Actas do Encontro Científico 3º Festival Nacional de Robótica - ROBOTICA2003 Lisboa, 9 de Maio de 2003.

Transcription

1 Actas do Enconto Científico 3º Festival Nacional de Robótica - ROBOTICA23 Lisboa, 9 de Maio de 23. PATH PLANNING OPTIMIZATION USING THE DIFFERENTIAL EVOLUTION ALGORITHM Hélde Santos, José Mendes, P. B. de Moua Oliveia and J. Boaventua Cunha Univesidade de Tás-os-Montes e Alto Douo Depatamento de Engenhaias Quinta dos Pados, 5 Vila Real Potugal opt_tajectoias@mail.pt Abstact: In this pape the Diffeential Evolution algoithm is deployed to the obotic path planning optimization poblem fo autonomous mobile vehicles. At this stage, the simulations conside the obot wold with level sufaces and static obstacles, in which the optimization is pefomed off-line. The global objective is to evaluate the diffeential evolutionay algoithm in solving path planning poblems, by finding the shotest path between stat and goal points. Results obtained fo thee case studies with static cicula and polygonal obstacles ae pesented. Keywods: Evolutionay Algoithms, Diffeential Evolution, Optimization, Path Planning, Obstacles Avoidance, Vehicle Routing. 1. INTRODUCTION One of the obotics main eseach field is the contol algoithms development which allows obot mobility in a envionment suounded by obstacles. In this context, the path planning poblem plays an impotant ole by poviding tajectoies to avoid obstacles. Thee ae seveal well known methods of path planning. Its choice depends on the comple xity and quality of the wold whee the vehicle will move, as well as the vehicle s qualities. Thee ae some citeia to evaluate the planning, such as path length, time spent, obstacle distance o the complexity and smoothness of the path. The elevance of applying efficient optimization algoithms to solve tajectoy optimization poblems is unquestionable. This is the case of evolutionay inspied algoithms, paticulaly the popula genetic algoithms (Goldbeg, 1989; Michalewicz, 1992). One moe ecently intoduced algoithm is the Diffeential Evolution poposed by Pice and Ston (1995) which povides good esults in the context of function optimization (Ston, 1996). This pape descibes the off-line optimization of mobile obots tajectoies using the Diffeential Evolution Algoithm, consideing a wold with level sufaces and obstacles epesented by ound and polygonal shapes. The path is fomed by stait line segments. The oveall objective of the expected wok is to acess the Diffeential Evolution algoithm in the context of obotic path planning optimization using evolutionay inspied techniques. The optimization objective is to achieve the shotest path, between two points, avoiding any obstacle in the way. The est of this pape is oganized as follows in section two an intoduction to the Diffeential Algoithm is povided. Section thee descibes the off-line path planning poblem consideed. In section fou simulation esults ae pesented and finally in section five some conclusions ae dawn. 2. DIFFERENTIAL EVOLUTION The Diffeential Evolution (DE) algoithm was oiginally developed by Pice and Ston (1995). The solution of this algoithm, to addess the path planning poblem, was based upon its efficiency, effectiveness and obustness in the optimization of functions (Ston, 1996). The DE can be easily implemented to solve seveal types of optimization poblems, such as path planning. Assuming that a vecto x epesented by (1) epesents a candidate solution to the path optimization poblem, in which t epesents the cuent evolutionay iteation, a new solution vecto x ( t +1) can be evaluated fom the pevious vecto by using equation (2), with δ (3), epesenting an

2 incemental vecto. v x ( t) x ( t) x ( t),, x ( t) = 1, 2 K n (1) x + 1 δ (2) ( t ) = x( t) + ( t) δ ( t) = δ 1( t), δ 2 ( t), K, δ n ( t) (3) Consideing a set pop of potential solutions with size m, epesented by (4), known as population (2) can be ewitten as (5) with d epesenting the dimension index. xid ( x ( t), x ( t),, x ( t ) i m popi( t) = i1 i2 K in 1 (4) ( t + 1) = ( t) + δ ( t + 1) 1 i m 1 d n x id id (5) A tial x n v R vecto is geneated in each iteation fo each population membe using (6) and the incement δ is evaluated using (7). In these expessions x, x and x ae vectos selected 1 2 andomly fom the population set pop, in each iteation, and F R+ is a constant defined pio to optimization. xvid 1 i δ id 1 i m ( t + 1) = x d ( t) + δ id ( t + 1) 1 m 1 d n [ 1, m] i 3 ( t + 1) = F x ( ) ( ) 2d t x 3d t 1 d n 2 [ 1, m] i 3 [ 1, m] i ( ) 1 3 (6) (7) The tial vecto obtained with (6) and (7) is then cossed with the cuent population element x accodingly to a cossove scheme, binomial o exponential (Ston and Pice, 1995), with a pedefined pobability defined by CR [,1 ] esulting in a new vecto x ci (8). If the value etuned by the objective function f fo the cossed tial vecto is bette o equal than the value obtained fo the cuent vecto, the latest is eplaced by the fome. This is epesented by (9) fo a minimization poblem. xi xci i vi ( t + 1) = cossove ( x ( t), x ( t + 1) ) 1 i m x c i x c i 1 i m i ( t + 1) = ( t + 1) if f ( ( t + 1) ) < f ( x ( t + 1) ) i (8) (9) [Initialization of the population vecto] Initialize pop_new; [matix with andom values] Initialize best; [best vecto] Initialize best_value; [best value] Fom j = until maximum numbe of iteations: [Copies the initialized matix to a geneal one] pop = pop_new; Fom i = until m: [Chooses andomly thee numbes between and m] 1 = a vecto index diffeent fom i; 2 = a vecto index diffeent fom i and 1; 3 = a vecto index diffeent fom i, 1 and 2; [Saves the vecto i ] temp = pop( i, :); [The following equation changes with the stategy] xv = pop( 1) + F*( pop( 2) - pop( 3) ); [Evaluate the new temp vecto] xc=cossove(xv,xi) Value_new = evaluation(xc); Value_old = evaluation(pop( i )) ; If (Value_new < Value_old) do: [It means that the new vecto is bette] pop_new( i, :) = xc; If (value_new < value_best) do: [Best value eve] value_best = value_new; best = temp; else [Keeps the old value] pop_new( i,:) = pop_old( i, :); i = i + 1; end j = j + 1; end Fig. 1. A Pseudo-code fo the DE algoithm. The diffeent stategies poposed fo this algoithm ae, basically, vey simila, diffeing in the evaluation of the x R n v vecto. Pice and Ston (1995) poposed ten diffeent options that can be classified using the notation DE/X/Y/Z, whee: - X: epesents the methodology used to select vecto 1, which can be andom selected (and) o the best population so fa (best). - Y: the numbe of vectos used to evaluate the diffeential petubation: - Z: Is the type of cossove opeation used which can eithe be of the exponential (exp) type o binay (bin) type. Thus, ten DE algoithm vaiations can be summaized by: 1) DE/best/1/exp; 2) DE/and/1/exp; 3) DE/and-to-best/1/exp; 4) DE/best/2/exp; 5) DE/and/2/exp; 6) DE/best/1/bin; 7) DE/and/1/bin; 8) DE/and-to-best/1/bin; 9) DE/best/2/bin and 1)DE/and/2/bin, and ae well descibed in ( Pice, 22). Figue 2 illustates the DE basic concept (Lampinem J. and Zelinka, 1999), and it is impotant to note that this algoithm does not use any type of coding scheme opeating diectly in the optimizing paametes. In Figue 1 a pseudo-code is shown that can be used to implement the DE algoithm, in which comments ae included between backets.

3 - cossove pobability (CR). Using a highe CR will incease the convegence speed. Values between and 1 ae usually used. At this stage, it is impotant to undeline the type of objective function used, which fo evey potential solution veifies if the coesponding tajectoy intesects obstacles. It etuns the length of the path connecting the stating and goal points. The global objective is to find the path which equies the minimum numbe of points and povides minimum length. The defined static obstacles ae defined using cicula and polygonal shapes. Fo the cicula case (Figue 3), the obstacles ae defined by thei cicle cente and adius. Using this, a simple way to avoid collisions, consists on detemining the minimum distance between each segment of the path and the cente of each obstacle. If this distance emains geate than the adius of each obstacle, thee ae no collisions. Fig. 2. Global scheme fo Diffeential Evolution. 3. OFF-LINE PATH PLANNING In the off-line case, the simulated wold is defined using static obstacles, and the optimization objective is to establish a full path, avoiding collisions. Thus, the DE algoithm is used as the optimization tool, and its pefomance evaluated. The obot wold is defined by a squae in the Catesian plan, limited by the coodinates of its vetices. The obot position is epesented by its coodinates coesponding to one point in the plane. The path between the stat and goal positions is defined by line segments which link all the diffeent points defined by the diffeential evolution. In ode to use the DE algoithm some paametes must be defined pio to the seach pocedue, such as: - the numbe of optimization points (d). The numbe of segments equals the numbe of points plus one. (Segments=d+1). Thus, path flexibility depends on the numbe of points, - initial and final positions fo the obot (Stat and Goal), - maximum numbe of iteations, - DE stategy, - population size (m). Fig. 3. Example of the defined wold, with cicula obstacles. The distance between a point and a line, is given by the pependicula segment that cosses the point. Howeve, it is necessay to detemine if the pependicula line intesects the segment, othewise the minimum distance is evaluated in elation to segment neaest exteme point. Assuming that the cente of the cicula obstacle is epesented by P and the line segment is epesented by P P 1, an easy way to solve this poblem is to conside the intenal poduct of the angles between vectos P P and P 1 P (Sunday, 21a). If the esult of this poduct is negative o positive, it means that the shotest distance between the cental point and the segment is connected to one of the exteme points of the latest. If the poduct esult is zeo it means that the pependicula line between the cicle cental point intesects the segment and it is the shotest distance (Figue 4). - amplification facto F. This paamete is multiplied by the diffeence between two vectos. It should be selected in ode to pevent pematue convegence (<F<2);

4 [Tests the intesection fom two segments, etuning: = none, 1 = intesec tion] Intesection (Segment P P 1; Segment P 2P 3) Test (Local) if P and P 1 ae in the left, ight o ove P 2P 3; If (P and P 1 ae in the left o P and P 1 ae in the ight) do: Retun ; Fig. 4. Angles between the segments, (Sunday, 21a). The pseudo-code used to evaluate the minimum distance between a cicula obstacle and a stait line segment is shown in Figue 5. (Point P, Segment P P 1) v = P 1 P ; w = P - P ; If ( (c 1 = w v ) <= ) Retun distance fom P to P ; If ( (c 2 = v v ) <= c 1 ) Retun distance fom P to P 1; b = c 1 / c 2; base point = P + b*v; Retun distance fom P to base point; Fig. 5. between a point and a line segment. In the case of polygonal obstacles, they ae defined by the connection of thei vetices by line segments, as it is illustated in Figue 6. Test (Local) if P 2 and P 3 ae in the left, ight o ove P P 1; If (P 2 and P 3 ae in the left o P 2 and P 3 ae in the ight) do: Retun ; Retun 1; [Checks if a point is in the left, ight o ove a segment, etuning: > fo left; = fo ove; < fo ight] Local (Point P a, Po int P b, Po int P c) Retun (P b.x P a.x) * (P c.y P a.y) - (P c.x P a.x) * (P b.y P a.y) Fig. 7. Intesection of two segments. The algoithm shown in Figue 8, fo each test point uses a paallel line to the x axis, detemining the numbe of intesections between the polygon and points to the ight of the test point. If the numbe of intesections is even it means that the point is outside the polygon, othewise it is inside. Checkpoint (Point P, Vecto with vetices fom polygon V) Counte = ; Do fo each segment of the polygon: If polygon segment intesects suppot line do: If P is left fom the segment: Counte = Counte + 1; If P is ight fom the segment: Counte = Counte 1; Cycle end; Retun Counte; Fig. 8. Algoithm that checks if a point is inside o outside a polygon. Anothe case is when one of the path segments and one of the polygon segments ae collinea. To solve this poblem the pseudo-code pesented in Figue 9 was used (Sunday, 21c). Fig. 6. Robot wold with polygonal walls. One way to avoid collisions is to check if evey line segment of the optimized tajectoy intesects the line segments defining the polygons (Sunday, 21d). The pseudo-code used fo this case is shown in Figue 7. It is clea that the optimized tajectoy can not include line segments with ae inside the polygonal shapes. To pevent this case it is necessay to detemine fo each solution if all the exteme points of all the defining segments ae not inside the polygons. In figue 8 a pseudo code (Sunday, 21b) is pesent that was used to solve this poblem. Point in a Segment (Point P, Segment S) If (S..x? S.1.x) do: If (S..x <= P.x) e (P.x <= S.1.x) do: Retun Tue; If (S..x >= P.x) e (P.x >= S.1.x) do: Retun Tue; If (S..y <= P.y) e (P.x <= S.1.y) do: Retun Tue; If (S..y >= P.y) e (P.x >= S.1.y) do: Retun Tue; Retun false; [If none of the conditions is possible] Fig. 9. Detects if two line segments ae collinea.

5 4.1 Case Study I 4. SIMULATION RESULTS In the fist case two cicula obstacles wee consideed located in between the stating and goal points, (Figue 1). The stategy numbe 6 was selected fo the DE algoithm and the cossove pobability and amplification paamete wee set to CR =.8 e F =.8, espectively. Diffeent population sizes and numbe of points defining the tajectoy ae used and the numbe of iteations is 1 pe test. The esults obtained with 2 uns fo each condition ae pesented in Table 1. d Table 1 Results fom the case study I m Best Aveage Value 4 3 3,7912 3, ,7911 3, ,7912 3, ,7718 3, ,7716 3, ,7725 3, ,7712 3, ,7741 3, ,7758 3, ,7667 3, ,7684 3, ,793 3,8562 The esults achieved fo diffeent combinations between the numbe of points and population size ae pactically identical. In Figue 1, the path obtained with minimum distance (with 7 points and population size of 3) and the path obtained with maximum distance (with 4 points and population size of 3) ae epesented. While diffeent, they coespond to vey simila cost values in tems of distances attained Best distance convegence Iteations Fig. 11. Best distance convegence gaphic with: 2 ound obstacles; stategy 6; population of 1 and 6 points Best distance convegence Iteations Fig. 12. Best distance convegence gaphic with: 4 ound obstacles; stategy 9; population of 5 and 7 points. Aveage Aveage convegence Iteations Fig. 13. Aveage convegence gaphic with: 2 ound obstacles; stategy 8; population of 3 and 5 points. Aveage convegence Aveage Fig. 1. Two optimized solutions fo the case study I. Figues 11 to 15 illustate convegence plots fo seveal optimization conditions Iteations Fig. 14. Aveage convegence gaphic with: 5 polygonal obstacles; stategy 6; population of 3 and 4 points.

6 Points Iteations X1 Y1 X2 Y2 X3 Y3 X4 Y4 Fig. 15. Point coodinates convegence gaphic with: 4 ound obstacles; stategy 6; population of 3 and 4 points. The esults clealy indicate that thee is no need to pefom 1 iteations, thus the simulation peiod is educed to 5 iteations pe test fo the case study II. 8 32, , , , ,815 37, , , ,74 43, , , , , , , ,859 35, , , , , ,947 37, , , ,129 45, In this case the diffeences between the diffeent test conditions ae moe significant. Stategy 6 is the one who deliveed best esults, as it can be veified by the best distances achieved. In Figue 16 some tajectoies ae plotted fo a population size of 1 and diffeent stategies and numbe of points. 4.2 Case Study II Case study I was not had enough to evaluate the DE algoithm. Thus in this case study the obot wold has fou cicula obstacles as it is shown in Figue 16. The simulated esults obtained fo this case ae pesented in Table 2, in which seveal combinations between the numbe of points defining the path, population size and DE stategies wee consideed. Table 2 Tests esults with fou case study II. N m St 1 Best aveage ,834 33, , , , , , , , , ,119 37, , , ,488 32, ,931 37, ,68 33, , , ,7624 4, , , , , ,991 4, ,617 34, ,466 33, ,881 4, , , , , , , , , Stategy. Fig. 16. Seveal paths in a wold with fou ound obstacles. 4.3 Case Study III In this case the simulated wold has polygonal obstacles epesented by walls, (Figue 17). Some peliminay esults ae pesented in Table 3 using stategy 6, CR=.8, F=.8 and 5 iteations pe test. A total of 2 uns pe combination wee made. The initial population was geneated andomly. The esults ae not conclusive at this stage. Table 3 Tests esults with polygonal obstacles Numbe of points Numbe of population Best distance aveage , , , , , , , , , ,943 While the DE algoithm managed to povide good path solutions fo this case, it was clea that allowing andomly geneated solutions which ae not viable to incopoate the initial population deteioate the algoithm pefomance. Some paths coesponding to the esults pesented in Table 3 ae plotted in Figue 17.

7 Sunday, Dan (21c). Intesections of Lines, Segments and Planes (2D and 3D),Geomety Algoithm Achive, Apil 21. Sunday, Dan (21d). The Intesections fo a Set of 2D Segments, and Testing Simple Polygons, Geomety Algoithm Achive, August 21. Pice K. e Ston R. (22), Diffeential evolution homepage: Lampinem J. e Zelinka I., (1999), Mixed Vaiable Non-Linea Optimization by Diffeential Evolution, Em Zelinka I. (Edito), Poceedings of Nostadamus'99, 2nd Intenational Pediction Confeence, Zlin, República Checa, pp Fig. 17. Seveal optimized paths fo a simulated wold with polygonal obstacles. 5. CONCLUSIONS AND FURTHER WORK The Diffeential Evolution Algoithm was poposed to addess the path planning optimization poblem. The obstacles wee consideed to be static and of cicula and polygonal shapes and the path optimization pefomed off-line. The esults indicated that the Diffeential Evolution Algoithm could solve this poblem satisfactoily, achieving the best esults in tems of the minimization of the path length with stategy numbe 6. Cuent eseach is being caied out in ode to complete the optimization with polygonal obstacles. Futue tests will implement the algoithms on-line assuming dynamic obstacles and compaisons with othe evolutionay based algoithms, such as genetic algoithm and the paticle swam optimization algoithm. REFERENCES Goldbeg, David E. (1989). Genetic Algoithms in seach, Optimization and Machine Leaning. Addison-Wesley, Reading (MA). Michalewicz Z, (1992), "Genetic Algoithms + Data Stuctues = Evolution Pogams", 2nd Edit., Spinge-Velag P.C.. Ston, Raine and Pice, Kenneth (1995). Diffeential Evolution A simple and efficient adaptive scheme fo global optimization ove continuous spaces. Technical Repot TR-95-12, ICSI, Mach Ston, Raine (1996). On the usage of Diffeential Evolution fo Function Optimization. NAFIPS 1996, Bekeley, pp Sunday, Dan (21a). About Lines and of a Point to a Line (2D & 3D),Geomety Algoithm Achive, Febuay 21. Sunday, Dan (21b). Fast Winding Numbe Inclusion of a Point in a Polygon, Geomety Algoithm Achive, Mach 21.