (10).

**Figure 7.** Images of Solution Process applied the ACO-GA Algorithm and parameters of table 1.

ACO

**Figure 10.** Example of a variation of epochs with (0 < α <1); β < γ or β > γ applied ACO and ACO-GA.

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67

**Figure 11.** Best value applied to the best ACO and the best ACO-GA

**Figure 8.** Images of Solution Process applied the ACO-GA Algorithm and parameters of table 1. Note that Different Start Node.

**Figure 9.** Solution to Labyrinth Applied ACO-algorithm.

Meta-Heuristic Optimization Techniques and Its Applications in Robotics http://dx.doi.org/10.5772/54460 67

**Figure 10.** Example of a variation of epochs with (0 < α <1); β < γ or β > γ applied ACO and ACO-GA.


**Figure 11.** Best value applied to the best ACO and the best ACO-GA

**Figure 8.** Images of Solution Process applied the ACO-GA Algorithm and parameters of table 1. Note that Different

**Pheromone**

Start Node.

**Figure 9.** Solution to Labyrinth Applied ACO-algorithm.

66 Recent Advances on Meta-Heuristics and Their Application to Real Scenarios

**Figure 13.** Besides using the ACO-Modified, is necessary calculate the distance of the first node through of de equa‐ tion (6). Because the initial pose of manipulator can be "above" of the piece or well to distance not detectable by ma‐

Meta-Heuristic Optimization Techniques and Its Applications in Robotics

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69

Polytechnic University of Guanajuato, Robotics Engineering Department, Community Juan

[1] Siegwart, R, & Nourbakhsh, I. R. (2004). Introduction to autonomous mobile robot.

[2] Borenstein, J, Everett, H. R, & Feng, L. Where am I? Sensors and Methods for Mobile Robot Positioning, Ann Arbor, University of Michigan,(1996). available at http://www-

Massachusetts Institute of Technologypress, Cambridge, U.S.A.

personal.engin.umich.edu/~johannb/position.htm), 184-186.

nipulator of the initial node, of manipulator makes it impossible of take.

Address all correspondence to: acruz@upgto.edu.mx

Alonso, Cortázar, Guanajuato, Mexico

**Author details**

**References**

Alejandra Cruz-Bernal\*

**Figure 12.** The robot manipulator select to piece of course to pheromone (green). Note, that the manipulator can't take the first piece. Graphics Displacement and Monitoring of the Manipulator respect to pheromone.

**Figure 13.** Besides using the ACO-Modified, is necessary calculate the distance of the first node through of de equa‐ tion (6). Because the initial pose of manipulator can be "above" of the piece or well to distance not detectable by ma‐ nipulator of the initial node, of manipulator makes it impossible of take.

## **Author details**

Alejandra Cruz-Bernal\*

Address all correspondence to: acruz@upgto.edu.mx

Polytechnic University of Guanajuato, Robotics Engineering Department, Community Juan Alonso, Cortázar, Guanajuato, Mexico

## **References**

**Figure 12.** The robot manipulator select to piece of course to pheromone (green). Note, that the manipulator can't

take the first piece. Graphics Displacement and Monitoring of the Manipulator respect to pheromone.

68 Recent Advances on Meta-Heuristics and Their Application to Real Scenarios


[3] Udupa, S. (1977). Collision detection and avoidance in computer controlled manipu‐ lator. PHD thesis, California Institute Technology, California, USA.

[20] Neira, J, & Tardos, J. D. Data association in stchocastic mapping using joint compati‐

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[21] Clark, S, Dissanayake, G, Newman, P, & Durrant-whyte, H. A Solution Localization and Map Building (SLAM) Problem. IEEE Journal of Robotics and Automation, June

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[23] Nebot, E. (2002). Simultaneous localization and mapping 2002 summer school. Australian centre field robotics. http://acfr.usyd.edu.au/homepages/academic/enebot/

[24] Duan, H. B, Zhang, X. Y, & Xu, C. F. Bio-Inspired Computing", Science Press, Beijing,

[25] Pehlivanoglu, Y. V. A new vibrational genetic algorithm enhanced with a Voronoi diagram for path planning of autonomous UAV", Aerosp. Sci. Technol., (2012). , 16,

[26] Ye, W, Ma, D. W, & Fan, H. D. Algorithm for low altitude penetration aircraft path planning with improved ant colony algorithm", Chinese J. Aeronaut., (2005). , 18,

[27] Duan, H. B, Yu, Y. X, Zhang, X. Y, & Shao, S. Three-dimension path planning for UCAV using hybrid meta-heuristic ACO-DE algorithm", Simul. Model. Pract. Th., (2010). , 18,

[28] Duan, H. B, Zhang, X. Y, Wu, J, & Ma, G. J. Max-min adaptive ant colony optimization approach to multi-UAVs coordinated trajectory replanning in dynamic and uncertain

[29] Xu, C. F, Duan, H. B, & Liu, F. Chaotic artificial bee colony approach to uninhabited combat air vehicle (UCAV) path planning", Aerosp. Sci. Technol., (2010). , 14, 535-541.

[30] Duan, H. B, Liu, S. Q, & Wu, J. Novel intelligent water drops optimization approach to single UCAV smooth trajectory planning", Aerosp. Sci. Technol., (2009). , 13, 442-449.

[31] Canny, J, & Reif, J. (1987). New lower bound techniques for robot motion planning problems. Proceedings of IEEE Symposium on the Foundations of Computer Science,

[32] Sugihara, K, & Smith, J. (1997). Genetic algorithms for adaptive motion planning of an autonomous mobile robot. Proceedings of the IEEE International Symposium on Computational Intelligence in Robotics and Automation, Monterey, California, ,

[33] Goss, S, Aron, S, Deneubourg, J. L, & Pasteels, J. M. (1989). Self-organized Shortcuts in the Argentine Ant. Naturwissenchaften pg. Springer-Verlag.(76), 579-581.

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[3] Udupa, S. (1977). Collision detection and avoidance in computer controlled manipu‐

[4] Latombe, J. C. (1991). Robot motion planning. Kluwer Academic Publisher, Boston. [5] Lozano-Perez T Wesley MA ((1979). An algorithm for planning collision-free paths

[6] Li, L, Ye, T, Tan, M, & Chen, X. (2002). Present state and future development of mobile

[7] Siegwart, R, & Nourbakhsh, I. R. (2004). Introduction to autonomus mobile robot.

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74 Recent Advances on Meta-Heuristics and Their Application to Real Scenarios


**Chapter 4**

**A Comparative Study on Meta Heuristic Algorithms for**

Material requirements planning (MRP) is an old field but still now plays an important role in coordinating replenishment decision for materials/components of complex product structure. As the key part of MRP system, the multilevel lot-sizing (MLLS) problem concerns how to determine the lot sizes forproducing/procuringmultiple items atdifferentlevels withquantita‐ tive interdependences, so as to minimize the total costs of production/procurement setup and inventory holding in the planning horizon. The problem is of great practical importance for the efficient operation of modern manufacturing and assembly processes and has been widely studied both in practice and in academic research over past half century. Optimal solution algorithms exist for the problem; however, only small instances can be solved in reasonable computationaltimebecausetheproblemisNP-hard(SteinbergandNapier,1980).Earlydynamic programmingformulationsusedanetworkrepresentationoftheproblemwithaseriesstructure (Zhangwill, 1968,1969) or an assembly structure (Crowston and Wagner,1973). Other optimal approaches involve the branch and bound algorithms (Afentakis et al., 1984, Afentakis and Gavish, 1986)thatuseda converting approach to change the classicalformulation ofthe general structure into a simple but expanded assembly structure. As the MLLS problem is so common in practice and plays a fundamental role in MRP system, many heuristic approaches have also beendeveloped,consistingfirstofthesequentialapplicationofthesingle-levellot-sizingmodels to each component of the product structure (Yelle,1979, Veral and LaForge,1985), and later, of the application of the multilevel lot-sizing models. The multilevel models quantify item interdependencies and thus perform better than the single-level based models (Blackburn and

Recently, meta-heuristic algorithms have been proposed to solve the MLLS problem with a low computational load. Examples of hybrid genetic algorithms (Dellaert and Jeunet, 2000, Dellaert et al., 2000), simulated annealing (Tang, 2004, Raza and Akgunduz, 2008), particle

> © 2013 Kaku et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 Kaku et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

**Solving Multilevel Lot-Sizing Problems**

Ikou Kaku, Yiyong Xiao and Yi Han

Millen, 1985, Coleman and McKnew, 1991).

http://dx.doi.org/10.5772/55279

**1. Introduction**

Additional information is available at the end of the chapter
