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A* Algorithm For Path Planning of Robotic SoccerPlayer

Tingsheng Liao Polat Utku Kayrak

Abstract—Path planning is one of the keys to win arobotic soccer game, and A* algorithm is one of thebest approach for path planning. Most of existing worksare using A* algorithm for static environment, but A*can also be applied to a dynamic system like a roboticsoccer game. This paper investigates the feasibility ofusing A* algorithm to generate a best path for roboticsoccer player to avoid it’s opponents in different scenarios.Our conclusion regarding the limit of performance in theextreme case to provide suggestions for robotic soccer pathplanning algorithm.

I. INTRODUCTION

In May, 1997. IBM Deep Blue defeated thehuman world champion in chess. Forty yearsof challenge in the AI community came to asuccessful conclusion. In the same year, NASAspathfinder mission made a successful landing andthe first autonomous robotics system, Sojourner,was deployed on the surface of Mars. Togetherwith these accomplishments, RoboCup made itsfirst steps toward the development of roboticsoccer players which can beat a human World Cupchampion team. [1]

The idea of robots playing soccer was firstmentioned by Professor Alan Mackworth(University of British Columbia, Canada) in apaper entitled On Seeing Robots presented atVI-92, 1992. [2] A long term goal for robots isto develop multi- robot adaptive, co-operative,autonomous systems solving common tasks.Professor Mackworth chose soccer playing as oneof the task in particular.

In this paper, we are only concerned with thepath planning for avoiding robotic soccer player’sopponents. Our algorithm takes for granted that the

position and goal of all opponents are correctly per-ceived. In the first part of this paper, an overview ofA* algorithm is presented. We review the conceptsof A* algorithm and propose our modification toallow A* algorithm to be used in dynamic envi-ronment. The second part of this paper present oursimulation in different scenarios, and we discusshow does the performance of our algorithm varieswith different scenarios. Finally, we give a brieflyconclusion of our simulation.

II. REVIEW OF A* ALGORITHM

In 1968, Hart, Nilsson, and Raphael [3] pro-posed a proven optimization of Dijkstra’s algorithmdubbed A∗ (after multiple earlier version A1, A2,etc)). A∗ is an informed search algorithm: it firstsearches the routes that appear to most likely leadto the destination. For this, A∗ uses a distance-plus-cost heuristic to determine the order in whichto visit potential nodes on the route. A∗ greatlyimproves the time of Dijkstra’s algorithm and workswell for path planning for robots navigating aroundfixed obstacles.A∗ algorithm is one of the most effective algo-

rithms to solve the path finding problem. It calcu-lates the costs of each solution to find an optimalsolution with minimal cost. The equation of totalcost is

F = G+H

where G is to the cost from the starting pointto current (visited) one, and H is the heuristicfunction estimating the cost from the current pointto the destination. Therefore, the design of G andH will determine the total cost F which can greatlyaffect the path A∗ finds.

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The principle of the A∗ algorithm is to expandeach possible node from start to destination andcompare the cost of each path. Once there is anobstacle that makes the current path cost more thanother available path, A∗ will go back to the newlowest path and expands it again. After repeatingthis finding process, A∗ will finally find a path thatcost least once it expands to the destination. Asshown in figure 1, the A∗ algorithm starts expandingnodes from A to the goal B, and the grids withcross marks are the obstacles. Each expanded gridcontains the information: F at the top left, G at thebottom left, H at the bottom right, and the arrowpointing its parent. In this example, the heuristicfunction is the Manhattan method:

The Manhattan method [4] calculates the totalvertical and horizontal distance between thecurrent cell and the destination, ignoring obstacles,discarding any diagonal movement.

Fig. 1. Illustration of A∗ path finding algorithm. [5]

Obviously, the heuristic function H is critical todetermine the performance of the A∗ algorithm. Byusing different heuristic functions, the result can becompletely different. Figure 2 shows how the pathsdiffer by using two different heuristic functions.

(a)

(b)

Fig. 2. (a) Path finding using classic A∗ method (b) Pathfinding using fudge method [6]

III. IMPLEMENTATION

The traditional A* algorithm mostly works instatic environment, but it is not necessary. Wecan predict the future position based on theircoordinates and bearing as reported by the sensoronboard. By designing a heuristic function thatcan describe a dynamic system like robotic soccergame, A* algorithm is capable to do the pathplanning to avoid moving opponents.

Figure 3 illustrates the concept of using A*algorithm to deal with dynamic environment.Instead of using traditional map, we use a mapthats varies with the increasing time step so thatA* algorithm can avoid the potential encountersbased on the prediction.

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Fig. 3. Illustration of danger grid of A∗ path findingalgorithm. [7]

In order to calculate the cost function in a chang-ing map, we introduce an additional equation todescribe the costs generated by other opponents.

H =n∑

i=1

C − (|xn − x0|+ |yn − y0|)

In this equation, C is a constant that usually chosenby the size of map. Therefore, when our playersare getting closer to other opponents, the cost ofheuristic function H becomes larger.

IV. SIMULATION

In our simulation, we consider three differentscenarios to see how A* algorithm perform in thesesituations. In each scenario, there are five opponentsto try to catch our player. Only our player isequipped the modified version of A* algorithm toallow it to find a best path to avoid it’s opponentsand reach the destination, other opponents are justequipped the general A* algorithm guiding themfrom one position to another.

A. Scenario:Surrounding OpponentsThe figure 4 shows that in this scenario, our

player is surrounded by the opponents from fourdirection, and it has to avoid them to reach thedestination on the top. For the opponents, they aremoving toward to our player and try to catch it.

The simulation result is shown in figure 5, andour player(blue star) just barely pass the opponentswithin one time step to be caught.

Fig. 4. Illustration of surrounding opponents.

0 5 10 15 20 250

5

10

15

20

25

Fig. 5. Simulation of surrounding opponents.

B. Scenario: Opponents in a row

In this scenario, the opponents start on the samerow and move forward to cut our player’s pathas shown in figure 6. Figure 7 shows that ourplayer can easily avoid the opponents to reach thedestination in the middle.

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Fig. 6. Illustration of opponents in a row.

−10 0 10 20 30 40−10

0

10

20

30

40

50

Fig. 7. Simulation of opponents in a row.

C. Scenario: Crowded Opponents

In this scenario, the opponents are more crowdedduring the movement because all of them will acrossthe original path of our player. Figure 8 shows theinitial position of all robots.

As a result, our player is more conservative in thisscenario. As shown in figure 9, instead of passingthrough the opponents our player choose the pathto keep a long distance from opponents.

Fig. 8. Illustration of crowded opponents.

−10 −5 0 5 10 15 200

5

10

15

20

25

30

35

40

Fig. 9. Simulation of crowded opponents.

V. CONCLUSION

Our simulation result shows that A* algorithmis capable to solve the path planning problem forrobotic soccer. However, in some scenario like thecrowded opponents, the path is not ideal. This isdue to the heuristic function we use would cumulatethe cost of each opponents. It means when theopponents are getting closer, the entire area aroundthem are block because of the cumulative costs.To solve this problem, a new heuristic functionis needed to compute the cost equation of eachopponent independently.

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REFERENCES

[1] “A brief history of robocup.” http://www.robocup.org/about-robocup/a-brief-history-of-robocup/.

[2] A. K. Mackworth, “On seeing robots,” 1993. Reprintedin P. Thagard (ed.), Mind Readings, MIT Press, 1998.

[3] P. E. Hart, N. J. Nilsson, and B. Raphael, “A formal basisfor the heuristic determination of minimum cost paths,”1968.

[4] P. Ballard and F. Vacherand, “The manhattan method:A fast cartesian elevation map reconstruction from rangedata,” 1993.

[5] P. Lester, “A* pathfinding for beginners.” http://www.policyalmanac.org/games/aStarTutorial.htm, 2005. This isan electronic document. Date of publication: [Date un-available]. Date retrieved: July 22, 2011. Date last modi-fied: July 18, 2005.

[6] J. Macgill, “A* demonstration.” http://www.vision.ee.ethz.ch/∼cvcourse/astar/AStar.html, 1999. This is an electronicdocument. Date of publication: [Date unavailable]. Dateretrieved: Oct 11, 2011. Last Updated January 18 1999.

[7] T. Crescenzi, A. Kaizer, T. Young, J. Holt, and S. Biaz,“Collision avoidance in uavs using dynamic sparse a*,”2011.

Tingsheng Liao is a graduate student at AuburnUniversity, and is presently majored in Electricaland Computer Engineering for M.S. degree. Hehave done research in the past on Loci Control ofCs-113 Robotic Arm for Plane Motion. His chiefresearch interests lie in the field of robotics andcontrol system, and he is working on UAV collisionavoidance as a research assistant.

Polat Utku Kayrak was born in Istanbul, Turkeyin 1987 and completed his undergraduate studies inKoc University, one of the best engineering schoolsin Istanbul. His graduation project was on ”Develop-ing Signal Processing Algorithms for MathematicalFinance”. He has been accepted as an undergraduateresearch assistant at Yale University in 2010 andworked on ”Bio-Inspired Synthetic Vision” in Prof.Eugenio Culurciello’s group. He’s currently pursu-ing his Master’s Degree in Auburn University inElectrical Computer Engineering Department.