Strategy-based Decision-making of a Soccer Robot System using a Real-time Self-Organizing Fuzzy Decision Tree

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Strategy-based decision making ofa soccer robot system using a

real-time self-organizing fuzzy decision tree

Han-Pang Huang

∗

_{, Chao-Chiun Liang}

Department of Mechanical Engineering, Robotics Laboratory, National Taiwan University, Taipei, Taiwan 10674, ROC

Abstract

In a soccer robot game, the environment is highly competitive and dynamic. In order to work in the dynamically changing environment, the decision-making system ofa soccer robot system should have the features of-exibility and on-line adaptation. This paper proposed a strategy-based decision-making system for a soccer robot system. The decision system is based on the proposed self-organizing fuzzy decision tree (SOFDT). Due to event-driven strategies, the proposed decision-making system has the advantage of-exibility. The proposed SOFDT possesses simple, apparent, and fast generation=reasoning processes. The resultant system is a self-organizing strategy-based decision-making system. According to decision results, SOFDT can on-line modify its parameters and structure for achieving better adaptability and improvement. The performance ofthe proposed systems was veri3ed by applying them to the simulation and experiment of3-to-3 robot soccer games. The results are satisfactory. c 2002 Elsevier Science B.V. All rights reserved.

Keywords: Fuzzy learning decision tree; Self-organizing; Strategy-based decision making; Soccer robots

1. Introduction

In a soccer robot game, the environment is highly competitive and dynamic. In order to work in the dynamically changing environment, the decision-making system ofa soccer robot system should have the features of -exibility and on-line adaptation. A typical multi-agent soccer robot system, such as NTU Formosa built by our laboratory, is shown in Fig. 1 [1,5–9,11,14,15]. This study used NTU-Formosa as the platform to develop a self-organizing strategy-based decision-making system. NTU-Formosa

∗_{Corresponding author. Tel.=fax: +886-2-2363-3875.} E-mail address: hanpang@ccms.ntu.edu.tw (H.-P. Huang).

consists ofmultiple mobile robots, a vision system, a wireless communication system, and a host computer. In Fig. 1, the image information of the entire soc-cer 3eld is captured by a CCD camera and sent to the corresponding host computer. The host computer then analyzes the image information to determine the situ-ation ofthe soccer 3eld. According to the determined situation, the host computer decides a strategy and plans the motion modes and the corresponding veloc-ity commands for every soccer robot of the same team. Each soccer robot ofNTU-Formosa then receives ve-locity command from the host computer and regulates the rotational velocities ofits right and left wheels.

A decision tree has several nodes arranged in a hier-archical structure. It implements decisions in a simple,

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Fig. 1. Graphical diagram oftwo competing teams ofNTU-Formosa.

apparent, multistage manner. Furthermore, since each node ofa decision tree uses only a simple splitting rule and a small subset ofall features, the entire deci-sion process is fast and eEcient [2,3,10,13]. This study proposed a self-organizing strategy-based decision-making system for NTU-Formosa to modify strategies and decide which strategy should be applied to the cur-rent situation. The decision system is based on the pro-posed self-organizing fuzzy decision tree (SOFDT). Due to event-driven strategies, the proposed decision-making system has the advantage of-exibility. It can be shown that the proposed SOFDT possesses simple, apparent, and fast generation=reasoning processes.

To generate a practical decision tree, numerous training data are necessary. The fuzzy learning deci-sion tree adopted in this study uses fuzzy statistics [12,16] to generate several fuzzy sets from the training data. This method eFectively compresses the training data and meaningfully represents its statistical distri-bution. This has the advantages ofincreasing

compu-tation speed, saving storage memory, and maintaining satisfactory performance. Given a set of training data, an optimal decision tree is inherently diEcult to ob-tain [10,13]. To cope with this diEculty, the adopted fuzzy learning decision tree uses a sub-optimal split-ting rule at each node to generate a decision tree. This is a cost-eFective method that quickly obtains a decision tree with satisfactory performance.

In order to adapt to the changing environment, the proposed SOFDT on-line tunes its parameters and adapts its structure to improve its performance in terms ofdecision results and the input data. To increase learning speed, the proposed SOFDT modi3es only a small subset ofthe decision tree by a local learning mechanism in each learning cycle. Ifthe decision re-sult is correct, SOFDT tunes its parameters to re-ect the new statistical distribution. Ifthe decision result is incorrect, SOFDT tunes its parameters for single-modal case and adapts its structure for multi-single-modal case. Ifthe decision result is unknown, SOFDT adapts

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its structure to cope with the multi-modal and new model cases. Finally, ifthe input data is unreasonable, SOFDT has a set ofrules to de3ne and identify them and then ignore them in learning process. SOFDT can perform on-line recognition and learning in real-time applications. To examine the performance of the pro-posed systems, the simulation and experiments of3-to-3 robot soccer games will be conducted.

The organization ofthis paper is as follows. The strategy-based decision-making system ofNTU-Formosa is developed in Section 2. In Section 3, a self-organizing fuzzy decision tree is proposed and its performance is analyzed. The proposed SOFDT is applied to the strategy-based decision-making system in Section 4. In Section 5, simulation and experiment of3-to-3 soccer robot games are given and discussed. Finally, several concluding remarks are drawn. 2. A strategy-based decision-making system of NTU-Formosa

The strategy-based decision-making system for se-lecting strategies is implemented by decision trees. The decision-making system 3rst recognizes the cur-rent situation ofthe soccer 3eld, including the ball lying in which zone and under which team’s control. A strategy is then selected, according to the current situation, to coordinate the multiple robots ofNTU-Formosa to play the soccer game. This section 3rst constructs the architecture ofthe decision-making sys-tem. A set ofpractical strategies is then developed to cope with the situations considered in the constructed decision-making system.

2.1. Architecture

In a soccer 3eld, the current situation depends on many factors. For two n-robot teams, the number of factors aFecting the current situation can be estimated by the following equation:

Nf= 4(2n + 1); (1)

where Nf is the number ofaFecting factors; the item

‘4’ represents two-dimensional position and velocity vectors; the item ‘1’ represents the ball. Note that Eq. (1) does not consider accelerations ofthe ball and the robots. It is only an estimation ofthe complexity

considering all kinds ofsituations. Ifall the factors are considered, the eEciency ofthe decision-making sys-tem will be poor. Decision trees have the advantage of considering only relevant factors to make a decision. To further improve the eEciency of the adopted deci-sion tree, this study 3rst considers the most important factor and places it in the root node of the decision tree. The second most important factor is then consid-ered and placed in the level next to the root node. This tree-generation process continues until all the avail-able and feasible factors are considered and placed in the proper nodes ofthe decision tree.

In a soccer game, the two most important issues are to bring the ball into the opponent’s goal and to prevent the ball from going into teammate’s goal. Hence, much attention should be paid to the ball. This study regards the position ofthe ball as the most important factor and places it into the root node of the decision tree. A soccer 3eld is divided into four zones in this study and hence the ball should lie in one of them. The four zones are oFense zone, for-mation zone, defense zone, and near-wall zone. The oFense zone is near the opponent’s goal and shoot-ing is the major goal. The formation zone is in the middle 3eld and making up the formation is the main goal. The defense zone is near one team’s own goal and preventing shooting is the 3rst duty. The near-wall zone is a region in which the ball is diEcult to shoot or pass directly. Since the root node decides in which zone the ball lies, there are four sub-nodes for the root node. Next, this study considers the con-trollability ofthe ball as the second most important factor and places it into each ofthe sub-nodes ofthe root node. This is because the team controlling the ball has the initiative to attack or to defend. These sub-nodes are non-terminal nodes deciding which team controls the ball. There are three cases about the controllability ofthe ball: controlled by teammate, controlled by opponent, and uncontrolled. Thus, there are three sub-nodes for each of these non-terminal nodes. The decision tree for the strategy-based decision-making system is shown in Fig. 2. In Fig. 2, there are 12 terminal nodes, and each represents a strategy coping with the corresponding situation. By experience and heuristic rules, the decision tree is optimized in its tree-generation process. This op-timization indirectly optimizes the recall process ofthe decision tree. When the number ofrobots

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Fig. 2. The decision tree which implements the strategy-based decision-making system.

increases, however, the decision tree will become more and more sophisticated in order to consider most ofthe important situations. According to Eq. (1), it is very diEcult or even impossible to manually list all kinds of situations. Therefore, a self-organizing deci-sion tree is necessary and will be developed in this study.

2.2. Practical strategies in the decision-making system

In a 3-to-3 soccer game, each team has three robots with diFerent duties. The robot in charge ofoFense zone is called oFense-robot, the robot in charge offor-mation zone is called foroffor-mation-robot, and the robot in charge of defense zone is called defense- robot. The strategies for the decision tree in Fig. 2 are de-veloped based on heuristic rules and game dynamics described by the concepts ofattack angle and pass angle [5,4,6,9]. The proposed attack angle is approx-imately proportional to the probability ofsuccessful shooting. Similarly, the proposed pass angle is derived from attack angle and is approximately proportional to the probability ofsuccessful passing. The devel-oped strategies for 3-to-3 soccer games are described as follows:

1. S OT: The o7ense-robot tries to 3nd a suitable

at-tack angle to shoot the ball. The formation-robot estimates the velocity ofthe ball and prepares to hold the rebounded ball. The defense-robot defends the goal.

2. S OU: The o7ense-robot tries to 3nd a suitable

at-tack angle to shoot the ball. The formation-robot blocks the pass angle ofthe ball. The defense-robot defends the goal.

3. S OO: The o7ense-robot tries to intercept the ball.

The formation-robot blocks the pass angle ofthe ball. The defense-robot defends the goal.

4. S FT: The o7ense-robot prepares to receive the

ball. The formation-robot tries to pass the ball to the robot. The defense-robot defends the goal.

5. S FU: The o7ense-robot prepares to receive the

ball. The formation-robot tries to inter-cept the ball. The defense-robot defends the goal.

6. S FO: The o7ense-robot blocks the pass angle of

the ball. The formation-robot tries to in-tercept the ball. The defense-robot defends the goal.

7. S NT: The o7ense-robot prepares to receive the

ball. The formation-robot clears the ball to oFense zone. The defense-robot defends the goal.

8. S NU: The o7ense-robot prepares to intercept the

ball. The formation-robot tries to clear the ball. The defense-robot defends the goal. 9. S NO: The o7ense-robot tries to intercept the ball.

The formation-robot blocks the pass angle. The defense-robot defends the goal. 10. S DT: The o7ense-robot and formation-robot

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defense-robot tries to pass the ball to the oFense-robot or formation-oFense-robot.

11. S DU: The o7ense-robot prepares to receive the

ball. The formation-robot tries to inter-cept the ball. The defense-robot defends the goal.

12. S DO: The o7ense-robot tries to intercept the ball.

The formation-robot blocks the attack an-gle ofthe ball. The defense-robot defends the goal.

In the 12 strategies developed above, the terms ‘shoot’, ‘block’, ‘pass’, ’defend’, ‘clear’, ‘intercept’, and ‘receive’ are all soccer skills. The reader can refer to [9] for the details.

3. A self-organizing fuzzy decision tree 3.1. Architecture of the self-organizing fuzzy decision tree

The concept and architecture ofthe proposed SOFDT (self-organizing fuzzy decision tree) can be

Fig. 3. The -owchart describing the concept and architecture ofSOFDT.

outlined in a -owchart, as shown in Fig. 3. Clearly, the SOFDT architecture is the kind ofdivide-and-conquer. The components ofSOFDT, including tree-generation algorithm, on-line learning mechanism, error-rejection rules, and performance analysis, are explained below.

3.2. Automatic generation of a fuzzy decision tree The tree-generation algorithm consists ofthree steps: (1) select a set offeatures that can well describe the objects and then massively measure every feature ofeach object to acquire suEcient training data, (2) apply fuzzy statistics to 3nd fuzzy sets for compress-ing and representcompress-ing the traincompress-ing data, (3) determine a decision tree from the fuzzy sets. The structure of a decision tree denotes the topology ofthe tree. The parameters ofa decision tree include the objects and the threshold in each non-terminal node. The tree-generation algorithm can be detailed in a -owchart, as shown in Fig. 4. The above-mentioned three steps and components ofthe -owchart are described in the subsequent context.

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3.2.1. Features selection

This study selected complex moment invariants as the describing features. The de3nition of a continuous complex moment is

Cpq=

(x + iy)p_{(x − iy)}q_{g(x; y) dx dy;} ₍₂₎

where p and q are non-negative integers; g(x; y) is the gray level function of the image. The corresponding discrete complex moments are:

Cd pq= n j=1 m k=1 (jMx + ikMy)p ×(jMx − ikMy)q_{g(jMx; kMy)MxMy:} ₍₃₎

The equation can be simpli3ed by setting Mx = My = 1 as Cds pq= n j=1 m k=1 (j + ik)p_{(j − ik)}q_{g(j; k);} ₍₄₎

where the ∗ represents a complex conjugate, and C.C. denotes the complex conjugate ofits previous term. 3.2.2. Fuzzy sets for compressing training data

This study adopted LR type fuzzy sets for com-pressing and representing training data. Fuzzy statis-tics [12,16] were then used to determine several points

Fig. 5. 2n + 1 points on a fuzzy set Fij.

ofthe membership functions ofthe fuzzy sets. Finally, the curve 3tting method and the pre-determined points were used to identify the LR type fuzzy sets. The pro-cedure for identifying the representative LR type fuzzy sets is described below.

Determining several points on a fuzzy set. The training data are denoted as T = {T1; T2; : : : ; Tn}, where

Ti= {Ti1; Ti2; : : : ; Tim} is the training data for the

ob-ject i, and Tij is the training data for the feature j of

the object i. The goal is to 3nd fuzzy sets Fij to

rep-resent Tij, i = 1; 2; : : : ; n, j = 1; 2; : : : ; m. The algorithm

for estimating several points on Fij for identi3cation

is outlined as follows:

(1) The standard deviation and the mean of Tij are

calculated as SDij and Mij, respectively.

(2) The number ofpoints, (2n + 1), to be identi3ed is determined.

(3) For the 2n + 1 points at Mij, Mij ± (1=n)SDij,

Mij ± (2=n)SDij; : : : ; Mij ± SDij, determine their

membership grades ij(k) with respect to Fij, as

shown in Fig. 5, using ij(k) = Sij(k)=Sij(0).

Sij(k) is the number ofdata located in the

inter-val [Mij + (k − 1)SDij=n; Mij + (k + 1)SDij=n],

k = 0; ±1; ±2; : : : ; ±n. From the above equation and Fig. 5, it is apparent that ij(k) is proportional to

Sij(k), the number ofdata located around the

identi-3ed point. Hence, the above equation possesses the physical meaning offuzzy sets and statistics and ij(k) is the desired point.

This algorithm is valid under the assumption that the training data probability distribution has the property Sij(0)¿Sij(k); ±1; ±2; : : : ; ±n. Under this

assumption, the identi3ed fuzzy sets are normal fuzzy sets. If the symmetrical fuzzy set assumption is further assumed, only n + 1 points need to be estimated.

Identifying a LR type fuzzy set. A fuzzy set M is ofthe LR-type if there exists reference functions L (for

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left) and R (for right), and scalars a¿0, b¿0 with M =        L m − x a for x 6 m; R x − m b for x ¿ m; (6) where m is the mean value of M. Ifthe reference functions are approximated by polynomials of order n, L and R can be formulated as

L = a0+ a1x + a2x2+ + anxn;

R = b0+ b1x + b2x2+ + bnxn: (7)

Since we have determined in the previous sub-section n + 1 points for L and R, respectively, the coeEcients a0; a1; : : : ; an and b0; b1; : : : ; bn can be computed by

curve 3tting method.

3.2.3. Generating a decision tree

The generating procedure ofthe fuzzy decision tree is stated in the following:

Step 1: Select a set ofaFecting factors suitable to partition the situation space, and then mas-sively measure each aFecting factor to get suEcient training data. The situation space is built on all the aFecting factors and con-tains all the situations in soccer games. Step 2: Use the algorithm proposed in Section 3.2.2

to 3nd the fuzzy sets Fij, i = 1–n, j = 1–

m, to represent the training data. Use the fuzzy sets to form fuzzy pattern vectors Fpvi= (Fi1; Fi2; : : : ; Fim; ) for representing

situation i, i = 1–n.

Step 3: Build the root node and start the tree-generation process. The root node con-tains all the situations represented by Fpvi,

i = 1–n.

Step 4: Find the most powerful aFecting factor for the present node. An aFecting factor is the most powerful factor if it can split the node into the greatest number ofsub-nodes. If more than one factor has the same power, the factor requiring the least computation time is chosen.

Step 5: Split the present node into several sub-nodes using the factor selected in step 4 and the splitting rule described below.

According to a heuristic threshold T and the distances between the situations, the splitting rule clusters the situations in the present node into several groups. These groups are the content ofthe desired sub-nodes. The heuristic threshold T is speci3c to the selected feature and the current tree level (root node is level 0). It is de3ned by the following equation:

T = MSD×LMAX− L₂ current; (8)

where MSDis the mean ofthe standard

de-viation ofthe selected factor; LMAX and

Lcurrent are the maximum allowable level

and current level ofthe fuzzy decision tree under design, respectively. The distance be-tween situations i and j can be viewed from any factor and is de3ned by the following equation:

dk(i; j) = |Mik−Mjk|+!|SDik−SDjk|;

+ ! = 1; and ! are real; (9) where dk(i; j) is the distance between

situa-tions i and j, viewed from factor k. Mijand

SDijare the mean and standard deviation of

the training set represented by Fij, and and

! are heuristically selected weighting fac-tors for indicating the relative importance between Mijand SDij. The above de3nition

has the advantage ofconsidering both Mij

and SDij. Several values for and ! may

be tried out to get a satisfactory result. Step 6: For each non-terminal node, repeat steps 4

and 5 until there are no non-terminal nodes. A node is a terminal-node ifit satis3es a prede3ned stopping criterion; otherwise it is a non-terminal node. A node satis3es the prede3ned stopping criterion ifit contains only one fuzzy pattern vector or more than one fuzzy pattern vector but are all too sim-ilar to be split.

Step 7: For every terminal node containing more than one fuzzy pattern vector, the next most powerful factor is linearly combined to split it. This step is repeated until each

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termi-Fig. 6. The detailed -owchart ofSOFDT without error-rejection.

nal node contains only one fuzzy pattern vector.

3.3. Learning law of the self-organizing fuzzy decision tree

The on-line real-time learning algorithm ofSOFDT can be detailed in a -owchart, as shown in Fig. 6. The proposed on-line learning algorithm consists of three parts: correct, incorrect, and unknown recogni-tion. The three parts are explained in the subsequent context. The term ‘class’ in Fig. 6 and subsequent context represents a situation in soccer games. For

clarity ofpresentation, Fig. 6 does not show the error-rejection part ofSOFDT and this part will be described in the next sub-section.

3.3.1. Part I: correct recognition

When the input data is correctly recognized, SOFDT adds it to the training data and uses the fol-lowing steps to modify its parameters.

Step 1: Form a new training data by adding the input data.

Step 2: Identify the fuzzy pattern vector to which the input data belongs.

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Step 3: Re-calculate the fuzzy pattern vector in step 2 with the new training data in step 1. Step 4: Modify the nodes containing the new fuzzy

pattern in step 3.

Step 5: Finish this case and restart to recognize an-other object.

The objective is to make the corresponding fuzzy sets more correctly represent the training data. Con-sidering the size ofthe training data, it is easy to see that the modi3cation scheme in this part only 3nely tunes the parameters.

3.3.2. Part II: incorrect recognition

When the input data is recognized as an incorrect class, SOFDT applies the following steps to tune its parameters or adapt its structure to accommodate itself to this kind ofinput data.

Step 1: Identify the terminal nodes of correct and incorrect classes ofthe input data.

Step 2: Find the nearest common ‘parent node’ of the nodes found in step 1. The de3nition of‘parent node’ is: A node Nf is a ‘parent

node’ ofa node Nsif, and only if, Nscan be

reached from the root node through Nf. The

nearest common parent node is regarded as the current node.

Step 3: In the current node, check whether or not the correct and incorrect classes are adja-cent. Ifthe answer is yes, go to step 4; oth-erwise go to step 6.

Step 4: In the current node, check that the input pattern is single-modal or multi-modal. If the input data is single-modal, go to step 5. Ifthe input data is multi-modal, go to step 6. The input data is single-modal if, and only if, the input data is in the ‘modi3cation zones’ ofboth the correct and incorrect classes, as shown in Fig. 7. The input data is multi-modal if, and only if, the input data is in the ‘modi3cation zones’ ofthe incor-rect class but not corincor-rect class, as shown in Fig. 8. The ‘modi3cation zone’ ofa cor-rect class is de3ned as a small region on the boundary but outside ofthe fuzzy set repre-senting this class. The ‘modi3cation zone’ ofan incorrect class is de3ned as a small

Fig. 7. The input data is single-modal.

region on the boundary but inside the fuzzy set representing this class. The fuzzy set is considered to be modi3ed if, and only if, the input data falls into the modi3cation zone. The de3nition of‘modi3cation zone’ is il-lustrated in Figs. 7 and 8, where the correct and incorrect classes are represented by tri-angular fuzzy sets Ccand Ci, respectively.

Step 5: The key situation in the current node is shown in Fig. 7. The fuzzy set Ccand Ci

are modi3ed in a way ofreducing error. After modi3cation, go to step 9. The mod-i3cation ofthe correct class is formulated by the following equation:

center(C

c) = center(Cc) + 0:5"c#c; (10)

right(C

c) = right(Cc) + "c#c; (11)

where C

c is the fuzzy set representing the

correct class after modi3cation; center(Cc)

is the center point of Cc; right(Cc) is the

right boundary point of Cc; "cis the

learn-ing rate ofthe correct class and in the range of(0; 1); #c is the distance from the input

data to the right boundary of Cc. Similarly,

the modi3cation ofthe incorrect class is formulated by the following equation:

center(C

i) = center(Ci) + 0:5"i#i; (12)

left(C

i) = left(Ci) + "i#i; (13)

where C

i is the fuzzy set representing the

incorrect class after modi3cation; left(Ci)

is the left boundary point of Ci; "i is the

learning rate ofthe incorrect class and in the range of(0,1); #iis the distance from the

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Fig. 8. The input data is multi-modal.

input data to the left boundary of Ci. If Cc

is in the right side of Ci, the modi3cation of

the correct class is changed to the following equations:

center(C

c) = center(Cc) − 0:5"c#c; (14)

left(C

c) = left(Cc) − "c#c: (15)

Similarly, the modi3cation ofthe incorrect class is changed to the following equations:

center(C

i) = center(Ci) − 0:5"i#i; (16)

right(C

i) = right(Ci) − "i#i: (17)

Step 6: The key situations in the current node are shown in Fig. 8, where the correct class needs not to be changed and the incorrect class is modi3ed to reduce the error. Af-ter modi3cation, go to step 7. The modi-3cation ofthe incorrect class follows Eqs. (12), (13), (16), and (17).

Step 7: Insert a fuzzy set Ccninto the current node,

as shown in Fig. 9, and go to step 8. Ccn

is a triangular fuzzy set and its shape is an isosceles. The center of Ccn is the

in-put data, and the base of Ccnis double the

modi3cation zone ofthe correct class. Since SOFDT can recognize the similar data, in-serting a fuzzy set is better than inin-serting a fuzzy singleton.

Step 8: Under the current node, generate a new ter-minal node to include the new fuzzy set in step 7 and go to step 9.

Fig. 9. The insertion of a fuzzy set for the input data in multi-modal case.

Step 9: Finish this case and restart to recognize an-other situation.

3.3.3. Part III: unknown object

SOFDT cannot recognize the input data when the representing fuzzy sets do not cover it. In this case, the input data can be single-modal, multi-modal, or represent a new situation. SOFDT applies the fol-lowing steps to adapt itselfto this kind ofinput data.

Step 1: Identify the node where the unknown case occurs and go to that node.

Step 2: Check whether or not the input data repre-sents a new situation. Ifthe answer is yes, go to step 5; otherwise go to step 3. Step 3: Check whether or not the input data is

multi-modal. Ifthe answer is yes, go to step 5; otherwise go to step 4.

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Step 4: Tune the parameters ofthe corresponding fuzzy sets of the input data in terms of Eqs. (10), (11), (14), and (15). After the parameter tuning, go to step 7.

Step 5: Insert a fuzzy set Ccninto the current node,

as shown in Fig. 9, and go to step 6. Step 6: Under the current node, generate a new

ter-minal node to include the new fuzzy set in step 5 and go to step 7.

Step 7: Finish this case and restart to recognize an-other situation.

It can be seen from Eqs. (10)–(17) that the pa-rameter tuning ofSOFDT is a gradual learning process because the learning rate " is in the range of(0,1). It has the advantage ofconsidering the general property ofthe training data. On the other hand, SOFDT needs to learn the training data sev-eral times to obtain a convergent learning result. To make SOFDT fast and accurately learn the train-ing data, the learntrain-ing rate " should be large at the beginning and gradually decrease as the training goes on.

3.4. Error rejection in learning process

Ifthe on-line input data is erroneous or unreason-able, it will not only decrease the performance of the decision tree but also waste learning time. On the other hand, ifthere is a restriction on the tree size, the multi-modal input data may not always be acceptable. To overcome these problems, SOFDT uses the follow-ing error-rejection rules to 3lter out the unreasonable data.

Rule I: For incorrect recognition case, ifthe input data is not in the modi3cation zone ofthe correct or incorrect class, ignore it in the cor-responding modi3cation.

Rule II: For unknown situation case, ifthe input data is single-modal and does not fall into the modi3cation zone ofthe correct class, ignore it in the corresponding modi3cation. Rule III: For unknown situation case, ifthe input data

is multi-modal or new model and adding a node will exceed the tree size restriction, ig-nore it in the corresponding modi3cation.

Fig. 10. The de3nition ofrecognition error.

3.5. Performance analysis

3.5.1. Convergence property of learning process The convergence property ofSOFDT can be under-stood by observing the recognition error oftraining data during the learning process. To well describe the convergence property, the recognition error Er is

de-3ned by the following equation and shown in Fig. 10:

Er = (ec=dc+ ei=di): (18)

The change ofrecognition error can be obtained by the following equation:

MEr= [(dcMec− ecMdc)=dc2+ (diMei− eiMdi)=d2i]

= [−Mdc(dc+ ec)=d2c− Mdi(di+ ei)=d2i]

* Mdc= −Mecand Mdi= −Mei: (19)

Therefore, MEr¡0 and the convergence oflearning

can be guaranteed.

3.5.2. Comparison with rule-based systems

In a decision tree, the splitting rules in a path must be simultaneously satis3ed so that the terminal node ofthe path can be reached from the root node. Using ‘and’ and ‘not’ operators, the splitting rules ofa path can be combined to form a new rule to represent the path. Repeating this procedure for every path in a de-cision tree, the dede-cision tree can then be mapped into a rule-based system with each path represented by a rule. According to this mapping ofstructure and pa-rameter, the learning mechanism ofSOFDT can 3nd its counterpart in its corresponding rule-based system. Adding a node in SOFDT will increase one decision path and hence one rule in its corresponding rule-based system. Tuning the parameters ofSOFDT will modify the parameters of the fuzzy sets in “if” part

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Table 1

Comparison between a decision tree and its corresponding rule-based system

Decision trees Rule-based systems

Speed Fast Moderate

Reliability Moderate Good

Storage space Small Large

Learning speed Fast Moderate Learning easiness Easy Moderate

ofits corresponding rule-based system. During learn-ing process, SOFDT 3nds the nodes to be modi3ed, whereas the corresponding rule-based system 3nds the rules to be tuned.

In every decision process ofa decision tree, only the splitting rules in one path will be checked. On the other hand, in the corresponding rule-based sys-tem, every rule will be checked in every decision pro-cess, where each rule is composed ofall ofthe split-ting rules in the corresponding path. Therefore, the inference speed of a decision tree is approximately n-times faster than that of its corresponding rule-based system, where ‘n’ is the number ofrules (or paths). SOFDT has an apparent hierarchical structure so that it can easily and fast 3nd the nodes and its parameters to be modi3ed. In the corresponding rule-based sys-tem, however, there are at least two rules containing any non-terminal node in SOFDT. Therefore, it is not easy for the corresponding rule-based system to 3nd the rules and its parameters to be tuned. The com-parison between a decision tree and its corresponding rule-based system is summarized in Table 1.

4. A self-organizing strategy-based decision-making system

The SOFDT presented in Section 3 is embedded in the decision-making system ofa multi-agent soccer robot system. However, the following transformations must be performed.

• Initial decision tree: The decision tree constructed in Section 2 is used as the initial decision tree in SOFDT. The tree-generation algorithm presented in Section 3 can then be applied to generate the desired initial decision tree from the obtained training data.

• Correct recognition: The correct recognition in SOFDT corresponds to the situation in the strategy-based decision-making system where the strategy works.

• Incorrect recognition: The incorrect recognition in SOFDT corresponds to the situation in the strategy-based decision-making system where the strategy does not work and the right strategy is known. • Unknown object: The unknown case in SOFDT

corresponds to the situation in the strategy-based decision-making system where the strategy does not work but the right strategy is not known.

• Multi-modal: The multi-modal in SOFDT cor-responds to the situation in the strategy-based decision-making system where some strategy works in separate regions ofthe situation space. • Single-modal: The single-modal in SOFDT

cor-responds to the situation in the strategy-based decision-making system where some strategy works only in one region.

• Parameter tuning: The parameter tuning in SOFDT corresponds to the situation in the strategy-based decision-making system where a diFerent strat-egy should be used in some terminal nodes. For example, “a robot’s action changes from shooting at middle goal to shooting at left-most goal” re-quires a diFerent strategy at some terminal nodes. • Structure adaptation: The structure adaptation in

SOFDT corresponds to the situation in the strategy-based decision-making system where a diFerent partition ofthe situation space should be performed. For example, the positions ofopponents in oFense zone will partition the original oFense zone into oFense zone with no opponent, oFense zone with one near-front opponent, oFense zone with one far-front opponent, etc.

• New object: The new object in SOFDT cor-responds to the situation in the strategy-based decision-making system where a new strategy is required in some terminal nodes. For example, “a robot’s action changes from shooting to passing” requires a new strategy at some terminal nodes. Using the basic decision tree in Fig. 2 as an ex-ample, the proposed self-organizing strategy-based decision-making system is illustrated as follows. In a soccer game, the basic decision tree encounters three problems. First, the oFense-robot controls the ball in

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Fig. 11. The improved decision tree for the strategy-based decision-making system.

oFense zone and then shoot the goal, but it fails. The reason is that one opponent in oFense zone blocks the ball. Second, the formation-robot controls the ball in formation zone and then passes it to the defense-robot in oFense zone, but it fails. The reason is that one opponent between the formation-robot and defense-robot intercepts the ball. Third, the defense-defense-robot in defense zone controls the ball and passes it to the formation-robot in formation zone, but it fails. The reason is that one opponent between the defense-robot and formation-robot intercepts the ball. According to the results ofthese strategies and encountered prob-lems, the proposed self-organizing strategy-based decision-making system adapts its structure from Figs. 2 to 11. In Fig. 11, U represents the ball is un-controlled, T represents the ball is under teammate’s control, and O represents the ball is under opponent’s control. Three extra strategies are self-generated in Fig. 11 to cope with the above-mentioned problems. The strategies S OO, S OT, S OU, S FO, S FT, S FU,

S NO, S NT, S NU, S DO, S DT, S DUin Fig. 11 are

the same as those in Fig. 2. The three self-generated strategies in Fig. 11 are listed in the following: 1. S OT1: The o7ense-robot passes the ball in

of-fense zone to the formation-robot. The

formation-robot rushes into oFense zone and tries to shoot the ball. The defense-robot defends the goal.

2. S FT1: The o7ense-robot prepares to receive the

ball. The formation-robot dribbles the ball into oFense zone. The defense-robot de-fends the goal.

3. S DT1: The o7ense-robot prepares to receive the

ball. The formation-robot prepares to inter-cept the ball. The defense-robot goal-keeps the ball to a free space.

The soccer skills ‘dribble’ and ‘goal-keep’ can refer to [9].

5. Simulations and experiments

The proposed self-organizing strategy-based deci-sion making system was applied to the 3-to-3 robot soccer games. Both simulations and experiments were performed to justify its performance. In simulation, 30 runs of3-to-3 robot soccer games were conducted in a soccer robot simulator developed by the authors [5,9]. The duration ofeach run was 5 minutes. The simula-tor was developed using Visual C++ with OpenGL as

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Fig. 12. Three-to-three soccer game where the ‘logo’ team is attacking.

Fig. 13. Experiment of3-to-3 soccer robot game.

the API for graphics hardware. The simulations were conducted on Pentium III 450 with TNT2 Ultra graph-ics engine as the video accelerator.

The snapshot ofthe conducted 3-to-3 robot soccer games is shown in Fig. 12, where the ‘logo’ team is in charge of defending the left goal and attacking the right goal while the ‘head’ team has the reverse duties as ‘logo’ team. In the 3gure, the ‘logo’ team is attack-ing the right goal. In the conducted simulations, two teams were equipped with the same decision-making system. The average time to goal is 12:5 s. EFec-tive oFense and defense were observed in each team. Fig. 13 shows the robots on the 3eld. The decision-making system oF-line learned the data collected from the image system. Based on the 100 training data, the

decision system generated 20 strategies. Then the de-cision system on-line performed dede-cision-making. A decision-making, including generating a new strategy, is less than 7 ms. These results are quite satisfactory. 6. Conclusions

A self-organizing fuzzy decision tree (SOFDT) was developed in this study. Comparing to traditional decision trees, the proposed SOFDT has less mem-ory space and faster generation speed and possesses the real-time on-line adapting capacity. The pro-posed SOFDT was integrated with a strategy-based decision-making system to select among the devel-oped strategies according to the situation ofthe soccer 3eld. The proposed decision-making system is more -exible and eFective than other type ofdecision-making systems. NTU-Formosa performs centralized coordination control to coordinate the multiple mo-bile robots in accordance with the adopted strategies. From the simulation and experiment of3-to-3 soccer games, the results are satisfactory.

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