Decision Algorithm for Pool Using Fuzzy System - CiteSeerX

Decision Algorithm for Pool Using Fuzzy System S.C.Chua 1

W.C.Tan2

E.K.Wong 3

V.C.Koo4

1

Faculty of Engineering & Technology Multimedia University, Jalan Ayer Keroh Lama, 75450 Bukit Beruang, Melaka, Malaysia Tel: +60-06-252-3007, Fax: +60-06-231-6552, E-mail: [email protected] 2 Faculty of Engineering & Technology Multimedia University, Jalan Ayer Keroh Lama, 75450 Bukit Beruang, Melaka, Malaysia Tel: +60-06-252-3007, Fax: +60-06-231-6552, E-mail: [email protected] 3

Faculty of Engineering & Technology Multimedia University, Jalan Ayer Keroh Lama, 75450 Bukit Beruang, Melaka, Malaysia Tel: +60-06-252-3251, Fax: +60-06-231-6552, E-mail: [email protected] 4

Faculty of Engineering & Technology Multimedia University, Jalan Ayer Keroh Lama, 75450 Bukit Beruang, Melaka, Malaysia Tel: +60-06-252-3004, Fax: +60-06-231-6552, E-mail: [email protected]

Abstract

computers and augmented reality has been demonstrated to enhance the game of billiards [4].

A decision algorithm using the concept of fuzzy logic is proposed in determining strategic shots in a game of pool. Each possible shot on the board is ranked in order to determine the level of difficulty based on two distance parameters and an angle parameter. The set of parameters in the fuzzy system can be tuned to suit the specific playing style (or player’s behavior) on the choice of the maneuver undertaken. The algorithm has been demonstrated on computer generated pool environments and results have shown acceptable trajectories based on a player’s choice.

Billiards is a game played by two players with cues and balls on a long table covered with green cloth (it is also refer to cue sports). It is a game which require precise geometry and strategy. The physics of the balls, cue and table together with the human interactions have made it an interesting model to implement an intelligent strategy-based automated system. In order to realize such a system, vision system, image extractions, geometry analysis, trajectory planning, pattern recognition, modeling and actuator system are among the various considerations and techniques which are needed to complete the automation process.

Keywords: Fuzzy system, Trajectory Planning, Robotics

This paper describes a decision algorithm to decide on a maneuver, that is, which ball to sink into which pocket in a game of pool (a specific type of billiards). This would be the part of trajectory planning of an automated system. If given an input image of a pool game situation from the vision system, the easiest (or best) shot to sink a ball is to be computed. This would involve a selection from a set of balls with different locations and relevant paths, which the relevant balls should follow to the pocket (or pockets). Computer generated pool environments would be used as the input before going on to the used of images from the real world. This is due to the fact that ball locations, pocket locations, colors and ball-type are known parameters in computer generated world, therefore releasing the hefty processing which needs to be carry out on the raw input image from the vision system. In the game of pool, there are two types of balls which each player would be playing: solid (the entire ball is one color) and stripe (the ball has white top and bottom, with a colored stripe painted around the middle).

1 Introduction Game playing has a long and distinguished history in mankind and if not by far the most suitable leisure activity. Over times, humans have made it possible to simulate games in virtual environments and even automate system of games. The integration of real and virtual environments of games made it possible to organize, plan and even improve the game accuracy since the virtual world provide the freedom for constraints and precision of the digital world. Researches have been carried out to promote intelligent robotics research in area of mobile robot, i.e. robot soccer [1] [2]. As of recent years, virtual reality has made it even possible to generate a totally synthetic environment for game [3]. Another potential area of development in game playing is in billiards. Wearable

α1 =α2

(2a)

d co1 =d op2

(2b)

2.1 Estimating Difficulty of A Shot

d co2 =d op1

(2c)

In order to estimate the difficulty of a shot on the board, there are three parameters of great interest: the distance traveled by the cue ball before the collision with the object ball, d co, the distance from the collided cue ball to the pocket, d op , and the angle between the line joining both the mentioned distances, α, as shown in Figure 1.

then ∆ for both cases would be the same. This shows the deficiency of this mathematical formula for such a situation. In practice, the difficulty is small if d co is large and d op is small, but not if d co is small and d op is large. Thus by relying on the use of mathematical formula to evaluate the difficulty of a shot is not really the best way. In the next section, fuzzy logic principle is applied to this problem.

2 Approach and Methods

cue object 1 d co

d co1

α1

cue

d co2

d op1 pocket 1

α d op object

α2

pocket

Figure 1 – Conventional Geometry System

pocket 2

If α≤90o , the shot is infeasible and it should be rejected. This is due to the fact that at such a condition, an attempt to sink the object using a direct shot into the pocket is never possible. A common solution to the problem of estimating the difficulty of the shot is to have a mathematical formula that expresses an output as a function of the inputs. Theoretically, the formula would represent an accurate model of the game behavior. An approximate difficulty function [5], is assigned as

∆=

d co d op b cos α 2

;

90 o < α ≤ 180 o

object 2 d op 2

(1)

to the shot. The factor b depends on the value of the ball: it is largest for yellow and smallest for the black (in snooker). One might easily guess that b=1 for pool game since there is no rules to which ball to sink before another except for the black which shall be sink only after the player has sink all his balls on the board. If the cue ball, object ball, and pocket lie in a straight line (easiest kind of shot), α=180o and cos2 α=1. The difficulty of the shot is then proportional to the two path lengths. Ifα≈90o , then cosα≈0, then ∆ is large. This is appropriate, because it correspond to a shot in which the cue ball just grazes the object ball, and such shots are hard to play accurately. By looking at the situation given in Figure 2 and using the revised ∆ for pool (with b=1), if

Figure 2 – Geometry Configurations for Two Object Balls 2.2 Estimating Difficulty Using Fuzzy Logic

cue

d co α object

d op pocket

Figure 3 – Modified Geometry System In this fuzzy logic approach, d co, d op and α are redefined. This is just an approximation made to simplify the geometry problem. d co will now be the length between the cue ball and object ball, d op will be the length between the object ball and the pocket, and α will be the angle between the mentioned lengths as shown in Figure 3. These 3 parameters will serve as inputs to the fuzzy system. Three fuzzy sets are defined for each of the inputs. Mapping is done for d co, d op , and α for all its fuzzy set to a set of rules. The rules follow the common sense behavior of the game. As such, these rules can be

altered to suit to different player’s behavior (beginner, intermediate, or advance), by changing the level of difficulty (D – Difficult, ND – Not Difficult, VD – Very Difficult, NA – Not Relevant: such combination does not exist) based on the combination of fuzzy sets (or combination of shot). This is shown in Table 1. In the previous section, it is shown that the evaluation of ∆ failed for a situation in Figure 2 with conditions given in Equation 2. In the approach using fuzzy logic principle, each combination of shot is given a level of difficulty (as defined above). Therefore, the situation encountered earlier will certainly give a different ∆ (in existing approach, ∆ is the difficulty function. In fuzzy logic approach, it is the output of the rule evaluation).

(b)

Table 1 – Rules Evaluation for Fuzzy System. α is easy d co

near average far

near ND ND ND

d op average D D D

far VD VD NA

(a) α is medium d co

near average far

d op near ND ND D

average D D VD

far VD VD NA

(b) α is hard d co

near average far

d op near D D VD

average VD VD VD

far VD VD NA

(c) Figure 4 shows the plot of each of the membership functions for the inputs. The reason behind the use of Gaussian Membership Function is the narrow peak which give a relatively high degree of membership within an acceptable limit, and also gradual fall after the limit.

(c) Figure 4 - Input membership functions for (a) cue ball to object ball length (cm), (b) object ball to pocket length (cm), and (c) the angle (degree) joining both lengths Fuzzy inference that is employed is the Sugeno-type with constant output membership functions. This is sufficient to the needs in this research. Furthermore, in terms of coding, it is definitely easier to be done. Sugeno-type inference also enhances the efficiency of the defuzzification process as one only need to compute the weighted average. 2.3 Decision Algorithm The decision algorithm employs the concept of fuzzy logic at its final stage to get the difficulty of each shot. The best shot is the easiest shot to sink the object ball into the pocket, and it is chosen from the fuzzy outputs. In general, the decision algorithm is shown in Figure 5. 2.4 Computing Relevant Paths For each shot, a determination of whether a clear path to target ball from cue ball (center to center) exist for a direct shot. If the ball must travel from A to B and that there is a ball at C, (refer to Figure 6) From the cosine rule, it can be shown that

(a)

u = b cos A =

b2 + c2 − a2 2c

(3a)

v = a cos B =

a2 + c2 − b2 2c

(3b)

then,

h = b2 −u 2 = a2 − v2

(4)

Ball C do not obstruct the path between A and B if h is more than the ball diameter. In the actual situation, the clear path depends on the point at which the cue ball collides with the object ball. Hence, this consideration is taken into account.

3 Simulation and Results Computer simulated environments were tested with the decision algorithm. The output from the algorithm is based on the rules in Table 1. Here, the playing area is of size 160x70 (cm2 ), ball radius of 1.75 cm and pocket radius of 4 cm. Appropriate scaling would need to be done if standard pool table and balls are to be simulated. Some priori information are: cue ball (white), 8-ball (black), ball number 1-7 (solid), ball number 9-15 (stripe). Few sets of virtual data are tested. One such situation is shown in Table 2. The graphical layout of the board is shown in Figure 7.

START Table 2 – Computer Simulated Data (a) Ball locations on the boards (b) Fixed Pockets Location

Generate locations for balls

Ball Cue 1 2 3 4 5 6 7

Compute possible shot and relevant paths For each possible shot and path, compute difficulty (Fuzzy System)

Location (x,y) (59,44) (114,48) (15,32) (71,25) (26,47) (111,50) (77,39) (21,32)

Ball 8 9 10 11 12 13 14 15

Location (x,y) (114,61) (44,19) (137,17) (128,62) (38,18) (10,7) (102,14) (134,13)

(a) Pocket 1 2 3 4 5 6

Find the best shot from the set of fuzzy outputs END

Location (x,y) (0,0) (80,0) (160,0) (0,70) (80,70) (160,70) (b)

Figure 5 - Decision Algorithm for Pool

C b

a

h

Table 3 – Results from the decision algorithm for (a) solid player (b) stripe player. The best shot (in bold italic) is ranked 1, which the difficulty value is the least from the set of shots.

c A

u

v

Figure 6 - Obstacle in a Path

The best shot to be chosen depends on the ball-type the current player is playing. The opponent, on the other hand, would have the opposite type (stripe if the current player is solid, and vice versa). Table 3 shows the results for both solid and stripe player following the data on Table 2.

B

Solid {Ball, Pocket}

d co

d op

α

Difficulty

Rank

{3,2} {4,4} {6,2} {7,1} {8,3}

22.4722 33.1361 18.6815 39.8497 57.5674

26.5707 34.7131 39.1152 38.2753 76.4003

167.5232 143.6980 109.9228 140.8005 109.8440

0.1419 0.2148 0.5322 0.2503 0.7659

1 2 4 3 5

(a)

Stripe {Ball, Pocket}

d co

d op

α

Difficulty

Rank

{13,1} {14,3} {14,6}

61.400. 52.4309 52.4309

12.2066 59.6657 80.6226

177.9355 158.6679 101.1026

0.0641 0.4076 0.8812

1 2 3

(b) Figure 8 shows the set of possible shot for solid and stripe player.

4 Discussion This paper has shown that, the decision making of sinking a ball from a set of pool balls on the board can be simulated using the concept of fuzzy logic. The fuzzy system is able to choose from a set of relevant paths, the easiest shot to sink a ball. Also, the fuzzy rules can be changed to suit the player’s preference and ability. This flexibility provides a practical, inexpensive solution. All that is really needed is a practical understanding of the overall game behavior. Further research is currently investigating combination shots and indirect shots (via a cushion) by the cue ball. Again, fuzzy logic is proposed to be used.

6 Acknowledgements Thanks to Prof. Peter Grogono from Department of Computer Science, Concordia University, for sharing his work on snooker (The Snooker Simulator and Mathematics for Snooker Simulation), to make this research on pool possible.

References [1]

Thomas Braunl. Dec. 1999. Research Relevance of Mobile Robot Competitions. IEEE Robotics & Automation Magazine: 32-37.

[2]

Hiroaki Kitano, Minoru Asada, Itsuki Noda and Hitoshi Matsubara. Sept. 1998. RoboCup: Robot World Cup. IEEE Robotics & Automation Magazine: 30-36. Daniel G. Aliaga. Virtual Objects in the Real World. Communications of the ACM (CACM), Vol.40. No.3: 49-54.,Mar. 1997. Tony Jebara, Cyrus Eyster, Josh Weaver, Thad Starner and Alex Pentland. "Stochasticks: Augmenting the Billiards Experience with Probabilistic Vision and Wearable Computers". In Proceedings of the International Symposium on Wearable Computers: 138-145, Cambridge, Massachusetts, October 1997. Peter Grogono. Jan. 1996. Mathematics for Snooker Simulation. Unpublished.

[3]

[4]

5 Conclusions A decision algorithm for pool playing has been tested possible in choosing appropriate shot given a set of player’s priorities in sinking the balls (rules). The algorithm has been demonstrated on computer generated pool environments and results have shown acceptable trajectories based on a player’s choice. Results presented in this paper also demonstrate that virtual environment of simulation is capable of scaling to the real world situation.

[5]

Figure 7 - Layout of the Board (Playing Area: Lower left (0,0), Upper right (160,70))

(a)

(b) Figure 8- A set of possible shot for (a) solid player (b) stripe player, of which the best shot is chosen