American Institute of Aeronautics and Astronautics

1

Intelligent Algorithms for Maze Exploration and Exploitation

Sydney Barker1 and Dr. Kelly Cohen.

2

University of Cincinnati, Cincinnati, OH, 45221

Cody Lafountain3

University of Cincinnati, Cincinnati, OH, 45221, United States

The purpose of the project is to develop maze exploration algorithms for a multi-agent

system, using autonomous robots, that allows the agents to successfully navigate through an

array of different mazes based on the game Theseus and the Minotaur. Theseus and the

Minotaur is a maze game in which Theseus tries to get to the exit of each maze without being

eaten by the Minotaur. For every one move Theseus makes, the Minotaur can make two.

The mazes become progressively harder as each maze is completed. A single intelligence

system is made up of algorithms for the robots to use in order successfully simulate the

Theseus and the Minotaur game. One of the robots will represent Theseus and the other

robot will represent the Minotaur. The Theseus robot will work to traverse the maze while

avoiding the Minotaur. The Minotaur robot will work to navigate through the maze in

order to catch the Theseus robot. The goal is to have the robots simulate the Theseus and

the Minotaur game without any human interaction. The next step is to validate the

developed algorithms in a laboratory experiment using programmable mobile robots.

Nomenclature

FIS = Fuzzy Intelligent System

I. Introduction

AZES have been of interest to humans for many years and have been used throughout history for scientific

research in which the use of small laboratory animals for testing is possible. Recently mazes have started to

be used to research the artificial intelligence of robots by examining the robots’ ability to traverse unfamiliar mazes.

Maze exploration algorithms have been studied since the mid-nineteenth century and are related to the Graph

Theory. Maze exploration algorithms and Graph Theory are used in mathematics and computer science and study

mathematical structures to model pair-wise relations between objects from a particular collection. Maze exploration

algorithms are usually applied to a single agent, most commonly a robot, traversing a maze, but recently the focus of

research is applying Maze exploration algorithms to multi-agent systems. The idea is to have the systems be used

autonomously in various situations to help humans, such as searching and working in hazardous or poisonous

environments (example: nuclear plants or waste sites), navigating through burning buildings, monotonous and tedius

tasks (example: carrying loads around a plant or warehouse), operations where the risk and cost of sending humans

are too high (example: voyages to other planets), and national defense operations. This research will focus on maze

algorithms for multi-agent systems, namely autonomous robots, using the Fuzzy Logic approach.

Fuzzy Logic is reasoning with fuzzy sets and allows a system to simultaneously handle numeric data as well as

linguistic knowledge. Fuzzy Logic deals with complex, nonlinear systems and uses decision making with estimated

values with incomplete, uncertain information. Fuzzy Logic Systems include inputs, outputs, membership

functions, and rules. Inputs and outputs are variables or concepts that are a part of a problem. Each input has a set

of membership functions that describe the input and a degree of membership that weighs the importance of the input

to the problem. The rules are set up into if-then statements. For example, if X is A then Y is B. X and Y are inputs

1 Student, Aerospace Engineering, 2705 Eden Avenue, Cincinnati, OH 45219, and AIAA Member Grade for third

author. 2 Professor, Aerospace Engineering, 735 Rhodes, Cincinnati, OH 45221, and and AIAA Member Grade for third

author. 3 Graduate Student, Aerospace Engineering, 221 E University Ave, Cincinnati, OH 45219, and AIAA Member

Grade for third author.

M

Infotech@Aerospace 201129 - 31 March 2011, St. Louis, Missouri

AIAA 2011-1510

Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Dow

nloa

ded

by U

NIV

ER

SIT

Y O

F C

INC

INN

AT

I on

Nov

embe

r 24

, 201

4 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

011-

1510

American Institute of Aeronautics and Astronautics

2

while A and B are membership functions. The rules can incorporate one of three possible operations: AND, OR,

and NOT. Fuzzy Systems are asked a question or are given an input and the system uses all of its rules to some

degree to get an array of outputs. The system takes the average of all the outputs using the centroid method in order

to develop a single final output. The centroid method is a defuzzification method that finds the “center of gravity”

of the area under the curve and outputs the value representing the “center of gravity” as the best answer. The

objective of the research is to create a Fuzzy Intelligent System in MATLAB that can be implemented to an

autonomous multi-agent system so that the multi-agent system (two robots) can autonomously traverse any maze.

II. Methodology

The research will begin by first playing the game Theseus and the Minotaur on the computer. A picture of

the game simulation is shown below in Figure 1.

Fig. 1 Theseus and the Minotaur Game Snap Shot

The game is a series of 87 mazes and the objective of the game is to get Theseus safely to the exit of the maze in the

least number of moves without being eaten by the Minotaur. Theseus can move one space at a time and for every

one move Theseus makes, the Minotaur can make two moves. There are obstacles set up throughout the maze that

Theseus can use to try and trap the Minotaur in order to proceed safely to the exit. The mazes get progressively

harder as you complete each maze. Playing the game will reveal the Minotaur’s tendencies, weaknesses, and

predictability. While working through each maze, Theseus’ moves and the Minotaur’s subsequent responses will be

tracked. By tracking the Minotaur’s responses, the tendencies and weaknesses of the Minotaur can be noted and

strategies for traversing the maze can be developed. The expertise gained from playing the game will be used to

formulate preliminary inputs, outputs, membership functions, and degrees of membership for the Theseus and the

Minotaur mazes. From the inputs, outputs, and membership functions, a rule base for my fuzzy inference system

can be formulated. To further refine the inputs, outputs, membership functions, and degrees of membership, an m-

file in MATLAB will be written in order to create an interactive maze game resembling Theseus and the Minotaur.

The interactive maze will prompt the user to choose where he or she wants Theseus to move next and the computer

will calculate where the Minotaur’s next two moves will be. At least ten to fifteen interactive mazes will be created

in MATLAB. The interactive mazes will help to create an m-file in MATLAB to define the moves the Minotaur

will make in response to Theseus’ moves and can be used for the FIS. Analyzing the interactive mazes will also

help in refining the inputs, outputs, membership functions, and degrees of membership. Once all the inputs, outputs,

and membership functions are completed, the rules can be completed. With the inputs, outputs, membership

functions, degrees of membership, and rule base, the FIS will be created in MATLAB using the Fuzzy Toolbox

function.

A single m-file was created which contains the FIS created for Theseus and a reactionary heuristic for the

Minotaur. The reactionary heuristic is the same as that used in the interactive games. Since the Minotaur’s moves

are predictable, a simple reactionary heuristic can be used rather than a Fuzzy Inference System. The m-file first

prompts the user to choose a maze design in which he or she would like to work with. MATLAB presents the

requested maze and plots the preliminary positions of Theseus and the Minotaur. The m-file runs the FIS to

Dow

nloa

ded

by U

NIV

ER

SIT

Y O

F C

INC

INN

AT

I on

Nov

embe

r 24

, 201

4 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

011-

1510

American Institute of Aeronautics and Astronautics

3

determine where Theseus’ next move should be. Once Theseus’ move is determined and executed, the m-file then

runs the reactionary heuristic to determine the Minotaur’s two subsequent moves. The m-file is set-up in a WHILE

loop and continues to run the same loop until either Theseus has reached the exit or the Minotaur has caught

Theseus. When Theseus reaches the exit, MATLAB presents the message “You Win”. If the Minotaur catches

Theseus, MATLAB shows the message “You Lose”. The FIS will first be tested in MATLAB using an interactive

simulation. The purpose of testing the FIS is to make sure that the FIS is fully functional and there are no problems

with the components of the FIS as well as to check if any information is missing in the FIS. Once the fuzzy

based decision making algorithm is tested and validated, it will be applied to the laboratory mobile robots (either

Khepera III robots or LEGO Mindstorms NXT 2.0). A representative maze environment will be built in the lab for

the robots to navigate in. One of the robots will play Theseus and the other robot will play the Minotaur. The

Theseus robot will utilize the above algorithm to navigate through the maze in order to reach the exit of the maze

without being caught by the Minotaur. The Minotaur robot will use a simpler reactionary heuristic to calculate its

moves in response to Theseus’ moves.

III. Results

From playing Theseus and the Minotaur game, I noticed three main weaknesses that the Minotaur has.

1. Minotaur’s tendency and preference to move horizontally first, then vertically. If there are no obstacles

blocking the Minotaur from moving horizontally and moving vertically does not get the Minotaur directly

closed to Theseus, then the Minotaur will always move horizontally.

2. Minotaur’s inability to move around obstacles.

3. Minotaur’s inability to look ahead more than one move and play strategically. The Minotaur will move in

immediate response to Theseus rather than strategically moving to prevent Theseus from getting to the exit.

I used the expertise gained from playing the game to formulate inputs, outputs, membership functions, and rules

for the Theseus and the Minotaur mazes. My first set of inputs, outputs, membership functions and rules were very

broad: therefore, the fuzzy system was very large and too complex. Some of the rules incorporated the same inputs

and membership functions as well as contradicted each other. Changes had to be made to my system. I found that it

would benefit my system more if I could come up with more specific inputs and a single output. Many of my

inputs were similar and could be combined into one input. I also eliminated inputs that I believed could be excluded

from the system. By trimming the number of inputs and outputs, the repetitive and unnecessary rules could be

eliminated. Having fewer rules would make my system more efficient and less complex. My final set of inputs,

outputs, and membership functions are displayed below in Figure 2.

Fig 2. Final Set of Fuzzy System Components

From the inputs, outputs, and membership functions in Figure 2, I was able to formulate a sufficient rule base for my

fuzzy inference system. The last step to creating a Fuzzy Inference System for Theseus and the Minotaur game was

to implement my inputs, outputs, membership functions, and rules to the Fuzzy MATLAB program. In order to

create a functional fuzzy inference system in MATLAB, I had to determine the degrees of membership for each

membership function corresponding to each input and output. Fifteen interactive mazes were created and tested in

MATLAB. One of the interactive maze designs is shown below in Figure 3.

Inputs Membership Functions

Theseus Position below, above, left, right

Theseus Position 2 below above, left, right

Number of Spaces one, multiple

Wall Position above, below, left, right

Wall Position 2 above, below, left, right

Wall Orientation vertical, horizontal, vertical and horizontal

Number of Walls one, multiple

Outputs Membership Functions

Theseus Action up, down, left, right, stay still

Dow

nloa

ded

by U

NIV

ER

SIT

Y O

F C

INC

INN

AT

I on

Nov

embe

r 24

, 201

4 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

011-

1510

American Institute of Aeronautics and Astronautics

4

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Fig 3. Interactive Maze Design in MATLAB

Creating and analyzing the interactive mazes helped to refine the current FIS and make a more functional system. A

new rule base was created using the expertise gained from creating the interactive mazes in MATLAB. The new

inputs, membership functions, and degrees of membership are shown below in Appendix A and the output is shown

in Appendix B. Examples of the fuzzy toolbox are shown in Figures 4, 5, and 6. Figure 4 is a picture of the

Membership Function editor, Figure 5 is a picture of the rule editor, and Figure 6 is a picture of the FIS editor.

Fig 4. Membership Function Editor of the Fuzzy Toolbox

Dow

nloa

ded

by U

NIV

ER

SIT

Y O

F C

INC

INN

AT

I on

Nov

embe

r 24

, 201

4 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

011-

1510

American Institute of Aeronautics and Astronautics

5

Fig 5. Rule Editor of Fuzzy Toolbox

Fig 6. FIS Editor of Fuzzy Toolbox

An m-file was created in MATLAB that would run the FIS for each interactive maze design created. This stage

of the research project is currently being finished. The m-file prompts the user to choose which maze design he or

she wants to work with. The m-file implements the FIS to the desired maze design. The m-file is set up to calculate

the value given to each input for Theseus’ current position and then calculates the output. Once the output is

determined, the m-file moves Theseus to the calculated new position. After Theseus’ move has been made, the m-

file then calculates the two subsequent moves that the Minotaur will make. The m-file is made up of a series of

FOR and WHILE loops so that the m-file continues this process for every time Theseus moves until Theseus has

either reached the exit and won or has been caught by the Minotaur. The single m-file and FIS can be applied to any

maze design created in MATLAB. Currently work is being done to refine the m-file. There are some problems with

repetitive input calculations and coding. Testing the FIS in MATLAB presented many problems with my Fuzzy

Inference System. The FIS had too many rules in the rule base and as a result some of the rules over lapped each

Dow

nloa

ded

by U

NIV

ER

SIT

Y O

F C

INC

INN

AT

I on

Nov

embe

r 24

, 201

4 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

011-

1510

American Institute of Aeronautics and Astronautics

6

other or contradicted each other. Due to overlapping and contradicting rules, the FIS was forced to arbitrarily

choose one of the rules when calculating the output and as a result the output was not correct. Overlapping rules

also cause the FIS to get “confused” about which rule to use and would present an error reading in MATLAB.

Another problem with the FIS was that it was too complex. There were too many inputs to consider when

calculating the output. In order to minimize the complexity of a single FIS, cascade learning was used. Cascade

learning takes a large system and splits the system up into different sets and sub-sets. In the case of the FIS in this

research project, in order to get the desired output (Theseus’ Move) multiple small, simple fuzzy inference systems

were created and then implemented together to get the final output (Theseus’ Move). The first cascade “tree” model

is shown in Figure 7.

The model had three sets to make up the entire system. The first cascade model proved to be too simple of a

maze algorithm to complete a full maze. The m-file was debugged in MATLAB in order to observe what rules in

the FIS were overlapping or missing in the rule base. Small changes and additions were made to the FIS sets as well

as to the m-file. The small changes that were made included changing and adding rules, changing the m-file script,

changing membership functions, adding inputs, and changing the order of my FIS sets. Apart from making small

changes, strategies needed to be developed for Theseus to use when he is close to the maze exit and when he is far

from the exit. The two strategies were named the “End Game” strategy and the “Minotaur Trap” strategy. The end

game strategy is used by Theseus when Theseus is close to the exit and the Minotaur is far from Theseus and the

exit. The end game strategy is comprised of a simple heuristic that does not require “fuzzy” thinking, but a straight

forward path to get to the exit. The “Minotaur Trap” strategy is used when Theseus and the Minotaur are close to

the exit or Theseus is far from the exit. In both cases Theseus is required to lure the Minotaur from the exit and trap

him so that Theseus can safely reach the exit. The second version of the cascade tree model is shown in Figure 8.

Fig 7. Cascade Learning Implementation to the Fuzzy Inference

Dow

nloa

ded

by U

NIV

ER

SIT

Y O

F C

INC

INN

AT

I on

Nov

embe

r 24

, 201

4 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

011-

1510

American Institute of Aeronautics and Astronautics

7

The second cascade model successfully solved six of the fifteen mazes in MATLAB. The cascade model did not

work for more complex mazes due to overlapping rules and missing information. In order to come up with another

FIS set to add to the cascade, the m-file had to be debugged again. The problems that were found during the

debugging process were analyzed in order to see what information was missing in the cascade model.

From the analysis, a new FIS set was created as well as an additional strategy. Three strategies were defined for

Theseus to use while in pursuit of the exit: the “End Game” strategy, “Minotaur Lure” strategy, and “Minotaur

Trap” strategy. The end game strategy and Minotaur Trap strategy were used in the same way as the second cascade

model. The Minotaur Lure strategy was used when the Minotaur is close to the exit and Theseus is far away from

the exit. In this case, Theseus must move towards the exit in a manner that will lure the Minotaur from the exit and

into the middle of the maze. Once the Minotaur is lured away from the exit, then Theseus and implements the

Minotaur Trap strategy to trap the Minotaur and then move towards the exit. The third cascade model is shown in

Figure 9.

Fig 8. Second Cascade Tree Model for the Fuzzy Inference

Dow

nloa

ded

by U

NIV

ER

SIT

Y O

F C

INC

INN

AT

I on

Nov

embe

r 24

, 201

4 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

011-

1510

American Institute of Aeronautics and Astronautics

8

Fig. 9 Final Cascade Tree Model for the Fuzzy Inference System

The third cascade model allowed for more complex maze designs to be traversed and solved by Theseus. The entire

cascade tree created for the Intelligent Algorithm contained four cascades.

Once the m-file for the FIS is fully functional in MATLAB, it will be validated in simulation and robustness will

be gauged by applying to logic to 10 additional “untrained cases” followed by the above described laboratory

experiment. Once the fuzzy based decision making algorithm is tested and validated, it will be applied to the

laboratory mobile robots (either Khepera III robots or LEGO Mindstorms NXT 2.0). A representative maze

environment will be built in the lab for the robots to navigate in. One of the robots will play Theseus and the other

robot will play the Minotaur. The Theseus robot will utilize the above algorithm to navigate through the maze in

order to reach the exit of the maze without being caught by the Minotaur. The Minotaur robot will use a simpler

reactionary heuristic to calculate its moves in response to Theseus’ moves.

IV. Conclusion

Creating maze exploration algorithms using fuzzy logic as an approach will be useful in traversing mazes where

there is no certain path to completing the maze. The maze can be completed using many different paths. In

problems where the completion of the maze can be done in many different ways, there is always an optimal path to

completing the maze. By implementing fuzzy logic to the exploration algorithms, the optimal path can be found.

The Fuzzy Inference System will be given the maze design and will use the inputs, outputs, membership functions,

and rules to calculate the best solution and outputs that solution to the multi-agent system: Theseus robot and

Minotaur robot. The best method to creating an algorithm using fuzzy logic is through a cascade learning concept.

By splitting the fuzzy inference system into smaller sets, the entire system is simplified and the output can be

calculated quicker.

After the robots successfully implement the exploration algorithms and autonomously traverse multiple maze

designs, the next step is to have the robots collaborate to solve the mazes rather than work against each other. A

intelligent algorithm would be implemented to both robots and the robots would communicate with other in order to

solve the maze. The robots would be placed in different location of the maze and they would share information

about their surroundings and failed or successful attempts. By sharing information, each robot can use its

observations as well as the other robots’ in order to traverse the maze. Communication between the two robots can

be done using Bluetooth. The collaboration between the two agents can be expanded to other applications. One

example is with Unmanned Air Vehicles (called UAVs). UAVs are autonomous systems. UAVs could collaborate

with each other when surveying an area or flying in swarms.

Dow

nloa

ded

by U

NIV

ER

SIT

Y O

F C

INC

INN

AT

I on

Nov

embe

r 24

, 201

4 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

011-

1510

American Institute of Aeronautics and Astronautics

9

Appendix

Inputs

Membership

Functions Degrees of Membership

Theseus Position Above Minotaur 0-180

Below Minotaur 180-360

Left of Minotaur 90-270

Right of Minotaur 0-90, 270-360

Distance from the Minotaur Close 0-1.4

Far 1.4-11

Wall 1 Position for Theseus Left of Theseus 90-270

Above Theseus 0-180

Below Theseus 180-360

Right of Theseus 0-90, 270-360

Wall Orientation Vertical 0

Horizontal 1

Wall 2 Position for Theseus Left of Theseus 90-270

Right of Theseus 0-90, 270-360

Above Theseus 0-180

Below Theseus 180-360

Wall Orientation Vertical 0

Horizontal 1

Wall Position 1 for Theseus Left of Theseus 90-270

Above Theseus 0-180

Below Theseus 180-360

Right of Theseus 0-90, 270-360

Wall Orientation Vertical 0

Horizontal 1

Wall Distance from Minotaur Close Less than 2

Far Greater than 2

Wall Position 2 from Minotaur Above Theseus 0-180

Right of Theseus 0-90, 270-360

Left of Theseus

Below Theseus

Wall Orientation Vertical 0

Horizontal 1

Wall Distance from Minotaur Close Less than 2

Far Greater than 2

APPENDIX A. Revised Inputs, Membership Functions, and Degrees of Membership

Dow

nloa

ded

by U

NIV

ER

SIT

Y O

F C

INC

INN

AT

I on

Nov

embe

r 24

, 201

4 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

011-

1510

American Institute of Aeronautics and Astronautics

10

Output Membership

Functions

Degrees of Membership

Theseus Move Up 0

Down 0.25

Left 0.5

Right 0.75

Stay 1

APPENDIX B. Revised Output, Membership Functions, and Degrees of Membership

Acknowledgments

The author of this paper would like to extend thanks to Dr. Kelly Cohen, Aerospace Professor at the University

of Cincinnati and Project Advisor. Dr. Cohen provided his lab, needed materials, reference reading materials, and

guidance for the current work and future work going into the research project. The author would also like to extend

thanks to Chelsea Sabo, an Aerospace Graduate student at the University of Cincinnati. Chelsea served as the

graduate assistant to the research project. Chelsea helped the Fuzzy Inference System get started and provided input

and guidance throughout the research period. The author would like to thank Pablo Mora for help in writing the

code for the interactive mazes. Thanks are extended to Cody Lafountain who helped correct the programming

mistakes in MATLAB. Thanks are given to the McNair Scholars program and the Ohio Space Grant Consortium for

grant that aided in developing the research project.

References 1Butler, Charles, and Caudill, Maureen, Naturally Intelligent Systems, The MIT Press, Cambridge, Mass, 2000, Chaps. 2- 5. 2Dixon, K. R., Khosla, P. R., and Malak, R. J., “Incorporating Prior Knowledge and Previously Learned Information into

Reinforcement Learning Agents,” Institute for Complex Engineered Systems Technical Report Series, 31 Jan. 2000. 3Huser, J., Peters, L., and Surmann, H., “A Fuzzy System for Indoor Mobile Robot Navigation,” Fourth IEEE International

Conference on Fuzzy Systems, FUZZ-IEEE 95, IEEE, 1995, pp. 83-88. 4Ishikawa, S., “A Method of Autonomous Mobile Robot Navigation by Using Fuzzy Control,” Advanced Robotics, 9th ed.,

1995, pp. 29-52. 5Kosko, B., Fuzzy Thinking: The New Science of Fuzzy Logic, Hyperion, New York, 1993.

Dow

nloa

ded

by U

NIV

ER

SIT

Y O

F C

INC

INN

AT

I on

Nov

embe

r 24

, 201

4 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

011-

1510