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Collision Free Trajectory Generation Using Artificial Potential Function and Stereo Vision System

Permanent Link: http://ufdc.ufl.edu/UFE0045466/00001

Material Information

Title: Collision Free Trajectory Generation Using Artificial Potential Function and Stereo Vision System
Physical Description: 1 online resource (77 p.)
Language: english
Creator: Yao, Ronghua
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: artificial -- function -- potential -- stereo -- trajectory -- vision
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Mobile robot path planning is one of the most important research domains of robot technology. Most self-navigating autonomous robots work in an environment that is filled with complex and unpredictable obstacles. The robot must work in this environment and avoid collision with the obstacles. The sensors for environment perception and robot self-status perception are implemented on the robot to avoid the obstacles. Usually the sensors used for perception are divided into two parts: one part is the passive sensors like passive sonar and stereo vision system, and the other is the active sensors like time of flight (TOF) optical sensors. In this thesis, stereo vision system is used for environment perception. The environment model can be built based on the position and local information of obstacles which is provided by the stereo vision system. In this research, artificial potential function (APF) is used as the path planning method. This is because the APF method has an advantage in real time obstacle avoidance and is convenient for practical use. However the APF method fails to take the specific robot dynamic model into consideration. Also there is the local minimal problem for this method. The improvement for the APF method is presented in this thesis. In this research a collision free path is generated by combining the stereo vision system and the APF method. The normal stereo vision system is improved by adding the memory to remember the detected obstacles. In this way image processing methods can be used to improve the drawbacks of the APF theory. In most practical implementation of robots, the robot is desired to finish its mission with minimal cost of time and fuel. Due to this requirement, optimal time or optimal fuel trajectory should be generated for the robot to track when it moves in the workspace. Also, the kinematic profile corresponding to the optimal trajectory should be generated to guide the motion of the robot. For further research, two kinematic smoother trajectory generation methods are discussed and compared. Finally the improved stereo vision system and time optimal trajectory generation method are evaluated by simulation.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ronghua Yao.
Thesis: Thesis (M.S.)--University of Florida, 2013.
Local: Adviser: Fitz-Coy, Norman G.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045466:00001

Permanent Link: http://ufdc.ufl.edu/UFE0045466/00001

Material Information

Title: Collision Free Trajectory Generation Using Artificial Potential Function and Stereo Vision System
Physical Description: 1 online resource (77 p.)
Language: english
Creator: Yao, Ronghua
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: artificial -- function -- potential -- stereo -- trajectory -- vision
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Mobile robot path planning is one of the most important research domains of robot technology. Most self-navigating autonomous robots work in an environment that is filled with complex and unpredictable obstacles. The robot must work in this environment and avoid collision with the obstacles. The sensors for environment perception and robot self-status perception are implemented on the robot to avoid the obstacles. Usually the sensors used for perception are divided into two parts: one part is the passive sensors like passive sonar and stereo vision system, and the other is the active sensors like time of flight (TOF) optical sensors. In this thesis, stereo vision system is used for environment perception. The environment model can be built based on the position and local information of obstacles which is provided by the stereo vision system. In this research, artificial potential function (APF) is used as the path planning method. This is because the APF method has an advantage in real time obstacle avoidance and is convenient for practical use. However the APF method fails to take the specific robot dynamic model into consideration. Also there is the local minimal problem for this method. The improvement for the APF method is presented in this thesis. In this research a collision free path is generated by combining the stereo vision system and the APF method. The normal stereo vision system is improved by adding the memory to remember the detected obstacles. In this way image processing methods can be used to improve the drawbacks of the APF theory. In most practical implementation of robots, the robot is desired to finish its mission with minimal cost of time and fuel. Due to this requirement, optimal time or optimal fuel trajectory should be generated for the robot to track when it moves in the workspace. Also, the kinematic profile corresponding to the optimal trajectory should be generated to guide the motion of the robot. For further research, two kinematic smoother trajectory generation methods are discussed and compared. Finally the improved stereo vision system and time optimal trajectory generation method are evaluated by simulation.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ronghua Yao.
Thesis: Thesis (M.S.)--University of Florida, 2013.
Local: Adviser: Fitz-Coy, Norman G.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045466:00001


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1 COLLISI ON FREE TRAJECTORY GENERATION USING ARTIFICIAL POTENTIAL FUNCTION AND STEREO VISION SYSTEM By RONGHUA YAO A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTOR OF SCIENCE UNIVERSITY OF FLORIDA 2013

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2 2013 R onghua Yao

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3 To my mother and father

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4 ACKNOWLEDGMENTS I would personally like to thank my supervisory committee chair (Dr. Norman Fitz Coy) for giving me the opportunity to do this research I would like to thank him for his patience and kindness in giv ing me advice and instruct ing me in finish ing my master thesis. I would like to thank the University of Florida and Mechanical and Aerospace Engineering Department for giving me the chance to study in the United State s I would also like to thank the professor s, friends and people who have give n me help while study ing abroa d. I would like to thank my supervisory committee members (Dr. Gloria Finally I would like to thank m y parents and family, without their support I would not have had the opportunity study ab r oa d and none of th is would be possible.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURE S ................................ ................................ ................................ .......... 8 LIST OF ABBREVIATIONS ................................ ................................ ........................... 10 ABSTRACT ................................ ................................ ................................ ................... 11 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 13 Motivation ................................ ................................ ................................ ............... 13 Mission Examples ................................ ................................ ................................ ... 14 Review ................................ ................................ ................................ .................... 16 Research Scope ................................ ................................ ................................ ..... 17 2 PATH PLANNIN G WITH ARTIFICIAL POTENTIAL FUNCTION ............................ 20 Theoretical Development ................................ ................................ ........................ 20 APF for Obstacles with Specific Configuration ................................ ........................ 21 Potential Field of Obstacles ................................ ................................ .............. 21 Limitation of the APF Method ................................ ................................ ........... 22 Static and Moving Obstacle Avoidance ................................ ................................ ... 23 3 TRAJECTORY GENERATION WITH OBTAINED PATH ................................ ....... 27 Interpolation Theory ................................ ................................ ................................ 27 Cubic Spline ................................ ................................ ................................ ..... 27 Fourth Order Spline ................................ ................................ .......................... 29 Features of Cubic and Fourth Order Spline ................................ ...................... 31 Trajectory Optimization with Limited Condition ................................ ....................... 35 Physical Limitation in Trajectory Generation ................................ .................... 35 Optimization for Feasible Trajectory ................................ ................................ 37 Time Optimal Trajectory Tracking ................................ ................................ ........... 38 4 STEREO VISION SYSTEM ................................ ................................ .................... 45 Depth Perception of Stereo Vision System ................................ ............................. 45 Pre Processing of Stereo Vision System ................................ .......................... 45 Depth Perception Algorithm ................................ ................................ .............. 46 De pth Perception Simulation ................................ ................................ ............ 48

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6 Improvement of Vision System ................................ ................................ ............... 49 Obstacle Location Memory with Cells Map ................................ ....................... 49 Step Estimation in Path Planning ................................ ................................ ..... 52 Image Processing for Vision System ................................ ................................ 53 Local minimal avoidance ................................ ................................ ............ 53 Obstacle segments connection ................................ ................................ .. 55 5 SIMULATION MODEL AND RESULTS ................................ ................................ .. 56 Introduction of Simulation Model ................................ ................................ ............. 56 Workspace Detection and Collision Free Path Generation ................................ ..... 58 Optimal Trajectory Generation ................................ ................................ ................ 61 Spline Interpolation Method ................................ ................................ .............. 61 Time Optimal Trajectory Generation Method ................................ .................... 65 Simulation Results Analysis ................................ ................................ .................... 70 6 CONCLUSION AND FUTURE RESEARCH ................................ ........................... 72 LIST OF REFERENCES ................................ ................................ ............................... 74 BIOGRAPHIC AL SKETCH ................................ ................................ ............................ 77

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7 LIST OF TABLES Table page 2 1 APF method simulation parameters. ................................ ................................ .. 24 4 1 Simulation parameters of stereo vision system. ................................ ................. 48 5 1 APF method simulation parameters ................................ ................................ ... 57

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8 LIST OF FIGURES Figure page 1 1 Examples of self navigation robots ................................ ................................ ..... 14 1 2 Bluefin 21 u nderwater vehicle de veloped by Bluefin Robotics [4] ...................... 15 1 3 Regolith Excavation Co mpetition sponsored by NASA [6] ................................ .. 16 2 1 Example showing the obstacle using APF ................................ .......................... 22 2 2 The local minimum problem in potential field [13] ................................ ............... 23 2 3 Simulation of obs tacle avoidance with APF method ................................ ........... 24 2 4 Path planning with APF method for previously known obstacles. ....................... 25 2 5 Path planning with APF method f or previously unknown obstacles .................... 25 3 1 Cubic spline in terpolation method illustration ................................ ...................... 28 3 2 Fourth order spline in terpolation method illustration ................................ ........... 30 3 3 Trajectories and kinematic profiles generated by the cubic and fourth order interpolation methods, trajectories are interpolated with 5 waypoints ................. 32 3 4 Trajectories and kinematic profiles generated by the cubic and fourth order interpolation methods, trajectories are interpolated with 15 waypoints ............... 34 3 5 Trajectories and kinematic profiles generated by the cubic and fourth order interpolation methods, trajectories are interpolated with different time inter vals ................................ ................................ ................................ .............. 35 3 6 Radial and tangential friction ellipse [24] ................................ ............................ 36 3 7 Velocity pr ofile with different time scale ................................ .............................. 38 3 8 Trajectory example to generate the time optimal kinematic profile ..................... 40 3 9 The corresponding time optimal velocity p rofile related with parameter u .......... 42 3 10 T he corresponding time optimal vel ocity profile related with time ....................... 43 3 11 The kinematic profile used to connect different kinematic profile segments ....... 44 4 1 Rectification of stereo image [21] ................................ ................................ ....... 46 4 2 Stereo vision system for depth perception. ................................ ......................... 47

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9 4 3 The simulation model and the stereo image of the two cameras ........................ 48 4 4 Simulation Results ................................ ................................ .............................. 49 4 5 Re presentation of obstacles in cells map when the robot is moving ................... 50 4 6 Comparing the APF path planning method with two vision sys tems ................... 51 4 7 The procedure of local minimum avoidance of the APF method by using the stereo image processing ................................ ................................ .................... 54 5 1 Scenario Description of Simulation Model ................................ .......................... 56 5 2 Obstacles which are located in travelli ng area in the simulation model .............. 57 5 3 Potential field and moving path in the first round trip ................................ .......... 59 5 4 Potential field and moving path in the second round trip ................................ .... 59 5 5 Potential field and moving path in the third round trip ................................ ......... 60 5 6 Potential field and moving path in the fourth round trip ................................ ....... 60 5 7 Trajectory generated by taking t wo spline interpolation methods ....................... 62 5 8 Kinematic profile of the two spline interpolation methods ................................ ... 63 5 9 Kinematic profile fr om the cubic interpolation method ................................ ........ 64 5 10 Kinematic profile from the fo urth order i nterpolation method .............................. 65 5 11 The trajectory used for time optim al trajectory generation method ..................... 66 5 12 The curvature of the interpolated trajectory ................................ ........................ 67 5 13 The maximum velocity profi le with variable of parameter u ................................ 68 5 14 The maximum velocity profile with variable of time. ................................ ............ 68 5 15 The time optimal kinematic profile of the interpolated trajectory. ........................ 69

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10 LIST OF ABBREVIATIONS APF Artificial P otential F unction DVL Doppler Velocity Log GPS Global Positioning System INS Inertial Navigation System NASA National Aeronautic & Space Administration SAD Sum of Absolute Difference SSD Sum of Squared Difference SVS Stereo Vision System TOF Time of Flight

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11 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master o f Science COLLISIO N FREE TRAJECTORY GENERATION USING ARTIFICIAL POTENTIAL FUNCTION AND STEREO VISION SYSTEM By Ronghua Yao May 2013 Chair: Norman Fitz Coy Major: Mechanical Engineering Mobile robot path planning is one of the most important research domains of robot technology. Most self navigati ng autonomous robot s work in an environment that is filled with complex and unpredictable obstacles The robot must work in this environment and avoid collision with the obstacles. T he sensor s for environment perception and robot self status perception are implemented on the robot to avoid the obstacles. Usually the sensors used for perception are divided into two parts: one part is the passive sensors like passive sonar and stereo visio n system, and the other is the active sensors like time of flight (TOF) optical sensors. In this thesis, stereo vision system is used for environment perception. The environment model can be built based on the position and local information of obstacle s wh ich is provided by the stereo vision system In this research, artificial potential function (APF) is used as the path planning method. T his is because the APF method has an advantage in real time obstacle avoidance and is convenient for practical use. Ho wever the APF method fails to take the specific robot dynamic m odel into consideration. A lso there is the local minimal problem for this method. The improvement for the APF method is presented in this thesis. In this

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12 research a collision free path is gener ated by combining the stereo vision system and the APF method. The normal stereo vision system is improved by add ing the memory to remember the detected obstacles I n this way image processing method s can be used to improve the drawbacks of the APF theory. In most practical implementation of robot s the robot is desired to finish its mission with minimal cost of time and fuel. Due to this requirement optimal time or optimal fuel trajectory should be generated for the robot to track when it moves in the workspace. Also the kinematic profile corresponding to the optimal trajectory should be generated to guide the motion of the robot. For further re search, two kinematic smoother trajectory generation methods are discuss ed and compared. Finally the improved stereo vision system and time optimal trajectory ge neration method are evaluated by simulation

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13 CHAPTER 1 INTRODUCTION Motivation The use of robotics for transportation a nd operation in agriculture, mining, industry, commerc e and home service has become more accessible with the development of robot technology In these domains, compare d to the work of people, robot s can offer more efficient and precise performance, improve safety reduce the cost for labor training and some other expenses Mobile robot ics is one branch of robotics that is commonly used in these areas U sually the mission for mobile robot s is to autonomously transit from a known launch site to a given target location. These robots always work in unknown environment s and terrains, so they have to plan the moving path on time with the information provided by perception sensors. As the unknown working environment is complex and unpredictable di fferent sensors must be added to the robots for various tasks. R obot s cannot directly use the global position system ( GPS ) for navigation in this kind of unknown environment. The self location of mobile robot s is provided by GPS or a self generated guiding map. Furthermor e, i n autonomous robot service the cost of time and energy should be considered as great ly influenc ing in the robot performance. Therefore minimizing time and fuel cost s during robot service and ensur ing robot s can finish its tasks while avoid ing collision with obstacles plays a crucial role in improv ing the efficiency of robot s This thesis provides the method for mobile robot s to plan the path in an unknown working environment the perception sensor in this thesis used for perception is the stere o vision system. Also a lot of research institutes and companies in these different domains work on mobile

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14 robot s work ing in unknown environment s. F ollowing section give s the examples of the application of the mobile robot in different area s Mission Exam ples One primary application of the mobile robot is the indoor delivery robot. The work environment for this kind of robot is usually in a home, restaurant or laboratory. T he situation and obstacles in these areas are complex and unpredictable, the deliver y robot has to go across these areas and travel around people and certain place s Figure 1 1 A) shows the Adapt SPC 4200 indoor robot and the Adapt Lynx self navigatin g indoor vehicle, these two robots are designed for delivering goods in challenging envir onment s that may include narrow and crowed areas as well as dynamic and peopled locations [1 2 ]. They are both self navigatin g autonomous robots that can generate path s and avoid collision s based on the sensors. Willow Garage has also develop ed the indoor delivery robot TurtleBot show n in Figure 1 1C ) T his robot is a st ereo vision based self navigati n g robot and can be improved as a platform for other tasks by combin in g its stereo vision system [3]. A B C Figure 1 1 Examples of self navigation robots. A ) T he Adapt SPC 4200 indoor robot, B ) Adapt Lynx self navigati ng indoor robot, C ) I ndoor robot TurtleBot produced by Willow Garage [1 3]

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15 In extreme environment s not accessible to people self navigatin g autonomous robot s are designed to replace people in do ing the work such as experiment s excavating and exploring. Usually in this environment there are unpredictable obstacles like rocks and pits, and a robot has to arrive at the destination to finish tasks without collision with these obstacles. Bluefin Robotics developed their Bluefin 21 underwater vehicle for underwater missions. The robot is relies on INS, DVL and SVS for navigation when working underwater [4]. FMC Technologies also developed the ir Ultra Heavy Duty robotic system for underwater tasks using automatic navigation and positioning system [5]. The automatic robots work ing in this environment not accessible to people highly improve d the safety and efficiency of the wor king environment In 2007 the National Aeronautic & Space Administration (NASA) began to sponsor the Regolith Excavation Competition, which provide s reference for further lunar mining robotics design s [6]. People cannot work openly on the lunar surface and robots work ing there cannot use GPS for navigation and localization. For these reasons, self navigati n g mining robots are needed to work in this environment They should be able to find a safe path to navigate, avoiding collisions with potentially dang erous obstacles using sensors. In order for this to work, path planning and obstacle avoid ing algorithm s for this self navigatin g robot should be developed to improve the work efficiency and safety. Figure 1 2 Bluefin 21 underwater vehicle developed b y Bluefin Robotics [4]

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16 Figure 1 3 Regolith Excavation Competition sponsored by NASA [6] Review The APF method is commonly used for real time robot path planning. Khatib appl ied the APF method on robot s to produce collision free path [ 7 ] However, it is very easy for a robot to be trapped into a local minim um when using the normal APF method, so Kim, J. O. and Khosla use d the h armonic f unction as the potential function for path planning [ 8 ] T his is because the harmonic function will not generate t he local minim um in the potential field. Mabrouk and McInnes, Min gyu Park and Jae hyun Jeon also provided some other method s to solve the local minimal problem [ 9 10 ] But the normal APF method is not perfect in avoid ing collision s with moving obstacles S.S. GE and Y.J. CUI improved the normal APF method for avoiding collision s with moving obstacles by taking the v elocity into consideration [11]. T his highly improved the robot performance in track ing the moving destination and avoid ing moving obstacles. One of the most important component s for robot s to avoid collision with obstacles is the ability to detect the environment. The s tereo vision system is one of the pa ssive sensors for environment perception. T he depth map generation algorithm of stereo

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17 vi sion system is introduced in the work of Ali Kilic and Maryum F. Ahmed [1 8 22 ] In autonomous robot localization and navigation a map is the essential component M. Hebert proposed the occupied grid map to describe the environment for robot self navigation [ 23 ] Now the stereo vision system is widely implemented on different kinds of robots for obstacle detecti on and navigation, Allan Eisenman and Steven B. Goldberg implemented the stereo vision system on mars rover [1 6 1 7 ] I n the work of Don Mu rray, the stereo vision system is used to build a grid map for real time navigation [ 20 ] The technology to use the stereo vision system for navigation is develop ing quickly and greatly promotes the development of robotics. When a robot moves in the worksp ace, its movement is constrained by the limitation of velocity, acceleration and jerk The path generated from the APF method is a series of discrete sets of points, the corresponding trajectory should be generated for the robot to track the corresponding kinematic profile also should be generated to guide the robot. V. M unoz prompt ed the trajectory generation method to generate the trajectory constrained by velocity and acceleration [ 24 ] K. Petrinec and Z. Kovacic provide d the trajectory generation method with consideration of jerk continuity and jerk limitation [2 5 ] In practical application s robot s are desired to finish the tasks in minim al time. Marko Lepeti c developed the time optimal trajectory generation method [2 6 ] Sonja Macfarlane [2 7 ] and Imran Waheed [2 8 ] provide two different methods for robot trajectory generation and implement ed their method on specific robot s for evaluation. Research Scope This thesis develops the collision free trajectory generation method b y combining the APF method and the improved stereo vision system. Chapter 2 explores the method that use s APF to avoid obstacles with specific contours. In C hapter 2 the obstacle

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18 configuration is discretized to build the APF potential field, and s imulation examples are provided to evaluate the performa nce of this method to avoid obstacle with specific contours T he drawbacks of the normal APF method in path planning are also discussed in C hapter 2 and these drawbacks can be solved by the improv ing stereo vision system which is discussed in later C hapter 4 Chapter 3 detail s the kinematic profile generation method in track ing the i nterpolated smooth collision free trajectory In the first part of C hapter 3 the positive and negative aspect s of t wo trajectory interpolation methods are discussed in ge nerat ing the smooth velocity and acceleration profile, as well as the method to rescale kinematic profiles to meet constrains of robot physical limitations. In the second part of C hapter 3 the time optimal trajectory generation method is developed. In this part the generation method of time optimal velocity and acceleration profiles is developed. C hapter 4 develops the improved stereo vision system to improve the limitation s of the APF method. In C hapter 4 the depth percepti on principle of the stereo vision system is discussed and evaluated by simulation. Furthermore a grid ded map is developed in order to me morize the obstacle information. The solution to the local minim um problem of the APF method and other improvements of the APF method are discovered based on the processing of this grid occupied map. In C hapter 5 a specific scenario is constructed to simulate the performance of this time optimal collision free trajectory generatio n method The improved stereo vision system is used in the simulation for environment p erception. T wo smooth trajectory generation methods and the corresponding kinematic profile s generation methods are

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19 compared by simulation to evaluate the time optimal t rajectory generation method. Finally in C hapter 6 the conclusion of the research in this thesis i s made and further research planning is prompted.

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20 CHAPTER 2 PATH PLANNING WITH ARTIFICIAL POTENTIAL FUNCTION Theoretical Development Artificial potential function (APF) is used to describe the working space for robot s In the APF method, the goal produces an attractive force on the robot and the obstacles produce repulsive force s on robot. With the force imposed by the environment, the robot can avoid collisions with obstacles and reach the goal from any initial position. By using the APF method, the obstacles, goal and robot are simplified as some points. T he robot working space can be described as a potential field by APF, and each position in the working space has a corresponding value. T he goal has the minimum value in the potential field and the obstacles have relative ly high value s The gradient on each positio n of the potential field gives the fastest direction to arrive at the adjacent relative ly small value position [14 15] So in the potential field, robot s can get to the destination from the initial position with the direction of the gradient. APF has two parts: the attractive potential function and the repulsive potential function. In this research the APF form proposed by Khatib [1] is used. The attractive potential function and attractive force are expressed as : (2 1) (2 2) Where is the rela tive distance between a certain position a nd the goal in the potential field. The attractive force is obtained by taking the derivative of the attractive potential function. The repulsive potential function and repulsive force are expressed as : (2 3)

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21 (2 4) Where is the relative distan ce between a certain position and obstacles in the potential field and is the maximum influence distance of the repulsive force. From Eq. (2 3) and Eq. (2 4), there is no repulsive force beyond the influence area. The repulsive force is obtained by taking the derivative of the repulsive potential function. The total force on the robot is expressed as : (2 5) The force imposed on the robot in the potential field i s the resultant force of attractive force and repuls ive force W ith this resultant force the robot moves from its initial point to its destination. APF for Obstacles with Specific C onfiguration Potential Field of Obstacles In the APF method, the obstacles are simplified as a series of points, but in practic al application the obstacles have specific contours. In some circumstance s by choosing appropriate attractive and repulsive coefficients the robots ca n avoid collision with obstacles and arrive at the destination T his may be appropriate in the condition in which the size of obstacle s is very small compare d to the relative distance [12] But in most practical problems, there are obstacles with complex contours in the workspace of the robot In this case robots have to work and move a round th ese obstacles. I n building the potential field of the robot working space, these obstacles must be modeled. Usually obstacle configuration is described as a combination of a series of splines, but these splines cannot be direct ly expressed in the APF, so it can be discretized into a series of points. Each of the discretized point s will generate the

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22 repulsive force in its area of influence The final repulsive force on a certain position is the sum of repulsive forces from different discretized points. The proper attractive and repulsive coefficient should be chosen to generate the potential field. Fig ure 2 1 A ) shows the potential field of a rectangular obstacle, the minimum value stands for the position of the goal the relative ly high value is under the influence of the rectangular obstacle. There is a threshold value for the repulsive APF, so the area inside the contour of the rectangular obstacle has the some value in the potential field. Fig ure 2 1 B ) gives the contour line map of the potential field. In this map, the gradient value, the position of the goal and obstacle s are clearly showed. A B Fig ure 2 1 Example showing the obstacle using APF. A ) P otential field of rectangular obstacle B ) T he corresponding Contour line map of rectangular obstacle Limitation of the APF Method The APF method is widely used because of its simple computation form. C ompare d to other methods for path planning, the APF method can generate a collision free trajectory with barely any computation burden. Therefore, APF can be used f or real time obstacle avoidance i n complex environment especially with moving obstacles. I t i s facilitated to use APF method for path planning [ 10 ]

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23 However, there are also drawbacks of the normal APF method T he obstacles wi th a concave contour may have local minimum in its potential field. Fig ure .2 2 shows a case where there is a local minimum. The potential value around the loca l minim um point is greater, so w hen the robot move s in this area it will be tra pp ed the resultant force on robot cannot lead the robot to arrive the destination. Therefore, in building the potential field, the local minimum must be avoided. In 1992, Kim, J. O. and Khosla use the h armonic f unction to solve local minimum problem [8], this is because the harmonic function has no local minimal in the potential map. Paraskevas Dunias also details the Harmonic Function in his research to solve the local minimal problem [ 13 ] In Chapter 5 the local minimum problem is solved by combin ing the APF method with stereo vision system. A B Figure 2 2 The local minimum problem in potential field [ 13 ] A) The right angle concave obstacle, B) The corresponding potential field of the obstacle. Static and Moving Obstacle Avoidance The numerical simulation examples are presented to demonstrate the APF method for path planning and obstacle avoidance. In the examples, the behavior of the APF method for static obstacles and moving obstacle s is evaluated. T wo assumption s

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24 are made for further research about the APF method O ne is that the robot already know s the contours of the obstacles in its work space before begins and the other is that the robot has no information about the obstacles in its work space, but it can detect obstacles with th e stereo vision system when it moves around. The parameters of the APF method used in the simulation are presented in Table 2 1. Figure 2 3 presents the wo rk space situation in the simulation t he blue line s stand for static obstacles, the yellow line s stan d for moving obstacles, and the robot is simplified as a green point and its detecting area is presented by the circular sector. In the simulation the robot can successfully reach the destination and avoid collision with the obstacles. The simulation resul ts are presented as two parts, one is the simulation with previous ly known obstacles, the other is with previous ly unknown obstacles. Table 2 1 APF method simulation parameters Parameter Value Unit Initial Position [0 0] m Finial Position [180 170] m Step Length 0.1 m Attractive Coefficient 0.05 N/A Repulsive Coefficient 5000 N/A De te c tion Radius 10 m Repulsive Force Influence Radius 30 m Figure 2 3 Simulation of obstacle avoidance with APF method

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25 Figure 2 4 Path planning with APF method for previous ly known obstacles As show n in F igure 2 4 the robot knows the contour and location of the obstacles before it starts moving, and the generated path is presented as a blue line. In this situation, the genera ted path is smooth and collision free. In F igure 2 5 the robot does not know the information of the obstacles before it starts, it detects the environment with each step. W ith the detected information of the obstacles, the robot calculates the resultant f orce at that position, and then decides where to go next. Figure 2 5 Path p lanning with APF method for p revious ly unknown obstacles

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26 From the results in Figure 2 5 as smooth as the situation with previous ly know n obstacles. In th is situation the robot does not know the information of the obstacles previously The robot calculates each move step by step, based on the information it gathers along the way. E ach time when the robot plan s the length an d direction of the next step it is only influenced by the repulsive force s from the obstacles which it has detected in that instant, but at that time the robot may still be located in an area influenced by previous ly detected obstacles. As the influence of the repulsive force on the robot does not change continuously so the path from the APF method is in a zigzag shape T he zigzag path is more obvious when the robot moves near obstacles Because of this, the robot consume s more time and fuel In C hapter 4 the improved stereo vision system is introduced to reform this zigzag path by combining the APF method with the stereo vision system. In these simulations, the robot moves step by step with constant length, but in practical situation s there must be a corresponding kinematic profile to make the robot move from one position to another. F urthermore, there is always a demand for minimum time and fuel consum ption in practical situations. I n Chapter 3 the smoother trajectory and time optimal trajectory generation method is presented.

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27 CHAPTER 3 TRAJECTORY GENERATION WITH OBTAINED PATH The path generated from the APF method is a set of discrete points for each step the robot move s from one point to another in straight line As the path is combined by a series of straight segments there will be sharp turning c orners on these discrete points. W hen the robot arrive s at this turning point the velocity in the moving direction has to decelerate to zero. This cost s much time and f uel for a robot moving in this kind of path. The kinematic profiles corresponding to the APF path are also discrete which makes it difficult for the robot to track the generated APF path precisely. The APF path is a collision free path, it s drawback is no t smooth I n this chapter the interpolated smooth spline is introduced to approximate the original APF path. This approximated trajectory will not go across the obstacles and can generate corresponding smooth kinematic profiles. T he interpolation control points are chosen from the original APF path. To guarantee precision of the approximation, the intervals between these control points and the intervals between the corresponding variables should be with in reasonable values T his is also an important requirement to make the generated new trajectory not cross the obstacles. In order to make the acceleration profile along the trajectory continuous, t he variable of the i nterpolation function on each segment must be at least three order s. Interpolation T heory Cubic Spline 2D Trajectory can be approximated by interpolating the X and Y coo rdinates of the original APF path separately with same interpolation variable. Cubic spline is described by the three order polynomial so there exi s ts the first order derivative and the

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28 second order derivative I n other word s, the smooth velocity and acceleration profile can be generated from cubic spline trajectory after interpolation with time Figure 3 1 shows the principle of cubic spline interpo lation method for the interpolation of the X coordinate Fit the given X coordinate of the waypoin ts during the time interval using a cubic polynomial (3 1) Where is the i nterpolation function in this interval and is the coefficient of the interpolation function T he cubic interpolation function in the left adjacent interval is defined as and the right is Figure 3 1 Cubic spline interpolation method illustration C onstrain t s that make the generated trajectory pass through the waypoints from the original APF path are shown in Eq. (3 2) which make s the interpolated trajectory pass along the original APF path. (3 2) The smooth ness in velocity and continuity in acceleration are regulated by two extra constrain t s that are im posed on the adjacent intervals. (3 3)

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29 (3 4) In practical application of trajectory generation, the initial conditi on of position and velocity usually have specific demands in different conditions So the Eq. (3 5) is used as the interpolation function for cubic sp line interpolation, the property of this function is detailed in the work of Imran Waheed [2 8 ] (3 5) With this cubic polynomial the time variable for each interval is restricted to T hus each parameter of the cubic polynomial stand s for the initial position and velocity as well as the final position and velocity. (3 6) (3 7) (3 8) (3 9) Furthermo re, no matter how many segments need to be interpolated, the initial and final velocity condition of the interpolated trajectory can be directly defined as the desired values by using the Eq. (3 5) Fourth Order Spline Fourth order spline is one order more than the cubic spline, so using fourth order spline for waypoints interpolation can make the acceleration profile smooth and the jerk profile continuous. Jerk continuity is important in jerk bounding circumstance, as t he jerk control is related with the control of torque rate. Thus the continuity of jerk will affect the original path tracking accuracy, the continuity of jerk and the smooth ness of

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30 acceleration will also affect the life of robotics. Fit the given X coordi nate of the waypoints during the time interval using a fourth order polynomial (3 1 0 ) Where is the interpolation function in this interval, and is the coefficient of the interpolation function. The cubic interpolation function in left adjacent interval is defined as and the right is the specific interpolation algorithm and simulation examp les are detailed in [2 5 ]. Figure 3 2 Fourth order spline interpolation method illustration Constrain t s that make the generated trajectory pass through the waypoints from the original APF path are show n in Eq. (3 1 1 ) which makes the interpolated traje ctory pass along the original APF path. (3 1 1 ) The smooth ness in velocity, acceleration and continuity in jerk are regulated by three extra constrain t s that are imposed on the adjacent intervals. (3 1 2 )

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31 (3 1 3 ) In most conditions, the robot starts moving from static statues and when it reaches its destination it should ke ep static at that point. So on the initial and final points of the trajectory, the velocity and acceleration should be zero. For this fourth order spline the initial condition is defined by the fourth order interpol ation function discussed before. However, the interpolation function on the final segment should be one order more than the other segments to make the final velocity and acceleration reach the desired value at the final point of the trajectory. The method to generate the final segment interpolati on function is presented in [2 5 ]. Features of C ubic and F o urth O rder S pline For the cubic and fourth order spline interpolation method, no matter which one is chosen to approximate the original path, the chosen waypoint is the only constrain t to regulate the generated trajectory pass ing through the same region as the original path does The generated trajectory must pass throu gh these waypoints, as the coordinates of the waypoints are expressed in the interpolation functions as constrain t s. Th e smooth ness of velocity profile is determined by the continuity of its corresponding acceleration profile, the smooth ness of acceleration profile is determined by the continuity of its corresponding jerk profile. As Fourth order spline is one order highe r than the cubic spline, so the third order derivative of fourth order spline is continuous but the third order derivative of cubic spline is discrete. So the jerk profile of cubic spline is discrete and the jerk profile of fourth order spline is continuou s. For this reason, the acceleration profile of fourth order

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32 spline is smoother than that of the cubic spline. Also as the fourth order polynomial is one order higher than the third order polynomial, the graph of fourth order polynomial oscillates much mor e than the graph of third order polynomial. As cubic spline is interpolated by the third order polynomial and the fourth order spline is interpolated by the fourth order polynomial, so the trajectory generated by the fourth order method oscillates much mor e than that generated by cubic spline method. Figure 3 3 shows that the trajectory generated by the fourth order oscillate s much more than the cubic spline trajectory and the kinematic profile corresponding to fourth order trajectory oscillates more than t hat from cubic trajectory. But the acceleration profile is smoother than the cubic spline method. Figure 3 3 Trajectories and kinematic profiles generated by the cubic and fourth order interpolation methods trajectories are interpolated with 5 waypoints. A) Trajectory, B) Velocity profile, C) Acceleration profile, D) Jerk profile. A C D B

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33 The interpolation function for each segment between two waypoints is different with each other and the only input to calculate the interpolation fu nction is the coordinate of the waypoint. In 2D trajectory generation the input s are the X coordinate and its corresponding time or the Y coordinate and its corresponding time So the changing of the time interval of each segment or the changing of chosen waypoints will have an effect on the property of the generated trajectory and the corresponding kinematic profile. The f ollowing discusses the influence of waypoint choosing and time interval in trajectory generation. If more waypoints with closer distanc e s are chosen from the original path, the generated trajectory is more similar to the original path. But in this way, the original path is divided into more segments, as the interpolation polynomial for each segment is different, when the number of polynomials used for interpolation increase s the oscillation times of the generated trajectory increase to o. Also t he total number of oscillation times of the kinematic profiles will increase. Usually the velocity and acceleration vibrate once in each segment. Sometime choosing more waypoints will connect the waypoints by high curvature curves Due to the limi ted turning radius of robot, this will lead to the robot being unable to track the trajectory. In Figure 3 4 t hree times waypoints are chosen to interpolate the same original path compared to the simulation in Figure 3 3 By compar ison the absolute inter polation error decreases with more waypoints, but the curvature along the generated trajectory increases. Likewise, the oscillation times along the trajectory and its corresponding kinematic profile increase. Oscillation of the trajectory can lead to the v ibration of the velocity and acceleration. As the vibration in acceleration, the robot has to continually change the

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34 states between acceleration and deceleration, which causes more fuel consum ption The vibration in velocity makes the robot unable to stay at a steady speed, so the robot can hardly stay in high speed for long time T his cause s the robot to spend more time to reach the destination. Figure 3 4 Trajectories and kinematic profiles generated by the cubic and fourth order interpolation methods trajector ies are interpolated with 15 waypoints. A) Trajectory, B) Velocity profile, C) Acceleration profile, D) Jerk profile. In Figure 3 5 the same waypoi nts are chosen for the interpolation of the original path as it does in Figure 3 3, but in Figure 3 3 the time interval for each segment is the same while in F igure 3 5 the time interval is different with each other. When the time interval changes the magn itude of the oscillation will increase So in order to generate the optimized trajectory, the distance between waypoints and time interval must be A C B D

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35 properly defined. The desired trajectory should make the robot consume the least amount of fuel or cost the s hortest time. Figure 3 5 Trajectories and kinematic profiles generated by the cubic and fourth order interpolation methods, trajector ies are interpolated with different time intervals. A) Trajectory, B) Velocity profile, C) Acceleration profile, D) Jerk profile. Trajectory Optimization with Limited Condition Physical Limitation in Trajectory Generation I nterpolat ing robot trajectory from the waypoints of original APF path can guarantee that the robot reaches its destination without collision with obstacles if the robot moves along it. A reasonable trajectory can ensure the robot tracks it perfectly, which means when the robot moves on this trajectory it will not slid e So the velocity, acceleration and jerk related to the tra jectory must meet some specific limitations. B A C D

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36 All the movements like acceleration deceleration and turn ing are caused by the friction force between wheels and the road. The tangential acceleration on the wheel make s the robot speed up and slow down along the tangential direction T he maximum frictional force is related to the friction coefficient and the mass of the robot. The radial acceleration ensure s that the robot turn s on the curve The maximum non sliding turning speed is restricted by both t he turning curve radius and the radial friction force on the wheel. So in order to make the robot tracking the generated trajectory perfectly, these non sliding constrain ts must be m et [2 4 ] Fig ure 3 6 shows the relation between radial acceleration and tangential acceleration, they should not be beyond the friction ellipse. Figure 3 6 Radial and tangential friction ellipse [2 4 ] W hen the robot moves in a complex obstacle figuration environment, the speed of robot cannot remain high in case of collision with suddenly appear ing obstacles. It should leave enough response time for the robot to decelerate. Usually the moving robot is controlled by several step motors or servo motors. T he velocity and acceleration of the robot are related to the sp eed and torque of the motor s s o there also must be some restriction of the velocity and acceleration from the

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37 motors. Furthermore, the limited torque rate in a certain range can extend the life of the robot as well as improve the tracking precision of mov ing robots [2 5 ] So the jerk must be limited and continuous, in this way the acceleration will become smoother. Optimization f or Feasible Trajectory From the constraints discussed in the previous section, the maximum and minimum of the velocity, acceleration and jerk can be obtained. So the corresponding velocity, acceleration and jerk files of the trajectory must meet the limitation s of these extreme values. For the interpolated trajectory the shape of the trajectory is determined. T o reform the restricted dynamic profile without changing the shape of the trajectory, the time interval during this segment must be rescaled. In this thesis the recalling method presented in V. Muiiozt 24 ]. Suppose is one trajectory segment in t he time interval and and are velocity, acceleration and jerk profile. The limitation s of these variable s are respectively defined as and (3 1 4 ) (3 1 5 ) (3 1 6 ) Define is the recalled time in the previous trajectory segment, so the new dynamic profile s will become and can be obtained from E q ( 3 18 ) (3 1 7 ) (3 1 8 ) Then we can obtain the new velocity, acceleration and jerk profile in the new scaled time. Figure 3 7 gives the simulation example of velocity profile rescalling.

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38 (3 19 ) Figure 3 7 Velocity profile with different time scale Time Optimal Trajectory Tracking After the path planning using the APF method, the generated path con sists of a series of waypoints. By simply connecting these way points with straight line s the robot can avoid collision s with the obstacles along the path However, there will always be sharp corners on the waypoints. So each time the robot meets these corners, it has to decelerate to zero velocity at these po ints in order to track the path. T hus the path should have continuous curvature to improve tracking performance and reduce time cost. For the generated continuous curvature trajectory, in practical condition s the robot should track this trajectory with minimal time or minimal fuel cost. In the followi ng section a method is introduced for obtaining the time optimal kinematic profile of the robot Using this method the robot can finish tracking certain given trajector ies with optimal time cost. As the fraction al force between the robot wheels and the road is limited, so the maximum central acceleration is also limited. Thus when the robot turns along the

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39 trajectory, the maximum turning speed is directly related to the curvature value at the turning point. The curvature along the trajectory should be c ontinuous and not oscillate too much, which will help to save time and energy of robot when it moves along the trajectory. From the equation for trajectory curvature shown in Eq. (3 2 2 ) the minimum polynomial order to interpolate the trajector y should be three. The interpolation order cannot be too high, when the order of interpolation function goes higher, the correspond ing curvature will oscillate much more. Marko Lepeti proposed a time optimal trajectory generation method [2 6 ], the following time optimal velocity profile generation method is based on Marko Lepeti [2 6 ] Cubic spline is used for trajectory interpolation. The X direction coordinate position and Y direction coordinate position are separately interpolated with variable parameter u. Then from this obtained continuous curvature trajectory, the time optimal kinematic profile can be generated for trajectory tracking. (3 2 0 ) (3 2 1 ) W hen the certain generated trajectory, a nd are determined, the time optimal velocity profile and can be obtained by using time optimal The curvature of the trajectory can be obtained from the interpolated function by using Eq. (3 2 2 ) As there are robot speed limitat ions from its mechanical parts and working environment the robot cannot move beyond its highest speed. Also the central acceleration provide d by the wheels is also limited I n order to guarantee that the robot does not slid e on the road, robot cannot move with a velocity beyond the curvature

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40 limitation of the road. The m axi mum velocity profile with variable u can be calculated out with Eq. (3 2 3 ) T he permitted maximum velocity on each point of the road can be obtained in this way. (3 2 2 ) (3 2 3 ) Where is the maximum central acceleration the robot can provide when tracking the path and is the max velocity on the corresponding parameter u to prevent sliding on path. A B Figure 3 8 Trajectory example to generate the time optimal kinematic profile. A ) I nterpolated trajectory with variable u, B ) T he curvature corresponding to the interpolated trajectory. As the central acceleration provided by the road is constant, so the maximum curvature value that can prevent robot sliding at maximum speed can be calculated If

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41 the robot will not slide on the highest speed permitted maximum curvature, it also will not slide on the trajectory with a curvature value less than this maximum curvature value In Figure 3 8 A ) the trajectory is presented, and in the top Figure 3 8 B ) is the curvature corresponding to this traject ory. T he bottom Figure 3 8 B) is the absolute curvature value with consideration of permitted maximum velocity, and the highest speed permitted maximum curvature is drawn instead of the curvature with smaller value. The relation between trajectory length and the interpol ation parameter u can be found through Eq. (3 2 4 ). The total cost of t ime for the trajectory is shown in Eq. (3 2 5 ). For certain interpolated trajectory, the distance cannot be changed, so in order to reduce the total time cost in tracking, the permitted h ighest velocity profile should be generated for the corresponding trajectory. (3 2 4 ) (3 2 5 ) The maximum velocity profile related to parameter u ca n be obtained by using Eq. (3 2 3 ). In Figure 3 10 the blue line shows the maximum velocity pro file corresponding to curvature. T his line indicates the permitted maximum velocity on each point of t he road related to parameter u. Sin ce the robot needs to get the maximum velocity profile it should accelerate with its maximum acceleration until the robot reach es the maximum velocity after which acceleration goes returns to zero. Each time the robot decelerates it should decelerate with maximum deceleration until it reaches the curvature permitted maximum velocity after which the acceleration returns to zero. For each extreme curvature the corresponding permitted maximum velocity value on this extreme point

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42 can be obtained. It means the velocity satisf ies the turni ng demand at this extreme point. B efore this extreme point the robot should decelerate with maximum deceleration and after this extreme point the robot should accelerate with the maximum acceleration. The generated velocity can not be beyond the curvature permitted maximum velocity W hen the velocity goes beyond the limitation it will go horizontal ly The velocities before and after this extreme p oint are obtained with Eq. (3 2 6 ) and Eq. (2 2 7 ). (3 2 6 ) (3 2 7 ) S u ppose the maximum robot acceleration is the maximum central acceleration provide by wheels is and the highest speed of the robot is Then using the method discussed a bove, the time optimal velocity profile can be generated. Figure 3 9 The corresponding time optimal velocity profile related with parameter u The simulated results are show n in Figure 3 9 T he maximum velocity profile is finally generated by choosing the minimum velocity value related to the same parameter u. But this velocity profile is related to the parameter u, by using Eq. (3 2 4 ) and Eq. (3 2 5 ) the velocity profile related to time will be obtained. The velocity profile related to

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43 time is show n in Figure 3 1 0 Although this vel ocity profile is continuous, it i s not continuous in acceleration and jerk. This is because the time optimal velocity profile is generated based on the assumption that the robot accelerate s with maximum acceleration and decelerate s with maximum deceleration. However in reality the robot cannot change the acceleration value directly from its minimum to maximum or change from the static statu s to its extreme acceleration statu s. Figure 3 1 0 The corresponding time optimal velocity profile related with time In order to make the acceleration profile become smooth, t he acceleration must change in certain algorithm but not chang e abruptly So each time the acceleration statu s changes, the acceleration should gradually change in certain algorithm. Sonja Macfarlane uses the sine wave as the acceleration changing algorithm in her work [2 7 ]. The acceleration changing function is written as: (3 2 8 ) Where is the robot maximum acceleration and is the cost of time for the robot to change its acceleration from one status to another status. Eq ( 3 2 8 ) shows the acceleration changing algorithm to connect the discrete acceleration segments. Figure 3 1 1 shows the behavior of distance, velocity, acceleration and jerk by using the sine

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44 wave changing algorithm. When the robot change s the acceleration with this algorithm the corresponding velocity profile will become smooth Bec ause the sine wave has an infinite number of derivatives, the corresponding jerk profile is also continuous. Figure 3 1 1 The kinematic profile used to connect different kinematic profile segments. A) Distance, B) Velocity profile, C) Acceleration profile, D) Jerk profile. A B C D

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45 CHAPTER 4 STEREO VISION SYSTEM In robot navigation, the APF method is used as the coll ision free navigation algorithm. T he application of this algorithm requires the obstacle information and r obot location In this chapter stereo vision system is used as th e environment perception sensor. The s tereo vision system is capable of offer ing the location of obstacles relative to the robot. By using the relative distance between robot and obstacles provided by the stereo vision system the robot can build an occupancy grid map for collision free path planning In a grid map, it shows the position of the robot and the instant relative position between the robot and obstacles. Depth Perceptio n of Stereo Vision System The stereo vision system simulates the perception algorithm of human eyes, it consists of two parallel cameras with a certain distance, just like the relative position of human eyes. In depth perception, the two cameras take an im age of the same scene B ecause of the different location of the two cameras, the image of each camera is different. The pixels stand for the same image element located in different position s on each image and by finding the pixels of one camera and the cor responding pixels of the other camera the disparity can be generated. The relative distance of obstacles in the workspace can be calculated from this disparity map. Pre Processing of Stereo Vision System Image pre processing is the step before stereo matching which includ es color adjustment, camera calibration, stereo image rectification and image crop operation [ 18 ]. The raw images from the two cameras have differences in contrast and brightness as well as noises and distortion in both of them. Throu gh pre processing the stereo

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46 matching precision will be improved The most important step before stereo matching is the image rectification. The stereo image is rectified by the epipolar constrain t line. E pipoplar constrain ts make s the corresponding pixel s of two images lies on the same horizontal line, this line is also called the epipolar line. Through the constrain t s of the epipolar line the searching area for stereo matching is reduced I n stereo matching it just needs to search the pixels along the sa me epipolar line. T his highly improve s the stereo matching computation speed, thus the real time obstacle detecting robot can move with higher speed. Figure 4 1 shows the example of stereo image rectification and the corresponding epipolar line. A B Figure 4 1 Rectification of stereo image A) Left image, B) Right image [ 21 ] Depth Perception Algorithm After the pre processing of two stereo images, the corresponding pixels lie on the same epipolar line. In order to get the d isparity map, the dis parity of a single pixel on one image with the corresponding pixel on the other image should be identified. Usually the area based algorithm is used for solving this single pixel stereo matching problem [1 8 ]. A block consist ing of one middle pixel and its surrounding pixels is taken into consideration and the color values or intensive values of this block are calculated to find

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47 the best matched block area in the other image. The most widely used methods for the calculation method for this correspondence pro blem are the Sum of Absolute Difference (SAD) and the Sum of Squared Difference (SSD) algorithm s [ 18 ]. Eq. (4 1) shows the SSD method to find the best matched pixel. (4 1) Where is the pixel value in the left image, is the pixel value in the right image, is the pixel coordinate in the X direction, is the pixel coordinate in the Y direction, is the disparity in the X direction, is the disparity in the Y direction, is the pixel enumerator in the X direction, is the pixel enumerator in the Y direction. With the disparity map and using the image forming principle and s imilar triangles the depth map can be generated. Figure 4 2 shows how to generate depth map from the disparity map, Eq ( 4 1 ) pr esents also presents the princi ple. Figure 4 2 Stereo vision system for depth perception (4 2 ) Where D is the distance of the point from camera base line, is the focus length of camera, B is camera offset on the camera base line, and is the corresponded pixel disparity of this point.

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48 Depth Perception Simulation Following is an example to simulate the stereo vision system for depth map generation. Table 4 1 provides the camera parameters and object position information in the simulation. The camera and the object monkey are built in 3D animation software Blender to simulate for stereo vision system. In this way the stereo image can be generated from each camera in the model. Table 4 1 Simulation parameters of stereo vision system Parameter Value Camera s Canon 60D Camera Resolution X:1920 Y:1080 Camera Sensor Size Width:28.7mm Length:19mm Left Camera Location [ 0.05 6 0], unit: m Right Camera Location [ 0.05 6 0], unit: m Object Location [0 2 0], unit: m Camera Focal Length 35mm Actual Size of Each Pixel width=28.7/1920mm height=19/1080mm Figure 4 3 presents the model built in Blender, two offset parallel cameras are set as the stereo vision system. Two images are generated from this system and the A B C Figure 4 3 The simulation model and the stereo image of the two cameras A) Object and stereo cameras model, B) Left image, C) Right image.

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49 After pre processing of the stereo images, the stereo matching is processed Figure 4 4 shows the generated disparity map and the corresponding dept h map. By comparing the results of t he depth map with the former defined relative position between camera base line and object, the stereo vision system can provide information about the precise location of obstacles with this method. A B Figure 4 4 Simulation Results. A ) T he disparity map, B ) T he depth map Improvement of Vision System Obstacle Location Memory with Cells Map Stereo vision system is used to provide the robot with obstacle information. W ith the obstacle information the ro bot can generate the resultant APF force imposed on i t. The s tereo vision system provide s obstacle information before the start of each step I f there are obstacles located in the detecting area, there will be a repulsive potential force imposed on the robot I f there are no obstacles located in the detecting area, the robot is only influenced by the attractive force from the destination. In the simulation presented in Figure 2 6 of Chapter 2, the generated path from the APF method with normal stereo vis ion system is zigzag shape it will cost the robot much more fuel and

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50 time when moving with this path. The reason it lead s to this kind of zigzag path is that the robot los t the repulsive force generated from the obstacles detected in the previous step whe n it starts path planning for the next step. This means that the potential force imposed on the robot is calculated only with the obstacles in the detecting area at that instant, but the robot may still be located at the position which is under the influen ce of previous ly detected obstacles at that time. So the vision system with memory of the obstacles location is generated to reduce the oscillation of the path. In order to improve the APF method, the map for navigation is divided into grid s to represent t he workspace of the robot. T he location of obstacles is represented by the occupied grid s o n the map With this occupied grid map which is also called the cells map, the location of obstacles is easily be remembered by the vision system. This is because t his cells map can be translated into a matrix form i n this way obstacle location can be easily remembered Figure 4 5 shows the principle of cells map for location memory. From Figure 4 5 A) to Figure 4 5 B) robot moves one step toward the right angle ob stacle. A B Figure 4 5 Representation of obstacles in cells map when the robot is moving A ) Cells map before the robot is moving, B ) R obot move s toward left bottom with one step. In a cells map, when the parts of an obstacle are located in the area represented by one grid that grid will be occupied. T he location of the robot is also represented in

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51 this way, but the grid that represents the location of the robot is always located in the middle of the map. If the r obot does not move out of the area represented by its location grid, the c ells map does not change. If it moves beyond this area the new location will be recalculated and the correspo nded change in grid is obtained. T he location of the robot is still at t he middle of the map and the location of the obstacles change instead. For each movement of the robot, new obstacles move in the map and the pre detected obstacles moves out. T he obstacles move in opposite direction s and the same length in the map compared to the movement of robot. Following in Figure 4 6 is the example to compare the performance of vision system in path planning between the normal vision system and the one with location memory. There is always a minimum turning radius limitation when the r obot turns along the path, so when the robot plan s for the collision free path the turning radius constrain t should also be taken into consideration. A B Figure 4 6 Comparing the APF path planning method with two vision system s, A ) T he path with normal stereo vision system, B ) T he path with improved stereo vision system

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52 Because the improved stereo vision system can remember the obstacles information aroun d the robot, s o long as the robot is located in the influence area of prev iously detected obsta c les the repulsive force will be always imposed on the robot In this way the robot get s a smoother path with less oscillation. Step Estimation in Path Planning In the APF method for path planning, the robot moves step by step F or ea ch step if the robot moves from static status and end s with static statu s, the robot will repeat edly accelerate and decelerate which cost s much fuel and time. In order to improve the efficiency of pat h planning, the initial and fin al condition should be defined for each step including velocity, acceleration and step length. When the vision system detect s no obstacles the robot should keep moving with high speed and long step distance. W hen the obstacles are close enough the robot should move with low speed. The kinematic profile to guide the mov ement of the robot must be generated for each step based on the circumstance s of the robot workspace. Based on the relative distance between obstacles and the robot, the final kinematic val ue after one step must ensure the robot has enough break distance to avoid collision s To make the robot mov e smoothly the final kinematic value of the pre step is the initial value of the next step. Also the trajectory function for each step should ha ve at least four th order. With the defined initial and final kinematic value, the trajectory for each step can be generated with Eq. (4 2) and Eq. (4 3) [27]. T he trajectory is expressed in a fifth order polynomial and in this way the corresponding jerk profi le of the robot is also smooth. (4 3 ) (4 4 )

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53 (4 5 ) Where is the time at the initial point, is the time at the final point and are the kinematic conditions at the initial point, and are the kinematic conditions at the final point, and is the coefficient of the trajectory function. Image Processing for Vision System The potential field of the worksp ace is generated based on the information provided by the stereo vision system. So when the obstacle information is changed the correspond ing potential field is also changed I n this way the APF path planning method can be improved by processing the stere o image of the stereo vision system. Local m inimal a voidance When the robot moves in a workspace with complex obstacle contours, especially with the concave configuration, it is easy to generate local minimum in potential filed with the APF method. So when the robot gets trap ped in the local minim um it can go out by changing the shape of the obsta cle. As the vision system memorizes the position and contour of the obstacles with cells map, the processing of stereo image is actually the processing of cells map. Figure 4 7 gives the example of local minimum avoidance by using stereo image processing it shows the procedure of coming out of the local minimum with this method In F igure 4 7 the obstacle is located in the middle, the blue line stands for the part of obstacle that is not detected by robot, and the red part is the detected part of the obsta cle The r obot start s moving from the

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54 initial point, as the obstacle has the concave contour, so the robot is trapped into the local minimum. When the robot is trapped, the vision system connects the endpoint of the concave obstacle part A fter image proce ssing the robot recalculates the potential field with this new obstacle contour. But the robot is still trapped in a new local minimum so the robot repeats the image processing method once more. In this way the robot finally goes out of the local minimum and reaches th e destination. The three image s of Figure 4 7 show the procedure of how the robot g e t s out of the local minimum. By combin ing the APF method and the stereo vision system, the theoretical defects can be overcome. A B C Figure 4 7 The procedure of l ocal minimum avoidance of the APF method by using the stereo image processing A) Robot is trapped in local minimal, B) Robot processed the detected obstacles using improved stereo vision system, C) Robot moves out from the local minimal.

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55 Obstacle s egments c onnection In a realistic problem, the robot has mass, volume, size in width and height, so it cannot pass the areas beyond constrain s of those physical limitations. So in path planning with the APF method, the actual size of the robot and obstacles should be taken into consideration Through the stereo vision system, if some area between the detected obstacles is smaller than the size of robot, these areas should be occupied in the cells map. Through this method the repulsive force will be generated from these areas to keep the robot away from these areas, so the generated path ca nnot pass through these limited areas. Furthermore, the contours of obstacles are usually no t smooth in a real workspace. If the robot moves along the obstacle with an undulated contour, the generated path along this obstacle will be oscillated. So the stereo vision system can connect the convex of the detected contour to m ake the contour become smoother. I n this way the oscillation of the generated robot moving path can be reduced.

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56 CHAPTER 5 SIMULATION MODEL AND RESULTS In prac tical application a self navigat ing robot needs to tra vel back and forth in its work space to finish tasks. When the robot travels between the base station and the destination, it always traverses an area with obstacles and unpermitted space which the robot must navigate without collision with these obstacles Also the robot is desired to work in high efficiency. I n other word s, it should work with minimal cost of time or fuel. In this chapter the simulation model is buil t to verify the collision free trajec tory generation m ethod s which are presented in this thesis Introduction of Simulation Model When the robot traverse s the area with obstacles, it can detect the enviro nment with stereo vision system. The robot then avoid s collision with these obstacles using the APF method. In Figure 5 1 it shows a scenario for simulation the robot is re presented by the blue rectan gle, and the detecting area is represented by the blue circle sector. The r obot under this situation can be used as the mining robot on the moon and may satisfy the requirement of Regolith Excavation Competition sponsored by NASA [6] as the robot gets the mineral from the goal and delivers it to the baseport. Figure 5 1 Scenario Description of Simulation Model

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57 The robot transports between the base station and the out port During the travers ing between the base station and out port, the robot must avoid collision s with the obstacles. Before the robot mov es in the working area, it only knows the probable location and contours of the obstacles which are represented as some circle obstacles The robot can detect its surrounding environment with its stereo vision syste m. Thus when robot mov es in the working area it can get the specific configuration of the obstacles. The traveling area i n this simulation is show n in Figure 5 2, the black patterns are obstacles, the base port is on the left bottom corner and the goal is on the up right corner. Figure 5 2 Obstacles which are located in travelling area in the simulation model Table 5 1 APF method simulation parameters Parameter Value Unit Initial Position [0 0] m Finial Position [180 170] m Step Length 0.1 m Attractive Coefficient 0.05 N/A Repulsive Coefficient 5000 N/A De te c tion Radius 10 m Repulsive Force Influence Radius 30 m In the simulation, the robot uses the stereo vision system for depth percepti on when it moves back and forth. W ith the obstacle information provided by the stereo

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58 vision system, the robot builds the cells map and uses the APF method to generate the potential field for navigation. The simulation parameters are listed in Table 5 1. Workspace Detection and Collision Free Path Generation Before the moving start s, the robot know s the proba ble location s and contours of the obstacles, but it do es not know the specific shape s and the location s of these obstacles During mov ement the robot u se s the APF algorithm as the guidance law to make su re it can reach the destination and come back withou t a collision with any obstacles. During the transport between the base station and the out port the robot detects the configuration of the obstacles encountered by it. But every time the robot can only detect limited parts of the obstacles in the detecti ng area, after new obstacles are detected the memory of the vision system can memorize the shape and location of these new obstacles. In this way after the robot goes back and forth several times, the location and the shape of the obstacles in travelling a rea can be gradually improve d and the moving path of the robot is gradually refined. After certain travelling time s there will not be new obstacles appearing in the detecting area and the moving of path of the robot becomes constant as the potential field of the travelling area will not be changed. Before the first travelling round trip the robot only knows the probable location and size of obstacle s. T hese obstacles are represented as circles in Figure 5 3 A ) while the actual contour s of the obstacles are show n in Figure 5 2. In Figure 5 3 A ) d uring the mov ement new obstacles were detected and memorized T he blue line is the APF path from the initial point to the goal In Figure 5 3 B ) when the robot arrives at the goal it has to come back to the initial point A t this time the new destination becomes the initial goal and the new pote ntial fi e ld is recalculated. T he source of the attractive force is changed to the initial point and th e robot comes back with the estimated obstacles and the

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59 detected obstacles. After the first round trip, the obstacles deleted along the moving path are stored in the memory of the stereo vision system, so in the next round trip in Figure 5 4 the robot can use this detected obstacle information instead of estimated information. In this way, after the fourth time travelling, there are no new obstacles detected, so the path generated in this potential field become s constant. Figure 5 3 to Figure 5 6 shows the procedure of gradually refined potential field. A B Figure 5 3 Potential field and moving path in t he first round trip A ) T he path from initial point to out port, B ) T he back path from out port to initial point. A B Figure 5 4 Potential field and moving path in the second round trip A ) T he path from initial point to out port, B ) T he back path from out port to initial point.

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60 A B Figure 5 5 Potential field and moving path in the third round trip A ) T he path from initial point to out port, B ) T he back path from out port to initial point. A B Figure 5 6 Potential field and moving path in the fourth round trip A ) T he path from initial point to out port, B ) T he back path from out port to initial point. In the generation of this collision free path, the robot detects obstacles with the stereo vision system in each step. At the beginning of each step, the robot decides the location of the end of this step based on the detected obstacle information and the kinematic information at the starting point. After a total of four round trips no new obstacles are detected and the path from APF method become s constant. So in this way the robot always switch es the status of ac celeration and deceleration A lso the

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61 generate d path oscillate s very much and it cost s a lot of time and fuel. But as the generated moving path is collision free, the time optimal trajectory can be generated based on this path generated by the APF method. Optimal Trajectory Generation As the collision free path generated by the APF method becom e s constant after the fourth travelling trip the robot can go along this generated path without detecting new obstacles for further tasks. But this generated APF pat h only provides the end point position of each moving step, so it is hard for the robot to track. Also there are sharp corners along the path, and the oscillation of the path highly reduce s the robot efficiency. With the interpolation method introduced in C hapter 3, new trajectory should be generated for the r obot to track in further tasks New trajectory should be easy for tracking and the desired trajectory makes the robot costs minimal time and fuel. The new generated trajectory is constrained by a serie s of waypoint from the APF path U nder constrain t of these waypoints the new trajectory will not cross with the obstacles. Two trajectory generation method s are used to generate the easy tracking collision free trajectory for the robot I n following content, the simulation results are also provided Spline Interpolation Method For this spline interpolation method, the position profile in the X axis and the Y axis are interpolated separately with the variable of time. Then the velocity, acceleration and jerk profile in each axis can be generated directly by taking the derivation of the corresponding axis position profile. ( 5 1) ( 5 2)

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62 The waypoints used for interpolation are extracted from the APF path with the same distance interval, and the cost of time for each interval is supposed to be 1s. Then the generated trajectory by using cubic spline and fourth order spli ne is show n in Figure 5 7. Two different interpolation methods are implemented and compared, one is the cubic spline method and the other is the fourth order interpolation method. Both method s are interpolated with time for the trajectory generation. S u ppose the maximum robot acceleration is the maximum central acceleration provide by wheels is and the highest speed of robot is The cost of time for e ach interpolated segment is 1s and the corresponding kinematic profil e is show n in F igure 5 8 G roup A is the kinematic profile of the X axis and group B is the kinematic profile of the Y axis. Figure 5 7 Trajectory generated by taking two spline interpolation method s As the robot trajectory is interpolated with time in two directions, so the kinematic profile show n in Figure 5 8 describe s the robot motion in two directions. The resultant velocity, acceleration and jerk profile can be obtained by using the data in these two directions. Figure 5 9 A ) shows the resultant kinematic profile of the cubic interpolation

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63 method, Figure 5 10 A ) shows the resultant kinematic profile of the fourth order interpolation method. A B Figure 5 8 Kinematic profile of the two spline interpola tion method s, A ) T he kinematic profile in X direction, B ) T he kinematic profile in Y direction. In Figure 5 9 A) and Figure 5 10 A) the kinematic profile is generated by taking the derivative of the interpolated trajectory with time T he maximum kinematic profile values are beyond the robot constrain t s. This is because the time interval for the robot to move between the two adjacent waypoints is too short. Figure 5 9 B) and Figure 5 10 B) is the kinematic profile after rescaling of the kinematic profile in group Figure 5 9 A) and Figure 5 10 A) After time rescaling, both the velocity and acceleration profile is under the robot physical limitation, but t he shape is still keep ing the same. I t takes

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64 almost 130s using the cubic spline method and takes more than 150s using the fourth order spline method. A B Figure 5 9 Kinematic profile from the cubic interpolation method A ) T he original kinematic profile from interpolated trajectory, B ) T he rescaled kinematic profile. The kinematic profile generated from the fourth order interpolation method oscillates much more than that from the cubic interpolation method. With the profile generated from the fourth order inte rpolation method, the robot can reach high velocity values However, it has to always change the status between acceleration and deceleration and there will be more time and fuel con sumption with this method. T his is show n in Figure 5 9 and Figure 5 10. Th e fo urth order interpolation method takes

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65 almost 30 second s more than the cubic interpolation method. The jerk profile obtained from the fourth order interpolation method is continuous but the amplitude is greater than the cubic interpolation method. A B Figure 5 10 Kinematic profile from the fourth order interpolation method, A ) T he original kinematic profile from interpolated trajectory, B ) T he rescaled kinematic profile. Time Optimal Trajectory Generation Method It is convenient to generate the kinematic profile by using the spline interpolation method, but this method cannot ensure that the robot track the generated trajectory with minimum cost of time. In most practical problem s the robot is desired to track the constant trajectory with in the least amount of time s o the time optimal trajectory

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66 generation method is used Cubic spline is used for trajectory interpolation. The X direction coordinate position and the Y direction coordinate position are separately in terpolated with variable parameter u. The generated tra jectory is show n in F igure 5 11 this trajectory is the as same as the trajectory from the cubic interpolation spline with variable of time. Figure 5 11 The trajectory use d for time optimal trajectory generation method The kinematic profile corresponding to the time optimal trajectory is generated from this interpolated trajectory with variable u. Because the traje ctory for tracking is chosen, and cannot be change d. T he proper should be chosen for the time optimal kinematic profile. (5 3) (5 4) The curvature of the trajectory can b e obtained from the trajectory interpolation function. Figure 5 1 2 shows the generated curvature. T he red line is the curvature generated directly from the trajectory, and the blue line take s the maximum velocity

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67 limitation into consideration. Because the robot maximum velocity is and the maximum central accelerati on provided by road is even the robot moves at the highest speed I f the curvature of the trajectory is smaller than the robot will not slide. To get the maximum velocity profile permitted by the curvature of the trajectory, only the c urvature greater than will be taken into consideration. F igure 5 12 The curvature of the interpolated trajectory. The maximum velocity profile can be generated base d on the modified curvature of the interpolated trajectory. In Figure 5 13 t he red line is the permitted maximum velocity on each position of the trajectory by the corresponding curvature at that position. The r obot starts from static status at the initial point and keeps static when it arrives at the final point I f the robot tra cks the generated trajectory with minimum cost of time it must accelerate with maximum acceleration and decelerate with maximum deceleration. The robot velocity is also limited by the extreme value of the curvature. W hen the robot moves on the point with t he extreme curvature value the velocity at this point must ensure the robot will not slide on the road. So before the robot arrive s at this point it must decelerate and after this point the robot can accelerate in order to sav e time. In Figure 5 13 the di fferent color lines except the initial and final ones are the maximum velocity profile under different extreme curvature constrain t s. Before and after the

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68 extreme curvature point the robot accelerate s and decelerate s with maximum acceleration, when the s peed goes beyond the limit of curvature it will become horizontal. Th er e will be relative ly smaller maximum velocity profile corresponding to this horizontal part. Figure 5 13 The maximum velocity profile with variable of parameter u Finally the maximum velocity profile is obtained by choosing the min i mum velocity value corresponding to a different parameter u. Then the velocity profile with variable of time will be obtained by taking the method in Chapter 3. The maximum velocity profile with variable of time is show n in Figure 5 14. Figure 5 1 4 The maximum velocity profile with variable of time This generated maximum velocity limitation is based on the maximum acceleration and deceleration, but in real conditions acceleration cannot change sharply.

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69 Suppose the acceleration is changed with the regulation presented in C hapter 3. The smoother velocity profile and t he corresponding acceleration and jerk profile s are show n in F igure 5 1 5 With this method the robot start s acceleration from the initial point with maximum acceleration, every time the robot reaches the highest speed the acceleration will go to zero, then the robot will keep moving at the high est speed. Before reaching the position with limited curvature the robot decelerate s with the previously mentioned maximum deceleration. Every time the robot changes the acceleration status it will cost little time, so the jerk changing magnitude is bigger than the interpolation method. Figure 5 1 5 The time optimal kinematic profile of the interpolated trajectory A) Distance, B) Velocity profile, C) Acceleration profile, D) Jerk profile. B D C A

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70 Simulation Results Analysis By comparing the two methods for optimal trajectory generation, the spline interpolation method is more convenient to generate the corresponding kinematic profile. B eca use the trajectory is interpolated with variable of time in this method the kinematic p rofile can be directly generated by taking the derivation of the trajectory with time. But this method cannot provide the time optimal kinematic profile for the generated trajectory. The time optimal trajectory generation method generates the kinematic pro file by considering the limitation of curvature, the robot move s at the highest speed anytime under constrain t of maximum velocity. But in this method, the work to make the velocity become smoother is difficult and need s more calculation. The time optimal method takes a total of 46s for robot to get to the destination, the cubic interpolation methods takes 120s and the fourth order interpolation method takes 154s. The velocity profile from time optimal method can stay on the highest speed f or most of the time T he velocity profile from cubic spline method can also stay in certain value f or most of the time, but due to the limitation of acceleration, the rescaled velocity can only reach as the highest speed. Because the kinematic profile from the fourth order interpolation method oscillates greatly the robot always accelerates and decelerates, so this method cost s the most of time. The jerk profile from the time optimal method and the fourth order interpolation method are continuous and the jerk profile from the cubic interpolation method is discrete. B ut the continuous jerk profiles have great er amplitude compare d to the jerk profile from the cubic interpolation method. Especially in the time optimal method, the jerk can go up to this will lead to the severe change of robot movement The

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71 jerk profile from the cubic interpolation method is discrete, and this will influence the trajectory tracking precision.

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72 CHAPTER 6 CONCLUSION AND FUTURE RESEARCH T his thesis discussed the path planning method for self navigati ng robot s in an unknown environment. The APF method is used to avoid the obstacles and the sensor for environment perception is the stereo vision system By study ing the depth perception algorithm of the stereo vision system and the principle of the APF method, an improved stereo vision system is presented in the thesis to solve the local minim um problem and to reduce the oscillation of the generated collision free path. The grid map is proposed for robot navigation B y combining the grid map and the depth map, an imaging processing method can be used to optimize the path generated from the normal APF method. The path from the APF method is connected by a serie s of straight line segments, this is hard for a robot to track and it will cost much time and energy. The corresponding trajectory generation method is proposed in this thesis to approximate the APF path. Two interpolation methods are compared and discusse d to find the best method for trajectory interpolation. Also the method to generate time optimal trajectory and the corresponding kinematic profile is discussed. Finally in the constructed scenario all of these new methods are evaluated by simulation T h e simulation results prove that the new stereo vision system and the time optimal trajectory are the most valid and efficient. The method provided in this thesis is suitable to the e nvironment with static obstacle s The APF method can be used to avoid moving obstacle s but if it is used directly the performance will not be very good Further work should be done in the further to improve the APF method for avoiding moving obstacle. Also the trajectory generation method is based on the constant APF path; if the robot encounters moving

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73 obstacles the original APF must be changed. I n future research the collision free trajectory generation method with moving obstacles should be discussed. F urthermore in this thesis the time optimal trajectory and corresponding time optimal kinematic profile generation method is used to optimize the constant APF path, so before the APF path become s constant each moving step of the robot is not time optimal M ore work should be done in the future to optimize the each step of the robot when using the APF method.

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74 LIST OF REFERENCES [1] ( 2013 February ) Adept Mobile Products Adept [Online]. Available: http://www.adept.com/products/mobile robots/mobile platforms/lynx/general [2] ( 2013 February ) Adept Mobile Products Adept [Online]. Available: http://www.adept.com/products/mobile robots/mobile transporters/spc 4200/general [3] ( 2013 February ) TurtleBot Willow Garage. [Online]. Available: http://www.willowgarage.com/turtlebot [4] ( 2013 February ) Bluefin 12D Bluefin Robotics [Online]. Available: http://www.bluefinrobotics.com/pro ducts/bluefin 12d/ [5] ( 2013 February ) Ultra Heavy Duty Work Class ROV System FMC Technologies [Online]. Available: http://www.schilling.com/products/ROVs/Pages/UHD.aspx [6] ( 2013 February ) s Mining Competition NASA. [Online]. Available: http://www.nasa.gov/offices/education/centers/kennedy/techn ology/lunabotics.html #Rules [ 7 ] O Khatib Time Obstacle Avoidance for Manipulators and Mobile Robots IEEE International Conference on Robotics and Automation Vol. 2 pp. 500 505 1985. [ 8 ] Jin Oh Kim and Pradeep K. Time Obstacle Avoid ance Using Vol.8, No.3, pp. 338 349, 1992. [ 9] M. H. Mabrouk and C. R. Autonomous Syst ems, Vol.56, No.12, pp. 1050 1060, 2008. [ 10 ] Min gyu Park, Jae hyun Jeon and Min cheol Lee, International Symposium on Industrial Electronics, Vol.3, pp.1530 1535 2001 [ 11 ] S. Vol. 13, pp. 207 222 2002. [ 12 ] Josue David Munoz, Rapid Path Planning Algorithm for Autonomous Proximity Operations of Satellites, PHD dissertation University of Florida, USA, 2011. [ 13 ] Paraskevas Dunias Autonomous Robots Using Artificial Poten tial Fields, PHD dissertation Eindhoven University of Technology, Netherlands, 1996

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75 [ 14 ] Nicholas S. Martinson, Obstacle Avoidance Guidance and Control for Autonomous Satellites, PHD dissertation University of Florida, USA, 2009. [ 15 ] Andrew R Tatsch, Artificial Potential Function Guidance for Autonomous In Space Operations, dissertat ion University of Florida, USA, 2006. [1 6 ] Allan Eisenman, Carl Christian Liebe, Mark W. Maimone, Mark A. Schwochert and Reg G. Willson, Mars Exploration Rover Engineering Cameras, Jet Propulsion Laboratory, 2003. [1 7 ] and Rover Navigation Software for Planetary e dings, Vol.5, No.5, pp. 2025 2036, 2002. [1 8 ] Ali Kilic, Navigation of Mobile Robot Using Stereo Vision, Master thesis, University of Gaziantep, Turkey, 2010 [1 9 ] Don Murray and Jim Little Using R eal time S tereo V is ion for M obile R obot N Autonomous Robots, Vol. 8 No.2, pp. 161 171 2000. [ 2 0] Don Murray and Cullen James Stereo Vision Based Mapping and Navigation for Mobile Robots IEEE International C onference on Robotics and Automation Vol.2, pp. 1694 1699 1997. [ 21 ] ( 2013 February ) Stereo Rectification F it.com [Online]. Available: http://fit.com.ru/Projects/stereo_rectification.htm [ 22 ] M aryum F A hmed Development of a Stereo Vision System for Outdoor Mobile Robots Master thesis, University of Florida, USA, 2006 [ 23 ] Based Range Processing for Autonomous IEEE International Conference on Robotics and Automation, Vol.4, pp.3362 2267 1994. [ 24 ] V. Mu n oz, A. Ollero, M. Prado and A. Simon, with Dynamic and Kinematic Constraints, IEEE International Conference on Robo tics and Automation, Vol.4, pp.2802 2807, 1994. [2 5 ] K. Petrinec and Z. Kovacic, P lanning A lgorithm B ased on the C ontinuity of J erk IEEE Mediter r a nean Conference on Control & Automation, pp.1 5. 2007. [2 6 ] Marko Lepet i c Gregor Klancar, Igor Skrjanc, Drago Matko and Bostjan Poto cnik O ptimal P ath P lanning C onsidering A cceleration L imits Robotics and Autonomous Systems, Vol.45, pp.199 210, 2003. [2 7 ] Sonja Macfarlane On Line Smooth Trajectory Planning for Manipulators Master thesis, The University of British Columbia, Canada, 2001.

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76 [2 8 ] Imran Waheed, Trajectory / Temporal Planning of a Wheeled Mobile Robot Master thesis, University of Saskatchewan, Canada, 2006.

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77 BIOGRAPHICAL SKETCH Ronghua Yao was born in Youyu, Shanxi, China, in 1988. In September 2007 he m echanical e ngineering. There he was honored excellent graduated stud ent after graduation. m echanical e ngineering at the University of Florida. While in his master ystem. His interest is focus on robotics, embedded system and mechatronics.