such as Marine Science Research, Robots in Mining Industry, and RoboCup ... simulated annealing based approach for both online and offline path ...... To the best of my knowledge and belief, the thesis contains no material ...... [40] Lecture Note, âRobot Motion Planningâ, Department of Computer Science, University.
Master by Research Thesis By
Hui Miao Student No.: 06478689 submitted to the
Faculty of Science and Technology Queensland University of Technology
Project Title: Robot Path Planning in Dynamic Environments using a Simulated Annealing Based Approach March 2009
Supervisor: Associate Professor Yu-Chu Tian Associate Supervisor: Associate Professor Yanming Feng
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Abstract Mobile robots are widely used in many industrial fields. Research on path planning for mobile robots is one of the most important aspects in mobile robots research. Path planning for a mobile robot is to find a collision-free route, through the robot’s environment with obstacles, from a specified start location to a desired goal destination while satisfying certain optimization criteria. Most of the existing path planning methods, such as the visibility graph, the cell decomposition, and the potential field are designed with the focus on static environments, in which there are only stationary obstacles. However, in practical systems such as Marine Science Research, Robots in Mining Industry, and RoboCup games, robots usually face dynamic environments, in which both moving and stationary obstacles exist. Because of the complexity of the dynamic environments, research on path planning in the environments with dynamic obstacles is limited. Limited numbers of papers have been published in this area in comparison with hundreds of reports on path planning in stationary environments in the open literature.
Recently, a genetic algorithm based approach has been introduced to plan the optimal path for a mobile robot in a dynamic environment with moving obstacles. However, with the increase of the number of the obstacles in the environment, and the changes of the moving speed and direction of the robot and obstacles, the size of the problem to be solved increases sharply. Consequently, the performance of the genetic algorithm based approach deteriorates significantly. This motivates the research of this work.
This research develops and implements a simulated annealing algorithm based approach to find the optimal path for a mobile robot in a dynamic environment with moving obstacles. The simulated annealing algorithm is an optimization algorithm similar to the genetic algorithm in principle. However, our investigation and simulations have indicated that the simulated annealing algorithm based approach is simpler and easier to implement. Its performance is also shown to be superior to that of the genetic algorithm based approach in both online and offline processing times as well as in obtaining the optimal solution for path
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planning of the robot in the dynamic environment.
The first step of many path planning methods is to search an initial feasible path for the robot. A commonly used method for searching the initial path is to randomly pick up some vertices of the obstacles in the search space. This is time consuming in both static and dynamic path planning, and has an important impact on the efficiency of the dynamic path planning. This research proposes a heuristic method to search the feasible initial path efficiently. Then, the heuristic method is incorporated into the proposed simulated annealing algorithm based approach for dynamic robot path planning. Simulation experiments have shown that with the incorporation of the heuristic method, the developed simulated annealing algorithm based approach requires much shorter processing time to get the optimal solutions in the dynamic path planning problem. Furthermore, the quality of the solution, as characterized by the length of the planned path, is also improved with the incorporated heuristic method in the simulated annealing based approach for both online and offline path planning.
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List of Tables Table 1: Four testing environments ----------------------------------------------------------------- 56 Table 2: Control parameters for the simulated annealing algorithm----------------------------- 56 Table 3: Results for environment one --------------------------------------------------------------- 58 Table 4: Results for environment two -------------------------------------------------------------- 59 Table 5: Results for environment three ------------------------------------------------------------- 61 Table 6: Results for environment four -------------------------------------------------------------- 63 Table 7: Processing time of online planning on each case --------------------------------------- 64 Table 8: Three environments for performance evaluation --------------------------------------- 68 Table 9: Comparison of execution time base on same vertices number ------------------------ 68 Table 10: Comparison of the length of the final path --------------------------------------------- 68 Table 11: Results of comparison in case one ------------------------------------------------------- 79 Table 12: Results of comparison in case two ------------------------------------------------------ 80 Table 13: Results of comparison in case three ----------------------------------------------------- 81 Table 14: Results of comparison in case four ------------------------------------------------------ 82 Table 15: Conclusion of the improvements -------------------------------------------------------- 84
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List of Figures 1. Classifications of the robot path planning methods -------------------------------------------- 12 2. Visibility graph -------------------------------------------------------------------------------------- 14 3. Complete visibility graph -------------------------------------------------------------------------- 14 4. Quadtree decomposition --------------------------------------------------------------------------- 15 5. Cell decomposition --------------------------------------------------------------------------------- 16 6. Potential field --------------------------------------------------------------------------------------- 18 7. Obstacle is out of the vision zone ---------------------------------------------------------------- 19 8. Obstacle partially occupies the vision zone ----------------------------------------------------- 20 9. Obstacle impedes the main line between robot and target ------------------------------------ 20 10. Obstacle fully occupies the vision zone -------------------------------------------------------- 20 11. Path planning rule 2 ------------------------------------------------------------------------------- 21 12. Path Planning Rule 3 ----------------------------------------------------------------------------- 22 13. The representation of the chromosome in GA planner --------------------------------------- 25 14. Genetic planners ----------------------------------------------------------------------------------- 26 15. Regular grids representation --------------------------------------------------------------------- 30 16. The D* algorithm --------------------------------------------------------------------------------- 31 17. The Focussed D* algorithm --------------------------------------------------------------------- 31 18. Framed-Quadatree Structure --------------------------------------------------------------------- 33 19. Procedure of the simulated annealing algorithm ---------------------------------------------- 36 20. Structure of Player -------------------------------------------------------------------------------- 38 21. Player/Stage architecture ------------------------------------------------------------------------- 39 22. Scenario one of moving obstacle ---------------------------------------------------------------- 43 23. Scenario two of moving obstacle --------------------------------------------------------------- 44 24. Online path planning ------------------------------------------------------------------------------ 44 25. State transformation diagram of the simulated annealing algorithm ----------------------- 44 26. Enlarged obstacles -------------------------------------------------------------------------------- 45 27. Procedure of simulated annealing approach for path planning ----------------------------- 49 28. The initial path selection process --------------------------------------------------------------- 51
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29. Deleting operator --------------------------------------------------------------------------------- 52 30. Offline path planning in case one --------------------------------------------------------------- 57 31. Online path planning in case one --------------------------------------------------------------- 57 32. Offline path planning in case two --------------------------------------------------------------- 58 33. Online path planning in case two 1 ------------------------------------------------------------- 58 34. Online path planning in case two 2 ------------------------------------------------------------- 59 35. Offline path planning in case three ------------------------------------------------------------- 60 36. Obstacles appears on different times ----------------------------------------------------------- 60 37. Online path planning in case three -------------------------------------------------------------- 60 38. Offline path planning in case four -------------------------------------------------------------- 62 39. Obstacles appears on different times ----------------------------------------------------------- 62 40. Online path planning in case four 1 ------------------------------------------------------------- 62 41. Online path planning in case four 2 ------------------------------------------------------------- 62 42. Convergence of the SA approach --------------------------------------------------------------- 63 43. Representation of the chromosome in GA approach ----------------------------------------- 65 44. The repair operator -------------------------------------------------------------------------------- 66 45. Comparison of execution time base on same vertices number ------------------------------ 69 46. Comparison of final path length base on same vertices number ---------------------------- 69 47. Convergence comparison in environment one ------------------------------------------------ 70 48. Convergence comparison in environment two ------------------------------------------------ 70 49. Convergence comparison in environment three ----------------------------------------------- 71 50. Environment one ---------------------------------------------------------------------------------- 72 51. Environment one from [17] ---------------------------------------------------------------------- 72 52. Process of the heuristic selection method ------------------------------------------------------ 76 53. Random picking method ------------------------------------------------------------------------- 77 54. Heuristic picking method ------------------------------------------------------------------------ 77 55. Random picking method in case one ----------------------------------------------------------- 78 56. Heuristic picking method in case one ---------------------------------------------------------- 78 57. Random picking method in case two ----------------------------------------------------------- 79 58. Heuristic picking method in case two ---------------------------------------------------------- 79 59. Random picking method in case three --------------------------------------------------------- 80
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60. Heuristic picking method in case three --------------------------------------------------------- 80 61. Random picking method in case four ---------------------------------------------------------- 81 62. Heuristic picking method in case four ---------------------------------------------------------- 81 63. Path solution comparison -------------------------------------------------------------------------84 64. Offline processing time comparison ------------------------------------------------------------ 84 65. Online processing time comparison ------------------------------------------------------------ 84 66. The comparison of the three methods in processing time ----------------------------------- 85 67. The comparison of the three methods in finding path length -------------------------------- 86
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Contents Chapter 1 Introduction
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1.1 Background ------------------- ------------------------------------------------------------ 1 1.2 Research Gap Statements and Motivation -------------------------------------------- 3 1.3 Aim of the Project ------------------------------------------------------------------------ 5 1.4 Significance of the Research ------------------------------------------------------------ 6 1.5 Contributions of the Thesis ------------------------------------------------------------- 7 1.6 Structure of the Thesis ------------------------------------------------------------------- 8 1.7 Related Publication ---------------------------------------------------------------------- 9
Chapter 2 Background and Literature Review
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2.1 The Classification of the Path Planning Methods ---------------------------------- 11 2.2 Path Planning in a Static and Known Environment -------------------------------- 13 2.2.1 Visibility Graph -------------------------------------------------------------------- 13 2.2.2 Cell Decomposition Method ----------------------------------------------------- 15 2.2.3 Artificial Potential Field -----------------------------------------------------------16 2.3 Path Planning in a Static and Unknown Environment ------------------------------18 2.4 Path Planning in a Dynamic and Known Environment ----------------------------23 2.5 Path Planning in a Dynamic and Unknown Environment -------------------------27 2.6 The Simulated Annealing Algorithm and SA Based Approach -------------------35 2.6.1 The Simulated Annealing Algorithm --------------------------------------------35 2.6.2 Simulated Annealing Based Approach ------------------------------------------36 2.7 Robot and Environment Simulators -------------------------------------------------- 38
Chapter 3 The Simulated Annealing Algorithm Based Approach
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3.1 Structure and Assumptions of the method ------------------------------------------ 42 3.2 Mathematic Modelling ----------------------------------------------------------------- 45 3.2.1 Environment Modelling ---------------------------------------------------------- 45 3.2.2 Algorithm Structure and Expressions --------------------------------------------46
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3.2.3 Initial Path Selection Process -----------------------------------------------------50 3.2.4 Random Path Planning ------------------------------------------------------------52 3.2.5 Online Path Planning --------------------------------------------------------------53 3.3 Platform for the Research --------------------------------------------------------------54
Chapter 4 Simulation Results and Performance Evaluation
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4.1 Simulation Environments and Algorithm Parameters ------------------------------55 4.2 Simulation Results ----------------------------------------------------------------------57 4.2.1 Case One ----------------------------------------------------------------------------57 4.2.2 Case Two ----------------------------------------------------------------------------58 4.2.3 Case Three -------------------------------------------------------------------------- 60 4.2.4 Case Four --------------------------------------------------------------------------- 61 4.3 Performance Evaluation --------------------------------------------------------------- 63 4.3.1 The Genetic Based Approach ---------------------------------------------------- 64 4.3.2 Results Comparison ----------------------------------------------------------------67
Chapter 5 Heuristic Search Method for the Simulated Annealing Approach
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5.1 The Structure and Implementation of the Heuristic Selecting Method ----------73 5.2 Performance Evaluation of the Heuristic Method ----------------------------------78 5.2.1 Environment One ----------------------------------------------------------------- 78 5.2.2 Environment Two ----------------------------------------------------------------- 79 5.2.3 Environment Three --------------------------------------------------------------- 80 5.2.4 Environment Four ---------------------------------------------------------------- 81 5.3 Discussion of the Evaluation Results ----------------------------------------------- 82 5.3.1 Offline Planning ------------------------------------------------------------------ 82 5.3.2 Online Planning ------------------------------------------------------------------- 83 5.3.3 The Length of the Path ----------------------------------------------------------- 83 5.3.4 Comparison with the Genetic Algorithm Based Approach ------------------ 85
Chapter 6 Conclusion
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6.1 Summary --------------------------------------------------------------------------------- 87
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6.2 Research Limitations ------------------------------------------------------------------- 88 6.3 Future Work ----------------------------------------------------------------------------- 89 Acknowledgement
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References
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Statement of Original Authorship The work contained in this thesis has not been previously submitted to meet requirements for an award at this or any other higher education institution. To the best of my knowledge and belief, the thesis contains no material previously published or written by another person except where due reference is made.
Signature:
Date:
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Acknowledgement I would like to express my gratitude to my supervisor Associate Professor Yu-Chu Tian and my associate supervisor Associate Professor Yanming Feng, who have given me excellent supervision and guidance. I would also like to thank them for helping me to gain experience on research study, which will be very helpful for my future study. I also wish to thank my friends who helped me through all the difficult times.
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Chapter 1 Introduction This thesis develops and implements a simulated annealing algorithm based approach with a heuristic search method for path planning for a mobile robot in dynamic environments. The approach uses the vertices of the static and dynamic obstacles as search space. Using the heuristic search method, it searches an initial feasible path for the robot in the environment. Then, the approach applies the simulated annealing algorithm to find the optimal path for the robot. When a moving obstacle is detected with possible collision with the robot, this two-step planning process is activated for dynamic path planning. This chapter introduces some background information about robot path planning, the motivation of the project, research gaps, and research significance. It also highlights the contributions of this thesis.
1.1 Background At present, research on various algorithms for mobile robot path planning is a hot topic. Mobile robots are widely used in many hazardous industrial fields where there may be dangers for people, such as aerospace research, the nuclear industry, and the mining industry.
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To find a safe path in a dangerous environment for the mobile robot is an essential requirement for the success of any mobile robotic systems. Therefore, research on path planning algorithms to make the robot move from the start point to the termination point without collision with obstacles is a fundamental requirement for the mobile robot safety in such environments. Moreover, to reduce the processing time, communication delay and energy consumption, the planned path is naturally required to be optimal with the shortest length.
At the initial stage of the robot industry, a robot was simple constituted by mechanical arms controlled by motor engines. Path planning for the robot was often in stationary obstacle environment. As an example of the robot, path planning in static environments were discussed in [1]. However, with the development of the robot technology, robots have been used in many industrial fields such as aerospace research, marine research, and mining industry, to just mention a few. A lobster-like underwater walking robot [2] is one of these new types of robots. Recently, Australian researchers have developed an unmanned underwater vehicle robot for reef surveying [3]. The robot is equipped with sonar and vision systems, and works at the platform of the sea. Thus, how to response quickly to the changing environment to avoid the stationary rocks in the seabed and big moving fish is a primary issue in the design and operation of the robot.
Recently, the genetic algorithm based approach has been developed for robot path planning in dynamic environments, but the efficiency is not sufficient for large-scale path planning problems. Therefore, more efficient path planning methods in dynamic obstacle environments need to be developed for adapting the development of robots research.
The simulated annealing algorithm is similar to the genetic algorithm in solving general optimization problems. It has been implemented for robot path planning in static environments. What’s the performance of the simulated annealing based approach in dynamic path planning? This is a question yet to be answered and this research gives a positive answer to the question.
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1.2 Research Gap Statements and Motivation Because the entire information of a dynamic environment will change along with the movement of obstacles, the complexity and uncertainty of the path planning problem increase greatly in dynamic environments. Therefore, traditional path planning methods, such as Visibility Graph [4], Voronoi diagrams [5] and Grids [6], are not suitable for planning path in dynamic environments. Recently, Wang, Sillitoe and Davide [14] introduced a genetic algorithm based navigation method to deal with the path planning problem in an environment with moving obstacles. However, this method still has drawbacks: local minima results may occur in some cases [14] and the calculating time for finding the first feasible path increases greatly while the number of the obstacles increases [14]. The result (feasible path) convergence rate is high using genetic algorithms, making the method lose the mechanism for exploration. If the method has a low convergence rate, it could raise the mutation probability and reduce the crossover probability [22]. Nevertheless, research in [9] shows that large mutation rates improve the quality of the algorithm. Therefore, an improved method for path planning in dynamic environment needs to be designed.
The A* algorithm [10] is another path planning method to help the robot to find the optimal path in grid decomposed static maps. The environment with free space and obstacles is presented by a set of uniformed regular grids. The A* algorithm uses heuristic based Dijkstra algorithm to obtain the optimal result for the robot. The drawback is that the A* uses uniformed grids representation which must allocate large amounts of memory for regions that may never be traversed or may not contain any obstacles. This drawback may affect the efficiency of the method. Quadtree representation can be used to enhance the efficiency of the method, but it sacrificed the optimality for the efficiency. The above drawbacks also happen in the D* method [11] which is the dynamic version of the A* method. The D* helps robot to find the optimal path in uncertain environments. Besides, in dynamic environments with moving obstacles, the environment changes the planning problem from fixed obstacles to moving ones; changes the problem from a geometric one to a dynamic one; equally the environment changes the problem from a deterministic problem to a stochastic problem [12].
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Therefore, a vertices based stochastic optimisation method can be applied to the dynamic environments for robots.
The simulated annealing algorithm is a generic probabilistic meta-algorithm for the global optimization problem, namely locating a good approximation to the global optimum of a given function in a large search space. It was independently invented by Kirkpatrick, Gelatt and Vecchi in 1983, and by Cerny in 1985 [13].
The reasons of choosing the simulated annealing algorithm to plan the shortest path for robots are listed below: 1) Recently, the genetic algorithm was implemented to find the optimal path for a robot in a dynamic environment [14]. According to [9], the simulated annealing algorithm is an optimization algorithm similar to the genetic algorithm, and can provide similar results in optimization searches. The simulated annealing algorithm has been already used in searching the optimal path in stationary path planning methods [15], [16] and [17]. Therefore, it is expected to work successfully in planning path for a robot in dynamic environments. 2) The simulated annealing algorithm based methods use vertices as search space. The reason to use the vertices as search space is the solution constituted by the vertices of obstacles can be the optimal result. If the search space is represented by quadtree grids, suboptimal result will be obtained. The robot can obtain the optimal path in a short time in a dynamic environment, if the developed simulated annealing based method is efficient. 3) According to [18], the performance of the genetic algorithm deteriorates greatly as the problem size increases. However, the simulated annealing algorithm performs better than the genetic algorithm in this case [18, 19]. Thus, it is worth researching this issue if the simulated annealing approach could perform better in searching the optimal path in an environment with large number of obstacles. Implementing the simulated annealing algorithm in dynamic environment can provide a convincing answer. Until now, limited work has been reported in the open literature on using the simulated annealing algorithm in path planning in dynamic environments. Therefore, the research in this area is original 4
and can make useful contribution to the research field. 4) The genetic algorithm uses a population that contains a collection of chromosomes rather than a single solution, implying that it is more complicated and harder to implement than the simulated annealing algorithm. The processing time of the simulated annealing algorithm based approach can be shorter than that of the genetic algorithm based approach. Compared with the genetic algorithm based approach, the simulated annealing algorithm based approach may give better trade-offs among simplicity, far-field accuracy and computational cost in some cases as those in [20].
1.3 Aims of the Project The research focuses on planning path for robots in dynamic environments. The aims of this project are summarized below:
To develop and implement a simulated annealing algorithm based approach for path planning for mobile robots in dynamic environments; the approach is expected to be able to determine the optimal feasible path quickly for a robot in an environment with dynamic obstacles and to give a result comparable or better than those results in the previous approach [21], which is a typical path planning method that uses the vertices of the obstacles as search space. 1) The simulated annealing based approach could quickly determine an alternative path for the robot if the sensor of the robot detects a moving obstacle. The alternative path generated should also be the optimal result for the robot. The online processing time is expected to be shorter than those of previous works. 2) To develop a heuristic method for searching the initial feasible path efficiently and to incorporate the heuristic method to the developed simulated annealing algorithm based approach. 3) To incorporate the heuristic method into the simulated annealing algorithm based approach for further improvement of the performance of the approach.
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1.4 Significance of the Research The project contributes novelty in implementing a simulated annealing based approach for robot path planning in dynamic environment that contains moving obstacles. The simulated annealing based approach solves drawbacks occurred in some previous methods such as:
1) The simulated annealing based approach takes the shape of every obstacle into calculation that can provide more precise result than other methods. Some methods like [22], [23] and [24] ignore the dimension of the obstacles. Obstacles in [22], [23] and [24] are considered as points or simple square blocks which could generate the repulsive force to robots. The methods are not suitable for optimal path searching in an environment with a variety of sharp obstacles. For searching the optimal path, the dimensions of the obstacles cannot be ignored in such a situation. In fact, most of the published researches that use artificial potential field neglect the dimensions of the static and dynamic obstacles. This simplification will generate imprecise route for the robot, and the optimal solution is hard to obtain in the artificial potential field based method. The simulated annealing based approach takes the vertices of any different sharp obstacles as search space, therefore the approach takes the shape of the obstacles into account that overcomes the drawback of the artificial potential field based method.
2) Compared with some methods like [11], [25], [26], [27] and [28], the simulated annealing based approach reduce the calculation consumption of updating the dynamic environment. The methods in [11], [25], [26], [27] and [28] focus on dynamic uncertain environments, in which the change of the environment caused by the absence of obstacles or the appearance of the unexpected obstacles detected by the sensor of the robot. D* based methods are used to re-plan path in dynamic environments where the unexpected obstacle appears or disappears caused by the incorrect information. However, the obstacles described in the environments are not purely dynamic with speed and moving directions. Compared with the grids representation methods, it is easier to use vertices to represent the moving obstacles. Updating the positions and directions of the vertices of the moving obstacles can be easier than updating every grids in the moving obstacles. In this project, 6
the simulated annealing based method is proposed to do one-time online calculation to generate an alternative optimal path for robot once a moving obstacle is detected. According to [29] and simulation result of [27], the computational time of the searching process in grids based method increases exponentially with the number of the vertices in the map. The large number of vertices in the map may affect the performance of the D* based methods. Therefore, the simulated annealing based approach can give a better performance than grids based methods in dynamic environments with moving obstacles.
3) The simulated annealing based approach provides a more efficient result than previous approaches which also use the vertices as a search space for path planning. The performance of the new simulated annealing based approach is all-better than the previous method [14] and [21] in calculating the optimal path for robot. As the method is able to quickly determine the optimal path for robots in dynamic environments, it could be used in industrial research robots such as [3] and [2]. Also, the heuristic method to search initial path in the simulated annealing based approach in dynamic environments can greatly improve the performance of the simulated annealing based approach in calculation time and convergence in online and offline calculation. The heuristic method could also be used in other optimisation methods which need to generate initial solutions for calculations.
1.5 Contributions of the Thesis With the completion of the project, the aims of the research have been fully achieved, and an integrated framework has been successfully developed for robot path planning in dynamic environments. Main contributions of the Thesis are: 1) Main contribution 1: The simulated annealing algorithm based approach to search feasible path for robot in static environments with improved performance over existing methods. Genetic algorithm based path planning method in [21] is a typical path planning method that uses the vertices of the obstacles as search space and it is the fundamental algorithm used by [14] in dynamic path planning. Because the genetic algorithm based method used in [14] and [21] uses the same algorithm mechanism, and the mechanism is
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stated clearly in [21], so the performance of the simulated annealing algorithm based approach is compared with that of the genetic algorithm based approach in [21]. The calculation time for obtaining the optimal path is all-better than genetic algorithm based approach in [21]. The simulated annealing based approach successfully generates collision free path for the robot in dynamic environments. The simulated annealing based approach can quickly determine an alternative path for the robot if the sensor of the robot detects a moving obstacle. The alternative path generated is also the optimal result for the robot in the dynamic environment. 2) Main contribution 2: A heuristic selection method for selecting the initial feasible path more efficiently. It can be used in different types of optimisation methods. Compared with the random picking method, the heuristic method uses a more intelligent way to search the initial feasible path for robots. The heuristic method is incorporated into the simulated annealing algorithm based approach to further improve the efficiency of the approach. The simulation results show that the new heuristic selection method can greatly improve the performance of the simulated annealing approach in searching the optimal path in dynamic environments.
1.6 Structure of the Thesis The following are the descriptions of the structure of the thesis. Chapter One states the main issues of the thesis, including a general introduction to the problem and aims and outcomes of the research. Chapter Two provides detailed literature review and background information for robot path planning. As one of the main contributions of this thesis, Chapter Three discusses the implementation details of the simulated annealing based approach. It illustrates the methodology of the simulated annealing based approach to calculating the optimal path for a robot in dynamic environments. In Chapter Four, the simulation results of the approach are compared with the genetic algorithm based approach to test the efficiency of the simulated annealing based approach. As another main contribution, Chapter Five proposes a heuristic method to search the initial feasible path. The method is incorporated into the simulated annealing based approach to further improve the performance of the approach. Simulation results are also given to demonstrate the effectiveness and high efficiency of the 8
approach with integrated heuristic initial path search method. Chapter Six concludes the thesis, and briefly discusses future directions on the research topic.
1.7 Related Publications The following papers are published or in preparation in support of this thesis: (1) “Robot Path Planning in Dynamic Environments using a Simulated Annealing based Approach”, accepted for presentation at IEEE 10th International Conference on Control, Automation, Robotics and Vision (ICARCV 2008), Hanoi, Vietnam, Dec 2008. This conference is ranked by Core (www.core.edu.au) as a Tier A+ conference in 2007 and a Tier A conference in 2008.
(2) A Journal paper “Path Planning in Dynamic Environment Using a Heuristic Simulated Annealing Approach” is being prepared to be submitted. It reports the outcomes of the entire Masters research project.
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Chapter 2 Background and Literature Review The mobile robot path planning task is to find a collision-free route, through an environment with obstacles, from a specified start location to a desired goal destination while satisfying certain optimization criteria [30]. This chapter classifies various robot path planning methods in different ways and gives some general information about traditional path planning methods in different environments such as the Visibility Graph method, Grid method, and Artificial Potential Field method.
2.1 Classifications of Robot Path Planning Methods Path planning for mobile robots is one of the most important aspects in robot navigation research. The robot path planning methods could be classified into different kinds based on different situations. Depending on the environment where the robot is located in, the path planning methods can be classified into the following two types:
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(1) Robot path planning in a static environment which only contains the static obstacles in the map; and (2) Robot path planning in a dynamic environment which has static and dynamic obstacles in the map.
Each of these two types could be further divided into two sub-groups depending on how much the robot knows about the entire information of the surrounding environment: (1) Robot path planning in a clearly known environment in which the robot already knows the location of the obstacles before it starts to move. The path for the robot could be the global optimised result because the entire environment is known. (2) Robot path planning in a partly known or uncertain environment in which the robot probes the environment using sensors to acquire the local information of the location, shape and size of obstacles, and then uses the information to proceed local path planning.
Fig 1 shows the hierarchy of the classifications of the robot path planning methods:
Fig 1: Classifications of the robot path planning methods.
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2.2
Path Planning in a Static and Known Environment
In a static and known environment, the robot knows the entire information of the environment before it starts travelling. Therefore the optimal path could be computed offline piror to the movement of the robot.
The path planning techniques for a static and known environment are relatively mature. Representative path planning methods for a clearly known static environment include the Visibility Graph method [4], Voronoi diagrams method [5], the Grids method [6], the Cell Decomposition method [31], and the Potential Field method [32].
Moreover, the genetic algorithm, the simulated annealing algorithm, and some other optimisation methods have also been used to obtain the optimal path for the robot. Davidor [33] developed a tailored genetic algorithm with a modified crossover operator to optimize robot path. Nearchou [29] used the number of vertices produced in visibility graphs to build fixed length chromosomes in which the presence of a vertex within the path is indicated by setting of a bit at the appropriate locus. The method applied a reordering operator for performance enhancement, and the algorithm was capable of determining a near-optimal solution. Cai and Peng [34] developed a fixed-length decimal encoding mechanism to replace the variable-length encoding mechanism and other fixed-length binary encoding mechanisms used in the genetic approach for robot path planning.
2.2.1 The Visibility Graph Method In this method, a visibility graph is used in robot motion planning when the geometry of the environment is known. The main idea of the visibility graph method is that if there is a collision-free path between two points, then there is a polygonal path that bends only at the obstacles vertices [31]. As Fig 2 shows, collision-free path (in curves) could be transformed into line segments (straight line).
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Fig 2: Visibility graph.
A visibility graph is constituted by nodes and edges. Nodes are the start point, destination point and the vertices of all obstacles. Edges are strait-line segment between two nodes which do not path through obstacles. Fig 3 shows the complete visibility graph base on Fig 2.
Fig 3: Complete visibility graph. The following is the pseudo-code for building a complete visibility graph [31]: Output: G (visibility graph) 1: 2: 3: 4: 5: 6: 7:
for every pair of nodes u, v if segment(u,v) is an obstacle edge then insert segment(u,v) into G; else for every obstacle edge e if segment(u,v) intersects e goto (1)
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8:
insert segment(u,v) into G
Fig 3 shows that there are multiplex paths that could lead the robot from the start point to the destination. Then, any optimisation algorithms such as the genetic algorithm [35] and the simulated annealing algorithm [36] could be used to calculate the optimal path for the robot. The defect of the visibility graph method is that the efficiency of the algorithm is low. From the pseudo-code for building a complete visibility graph, it can be seen that the visibility algorithm requires O(n3) time. Furthermore, the obtained path is often very close to obstacles and thus, may lead to crashing of the robot. However, the problem can be fixed by enlarging the obstacles by a value according to the dimension of the robot. In this way, the robot can approaches obstacles without collision.
2.2.2 The Quadatree Method
The Quadatree method is another algorithm for searching the collision free path for a robot. It uses small non-overlapping grid cells to represent the entire environment. The cells usually are simple squares. There are three types of cells: empty cell, mixed cell and full cell. An empty cell is a free space, where the robot could go through, in the environment. A mixed cell contains obstacles and free space. A full cell is the block of the obstacles. In a two-dimensional map, a Quadtree is used to decomposition the map, as shown in Fig 4:
Fig 4 [16]: Quadtree decomposition.
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The cell decomposition method is briefly outlined below [31]: 1. Decompose the map into cells. 2. Search for a sequence of mixed or free cells that connect the initial and goal positions. 3. Further decompose the mixed cells. 4. Repeat (2) and (3) until a sequence of free cells is found.
Then, the method uses an optimisation algorithm, such as A* algorithm, to find the optimal path for the robot. The A* algorithm [37] is first described early in 1968 by Hart, Nilsson and Raphael. The algorithm is a best-first, tree search algorithm, and could find the shortest path from the start point to the destination point. Lingelbach in [38] used the cell decomposition method combined with probabilistic sampling to plan path for a robot in high-dimensional static configuration spaces. In [39], a combined artificial potential field and probabilistic sampling cell decomposition method was proposed. The potential field approach based on harmonic function is computed over a non-regular grid decomposition of a high-dimensional space obtained with the probabilistic sampling of cells.
Fig 5 [40]: Cell decomposition
2.2.3 The Artificial Potential Field Method The potential field method was initially proposed by Khatib in 1986 for mobile robot path planning. The main idea of the method is to imagine that all obstacles can generate repulsive force to the robot, while the destination point has attractive force to the robot. The robot applies a force proportional to the negated gradient of the composition of the attractive and repulsive force. The position and direction vector X = [ x, y ]
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composite of attractive force and repulsive force. 16
of robot are fixed on by
The attractive potential field function is given by U att ( X ) =
1 2 kρ ( X , X g ) 2
k : positive scaling factor X : position of the robot X g : goal of the robot
ρ ( X , X g ) = X g − X : distance from robot to goal Attractive force Fatt is negative grads of the attractive potential field function: Fatt = −∇ ⎡⎣U att ( X ) ⎤⎦ = k ρ ( X , X g ) The repulsive potential field function is described by ⎧ ⎛ 1 1 ⎞ − ⎟⎟ ρ ( X , X o ) ≤ ρ0 ⎪0.5η ⎜⎜ U rep ( X ) = ⎨ ⎝ ρ ( X , X o ) ρ0 ⎠ ⎪ ρ ( X , X o ) > ρ0 ⎩0 η : positive scaling factor ρ ( X , X o ) : the shortest distance between the robot and obstacles Constant ρ 0 : distance of influence imposed by the obstacle; its value depends on the condition of the obstacle and the goal of the robot, and is usually less than half distances between the obstacles or shortest length from the destination to the obstacles.
When the robot is not at the goal, the repulsive force is Frep ( X ) = −∇ ⎡⎣U rep ( X ) ⎤⎦ ⎧ ⎛ 1 1⎞ 1 − ⎟⎟ 2 ⎪η ⎜⎜ = ⎨ ⎝ ρ ( X , Xo ) ρ ⎠ ρ ( X , Xo ) ⎪ ⎩ 0
ρ ( X , X o ) ≤ ρ0 ρ ( X , X o ) > ρ0
The resultant force is F = Fatt + Frep
F navigates the movement of the robot [23]. This is illustrated in Fig 6:
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Fig 6: Potential field. The artificial potential field method provides a simple yet effective method to plan paths for a robot. However, it also has some drawbacks. The major problem is that robots are often trapped into a local minimum before reaching the destination. Nowadays, the artificial potential field method is combined with many other computational methods to improve its efficiency. Park and Lee [41] have integrated the artificial potential field method with the simulated annealing algorithm to escape local minimum. Zhu, Yan and Xing [15] also use the artificial potential field method in combination with the simulated annealing algorithm for similar scenarios. In addition, as will be discussed later, the potential field method is also used in planning path in dynamic environment, as demonstrated in [8] and [23].
2.3
Path Planning in Static and Unknown Environment
In a static and unknown environment, robot navigation is more difficult than that in static and known environment. This is due to the uncertainty of the information in the environment, such that the optimal result is difficult to obtain and the robot has to use local information to calculate the path. Some path planning methods in static and unknown environment will be reviewed below.
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In Ying and Eicher [42], a sensor-based path planning method was proposed to help underwater robotic vehicles perform real-time path planning in a static and unknown environment. The environment is constituted by various shaped static obstacles. A three-dimensional path planning algorithm is developed for the robot in such an environment. Using sonar to detect the static obstacles in the environment, the robot is controlled by a group of thrusters. The simulation results show that the on-line path planning method can plan a collision free path for the robot. The calculation time in the simulations is acceptable for real-time online planning application.
In [42], the relationships among the robot position, vision zone, obstacles and target position are described as the following situations: 1) No obstacle is inside the vision zone of the robot, shown as Fig 7:
Fig 7: Obstacle is out of the vision zone. uuur 2) An obstacle partially occupies the vision zone but it does not impede the main line VT
between robot and target. This is shown in Fig 8.
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Fig 8: Obstacle partially occupies the vision zone. 3) As shown in Fig 9, obstacle partially occupies the vision zone and it impedes the main uuur line VT between the robot and the target.
Fig 9: Obstacle impedes the main line between robot and target. 4) Obstacle fully occupies the vision zone, as shown in Fig 10:
Fig 10: Obstacle fully occupies the vision zone. 20
Several path planning rules are used to plan path against above mentioned situations:
(1) Path planning rule 1: If no obstacles exist inside the vision zone or the obstacles uuur partially occupy the vision zone but do not impede the main line VT , the robot will uuur follow the main line VT to the target. (2) Path planning rule 2: The best gate rule. If an obstacle partially occupies the vision uuur zone and it impedes the main line VT between robot and target, then the robot search for the best path based on the currently known information. The path is obtained by the current robot position and the edge point of intersected surface produced by obstacle surface and vision zone of the robot. The best gate A and the uur uuur uuuur shortest VAT are selected, which can be determined by MIN (VA + AT ) . The following figure demonstrates that:
Fig 11: Path planning rule 2.
(3) Path planning rule 3: The possible shortest path rule. If Obstacle fully occupy the vision zone, the current location of the robot V, the target position T and the puncture point G are used for calculating the feasible path: uuur a): Find puncture point G produced by the VT and the intersected surface of the obstacle. uuur b): Find the minimum gradient direction PG at the puncture point P on the
intersected surface. G is on the edge of intersected surface.
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uuur uuur uuuur c): The path VGT = MIN (VG + GT ) .
Fig 12: Path planning rule 3.
The path planning algorithm applies each planning rule against each situation robot encounter until the robot reaches the destination point T.
In [43], Park and Lee proposed a modified virtual hill concept based on the artificial potential field method to escape local minimums in the local environment. A virtual hill with extra potential is added at the trapping obstacle when the robot is trapped in a local minimum position. The extra potential is added to the global potential, which can repel the robot from the local minimum. However, due to the insufficient information of the obstacles in the environment, the global optimization result can not be acquired and only the local optimal result is attained. Recently, Chang and Yamamoto [44] combine the Voronoi diagram approach with the potential field method to avoid dead-lock problem in the potential field method. The problem is solved by defining necessary sub-goals between targets on the constructed map. Xie and Chen [45] use a virtual water-flow concept combine with potential field method to avoid the local minimum problem in potential field method. Vanualailai and Sharma [46] developed an asymptotically stable point-mass system to avoid the local minimum in artificial potential field method. The Lyapunov function in the asymptotically stable system produces artificial potential fields around the goal and the obstacle, has no local minimum other than the target.
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In [47], Wan, Chen and Earnshaw used a robot version hypothesis similar to that in [42] and proposed an A* based path finding algorithm. The method allocates the points of visibility dynamically and decentralise the view angle in each step to build state node. Paper [48] proposed a more practical method using a virtual window in the front of the robot to calculate the space ahead and any intersections that are done. The robot acquires a two-dimensional image of the space ahead and processes the information to find a collision free path. The simulation results show that path planning can be successfully implemented using the virtual window method and that the computational time limitation can be overcome.
In [49], an artificial immune network based path planning method is proposed. The method is capable of obtaining an optimal or near-optimal collision free path in unknown environments that is presented by grids. According to the simulation results, the method can give a better performance on complex environments than the Knowledge-based Genetic Algorithm [58] does.
Nowadays, the path planning problem is still a favorite research topic in robotic and artificial intelligence studies. The path searching for robot in unknown static environment will still be study in the future.
2.4 Path Planning in Dynamic and Known Environment Research on methods that deal with the static environment path planning has been introduced in previous sections. Currently, the path planning methods to find paths in static environment have been well developed with hundreds of published papers. Given the entire information of the environment, the global optimal or near-optimal result path could be found by these algorithms, such as the genetic algorithm [50].
However, in practical applications, robots such as those described in [2] and [3], often face obstacles that are not all static in the environment; the status and the movement of the obstacles change continuously in the map. Moving obstacles in a dynamic environment 23
increase the difficulty of planning path for the robots in the map.
Unlike the situation for path planning in a static environment, limited reports have been found in the open literature to discuss optimal path planning in dynamic environments. Complexity and uncertainty increase with the number of the dynamic obstacles. Therefore, traditional path planning algorithms, such as the visibility graph, the voronoi diagrams and the grids method, do not perform well in dynamic environments. It is also difficult to gain the optimal path for the robot using these methods. Robot path planning in a dynamic environment is thereby an issue for further research. In a dynamic environment, how to manipulate the robot so that it can travel to the destination safely and optimally without collision is an important issue of concern.
In 2005, Chestnutt, Lau and Cheung [51] used a modified A* algorithm to calculate path for a Honda ASIMO humanoid robot. The path planning method is applied to real robots rather than simulated on software. A grid of cells is employed to represent the environment. Colour cells represent the obstacles. The cells create a bitmap representing the free spaces and obstacles in the map. The algorithm plans a sequence of footstep positions to navigate toward a goal location based on known static and moving obstacles with predictable trajectories. It uses three cost functions to constrain the step nodes used by the robot. (1): The first cost is the location cost, which evaluates the footstep location with regard to the environment to determine if the location is safe place to step. (2): The second cost is the step cost, which calculates the cost to the robot to make the desired step. (3): The last cost is the estimated remaining cost-to-go, which is calculated by planning backwards from the goal with a standard mobile robot planner as a precomputation step. This cost is used to avoid the local minimums.
Wang and Sillitoe [14] proposed a vertices genetic algorithm planner in 2007. The planner is able to rapidly determine optimal or near-optimal solutions for a mobile robot in an environment with moving obstacles. The method uses the vertices of the obstacles as search space and produces off-line path planning through the environment with dynamic obstacles.
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It firstly incorporates the robot speed into the genetic genes, which could optimise both the travel time and distance of the robot. Before the robot starts movement, the complete motion knowledge of the moving obstacles in the observed region is available for the robot. The robot uses the genetic algorithm based planner to calculate the time or distance-optimised solution and then starts to travel. The assumptions of this method are: •
Obstacles are bounding polygons with vertices.
•
The speed of the moving obstacles is fixed value, and
•
Physical dimensions of the robot are neglected and regarded as a single point.
The genetic representation of the method is outlined in Fig 13:
Fig 13: The representation of the chromosome in GA planner.
As shown in Fig 13, the possible solution paths are represented by a chromosome which is constituted by a bunch of genes. The minimum number of genes for a chromosome is two corresponding to the start point and the end point, and the maximum number of genes is thus the total number of whole vertices N plus the start and the end points.
Each gene contains four parts with the first two being the x and y coordination positions of the vertex. The third part is the robot’s speed in the segment of each gene. The robot speed is selected from a group of predefined discrete speeds. The last part indicates the feasibility of the path that originates from the genes; if the path segment connecting two consecutive vertices intersects one or more obstacles, then the infeasibility bit of the gene representing the originating node is assigned 1 to mark the infeasibility of this segment. The evaluation functions have two types: the path length and the travel time.
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Four genetic operators are used in the method; they are crossover operator, mutation operator, repair operator and robot speed mutation operator. Fig 14 illustrates the first three genetic operators.
Fig 14: Genetic planners.
The generational operation begins with the random selection of a genetic operator and a quadratic ranking scheme is used to retain the constant selection differential after evaluation. The parent (or parents for the crossover operation) involved in the genetic operation is determined by a roulette wheel whose slots are sized in proportion to the fitness as scaled by a ranking technique. To form a new generation, the newly generated offspring replace the worst individual (or pair of individuals if crossover is applied) in terms of fitness in the existing population. The evolutionary process continues until a termination condition is satisfied, which can be defined to be a number of generations specified by the user or determined by monitoring against a specified performance criterion. When the evolution terminates, the best individual is selected as the path planning solution. The simulation results show that the planner changes in robot speed significantly improves the efficiency of the method especially in time management for robot in dynamic environment.
A hybrid navigation method [52] was proposed in 2003. The method consists of two modules: global and local modules, which could combine two robot path methods to deal with the global map information and local sensor information.
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(1) The global module specifies the global route positions. It uses prior information on the navigation environment and chooses critical points to pass through before actual navigation takes place. This module uses the A* algorithm to determine the optimal route to the specified goal position. Using the algorithm, it is possible to find the optimal route to the goal.
(2) The local module carries out the navigation itself, relying on current sensor data, thus making it easier to avoid static or dynamic obstacles. It uses a fuzzy neural representation of the potential field based navigation method.
Firstly, the global planning module finds the optimal route to the goal and proposes the positions to pass through as intermediary points. These intermediary points are then passed one by one to the local navigator, which makes the robot reach them while reactively avoiding the obstacles present in the environment, according to the potential function previously supplied to the local navigator.
A dynamic environment is more complicated than a static environment in the robot path planning issue. Several methods were proposed to solve the problem. Because the moving information and the obstacle information can be known in advance of movement commencing, the optimal solution can still be obtained.
2.5 Path Planning in Dynamic and Unknown Environment Path planning in dynamic and unknown environment is the most complicated case in robot path planning, and is also the most common situation that mobile robots will confront. In the real world, mobile robots such as undersea unmanned vehicles often need not only to avoid static rocks in the seabed, but also to avoid colliding with the large sea lives. Due to the complicated and unknown environment, the robot cannot adopt one time global path planning for the environment. The global optimised solution is thereby difficult to be obtained. The robot has to use sensors acquiring the information of surrounding environment and do online real-time path planning. The planning time for the robot should be short 27
because the robot needs a sufficient time interval to adjust its movement in order to avoid the coming obstacle. In next few paragraphs, recent research in this area will be reviewed.
In 2006, Lv and Feng [53] introduced a numerical potential field for finding the path in a dynamic environment for a robot. This research used an ant colony optimisation algorithm to perform optimal path searching. The movement of the obstacles is taken into considerations by changing the local potential values. The method firstly uses dynamic valued potential field to model the dynamic environment, then employs an ant colony optimisation method to search the path for local path planning.
In 2007, Wang, Sillitoe and Mulvaney [14] introduced a genetic algorithm planner to determine optimal or near-optimal solutions for a mobile robot in dynamic environments. The obstacles are described as bounding polygons and the robot dimension is neglected and regarded as a point. The method firstly uses offline planning to generate a path for the robot, then the robot starts to travel with its sensors. The online path planning method is activated to calculate an alternative path for the robot if obstacles are detected. Because of the complexity and uncertainty involved, very limited work has been conducted to develop complete solutions for a purely uncertain dynamic environment.
Furthermore, implementing path planning algorithms in real robots in dynamic cases is also very limited. Cao, Huang and Zhou [23] provided an evolutionary artificial potential field method for dynamic path planning for soccer robots in RoboCup 2005. The new potential field method is intended for path planning of mobile robots in a dynamic environment where the target and obstacles are moving. Firstly, the new force function and the relative threat coefficient function are defined. Then, a new potential field path planning algorithm based on the relative threat coefficient is presented. Finally, computer simulation and experiment are used to demonstrate the effectiveness of the dynamic path planning scheme [32].
Zheng and Zhao [54] proposed an artificial potential field based approach in 2006. The method could handle path planning for five mobile robots in a dynamic unknown environment. The robot will pass the intersection area in order according to their priority,
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which is predefined before movement. The method assumes that the obstacles in the map have identical sizes. It successfully coordinates the movements of multiple robots, and makes the robots pass through the static obstacles and other robots without collision. The novelty of the research is the application of the artificial potential field method to multi-robot path planning in dynamic environment.
Tarokh [55] develops a hybrid intelligent approach to path planning for high mobility robots operating in rough environments. Path planning consists of characterization of the environment using a fuzzy logic framework, and a two-stage genetic algorithm planner. A global planner determines the path that optimizes a combination of terrain roughness and path curvature. A local planner uses sensory information, and in case of detection of previously unknown and unaccounted for obstacles, performs an on-line re-planning to get around the newly discovered obstacle.
Hu and Gu [56] developed an improved algorithm to solve the problem of optimum route planning in vehicle navigation systems. The algorithm is based on the standard genetic algorithm and the lambda-interchange local search method and has evolved from the improved A* shortest-path algorithm. The algorithm can find the optimum route more efficiently and quickly without any network constraint condition and can work well either in continuous or in discrete networks.
In [57], potential field method is applied for both path and speed planning for a mobile robot in a dynamic environment where the target and obstacles are moving. The robot’s velocity and trajectory are determined by the relative speed and moving directions of the obstacles and target. The simulation results verify that the method can efficiently tracking the moving target while avoiding the obstacles along its path.
The D* (Dynamic A*) [11] method is a typical method used in path planning in dynamic unknown environments. It plans optimal traverses in real-time by incrementally repairing paths to the robot’s state as new environment information is known, which makes it possible to greatly reduce the computational cost. When the robot gathers new information about the
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environment, it will replan new paths based on the new information and produce a path for the robot. D* does not require complete re-planning of the path every time when new information comes in. It updates the path arc cost locally when environment changes. However, D* still try to obtain the globally optimal path when possible.
Like A*, D* operates on a cost graph. The environment with the obstacles is represented by a uniform grids map as in Fig 15. The main idea of the method is illustrated as following: From the initial state, the method repeatedly selects the neighbour with the minimum cost until propagates to the goal. Each small cell in the map is called state. Each state X has the arc cost from the state X to the goal given by the path cost function h(X,G). From the start point (start state), all neighbour states of the current state are listed on the open list. From the open list, the method calculates the arc cost of the states by h(X,G). Then, select the state with the minimum h(X,G), go to this state, and new neighbours are added to the open list. As Fig 16 shows, from the start point, the method repeatedly propagating to neighbours and selecting the state with the minimum h(X,G) until the goal state is propagated.
In the dynamic uncertain environment, when the robot detects new obstacles or the absence of expected obstacles, the cost values of the states in the area change. And the adjoining states are put on the open list for cost correction. Encountering unexpected obstacles, D* will set off a “raise” wave, a wave of increasing cost, through neighboring states. When this wave meets with the states that are able to lower path costs, these “lower” states are put on the open list to recalculate new optimal paths. When it detects the absence of an expected obstacle, the arc of the path passing through this “missing” obstacle is assigned a small cost, denoting an empty space, and the adjoining state is put on the open list as a lower state, setting off a “lower” wave, a wave of decreasing cost. D* is able to determine how far the raise and lower waves need to propagate while providing the optimal path for robot traverse continuously [26].
The basic D* method was originally proposed in 1994. However, the cost change propagation method used in basic D* does not consider which expansions will benefit the robot at its current location. Thus, the performance of the algorithm [25] can be affected.
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Therefore, like A*, D* can use heuristic function to focus on the search in the direction of the robot and reduce the total number of the state (grids) expansion. Focussed D* method [26] uses the focussing heuristic function to estimate the estimated path cost from the current location to the goal to help the robot to minimise its search space. In comparison with Fig 16, Fig 17 shows that with focussed D*, robot uses fewer grids in search the optimal path.
Fig 15: Regular grids representation.
Fig 16: D* Algorithm [25]
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Fig 17: Focussed D* Algorithm [25]
To enhance the performance of the D* algorithm, framed-quadatree D* [27] and Field D* [28] methods were proposed. •
The Framed-quadatree method uses quadatree structure to represent the environment. It uses different dimensions of grids to represent the environment in order to minimize the search space, and then adding the border-cells to connect each grid, as shown in Fig 18. The framed-quadatree structure can reduced the search space for the robot and also the framed-quadatree uses border-cells to over come the disadvantage of the suboptimal result in using the quadatree structure. However, because the method has to create more sub-cells for calculation, the performance of the framed-quadatree method deteriorates greatly as the fractal gain increases, especially in offline planning [27].
•
The Field D* method [28] uses an interpolation-based planning and re-planning algorithm to generate smooth paths for robots through non-uniform cost grids. It uses linear interpolation during planning to calculate accurate path cost estimates 32
for arbitrary positions within each grid cell and to produce paths with a continuous range of headings. The method can produce a smooth optimal path for a robot to over come the suboptimal problem in using non-uniformed girds method. However, the method scarifies the performance for the accuracy. It reduces the path length of the robot in offline and online planning but takes 1.8 times longer in calculation time than previous D* method [28].
Fig 18: Framed-quadatree structure.
Recently, a grid-based distance-propagating dynamic system is proposed in [58]. The algorithm is similar to the D* algorithm, but it does not maintain a sorted queue of points to update. The algorithm uses only local information when computing an update at each point, and the order of updating is predetermined. This makes the algorithm exceedingly easy to parallelize by simply assigning a subset of points to each processor. Consequently, in situations where obstacles and targets are moving at substantial distances from the current robot location, the grid-based distance-propagating dynamic algorithm is more efficient than the D* algorithm.
Until now, most research activities on robot path planning in dynamic environments have been theoretical in focus. Some methods such as those in [11], [25], [26], [27] and [58] are
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proposed to deal with the issue of robot path planning in uncertain environment. However, due to the complication of dynamic environment, the methods still have limitations concerning their assumptions, algorithm efficiency and other issues:
1) The methods in [11], [25], [26], [27], [28] and [58] focus on dynamic uncertain environments, in which the change of the environment caused by the absence of obstacles or the appearance of the unexpected obstacles detected by the sensor of the robot. The robot uses D* or algorithm similar to D* based methods to re-plan path in dynamic environments where the unexpected obstacle appears or disappears caused by the incorrect information. However, obstacles described in the environments are not purely dynamic with speed and moving directions. Compared with the grids representation, it is easier to use vertices to represent the moving obstacle. Updating the positions of the vertices of the moving obstacle can be easier than updating every grids in the moving obstacle. In this project, a simulated annealing based method is proposed to do one-time online calculation to generate an alternative optimal path for robot once a moving obstacle is detected. According to [29] and simulation result of [27], the computational time of the searching process in grids based method increases exponentially with the number of the vertices in the map. The large number of vertices in the map may affect the performance of the D* based methods. Therefore, it is worth implementing efficient vertices based simulated annealing approach for robot path planning.
2) The efficiency of some methods such as that in [14] is not practically acceptable when applying the method in real-world systems. The methods assume that in order to allow the robot to be guided so as to avoid any potential collisions with obstacles, there is an adequately large time interval between the detection of obstacle movements and the implementation of new generated actions. When dealing with robot path planning in complex environments, genetic based approach is time consuming. According to the simulation result of [14], it requires nearly 30 seconds to find the first feasible path for the robot in an environment with 14 static obstacles and 5 dynamic obstacles. It seems not practical applying the approach to real robot application.
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Addressing those drawbacks, this project develops and implements a new approach for more efficient robot path planning in dynamic unknown environments. The proposed approach could provide a near or better result than previous solutions like [14] and [21]. The algorithm is able to quickly determine the optimal path for a robot in dynamic unknown environments. The robot uses sensors to gain the information of moving obstacle, and compounding it with the information of static obstacles to acquire the optimal path for the robot in dynamic environments.
With the time limitation of this research, we will use a robot and environment simulator to investigate the robot path planning problem, and, implement our algorithms in the simulator. In this project, we use Matlab to simulate the robot with sensors and the unknown dynamic environments. For simulating the algorithms, Matlab could be the suitable application, because of its sophisticate math toolbox. In this paper, the simulation results show that the approach can give a better performance in path planning than [21], and takes the dimensions of the static and dynamic obstacles into account comparing with the artificial potential base methods. The simulated annealing based approach with heuristic search method improves the searching algorithm in finding the optimal solution in dynamic unknown environments. The detailed methodology of the approach will be explained in Chapter 3.
2.6 The Simulated Annealing Algorithm and SA Based Approach 2.6.1 The Simulated Annealing Algorithm
The simulated annealing algorithm is a generic probabilistic meta-algorithm for the global optimization problem, namely locating a good approximation to the global optimum of a given function in a large search space. It is widely applied in many industrial fields. The well-described application of the simulated annealing algorithm is for the Traveling Salesman Problem (TSP) [59]. The TSP problem is that given a number of cities and the cost of traveling form one city to another, to get the least cost round trip to visit each city exactly once and then return to the start city.
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The simulated annealing algorithm is analogous to metal physical cooling process, with each step of the simulated annealing algorithm replacing the current solution by a random "nearby" solution. The probability of choosing the random "nearby" solution depends on the difference between the corresponding function values and on a global parameter T (called the temperature) that is gradually decreased during the process. The dependency is such that the current solution changes almost randomly when T is large, but increasingly "downhill" as T goes to zero [60]. Fig 19 is the flow chart of the general steps of the simulated annealing algorithm:
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Fig 19: Procedure of the simulated annealing algorithm.
2.6.2 Simulated Annealing Based Approach
Before this project, the simulated annealing algorithm approach had already been applied to the robot path planning method to deal with finding path for robot. Zhu, Yan and Xing [61] proposed a simulated annealing based artificial potential method to deal with the robot path planning issue in static environments. The artificial potential method is firstly used to map the environment and find a route for the robot. The simulated annealing algorithm is applied to the artificial potential method to overcome the local minimum problem that is caused by obstacles in the map. Zhang and Lu [62] integrate the simulated annealing algorithm into the artificial potential field method to solve the GNRON and local minima problems. Neachor [19] used vertices of the static obstacles as search space and implemented the simulated annealing based approach to find optimal path for robot. The simulated annealing based approach is compared to genetic based approach in his paper. The performance of the simulated annealing approach is slightly lower than the genetic approach in his experiment.
There might be two reasons that affect the performance of simulated annealing approach in Neachor’s model.
(1) The simulated annealing algorithm is more sensitive to the control parameters such as initial temperature and cooling rate. In Neachor’s research, only one group of control parameters is set in dealing with three different situations. The performance of simulated approach could be improved if suitable parameters are set.
(2) The operator that generates the new solutions is too simple. Neachor only used a swapping operator to generate the new solution. The swapping operator only flips some bits of the result to generate a new result. In this way, the possibility of jumping out of the local optimal result is small. The simulation results in this Thesis proves that adding more algorithm operators to generate the new result gives the simulated annealing approach better performance. In this thesis, more than one operator were
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used, the simulation results show that the performance is improved by the multi-operator in the approach. The detailed approach description will be discussed in Chapter 3.
Path planning in a dynamic unknown environment is the most common case in the real world. Therefore, a more effective robot path planning method should be proposed, which can satisfy the accuracy and the efficiency simultaneously. Our project is trying to apply the simulated annealing method to quickly produce online calculation and efficiently defines the optimal path for robot in a dynamic environment.
2.7 Robot and Environment Simulators Implementing a new method on real robots is not only complicated and time consuming, but also expensive. For the purpose of this Master research project, we use software simulators to model both the robot and the environment. Then, the developed approach is implemented in the simulators. There are many kinds of robot and environment simulators, such as multiple mobile robot simulators EyeSim [63] and Player/Stage [64]. The following is an introduction to the Player/Stage appliance.
The Player/Stage Project is an open source project for multiple, distributed robots control and sensor network system, in which Player provides as robot device server. Stage works as a multiple robotics simulator, plus some supporting tools. Since that Player/Stage can work under different platforms and be easily programmed, they have been adopted, modified and extended by researchers all around the world. In fact, Player can actually control and work in many popular robot devices, such as Pioneer 2-DX mobile robot and etc [64]. The Player/Stage project is carried out in Robotics Research Lab of University of Southern California. Player is an open sourced middleware server that could run on variety of robots, sensors and actuators. It hides the hardware detail of robots for users (like an operating system on robots). Users could write their own control program at client to control the robot through TCP channel.
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Fig 20: Structure of Player Researchers do not need to access to the real hardware and environments when they are focusing on Player and robot research. Stage is used to simulate different kind of robots, sensors and object for the Player. It provides two-dimensional bitmapped environments, and also could let research to do experiments with novel devices that do not exist (By adding library files). Stage simulates a population of simulated robots and sensors operating in a two-dimensional bitmapped environment. The devices provided by the Stage are accessed through Player, as if they were real hardware. Stage aims to be efficient and configurable rather than highly accurate. In practice this means that Stage can simulate tens or hundreds of robots on a desktop PC, and the simulated robots commonly work similarly to the real robots in the world [65].
Fig 21: Player/Stage architecture. 39
The implementation of building the new simulated annealing approach on Player/Stage will not only demonstrates the feasibility of approach but also provide a more practical simulated environment for the new method. The simulated annealing based approach could be tested by simulated robot hardware, not only exist in theoretics.
The Stage Library (libstage) provides a C code library for simulating a population of mobile robots, sensors and environments. Stage is one product of Player/Stage project, which is a simulating tool used in the field of mobile robot and intelligence sensor system. Stage simulates mobile robot, sensor, obstacles and other objectives in two-dimensional bit map. Stage could simulate different equipments and models. It could also simulate virtual robots and sensors that have not yet been developed. Stage could simulate variety of sensors and executor including sonar, laser scanning distance measurer, splash display, blob gripper, mobile robot and so on.
Equipments simulated by Stage can be controlled by Player which is a network robot server. Interface program provided by Player drives equipments like robot and sensor, and Stage simulates these equipments. The TCP ports for controlling the real robots provided by Player are identical with the TCP ports to control the simulated models in Stage. Therefore, Stage could totally provide comprehensive real environments and robots for the Player and its clients. It is expected that Stage builds the dynamic environment and simulates real robot and real sensors for the player sever. The simulated annealing approach is implemented in client machine, the client acquires the useful information from Player server, and then use simulated annealing approach to acquire the optimal path.
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Chapter 3 The Simulated Annealing Algorithm Based Approach This chapter explains the methodological approach in details, including the modeling of environments, structure of the approach/algorithm, the generation of the initial feasible path, the new planner for generating the random path and the procedure of online calculation. One of the aims of this project is to implement a simulated annealing based approach into robot path planning in a dynamic environment. The simulated annealing based approach is expected can quickly determine the optimal feasible path for robot in the environment with moving obstacles. The research uses the mathematic software Matlab to simulate the dynamic environment, real world robots and sensors. The research then implements the simulated annealing path planning method in the environment. Therefore, the entire research project could be separated into the following three phases: 1) Design the algorithms that could find paths and optimising the paths for the robot in dynamic environments.
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2) Use Matlab to build mathematic model and obtaining the simulation results proving the feasibility of algorithm. 3) Compare the results of the new approach with the those from existing methods, and discuss the performance and the efficiency of the new approach. .
3.1 Structure and Assumptions of the Method First, as in the development of many existing methods, some assumptions are made before designing the algorithm. 1) There are stationary and moving obstacles in the dynamic environment. 2) As a traditional assumption, both the stationary and dynamic obstacles are bounding polygons. 3) The search space is the vertices of the static obstacles plus the vertices of the detected moving obstacles. 4) The movement trajectory of the dynamic obstacles is constituted by a series of line segments, and is constrained by a straight line. 5) Any changes in motion parameter of the dynamic obstacle are immediately available to the robot, if the obstacle is in the range of the sensor. The time of the robot obtaining the moving parameter (such as speed and direction) is minimum that does not affect the path planning calculation. 6) The robot could change its speed and direction at any time. 7) After the sensor of the robot acquires the motion information of the moving obstacle. It is assumed that a separated module is used to monitor the movement of the obstacles for the robot. The module will predict the next movement for the robot based on the previous movement of the obstacle. It will report any change of direction and speed of the obstacle. The module is separated from the robot, which can be simply implemented by using extern sensors in real robot implementation.
The designed structure of the simulated annealing algorithm in dynamic obstacles environments is divided into two stages: off-line calculation for stationary obstacles, and online calculation when moving obstacles are detected. 42
Stage 1: Offline calculation for the path on stationary obstacle based on the vertices of the static obstacles. The simulated annealing based approach firstly calculates the optimised path for the robot base on the positions of the stationary obstacles. Once the path is ready, the robot starts to travel through the stationary obstacles with the sensor that could detect the 360 degree direction of the robot. Stage 2: Online path calculation once the moving obstacles are detected by the sensor. As moving obstacle enters the detection range of the robot, the sensor will detect the obstacle and the robot acquire the moving information such as speed and moving direction of the obstacle from the sensor of the robot. Then the robot uses the motion information to calculate the possibility of the moving obstacle clashing with the robot. If it is calculated that the moving obstacle will not collide with the robot, the robot will use the current path plan to travel through the map. As Fig 22 shows, the sensor of robot calculates out that the L shaped moving obstacle will not collide with the robot. Therefore, the robot will not change the current path plan. If the moving obstacle will clash with the robot as Fig 23 shows, the robot will activate the algorithm again to calculate a new path online for the robot (Fig 24). Fig 25 shows that state change of the robot in path planning approach in dynamic environments.
Fig 22: Scenario 1 of moving obstacle.
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Fig 23: Scenario 2 of moving obstacle
Fig 24: Online path planning
Fig 25: State transformation diagram of simulated annealing approach.
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In the actual calculation for either offline or online path planning, the algorithm consists of two procedures: •
Firstly, using randomly selection to find one initial feasible path which could lead robot from start point to destination.
•
Secondly, based on the initial feasible route, using the optimization algorithm to search for the optimal or near-optimal path for the robot in the static environment. In this Thesis, the optimization algorithm used for searching the optimal path is a simulated annealing algorithm.
3.2 Mathematic Modelling 3.2.1 Environment Modelling
The environment is represented by polygons with vertices and edges. The mathematic model of locating the edges and vertices of the obstacles in the environment is not complicated, use the model mentioned in Chapter 2 about visibility graph. For realistic simulations, as in [21], all obstacles in the map are enlarged by a fixed value that the robot could approach obstacles without collision. The dimension of the robot is neglected, and consequently the robot is regarded as a single point. In Fig 26, the black polygons represent the static obstacles and the hollow polygons are moving obstacles. The vertices of the enlarged polygons form the search space for the robot.
Fig 26: Enlarged obstacles 45
A mathematic model can be developed to determine the possibility of robot colliding with a moving obstacle. The model is described as follows: 1) The first crossing point between the robot path proposed by the planner and the trajectory of a moving obstacle is calculated before examining the possibility of collision; 2) Based on the time t required for the robot to cover the distance from the current position to the first crossing point, the instantaneous location of the moving obstacle can be calculated and. Consequently, the exclusion area for this obstacle can be obtained; 3) If the robot path between the vertex and the crossing point across the edges of the moving obstacle in odd times, then a collision would occur between the robot and the moving obstacle, otherwise no collision will happen between the robot and the obstacle;
3.2.2 Algorithm Structure and Expressions
Traditionally, the path length Ef is the evaluation criteria for the quality of the solution derived from the algorithm. The shorter the path, the better the solution. A feasible solution is expressed by a series of vertices linking the start point through to the end point. Each vertex of the obstacle has its series number. The path for the robot is represented by a sequence of vertex numbers. Thus, the feasible solution X is given by:
X = {Vstart, Vstart+1, Vstart+2…Vend-1, Vend}; The evaluation function Ef is given by:
Ef
=
∑
i = end −1 i = start
D(Vi, Vi + 1)
Where D(Vi, Vi+1) represents the direct distance from vertex Vi to Vi+1.
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The pseudo-code of
the algorithm is as following:
Step 1: Set an initial temperature T; Step 2: Generate a initial path order randomly from the start to the destination and calculate the path length L(initial); Step 3: Generate a new path which also from the start to the end and calculate path length L(new); Step 4: If L(new) < L(initial), accept new path. Else possibly accept new order according to some scheduling. Step 5: Repeat step 3 and step 4 until the temperature gets down. Step 6: Down the temperature and return to step 1.
The following are pseudo-codes for the simulated annealing algorithm:
T = Tinit; while (T>Tterminate) randomly generate one feasible solution Xs; evaluate Xs, Ef = f(Xs); count = 1; while (count < Threshold) generate a new feasible solution Xn base on Xs; evaluate Xn, En = f(Xn); if f(Xn) < f(Xs) Xs = Xn; else if rand(1) < (exp((f(Xs)- f(Xn))/T) Xs = Xn; count =count +1; endwhile T = cool_rate * T; update Xs at each reduction of temperature T endwhile
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Xs is the optimal or near-optimal solution for the robot. Fig 27 is the flow chart of the simulated annealing approach for searching the path for robot in the environment.
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Fig 27: Procedure of simulated annealing approach for path planning.
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3.2.3 Initial Path Selection Process
It is known from Fig 27 that after each reduction of initial temperature T, a new feasible solution Xn is selected in each new round. It is essential for the algorithm to quickly and correctly generates a random feasible path in each round. For this purpose, the edges of the static obstacles should be specified first. An edge is any straight-line between two points on the edge of or within the obstacles. Any path section crosses the edge is defined as an invalid path. In the proposed program, a separate array is used to store all the edges of the map.
At the initial stage of the program, except dynamic objects, starting point and end point; a vertex is chosen randomly from the map, then use a strait-line to connect the start point and the selected vertex. If the path line intersects with any edges of the obstacle in the map, the path is recognized as invalid path. Another vertex will be randomly selected again for testing. If the strait-line to the selected vertex does not intersect with any edges in the map, the strait-line is recognized as part of a valid path. Then add the vertex into the path. After that, start from the selected vertex to find next valid vertex. Keep doing the above procedure until the end point is selected and also the path line to the end point is a valid path, i.e., the path to the end point does not intersects with any edges. The following pseudo-codes and Fig 28 illustrate the process of the initial path selection process.
initial path IP = {start}; while (1) randomly select Vs; if path{IPend, Vs} intersects any obstacles; continue; else IP = {IP,Vs}; endif if Vs == destination point; break;
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else continue; endif endwhile
Fig 28: The initial path selection process.
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3.2.4 Random Path Planner
As discussed in Chapter 2 (Literature Review), a simpler random path planner directly affects the performance of the path planning algorithm. Therefore, advanced and more efficient path planner needs to be implemented. Different from the simple path planner used in [30], a more efficient random path planner is developed in this thesis.
Additional deleting and switching operators in the planner are used to generate a new solution by flipping some bits of the Xs. The switching operator randomly switches two vertices in the feasible path, then check if the new generated path is a feasible path (which does not intersect with any edge). If the new path is a feasible path, accept the new path; otherwise, discard it. The algorithm randomly chooses one operator to generate the new path. Fig 29 shows that the deleting operator randomly deletes one vertex from the initial Xs to generate a new solution, while the switching operator randomly swaps two vertices in Xs. As the same selection criteria in generating initial path, each line section generated by the operators should be firstly tested against the edges in the map in order to generate a valid path line. This means that the line section created does not intersect with any edges in the map.
Fig 29: Deleting operation.
When the path length is the evaluation criterion, randomly deleting vertices could contribute more improving the performance of the solution. Therefore, the possibility of choosing the deleting operator is set to be higher than the possibility of selecting the switching operator. In our simulation program that will be discussed later, the possibility of choosing the deleting operator is set to be 0.78. After generating a new path solution by deleting or switching
52
operator, the new solution will be evaluated using the evaluation function. Ether accept the new solution if the new solution is better than the previous one, or accept the solution in a certain probability defined by the current temperature.
3.2.5 Online Path Planning
As stated in Chapter 1 (Introduction), while a robot uses the route generated by offline planning to travel through static obstacles, the online path planner is triggered automatically to calculate an alternative optimal path when a dynamic obstacle is detected. As no particular brand or configuration of the sensor is specified, the sensor range of the robot can be simulated by a fixed value. The distances between the robot and every vertices of the moving obstacle real-timely get updated. If the distance between the robot and any vertex of the moving obstacle is shorter than the fixed value, it can be defined that the moving obstacle enters the range of the sensor, and can be detected by the sensor of the robot. Then, the separate module as described in assumption (7) (Section 3.1) can monitor the trajectory and the shape of the moving obstacle. The moving information of the dynamic obstacles gathered by the sensor of the robot includes speed and moving direction. After the robot combines the information of the dynamic obstacle with the moving parameter of itself, the robot could infer the possibility of collision with the moving obstacles.
The simulated annealing optimization algorithm for finding optimal path is triggered when it is calculated that the robot will collide with the moving obstacle if no change of movement will be made in the future. The simulated annealing algorithm will be activated and reloaded with the updated search space. The search space for the algorithm will be updated. The current status of the robot, the current location information of the dynamic obstacle, the location information of the dynamic obstacles that could cause the collision and the location information of the static obstacles are all combined as a new search space for the robot. In a realistic case, the planning time should be short enough for the robot to implement motion changes to avoid collision. With the search space becomes larger, as a result, the time required of planning a path becomes longer. Therefore, an efficient algorithm is required that its planning time is relatively short for the robot to change the directions to avoid the
53
collision with the obstacles, which is one main aim of our Thesis.
3.3 Platforms for the Research Algorithm simulating software and system: Matlab 7 and Windows XP Operating System Hardware equipment: Pentium Core 2 Duo 1.6Ghz Process, 1G RAM
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Chapter 4 Simulation Results and Performance Evaluation This chapter presents the simulation results. With these results, the performance of the simulated annealing based approach is evaluated. Four environments are tested using the developed approach. The simulation results are compared with those presented in [21] to see weather the new approach is efficient for path planning in dynamic environments. The evaluation results clearly illustrate the first main contribution of the Thesis, that the simulated annealing based approach successfully generates collision free path for the robot in a dynamic environment. The approach gives improved performance over existing methods. The calculation time for obtaining the optimal path all-better than genetic algorithm based approach in [21].
4.1 Simulation Environments and Algorithm Parameters The approach is tested in four different environments. Each environment contains static and dynamic obstacles. The path is optimized for length. The solution derived from the algorithm
55
is optimal or near-optimal. The numbers of static and dynamic obstacles in the four testing environments are summarized in Table 1. The numbers of the vertices for offline planning are also given in Table 1. The dynamic obstacles in all four cases have random shapes. The first two environments simulate simple scenarios where the dynamic obstacles appear simultaneously and simply travel forward in the same direction. The last two environments are more complicated scenarios where the dynamic objects do not appear simultaneously but each appears at a random time and moves forward or backward. Also the numbers of the moving objects and static objects are larger than those of the simple environments. All the position information of the static obstacles in the map is known to the robot for offline planning before it starts to travel.
Table 1: Four testing environments Number of Static
Number of
Number of Static
Obstacles
Dynamic Obstacles
Vertices
1
3
2
10
2
6
2
25
3
9
4
56
4
14
6
90
Environment
The control parameters for the simulated annealing algorithm are set in a traditional way. According to [18], with bigger initial temperature and smaller cooling rate, it is much more likely to find the optimal solution, but it will require longer processing time for running. After many times of testing, we set the control parameters of the algorithm as shown in table 2:
Table 2:Control parameters for the simulated annealing algorithm. Initial Temperature
Terminate Temperature
Cooling Rate
Deleting Operator Rate
Switching Operator Rate
999999999
555555555
0.97
78%
22%
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4.2 Simulation Results 4.2.1 Case One (Three static obstacles with two dynamic obstacles)
A simple environment that contains three static and two dynamic obstacles is tested firstly. The total number of vertices of the static environment is ten. Fig 30 shows that the robot firstly uses offline path planning to obtain the optimal path based on static obstacles. Fig 31 shows that if moving obstacle is detected, an alternative path is generated by the online planner.
Fig 30: Offline Path Planning
Fig 31: Online Path Planning
The black full filled blocks in the map above are the static obstacles in the environment, the hollow triangles are the trajectories of the dynamic obstacles, the sequence of the points are the travel trajectory of the robot from start to end point. The above figures illustrate the optimal path has been found and the robot can perform online path planning to find alternative path for the robot in online path planning mode. Table 3 concludes the calculation time of online and offline planning and the optimal result for environment one. The simulation is run for ten times, the results in the table are the median results of the simulation. Optimised path length and offline and online calculation time are recorded since the results will be compared with previous results for discussion.
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Table 3: Results for environment one Search Space (Vertices Numbers) Number of Dynamic Obstacles Number of
10
2
3
Static Obstacles Optimised Path Length (Base on Static Obstacles) Calculation Time for Offline Planning (Seconds) Calculation Time for Online Planning (Seconds)
145.78
0.453
0.573
4.2.2 Case Two (Six static obstacles with two dynamic obstacles)
In case two, a more complex case is presented; the environment contains six static obstacles with two dynamic obstacles. The same strategy as case one, Fig 32 shows that the robot firstly uses offline path planning to obtain the optimal path based on static obstacles. Fig 33 shows that if a moving obstacle is detected, an alternative path is generated by the online planner.
Fig 32: Offline Planning
Fig 33: Online Planning 1 58
Fig 34: Online Planning 2
The above figures illustrate the optimal path has been found and the robot can perform online path planning to find alternative path for the robot in online path planning mode. Fig 34 shows that the robot will use the current route to travel if it is calculated that no collision will happen between robot and dynamic obstacle. Results in Table 4 show that due to the increasing search space, the approach needs more time to calculate the optimal path for the robot. Table 4: Results for environment two Search Space (Vertices Numbers) Number of Dynamic Obstacles Number of Static Obstacles Optimised Path Length (Base on Static Obstacles) Calculation Time for Offline Planning (Seconds) Calculation Time for Online Planning (Seconds)
59
25
2
6
257.25
1.201
1.501
4.2.3 Case Three (Nine static obstacles with four dynamic obstacles)
In case three, a more complicated environment was involved. Additional four static and two dynamic obstacles were added in the map. Other than the last two cases above, the dynamic obstacles in the map do not appear simultaneously and the trajectory of one dynamic obstacle is not the strait line. The dynamic obstacle can change moving direction during its movement. Fig 36 shows that the dynamic obstacles do not appear simultaneously in the map. Fig 37 shows that one dynamic obstacle change its trajectory when it is moving.
Fig 35: Offline Planning
Fig 36: Obstacles appear on different times.
Fig 37: Online Planning
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Fig 37 shows the robot uses the online path planner twice to adjust its route safely travelling to the final destination. Because the map is more complex, the results obtained by the simulated annealing based approach are not the same each time. However, all the results obtained by the approach are the optimal or near-optimal results. The following table shows the median result of ten runs.
Table 5: Results for environment three Search Space (Vertices Numbers) Number of Dynamic Obstacles Number of Static Obstacles Optimised Path Length (Base on Static Obstacles) Calculation Time for Offline Planning (Seconds) Calculation Time for Online Planning (Seconds)
56
4
9
288.01
3. 418
4. 784
4.2.4 Case Four (Fourteen static obstacles with six dynamic obstacles)
Case four is the most complicated case in all four cases. Fourteen static obstacles and six dynamic obstacles are included in the map. Dynamic obstacles appear randomly on different times and move in different directions. Also, the dynamic obstacles could change their moving directions during the movement.
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Fig 38:Offline Planning
Fig 39: Obstacles appear on different times
Fig 40: Online Planning 1
Fig 41: Online Planning 2
As in case three, Fig 39 shows that the dynamic obstacles do not appear simultaneously in the map. Fig 40 and Fig 41 shows that two dynamic obstacles change their trajectory and the robot uses the online path planner twice to adjust its rout safely travelling to the final destination. Table 6 below concludes the obtained path length and calculation time for case four.
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Table 6: Results of environment four Search Space (Vertices Numbers) Number of Dynamic Obstacles Number of Static Obstacles Optimised Path Length (Base on Static Obstacles) Calculation Time for Offline Planning (Seconds) Calculation Time for Online Planning (Seconds)
90
6
14
434.2397
10.98
13.57
Fig 42 below illustrates the convergence of the algorithm. It is seen from Fig 42 that the result converges rapidly in each round of temperature reduction. The algorithm could jump out of the local minimum and approaches to the global minimum. Table 7 concludes the calculation time of online path planning in four environments.
Fig 42: Convergence of the simulated annealing approach.
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Table 7: Processing time for online planning for each case.
Environment
1 (10 vertices)
2 (25 vertices)
3 (56 vertices)
4 (90 vertices)
Processing Time (Seconds)
0.57
1.201
4.784
13.57
4.3 Performance Evaluation The efficiency of the simulated annealing based approach is discussed in this section. The simulated annealing based approach is compared with the previous results obtained based on genetic algorithm in [21]. The new genetic based approach in [21] will be introduced firstly in this section. The genetic based approach is re-implemented in the same hardware on which simulated annealing based approach is implemented. The comparison results are also included in this section.
4.3.1 The Genetic Based Approach
Wang, Mulvaney and Sillitoe [21] proposed a genetic based path planning in 2006. The approach also uses the vertices of the obstacles as search space. Similar to the simulated annealing based approach, obstacles are described as polygons. The proposed genetic vertex planning method is compared with the Evolutionary Navigator/Planner [66], and it is claimed that the performance is better than the Evolutionary Navigator/Planner results. The rest part of this section introduces the genetic approach; most of the details are from [21]. In genetic approach, each gene represents a single obstacle vertex selected as an intermediate node. The solutions are described as chromosomes in the genetic approach. The chromosome in the genetic approach contains a total number of genes N, whose minimum value is two (a path containing only the start and goal nodes) and whose maximum value is L + 2, where L is the total number of vertices in the static map. The gene in the chromosome has two parts, one is the vertices reference, and the other is the feasibility bit. The feasibility bit for each gene
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indicates whether the path segment originating from the vertex referenced by the node is feasible. If the path segment connecting two consecutive vertices intersects any one or more obstacles, then the feasibility bit of the first node of the path segment is assigned to 1 to mark this segment as infeasible. If there is no intersection, 0 is assigned to indicate the segment is feasible. Each population contains 30 of chromosomes. The following figure [21] shows the structure of a chromosome:
Fig 43: Representation of chromosome in GA approach [21].
The approach use two evaluation functions, one is Ef which is simply the lengths of the generated paths. Ef is given by:
where d(Vi, Vi+1) demotes the distance between the referenced pair of vertices. The other evaluation function Ei indicates how deeply an infeasible path segment intersects with an obstacle and is given by:
Where
denotes the number of the obstacle intersections along the entire path and
is the mean number of intersections per infeasible segment. When the population contains both feasible and infeasible paths, the infeasible paths are all assumed to be worse than the worst feasible path.
Three genetic operators are used in the genetic approach, crossover, mutation and repair. The crossover operator randomly selects two crossover points; the parts after the crossover points of the two parent individuals are swapped. The mutation operator uses another vertex to
65
replace the selected vertex. The repair operator adjusts a randomly selected infeasible segment of an infeasible path. Fig 44 illustrates the repair operation:
Fig 44: The Repair Operator [21]
The evaluation process is as the standard GA approach; only one genetic operator is selected to produce the offspring for the approach. Each generation contains 30 chromosomes and evolution was terminated when there was no further improvement in the fitness of the best individual over 300 generations. The following are the copy of the pseudo-code from [21]:
Procedure vertex planning algorithm begin t=0 enlarge the obstacles encode the vertices of the obstacles initialise P(t) decode P(t) evaluate P(t) while (not terminating condition) do
66
t = t+1 select operator Oi with probability Pj select parent (s) from P(t) apply the operator Oj to produce offspring decode offspring evaluate new offspring replace worst member in P(t) by offspring end select the best individual p from P(t) end
4.3.2 Results Comparison
Three environments are used to make comparison of the approaches. One environment is selected from the cases in [21], and other two cases are the case 3 and case 4 in the simulated annealing approach simulation in last chapter. The search space for the two approaches is the number of the vertices in the map. First environment has 5 static obstacles which has total number of 22 vertices in the map. Second environment has 9 static obstacles which has 56 vertices in the map. The last environment has 14 obstacles which are constituted by 90 vertices. The termination conditions of the two approaches are: (1) Termination condition of the simulated annealing approach is the parameter termination temperature terminT which is predefined at the start of the approach. (2) As in [21], the genetic based approach terminates when noticing that there is no improvement on path length after 300 generations.
Every approach was run 10 times on each environment. The results in the Table 9 and Table 10 are the medians obtained over 10 runs on each environment. Table 9 and Table 10 present the comparison results of the execution time to get the final path and the final path length from the two approaches:
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Table 8: Three environments for performance evaluation. Number of Static
Number of Total
Obstacles
Vertices
1
5
22
2
9
56
3
14
90
Environment
Table 9: Comparison of execution time of getting the final path (Seconds). Simulated Annealing
Genetic Algorithm Based
Algorithm Based Approach
Approach
1
1.01
6.90
2
3.69
21.78
3
12.01
29.78
Environment
Table 10: Comparison of the length of the final path. Simulated Annealing
Genetic Algorithm Based
Algorithm Based Approach
Approach
1
258.477
259.455
2
291.690
351.238
3
429.022
469.580
Environment
Fig 45 compares the offline execution time of two approaches base on same vertices numbers. It shows that the simulated annealing algorithm based approach consumes a shorter time in path planning than the genetic algorithm based approach.
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Fig 45:Comparism of execution time based on same vertices number.
Fig 46 compares the final optimal path lengths obtained by two approaches base on same vertices numbers. It shows the solutions are improved in the simulated annealing algorithm based approach.
Fig 46: Comparison of final path length based on same vertices number.
For comparing the convergence and the optimisation efficiency of the two approaches, from each case, selecting the individual runs that produce similar path length results. And comparing the convergence and efficiency base on the time elapses in two approaches.
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Environment 1 (22 Vertices): Final Path Length: 259.27 GA Processing Time: 6.969 Seconds (Dashed Line) SA Processing Time: 1.107 Seconds (Solid Line)
Fig 47: Convergence comparison in environment one.
Environment 2 (56 Vertices): Final Path Length: 365.18 GA Processing Time: 13.422 Seconds (Dashed Line) SA Processing Time: 3.437 Seconds (Solid Line)
Fig 48: Convergence comparison in environment two.
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Environment 3 (90 Vertices): Final Path Length: 516.3911 GA Processing Time: 39.015 Seconds (Dashed Line) SA Processing Time: 10.001 Seconds (Solid Line)
Fig 49: Convergence comparison in environment three.
In paper [21], insufficient information is provided about the implementation of the genetic operators in the approach. For example, in [21], how to implement the repair operator in the approach is not clearly specified. Therefore, the differences of implementing the genetic operators between this paper and [21] may slightly affect the simulation results on our hardware. Thus, the simulation results of genetic based approach in our hardware may be slightly different than the simulation results in [21]. However, as Fig 50 shows: environment one with 25 vertices is established to build a mimic environment one in [21], and the simulation result in environment one could produce similar results as in [21]. Therefore the genetic approach implemented on our hardware could illustrate a comparative performance to [21]. Therefore, the comparison could show that the performance of the simulated annealing based approach is comparable or even more efficient than the genetic based approach in [21]. The simulated annealing based approach is able to use shorter processing time to determine the optimal path in the environment.
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200 180
Start Point
160 140 120 100 80 60 40
End Point
20 0
0
20
40
60
80
100
120
140
160
180
200
Fig 50: Environment One
Fig 51: Environment One From [21]
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Chapter 5 Heuristic Search Method for the Simulated Annealing Approach This chapter proposes a heuristic search method to further improve the simulated annealing approach for robot in dynamic environment. The simulation results show that the heuristic method can greatly improve the efficiency of simulated annealing approach in offline and online planning for robots in dynamic environments. The simulation results in this chapter can totally illustrate the second main contribution of this Thesis.
5.1 The Structure and Implementation of the Heuristic Selecting Method From Chapter 3 (Implementation of the Simulated Annealing Algorithm Approach), it is known that in finding the optimal path, the simulated annealing approach needs firstly selecting a random feasible path, then apply a mathematic operator to generate a new path base on the selected feasible path. The initial selected path is the beginning of the search point; the initial solution is replaced by the randomly generated neighbour solution. The search goes along “down hill” movement from the initial solution. Then, as temperature goes
73
down, the algorithm accepts the “up hill” movement to avoid being stuck at local minimus. Therefore, a better initial solution means a better initial searching point, which could enhance the efficiency and the performance of the approach.
From Chapter 3 (Implementation of the Simulated Annealing Algorithm Approach), the way of generating the initial path was illustrated as follows: •
Firstly, except dynamic objects, starting point and end point, a vertex is chosen randomly from the map, use strait-line to connect the start point and the selected vertex. o
If the path line intersects with any edges of the obstacle in the map, the path is recognized as invalid path. Another vertex will be randomly selected again for testing.
o
If the strait-line to the selected vertex does not intersect with any edges in the map, the strait line is recognized as part of a valid path. Then add the vertex into the path.
•
Then, start from the selected vertex to find the next valid vertex. Keep performing the above procedure until the end point is selected and also the path line to the end point is a valid path (i.e., the path to the end point does not intersects with any edges).
The method just randomly picks a feasible vertex until the end point is picked. The blindly picking is not an efficient way of producing the initial path.
To improve the efficiency of the initial solution selecting process, a heuristic process is proposed in this Chapter. The process makes a modification on the current method. In current method, a random vertex is selected after one feasible vertex is select, and then tests the feasibility of the line segment between randomly selected vertex and the feasible vertex.
In the proposed heuristic method, after one feasible vertex is select, an end point feasibility test will be carried out before selecting another vertex. Before selecting next vertex, the method will firstly test whether the line segment between the feasible vertex and the end point is feasible (does not interest with any obstacles). If the segment between the feasible 74
vertex and the end point is feasible, then put the end point as the next vertex and complete the selection process. The following pseudo-codes and Fig 52 illustrates the process of the heuristic initial path selection method.
initial path IP = {start}; while (1) if path{IPend, Destination Point} is a feasible path IP = {IP, Destination Point}; break; endif; randomly select Vs; if path{IPend, Vs} intersects any obstacles; continue; else IP = {IP,Vs}; endif if Vs == destination point; break; else continue; endif endwhile
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Fig 52: Process of the heuristic selection method.
Fig 53 and Fig 54 show the difference between the random picking method in normal simulated annealing algorithm and the proposed heuristic method. Fig 53 shows it is possible that it requires five times of random picking to find a feasible path to the end point. And Fig 54 shows that if testing the end point feasibility before random selection, a better feasible path may be obtained.
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Fig 53: Random picking method.
Fig 54: Heuristic picking method.
The heuristic method could improve the approach in two aspects: •
One is that the method could enhance the quality of the solution, i.e., a shorter path length solution, obtained by the approach. Fig 54 above shows that the heuristic method may generate a better initial path than the randomly picking method. The better initial path means the better searching beginning, which may lead a better solution.
•
The other improved aspect is the processing time of getting the solution. After each feasible vertex is selected, the heuristic method firstly checks the feasibility of the line segment to the end point rather than randomly select next vertex. Therefore, the processing time of selecting vertices could be reduced by the heuristic method.
In fact, the heuristic method could be used not only in the simulated annealing approach, but also in any other mathematic methods which generate initial solution for calculation.
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5.2 Performance Evaluation of the Heuristic Method To test the performance of the heuristic method, the random picking process is replaced by the heuristic method in the simulated annealing approach, and the modified simulated annealing approach is tested by the same four environments as those in Chapter 4 (Simulation Results and Performance Evaluation). The simulation results of the modified simulated annealing based approach are compared with those obtained in Chapter 4. The following are the compared results of the two methods:
5.2.1 Environment One (Three static obstacles with two dynamic obstacles):
Fig 55: Random picking method.
Fig 56: Heuristic picking method.
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Table 11: Results of comparison in case one (3 static and 2 dynamic obstacles). Random Environment One
Picking Method
Optimal Path Length
Heuristic Method
Improvement
145.78
145.78
0%
0.453
0. 447
1.34%
0.573
0. 335
41.54%
Calculation Time (Offline Planning) (Seconds) Calculation Time (Online Planning) (Seconds)
5.2.2 Environment Two (Five static obstacles with two dynamic obstacles):
Fig 57: Random picking method.
Fig 58: Heuristic picking method.
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Table 12: Results of comparison in case two (5 static and 2 dynamic obstacles). Random Environment Two
Picking Method
Optimal Path Length
Heuristic Method
Improvement
257.25
246.28
4.26%
2.453
0.793
67.67%
2.573
0.822
68.15%
Calculation Time (Offline Planning) (Seconds) Calculation Time (Online Planning) (Seconds)
5.2.3 Environment Three (Nine static obstacles with four dynamic obstacles):
Fig 59: Random picking method.
Fig 60: Heuristic picking method.
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Table 13: Results of comparison in case three (9 static and 4 dynamic obstacles). Random Environment Three
Picking Method
Optimal Path Length
Heuristic Method
Improvement
288.01
277.60
3.62%
3.418
1.411
58.72%
4.784
1.562
67.35%
Calculation Time (Offline Planning) (Seconds) Calculation Time (Online Planning) (Seconds)
5.2.4 Environment Four (Fourteen static obstacles with six dynamic obstacles):
Fig 61: Random picking method.
Fig 62: Heuristic picking method
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Table 14: Results of comparison in case four (14 static and 6 dynamic obstacles). Random Environment Four
Picking Method
Optimal Path Length
Heuristic Method
Improvement
434.24
416.36
4.12%
10.98
2.796
74.54%
13.57
2.96
78.19%
Calculation Time (Offline Planning) (Seconds) Calculation Time (Online Planning) (Seconds)
5.3 Discussions of the Evaluation Results From the comparison results presented above in Section 5.2, it can be seen that the heuristic search method can greatly enhance the performance of the simulated annealing based approach in dynamic environments. The heuristic method can accelerate the calculation in online and offline planning modes and can reduce the optimal path length obtained, especially in complex environments. The evaluation results clearly show the second main contribution of the Thesis, the heuristic method is successfully incorporated in the simulated annealing algorithm based approach and can boost the efficiency of the approach.
5.3.1 Offline Planning
In offline planning calculation, •
In the simple environment with 3 static obstacles, the heuristic method improves the offline processing time by 1.34%.
•
In the environment with 5 static obstacles, the heuristic method improves the offline
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processing time by 67.67%. •
In the environment with 9 static obstacles, the heuristic method improves the offline processing time by 58.72%.
•
The heuristic method gives nearly 74.54% improvement in offline processing time in finding the optimal path in the environment with 14 static obstacles.
5.3.2 Online Planning
In online planning mode, the heuristic method improves the online processing time by •
41.54% in environment 1;
•
68.15% in environment 2;
•
67.35% in environment 3 with 9 static obstacles; and
•
78.19% in environment 4 with 14 static obstacles.
5.3.3 The Length of the Path
The heuristic method improves the obtained optimal path by 4.26% in environment 2, and it improves the optimal path by 3.62% and 4.12% in environment 3 and environment 4 respectively.
Fig 63 compares the optimised path length solution on each case. Fig 64 and Fig 65 compare the offline and online processing time of the two methods on each environment. Table 15 summarizes the new heuristic method improvements on the processing time and the optimal path length base on four environments. It can be seen that the heuristic method is far superior to the random picking method and should be used by the simulated annealing algorithm based approach in complex environment especially in online path planning calculations.
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Fig 63: Path solution comparison.
Fig 64:Offline processing time comparison.
Fig 65: Online processing time comparison.
Table 15: Conclusion of the improvements. Improvements
Improvements
Improvements
(Offline Planning)
(Online Planning)
(Optimal Path)
1 (10 Vertices)
1.34%
41.54%
0%
2 (25 Vertices)
67.67%
68.15%
4.26%
3 (56 Vertices)
58.73%
67.15%
3.62%
4 (90 Vertices)
74.54%
78.47%
4.12%
Environment
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5.3.4 Comparison with the Genetic Algorithm Based Approach
Finally, the performance of the heuristic simulated annealing based approach is compared with the performance of the normal simulated annealing approach and the genetic algorithm based approach. The heuristic based simulated annealing approach is implemented on the three test environments in Section 4.3.2 to compare the offline processing time of three approaches. Compared with the genetic algorithm based approach; the heuristic method improves the offline processing time by 88.69% in environment 1 with 5 static obstacles. The heuristic method improves the offline processing time by 93.07% and 90.03% in environment 2 with 9 obstacles and environment 3 with 14 obstacles respectively. Fig 66 concludes the calculation time performance of the heuristic method based simulated annealing approach, the basic simulated annealing approach, and the genetic algorithm based method.
Compared with the genetic algorithm based approach; the heuristic method improves the obtained optimal path by 0.38% in environment 1 with 22 vertices, and it improves the optimal path by 20.96% and 11.33% in environment 2 and environment 3 respectively. Fig 67 concludes the performance of obtaining the optimal path in the three methods. Fig 66 and Fig 67 clearly show that the heuristic method based simulated annealing approach gives the best performance. Therefore, it can be concluded that the new heuristic method for simulated annealing algorithm gives superiority to other methods in path planning on vertices based complicate environment, and it should be used in simulated annealing based approach to deal with the path planning issue in dynamic environment.
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Fig 66: The comparison of the three methods in processing time.
Fig 67: The comparison of the three methods in final path length.
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Chapter 6 Conclusion This chapter summarizes the Thesis, and discusses the research limitations of the project. Future research on the simulated annealing based approach will also be discussed.
6.1 Summary This thesis has developed and implemented a simulated annealing based approach in Chapter 3 to deal with path planning issue for robots in dynamic environment – this is one of the main contributions of the thesis. The approach uses the vertices of the static and dynamic obstacles as search space to obtain the optimal path for robots, the approach searches the initial feasible path for robot in dynamic environment which contains static and dynamic obstacles, then uses simulated annealing algorithm to obtain the optimal path for the robot in the dynamic map. Compared with the genetic based approach in [21], the simulation results in Chapter 4 show that the simulated annealing based approach provides a better performance in processing time, and is crucial for a robot to quickly response to avoid
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collision with the dynamic obstacles.
The thesis has also proposed a heuristic search method in Chapter 5 to search the initial path – this is another main contribution of the thesis. The heuristic method has been incorporated into the developed simulated annealing algorithm based approach to greatly enhance the performance of the path planning in both online and offline calculations. The experiment results show that the heuristic search method can improve not only the processing time of offline and online planning but also the quality (path length of the solution) of the solution. The percentage of the improvement in online processing time ranges from 41% to 78% in our simulated environments for dynamic path planning.
The novelty of the project is the implementation of a simulated annealing based approach incorporated with a heuristic selecting method for robot path planning in dynamic environments with moving obstacles. The approach is simpler and easier to implement than existing genetic algorithm based methods. As the method is able to quickly determine the optimal feasible path for robot in dynamic environments, it could be used in marine research robot such as [3] and [2]. Also, the heuristic selecting method can enhance the efficiency of the simulated annealing based approach. Furthermore, the heuristic selecting method can also be used in other optimisation methods which need to generate an initial feasible solution for calculation.
6.2 Research Limitations The project successfully implements the simulated annealing based approach incorporated with a heuristic selecting method to plan path for mobile robots in dynamic environments. However, there are still some research limitations in the project. The limitations of the project are described below:
1) The shape of the robot is ignored in the project though the dimensions of the obstacles are considered for calculation. This is a topic that has not been systematically investigated in the dynamic path planning for mobile robots. Saboori and Menhaj [67] proposed a robot 88
path planning method that is based on fuzzy model. The method implements Dijkstra algorithm to find an optimum path for in an area with static obstacles for the robot. The dimensions of the robot and the obstacles have been taken into consideration to have a more realistic and efficient technique. However, the research and the simulation experiments are constrained in static environments. Therefore, taking the dimension of the robot into consideration in dynamic environments is a future issue.
2) In the project, no particular brand or configuration of the sensor is specified. The sensor is assumed to be able to acquire the motion information such as the speed and direction of the moving obstacles once the obstacles are in the sensor range. For a more realistic modelling, how the sensor obtains the motion information of the dynamic obstacle should be modelled and discussed.
6.3 Future Work Though the simulated annealing based approach solves the issue of obtaining the optimal path in dynamic environments, some work is still expected to be done in the future:
1) For a more realistic modelling, a more complex and realistic environment shall be used and tested using the simulated annealing approach. The environments in the project are described in a two-dimensional surface and the trajectories of the dynamic obstacles are constituted by a series of line segments. For modelling a more realistic environment for robot, a three-dimensional modelling is expected and the trajectories of the dynamic obstacles can be curves instead of strait line segments. The three-dimensional environment still uses vertices as search space; the only difference is that the calculation should add Z coordinate.
2) As stated in Section 6.2 about research limitations, for a more realistic modelling, the method used by the sensor to obtain the motion information of the dynamic obstacle should be further investigated. More research should be done to discuss the detailed method on how the sensor acquires the motion information of dynamic obstacles (e.g. 89
using which sensor? What is the appropriate mathematical method?).
3) Implementing the simulated annealing approach on a more realistic robot simulator, such as the Player/Stage [65] application. Successfully implementing the simulated annealing approach into Player/Stage provides a more practical environment than Matlab which only uses mathematic model to demonstrate the ability of algorithm. The implementation of the new method on Player/Stage not only demonstrates the feasibility of method but also provides a more practical testing environment for the new method.
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