International Conference on Artificial Neural Networks and Genetic Algorithms. 01/01/1997. NorwichA method based on genetic algorithms for obtaining coordinated motion plans of manipulator robots is presented. A decoupled planning approach has been used; that is, the problem has been decomposed into two subproblems: path planning and trajectory planning. This paper focuses on the second problem. The generated plans minimize the total motion time of the robots along their paths. The optimization problem is solved by evolutionary algorithms using a variable-length individuals codification and specific genetic operators

Camacho, Eduardo F.

Ridao Carlini, Miguel Ángel

Riquelme Santos, José Cristóbal

Toro Bonilla, Miguel

English

idUS. Depósito de Investigación Universidad de Sevilla

Using Genetic Algorithm With Variable-lengthIndividuals For Planning Two Manipulators MotionJ. RIQUELME‡, M.A. RIDAO* , E.F. CAMACHO* , M. TORO‡‡  Dpto. Lenguajes y Sistemas Informáticos. Facultad de Informática y Estadística.*  Dpto. Ingeniería de Sistemas y Automática. Escuela Superior de Ingenieros.Universidad de Sevilla (SPAIN).AbstractA method based on genetic algorithms for obtainingcoordinated motion plans of manipulator robots ispresented. A Decoupled Planning Approach has beenused; that is, the problem has been decomposed intotwo subproblems: path planning and trajectoryplanning. This paper focuses on the second problem.The generated plans minimize the total motion time ofthe robots along their paths. The optimization problemis solved by evolutionary algorithms using a variable-length individuals codification and specific geneticoperators.1 IntroductionThe problem is to plan a collision-free motion(obstacles and other robots), from an initialconfiguration to a goal configuration. The mostextended approach to this problem is to decompose itinto two subproblems: path planning and trajectoryplanning. Many algorithms to solve this problem canbe found in the literature [1,2,3,4].The solution obtained by most of these algorithms isa robot trajectory. These trajectories are very difficultto implement in most industrial robots, because theyrequire the internal controller of each articulation to befully available to the user.A method is presented in [5,6] to minimize the totalmotion time of the robots along their paths. This method is used in this paper to find a collision-freemotion plan for two robots, and evolutionaryalgorithms with three different chromosomecodification are presented to solve the optimizationproblem. In this paper a genetic algorithm where thelength of the individuals is variable [7] is proposed.                                                            Supported by DGCYT from the Spanish Ministerio deEducacion under grants TAP-95-0370 and TIC-94-0282Also, new genetic operators adapted to this codificationare presented.2 Problem StatementThe problem can be stated as: Given two robots R1 andR2, a set of known fixed obstacles and the initial andfinal configurations of R1 and R2; find a coordinatedmotion plan for the robots from their initialconfiguration to their final configuration avoidingcollisions with environments obstacles and themselvesThe use of a Decoupled Planning approach needs afixed obstacle collision-free path to be previouslyobtained for each of the robots. The paths which therobots are expected to follow are assumed to be givenas a parametrized curve in the joint space, where λparameter is the distance along the path. TheCoordination Space (CS) is defined as the R2 region{ }CS = ( , ) /  0   1 j 21 2 j jλ λ λ λ≤ ≤ ≤ ≤max withAnypath from (0,0) to (λ1,max,λ2,max) determines acoordinated execution of the two paths, and is called aCoordination Path (CP). The Collision Region (CR) isdefined as the set of points in CS where a collisionbetween two manipulators is produced. In order toreduce the search space in CS, a discretization of eachpath has to be made, so the path is divided into severalequal intervals. Let's number the intervals of each pathfrom 1 to maxj and the ordered set of intervals is calledΩj . A cell is defined as the subspace formed by oneinterval of the paths of each of the robots and isrepresented as the pair (n1,n2). With these discretizedpaths, CS is transformed into an array of cells, theCoordination Diagram (CD). Cell. Let's notate C0=(0,0)and Cmax=(max1,max2). A cell (n1,n2) is consideredcollision FREE if every point inside the cell does notbelong to the collision region. Otherwise, it isconsidered an OBSTACLE cell.Robots can be synchronized using SynchronizationPoints (SP), that is, a point in CD, which any CP willnecessarily pass through.  When the robot arrives atthat place on its path, it will stop until the other robotarrives at its respective points. To avoid a collision it ispossible to alter the CP defining the number andposition of the SP, determining the total motion time.Let's consider a rectangle formed by free cells in CDand let's consider the motion of the robots from thelower left corner cell to the upper right corner cell. Anytrajectory defined for each robot between these twopoints in CD will always be a collision-free CP. Thisclass of rectangles is going to be called FreeRectangles.Let's consider a set of Free Rectangles, connected insuch a way that the upper right corner of one rectangleis the lower left corner of the next. Furthermore, thelower left corner of the first one is the lower left cornerof the whole CD, and the upper right corner of the lastrectangle is the upper right corner of CD. This set ofrectangles is a Free Rectangle Sequence, and theintersection points between two rectangles will be theSP. This constraint is very easy to implement using anyrobot programming language.The problem can be stated as that of finding a FreeRectangle Sequence that minimizes the total executiontime necessary for the robots to complete their wholepath. The main variables used to find this sequence arethe number of SP and the position of these points inCD. A complete description of the method can befound in [5,6].3.- The Proposed Evolutionary AlgorithmsThree different evolutionary algorithms to solve theoptimization problem are presented in this paper. Themain differences among them are the codification ofthe individuals and the genetic operators. While thefirst one uses the classical binary codification ofgenetic algorithm, the others use a specific integercodification with new genetic operators. One of themuses fixed-length chromosomes and the other variable-length individuals.3.1.- Fixed-length integer codification.In the solution proposed in [5], each individual isformed by a fixed number of points represented bytheir absolute coordinates in Ω. Different crossover,mutations and specific genetic operators were studiedin the same paper, obtaining good results. The maindrawback of this codification is the use of fixedchromosome length. In order to avoid thisdisadvantage, a new variable length codification isproposed in this paper.3.2.- Variable-length integer codification.Chromosome representation of the individuals.Eachindividual is represented by a increasing SP sequence.The chromosomes that represent each solution havevariable lengths. If n is the length of a chromosome,then a SP sequence will be determined by n+2 points,where (x0,y0)=(0,0) and (xn+1,yn+1)=(max1,max2). Anindividual is valid if it forms an increasing Sequence ofFree Rectangles. Finally, when a genetic operator isapplied, it is possible for a non-increasing SP Sequenceto be obtained (non-acceptable individual).Generation of the initial population. The initialpopulation is selected randomly.A uniform distributionof the number of points between 1 and NMAX hasshown to be  inappropriate in the tests to represent thelenght of the individuals. The reason is that non-optimal solutions, with few points, can constitute localminima. For this, an increasing probability distributionfrom 1 (minimum) to NMAX (maximum) was selected.Once the number of points of an individual n isselected, its values are created as follows: two sets of nrandom values in [0,1] are generated, then they areordered in an increasing way and projected on [0,max1]and [0,max2] respectively.Fitness measure The evaluation of the fitness measurewill consider valid and non-valid individuals. For validones, the fitness function gives the total execution timeneeded by the robots to complete their paths, when SPare placed in the positions defined by the individualspecifications (See [5] for a more detailed description).The fitness function for non-valid individuals iscompletely different. The function must measure howfar it is from a valid individual. The functionconsidered is f(N)=K+nco, where K is an offset, i.e. ahigh value with respect to the value associated with thevalid individuals, and nco is the number of obstaclecells inside the rectangle sequence.Genetic operators.Crossover Operator: Given two individuals S1 and S2formed by sequences of n and m SP respectively, theidea is to obtain another SP sequence through geneticinformation exchange from parents S1 and S2. Thefollowing method is proposed:A SP of S1 is randomly selected, called P. Then, Q, aSP of S2 is selected. Q is the first SP that has bothcoordinates x and y greater than the respectivecoordinates of P, so the resulting child always will bean increasing SP sequence. It no point of S2 verifies thislimitation, then the child is returned as a copy of S1. Ifpoint Q exists, a new individual is formed by the firstpoints of S1 (P inclusive), followed by the points of S2from Q (inclusive) to the end of S2 (see fig. 1).Mutation Operator: . Two groups of mutationoperators are proposed in this paper:• Slight Operators, which vary the genetic informationsightly. These operators accomplish a local search,that is, in the neighbourhood of a solution.• Strong Operators, which modify the structure of asolution. The objective is to extend the search to newregions, permitting the algorithm to escape fromlocal minima.Experimental results show that the most importantimprovement are obtained with the slight mutation.Nevertheless, the strong mutations play a veryimportant role in the process, avoiding the algorithm toremain trapped in a local minima, even when thesemutations rarely improve the individuals. The Slightmutations are defined as follows: A SP (xk,yk) ofindividual S is selected randomly and also an integer mbetween -MUTMAX and MUTMAX is chosen too,where MUTMAX is the maximum permitted mutation.The mutation consists of substituting the point (xk,yk)by (xk+m,yk+m), where the new point will be placed inthe grey rectangle.On the other hand, the strong mutations can bedefined, for example, in the following way: Twoconsecutives SP point P=(xk,yk) and P'=(xk+1,yk+1) arechosen, and applies them the slight mutation with agiven probability. Thereafter, this mutation adds a newSP between both of them. For this, the differencesdx=xk+1-xk and dy=yk+1-yk are obtained. Then, another twointegers are chosen randomly: vx between 1 and dx-1and vy between 1 and dy-1. The new point is located in Sbelow P with coordinates (xk+vx,yk+vy). (Fig. 2)Restrictions to the individuals: When these operatorsare applied, the resulting individual may not verify thecondition of being an increasing sequence. To solvethis problem, the conflicting coordinates are madeequal to the same coordinates of the following point.For example, given two consecutive points P=(xk,yk)and P'=(xk+1,yk+1) , if yk>yk+1 then yk=yk+1 is made.Evolutionary algorithm characteristics. An elitistevolutionary algorithm has been used, where the bestindividual of each generation is replicated in thefollowing one. A percentage of the offspring isobtained through parents mutations selected with aprobability proportional to its fitness. The rest of theoffspring is obtained through parents crossover (alsoselected as a function of its fitness), and after, one ofthe above defined mutation operators is applied with arandom probability.4 Application ExamplesThe proposed algorithm has been implemented andapplied to several examples in order to study itsefficiency. Only one example is presented in this paper.The example corresponds to the motion of twoSCORBOT and sixteen collision regions and 180×180cells (Fig. 3). It is an iterative motion represented inFig. 4 The motion from Fig. 4-1 to 4-2 and again to theinitial configuration 4-3, is repeated twice.To compare the different evolutionary algorithms, 40ChildS   parentS   parentPQΩΩ2121Fig. 1 - Crossover operator2ΩΩ 1After mutationBefore mutationFig. 2 - Segment mutationsimulations have been executed  with a 100 individualspopulation and 300 generations. Individuals in binarycodification and fixed-integer are formed by 20 SP.Also, the maximum number of SP of the initialpopulation is 20 in integer-variable codification, inorder to use similar initial information.The best results are obtained with integer variablecodifications.  Also, fixed codification reached a lowerminimum value, but standard deviation is three timesgreater.Finally, notice two other advantages in usingvariable codification. First, its computational cost ofcrossover and mutation operations may be even 60% ofthe fixed codification cost, because normally, thenumber of SP is smaller in variable codification. Also,it is not necessary to previously know an estimation ofthe number of SP of the optimal solution.5  ConclusionsThis paper describes a method to generate collision freecoordinated motion plans in multirobots systems. Themethod tries to find a SP sequence that minimizes thetotal execution motion time. This optimization problemhas been solved using different evolutionaryalgorithms. Three chromosome codifications anddifferent genetic operators have been implemented.Better results are obtained in the application exampleswhen using an integer variable codification.Table I - Results (Time in seconds)References[1]  J.C. Latombe, Robot Motion Planning, KlewerAcademic Publishers, 1991.[2] K. Kant and S.W. Zucker, Toward EfficientTrajectory Planning: The Path-Velocity Decompo-sition. Int. J. of Robotics Research, 5 (3).1986.[3] B.H. Lee and C.S.G. Lee, Collision-Free MotionPlanning of Two Robots, IEEE Trans. on SMC, pp21-32,Vol 17 no 1, Jan-Feb 1987.[4] P.A. O´Donnell and T. Lozano-Pérez, Deadlock-Free and Collision-Free Coordination of TwoRobots Manipulators, Proc. of the IEEE Int. Conf.on Robotics and Automation, pp 484-489, 1989.[5] M.A. Ridao, Generación Automática de Trayec-torias Libres de Colisiones para Múltiples RobotsManipuladores. Ph. D. Thesis. Universidad deSevilla. 1995.[6] M.A. Ridao, J. Riquelme, E.Camacho and M. Toro,Coordinated Motion Planning of Manipulators byEvolution Strategies. Proc. of the X Int. Conf. onApplications of AI in Engineering. Udine.1995.[7] S.F. Smith, A Learning System Based on GeneticAdaptive Algorithms. Ph. D. Thesis. University ofPittsburgh. 1980.Ω 1Ω 2Fig. 3 - Coordination Diagram and the best SP sequenceFig. 4 - Initial position (1), intermediate (2) and goal position(3)Codification Avg.   σ Min.  Binary   54.78  0.56  53.25 Integer fixed  42.23  2.78  37.63 Integer Var.  41.19  1.16  38.63

Using Genetic Algorithms with Variable-length Individuals for Planning Two-Manipulators Motion

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Using Genetic Algorithms with Variable-length Individuals for Planning Two-Manipulators Motion

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