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IOSR Journal of Computer Engineering (IOSR-JCE)
e-ISSN: 2278-0661,p-ISSN: 2278-8727, Volume 17, Issue 6, Ver. I (Nov ā€“ Dec. 2015), PP 147-151
www.iosrjournals.org
DOI: 10.9790/0661-1761147151 www.iosrjournals.org 147 | Page
Planning the Path and Avoidance Obstacles for Visually
Impaired/Blind People
Mohammed N. Abdullah 1
, Ayad E. Korial1*
1 Department of computer engineering, University of Technology,Baghdad, Iraq
Abstract: This paper suggests a navigation structure for visually impaired/blind people using A star algorithm.
The proposed structure is consisted of two parts. In the first part planning the path and in the second part
avoiding obstacles. The aim is to guidevisually impaired/blind people from a required location to a desired
location. Two tests are run in each part and modeled in C sharp language and MATLABprogram. Simulation
results show good performance for the proposed scheme that the visually impaired/blind people reach the
desired location successfully without error. In conclusion, A star path finding algorithm using C sharp language
was a valid and reliable method to guide the visually impaired / blind people from a required location to a
desired location in indoor environment with or without obstacles.
Keywords: Planning the path, avoidance obstacles, visually impaired/blind people, A star algorithm, path
finding, and indoor navigation.
I. Introduction
In the difficult context of day to day life of visually impaired/blind persons, the white cane still
represents the most popular tool used for obstacle detection, provided that the encountered locations have been
visited before and memorized. In an unfamiliar setting, they completely depend on other humans to reach the
desired destination [1].Sentient computing is the proposition that applications can be made more responsive and
useful by observing and reacting to the physical world. It is particularly attractive in a world of mobile users
and ubiquitous computers [2].Blind and visually impaired people have different important daily needs including
navigation. Navigation and way finding is crucial for the blind population, as for any other. Visually impaired
people in new environments may feel totally disoriented or isolated. These people can easily end up in
dangerous and confusing situations as they move in unknown places. Pressing navigation-related problems blind
people face include determining oneā€™s position, heading or moving direction, and the detection of close objects
(most probably mobility barriers) [3].
Visually impaired people rely on different tools and skills to navigate. They usually rely on their white
canes, seeing-eye-dogs and other skills acquired over time to aid their navigation. Visually impaired people, that
are able to perceive sounds, tend to learn how to identify audible landmarks. In general, people supplement their
loss of sight by becoming better with other senses, even smell [4].
Cognitive mapping skills are flexible enough to adapt to the sensory loss. Even people who are blind
from birth can deal with spatial concepts and can find their way through different spaces [3].
Many technologies have been developed to assist blind users with different navigation needs. This
includes avoiding obstacles like laser canes and other handheld devices that transmit laser or ultrasound beams
to detect objects ahead of the user and give audio or vibration feedback that varies according to how close the
objects are [5, 6].In indoor navigation, we need two things, positioning system and suitable path finding
algorithm so research into indoor positioning systems has identified some possible technologies, but none of
these has been developed and distributed to consumers [7].
Guerrero et al., (2012) [8] demonstrated an indoor navigation system for the visually impaired. They
reported that a navigation in indoor environments was highly challenging for the severely visually impaired,
particularly in spaces visited for the first time. Several solutions have been proposed to deal with this challenge.
Kim and Chang (2013) [9] presented high-resolution touch floor system using particle swarm optimization
neural network. A touch floor system, based on force sensitive resistors, capable of identifying user position and
motion with high resolution, was proposed in this paper. Sun et al. (2014) [10] presented a new wearable
computer called eButton. The concepts of its design and electronic implementation were described. Zhang
et al. (2014) [11] described an indoor positioning algorithm for mobile objects based on track smoothing. A new
indoor positioning algorithm was proposed for mobile objects based on track smoothing. Abou Haidar and
Roger Achkar (2014)[12] proposed smart walker. Elderlies, blind or visually impaired people recur to a cane as
a traditional ambulation aid.
In this paper, planning the path part and avoiding obstacles part using A star algorithm is presented for
a visually impaired/blind people operating in two runs in each part using C sharp and MATLAB program.
Planning the Path and Avoidance Obstacles for Visually Impaired/Blind People
DOI: 10.9790/0661-1761147151 www.iosrjournals.org 148 | Page
II. Path Finding Algorithms
Path finding in the basic definition is moving an object from its initial position to the final location.
Path Finding Algorithms (PFA) are used in many different application areas such as: -Games and Virtual Tours,
Robot Motion and Navigation, Driverless Vehicles, Transportation Networks, and Human Navigation. Path
finding algorithms have many usage areas. These algorithms are useful in the field of roboticsbecause they can
be used to guide a robot around difficult terrain without human intervention [13].
There are many algorithms that are commonly known and used, ranging from simplex to
complex, in order to be able to solve the path finding problem. There also exist some algorithms that plan
the whole path before moving anywhere. Best-first algorithm expands nodes based on a heuristic estimate of the
cost to the goal. Nodes, which are estimated to give the best cost, are expanded first. The most commonly used
algorithm is A* algorithm, which is a combination of the Dijkstra algorithm and the best-first algorithm [14].
A* is one of the many search algorithms that takes an input, evaluates a number of possible paths and
returns a solution. In computer science, A* (as ā€•A starā€–) is a computer algorithm that is widely used in
pathfinding and graph traversal, the process of plotting an efficiently traversable path between points, called
nodes [15].
III. System Design Navigation
Provided that the visually impaired/blind people (users) has traversed the path, our navigational system
should determine a userā€™s location, ļ¬nd the best path to a destination, and give meaningful instructions along
that path.While attempting to navigate an already constructed map, the system must track its belief about the
visually impaired/blind people current location on that map. Figure 1 shows in indoor navigation system.
Figure 1: In indoor navigation system.
IV. Overall System Navigation
In the case of a visually impaired/blind people, the navigation intelligence and local path planning is provided
by the person. Hence, our proposal focuses on the planning the path and the obstacles detection system to assist
visually impaired/blind people as shown in the Figure 2 below.
Figure 2: The structure of planning the path of visually impaired/blind people
In the same concept our proposed system of visually impaired/blind people when the environment or indoor
navigation contains several obstacles, our structure is shown in Figure 3 below.
Figure 3: The structure of avoiding the obstacles of visually impaired/blind people
1.1 Planned path results and discussion
Increasing sources of sensor measurements and prior knowledge have become available for indoor
localization on smartphones. How to effectively utilize these sources for enhancing localization accuracy is an
important yet challenging problem. In this section, we present planning the path of visually impaired/blind
Planning the Path and Avoidance Obstacles for Visually Impaired/Blind People
DOI: 10.9790/0661-1761147151 www.iosrjournals.org 149 | Page
people localization algorithm that exploits various sources of information. Speciļ¬cally, we also introduce the
concept of A star algorithm to indicate the area where the visually impaired/blind people is in an indoor map.
In run 1 where the visually impaired/blind people is moved from start position (2,3) to goal position (20,10), and
in run 2 where the visually impaired/blind people is moved from start position (3,13) to goal position (22,1) the
results are shown in Figures 4 a, b and Figures 5 a, b respectively.
Figure 4 a: Indication of start and goal position of run 1
Figure 4 b: Visually impaired/blind people path planning using A star algorithm of run 1
Figure 5 a: Indication of start and goal position of run 2
5 b: Visually impaired/blind people path planning using A star algorithm of run 2
Planning the Path and Avoidance Obstacles for Visually Impaired/Blind People
DOI: 10.9790/0661-1761147151 www.iosrjournals.org 150 | Page
The evaluation of path points are done in distance unit using Matlab as shown in Figure 6, 7 for run 1 and run 2
respectively.
Figure 6: The evaluation of path points of run 1 Figure 7: The evaluation of path points of run 2
1.2 Avoiding Obstacles Results and Discussion
In this section, we present avoiding obstacles of visually impaired/blind people localization algorithm
that exploits various sources of information. Speciļ¬cally, we also introduce the concept of A star algorithm to
indicate the area where the visually impaired/blind people is in an indoor map.
In run 1 where the visually impaired/blind people is moved from start position (1,7) to goal position (20,3), and
in run 2 where the visually impaired/blind people is moved from start position (6,4) to goal position (19,12) the
results are shown in Figures 8 a, b below Figure 9 a, b respectively.
Figure 8 a: Indication of start and goal position of run 1
Figure 8 b: Visually impaired/blind people obstacles avoidance using A star algorithm of run 1
Figure 9 a: Indication of start and goal position of run 2
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
X
Y
0 5 10 15 20 25
0
2
4
6
8
10
12
14
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Planning the Path and Avoidance Obstacles for Visually Impaired/Blind People
DOI: 10.9790/0661-1761147151 www.iosrjournals.org 151 | Page
Figure 9 b: Visually impaired/blind people obstacles avoidance using A star algorithm of run 2
The evaluation of path points are done in distance unit using Matlab as shown in Figure 10, 11 for run
1 and run 2 respectively.
V. Conclusion
A star path finding algorithm using C sharp language guided the visually impaired / blind people from
a required location to a desired location in indoor environment with or without obstacles. This method was valid
and reliable.
References
[1]. Shah C., Bouzit M., Youssef M., and Vasquez L., ā€•Evaluation of RU-Netra Tactile Feedback Navigation System for the Visually
Impairedā€–, International Workshop on Virtual Rehabilitation, pp. 72-77, 2006.
[2]. Hopper A. ā€•Sentient Computing?ā€–, Phil. Trans. R. Soc. Lond., A, (2000), Vol. 358, pp. 2349-2358, August 2000.
[3]. Hub A., Diepstraten J., Ertl T., ā€•Design and development of an indoor navigation and object identification system for the blindā€–.
Assets '04 Proceedings of the 6th international ACM SIGACCESS conference on Computers and accessibility, pp. 147 - 152 ,
October 18 ā€“ 20, 2004.
[4]. Gardner J., ā€•Seeing with Other Senses: Gardens for the Blindā€–. The Herb Companion, February 3, 2007.
http://www.perkins.org/news-events/press/seeing-with-other- senses.html, last visit November, 2015.
[5]. Lakde C. K., Prasad P. S. ā€•Review Paper on Navigation System for Visually Impaired Peopleā€–, IJARCCE, Vol. 4, Issue 1, 2015
[6]. Agarwal, N, Iyer, A. , Naidu, S. , Rodrigues, S. ā€•Electronic guidance system for the visually impaired ā€” A frameworkā€–,
Technologies for Sustainable Development (ICTSD), 2015 International Conference on Mumbai, IEEE, PP 1-5, 2015 DOI
10.1109/ICTSD.2015.7095872
[7]. Changdon Kee et al., ā€•Centimeter-Accuracy Indoor Navigationā€–, GPS World, November 2001. ā€Ø
[8]. Guerrero, L.A.; Vasquez, F.; Ochoa, S.F., ā€•An Indoor Navigation System for the Visually Impairedā€–, .Sensors, Vol.12, pp. 8236-
825, 2012.
[9]. Kim H.,Chang S., ā€•High-Resolution Touch Floor System Using Particle Swarm Optimization Neural Networkā€–,Sensors Journal,
IEEE, Vol. 13, pp. 2084-- 2093, June 2013.
[10]. Sun M., Bruke L., Mao et al., ā€•eButton: A Wearable Computer for Health Monitoring and Personal Assistanceā€–, Proceeding
DAC '14 Proceedings of the 51st Annual Design Automation Conference, , pp. 1-6, 2014.
[11]. Zhang Y. ; Zhang R.; Miao W. ; Xia W., Jia Z. ; Shen L., ā€•An Indoor Positioning Algorithm for Mobile Objects Based on Track
Smoothingā€–,Advanced Infocomm Technology (ICAIT), 2014 IEEE 7th International Conference , pp. 252, 14-16Nov. 2014.
[12]. Abou Haidar G., Achkar R., Maalouf R., et al. ā€•SMART WALKERā€–,2014 International Conference on Future Internet of Things
and Cloud, pp. 415, 2014.
[13]. Carsten J., ā€•Global Path Planning on board the Mars Exploration Roversā€–, IEEE Aerospace Conference, 2007.
[14]. Mustafa KilinƧarslan, ā€•Implementation Of A Path Finding Algorithm For the Navigation of visually Impaired Peopleā€–, Master
thesis. CH 2, PP26, 2007
[15]. Reddy H, ā€•Path Finding ā€“Dijkstraā€™s and A* Algorithmā€™sā€–, International Journal in IT and Engineering, pp 1-15, 2013.
Figure 10: The evaluation of path points of run 1
Figure 11: The evaluation of path points of run 2
0 5 10 15 20
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U01761147151

  • 1. IOSR Journal of Computer Engineering (IOSR-JCE) e-ISSN: 2278-0661,p-ISSN: 2278-8727, Volume 17, Issue 6, Ver. I (Nov ā€“ Dec. 2015), PP 147-151 www.iosrjournals.org DOI: 10.9790/0661-1761147151 www.iosrjournals.org 147 | Page Planning the Path and Avoidance Obstacles for Visually Impaired/Blind People Mohammed N. Abdullah 1 , Ayad E. Korial1* 1 Department of computer engineering, University of Technology,Baghdad, Iraq Abstract: This paper suggests a navigation structure for visually impaired/blind people using A star algorithm. The proposed structure is consisted of two parts. In the first part planning the path and in the second part avoiding obstacles. The aim is to guidevisually impaired/blind people from a required location to a desired location. Two tests are run in each part and modeled in C sharp language and MATLABprogram. Simulation results show good performance for the proposed scheme that the visually impaired/blind people reach the desired location successfully without error. In conclusion, A star path finding algorithm using C sharp language was a valid and reliable method to guide the visually impaired / blind people from a required location to a desired location in indoor environment with or without obstacles. Keywords: Planning the path, avoidance obstacles, visually impaired/blind people, A star algorithm, path finding, and indoor navigation. I. Introduction In the difficult context of day to day life of visually impaired/blind persons, the white cane still represents the most popular tool used for obstacle detection, provided that the encountered locations have been visited before and memorized. In an unfamiliar setting, they completely depend on other humans to reach the desired destination [1].Sentient computing is the proposition that applications can be made more responsive and useful by observing and reacting to the physical world. It is particularly attractive in a world of mobile users and ubiquitous computers [2].Blind and visually impaired people have different important daily needs including navigation. Navigation and way finding is crucial for the blind population, as for any other. Visually impaired people in new environments may feel totally disoriented or isolated. These people can easily end up in dangerous and confusing situations as they move in unknown places. Pressing navigation-related problems blind people face include determining oneā€™s position, heading or moving direction, and the detection of close objects (most probably mobility barriers) [3]. Visually impaired people rely on different tools and skills to navigate. They usually rely on their white canes, seeing-eye-dogs and other skills acquired over time to aid their navigation. Visually impaired people, that are able to perceive sounds, tend to learn how to identify audible landmarks. In general, people supplement their loss of sight by becoming better with other senses, even smell [4]. Cognitive mapping skills are flexible enough to adapt to the sensory loss. Even people who are blind from birth can deal with spatial concepts and can find their way through different spaces [3]. Many technologies have been developed to assist blind users with different navigation needs. This includes avoiding obstacles like laser canes and other handheld devices that transmit laser or ultrasound beams to detect objects ahead of the user and give audio or vibration feedback that varies according to how close the objects are [5, 6].In indoor navigation, we need two things, positioning system and suitable path finding algorithm so research into indoor positioning systems has identified some possible technologies, but none of these has been developed and distributed to consumers [7]. Guerrero et al., (2012) [8] demonstrated an indoor navigation system for the visually impaired. They reported that a navigation in indoor environments was highly challenging for the severely visually impaired, particularly in spaces visited for the first time. Several solutions have been proposed to deal with this challenge. Kim and Chang (2013) [9] presented high-resolution touch floor system using particle swarm optimization neural network. A touch floor system, based on force sensitive resistors, capable of identifying user position and motion with high resolution, was proposed in this paper. Sun et al. (2014) [10] presented a new wearable computer called eButton. The concepts of its design and electronic implementation were described. Zhang et al. (2014) [11] described an indoor positioning algorithm for mobile objects based on track smoothing. A new indoor positioning algorithm was proposed for mobile objects based on track smoothing. Abou Haidar and Roger Achkar (2014)[12] proposed smart walker. Elderlies, blind or visually impaired people recur to a cane as a traditional ambulation aid. In this paper, planning the path part and avoiding obstacles part using A star algorithm is presented for a visually impaired/blind people operating in two runs in each part using C sharp and MATLAB program.
  • 2. Planning the Path and Avoidance Obstacles for Visually Impaired/Blind People DOI: 10.9790/0661-1761147151 www.iosrjournals.org 148 | Page II. Path Finding Algorithms Path finding in the basic definition is moving an object from its initial position to the final location. Path Finding Algorithms (PFA) are used in many different application areas such as: -Games and Virtual Tours, Robot Motion and Navigation, Driverless Vehicles, Transportation Networks, and Human Navigation. Path finding algorithms have many usage areas. These algorithms are useful in the field of roboticsbecause they can be used to guide a robot around difficult terrain without human intervention [13]. There are many algorithms that are commonly known and used, ranging from simplex to complex, in order to be able to solve the path finding problem. There also exist some algorithms that plan the whole path before moving anywhere. Best-first algorithm expands nodes based on a heuristic estimate of the cost to the goal. Nodes, which are estimated to give the best cost, are expanded first. The most commonly used algorithm is A* algorithm, which is a combination of the Dijkstra algorithm and the best-first algorithm [14]. A* is one of the many search algorithms that takes an input, evaluates a number of possible paths and returns a solution. In computer science, A* (as ā€•A starā€–) is a computer algorithm that is widely used in pathfinding and graph traversal, the process of plotting an efficiently traversable path between points, called nodes [15]. III. System Design Navigation Provided that the visually impaired/blind people (users) has traversed the path, our navigational system should determine a userā€™s location, ļ¬nd the best path to a destination, and give meaningful instructions along that path.While attempting to navigate an already constructed map, the system must track its belief about the visually impaired/blind people current location on that map. Figure 1 shows in indoor navigation system. Figure 1: In indoor navigation system. IV. Overall System Navigation In the case of a visually impaired/blind people, the navigation intelligence and local path planning is provided by the person. Hence, our proposal focuses on the planning the path and the obstacles detection system to assist visually impaired/blind people as shown in the Figure 2 below. Figure 2: The structure of planning the path of visually impaired/blind people In the same concept our proposed system of visually impaired/blind people when the environment or indoor navigation contains several obstacles, our structure is shown in Figure 3 below. Figure 3: The structure of avoiding the obstacles of visually impaired/blind people 1.1 Planned path results and discussion Increasing sources of sensor measurements and prior knowledge have become available for indoor localization on smartphones. How to effectively utilize these sources for enhancing localization accuracy is an important yet challenging problem. In this section, we present planning the path of visually impaired/blind
  • 3. Planning the Path and Avoidance Obstacles for Visually Impaired/Blind People DOI: 10.9790/0661-1761147151 www.iosrjournals.org 149 | Page people localization algorithm that exploits various sources of information. Speciļ¬cally, we also introduce the concept of A star algorithm to indicate the area where the visually impaired/blind people is in an indoor map. In run 1 where the visually impaired/blind people is moved from start position (2,3) to goal position (20,10), and in run 2 where the visually impaired/blind people is moved from start position (3,13) to goal position (22,1) the results are shown in Figures 4 a, b and Figures 5 a, b respectively. Figure 4 a: Indication of start and goal position of run 1 Figure 4 b: Visually impaired/blind people path planning using A star algorithm of run 1 Figure 5 a: Indication of start and goal position of run 2 5 b: Visually impaired/blind people path planning using A star algorithm of run 2
  • 4. Planning the Path and Avoidance Obstacles for Visually Impaired/Blind People DOI: 10.9790/0661-1761147151 www.iosrjournals.org 150 | Page The evaluation of path points are done in distance unit using Matlab as shown in Figure 6, 7 for run 1 and run 2 respectively. Figure 6: The evaluation of path points of run 1 Figure 7: The evaluation of path points of run 2 1.2 Avoiding Obstacles Results and Discussion In this section, we present avoiding obstacles of visually impaired/blind people localization algorithm that exploits various sources of information. Speciļ¬cally, we also introduce the concept of A star algorithm to indicate the area where the visually impaired/blind people is in an indoor map. In run 1 where the visually impaired/blind people is moved from start position (1,7) to goal position (20,3), and in run 2 where the visually impaired/blind people is moved from start position (6,4) to goal position (19,12) the results are shown in Figures 8 a, b below Figure 9 a, b respectively. Figure 8 a: Indication of start and goal position of run 1 Figure 8 b: Visually impaired/blind people obstacles avoidance using A star algorithm of run 1 Figure 9 a: Indication of start and goal position of run 2 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 X Y 0 5 10 15 20 25 0 2 4 6 8 10 12 14 X Y
  • 5. Planning the Path and Avoidance Obstacles for Visually Impaired/Blind People DOI: 10.9790/0661-1761147151 www.iosrjournals.org 151 | Page Figure 9 b: Visually impaired/blind people obstacles avoidance using A star algorithm of run 2 The evaluation of path points are done in distance unit using Matlab as shown in Figure 10, 11 for run 1 and run 2 respectively. V. Conclusion A star path finding algorithm using C sharp language guided the visually impaired / blind people from a required location to a desired location in indoor environment with or without obstacles. This method was valid and reliable. References [1]. Shah C., Bouzit M., Youssef M., and Vasquez L., ā€•Evaluation of RU-Netra Tactile Feedback Navigation System for the Visually Impairedā€–, International Workshop on Virtual Rehabilitation, pp. 72-77, 2006. [2]. Hopper A. ā€•Sentient Computing?ā€–, Phil. Trans. R. Soc. Lond., A, (2000), Vol. 358, pp. 2349-2358, August 2000. [3]. Hub A., Diepstraten J., Ertl T., ā€•Design and development of an indoor navigation and object identification system for the blindā€–. Assets '04 Proceedings of the 6th international ACM SIGACCESS conference on Computers and accessibility, pp. 147 - 152 , October 18 ā€“ 20, 2004. [4]. Gardner J., ā€•Seeing with Other Senses: Gardens for the Blindā€–. The Herb Companion, February 3, 2007. http://www.perkins.org/news-events/press/seeing-with-other- senses.html, last visit November, 2015. [5]. Lakde C. K., Prasad P. S. ā€•Review Paper on Navigation System for Visually Impaired Peopleā€–, IJARCCE, Vol. 4, Issue 1, 2015 [6]. Agarwal, N, Iyer, A. , Naidu, S. , Rodrigues, S. ā€•Electronic guidance system for the visually impaired ā€” A frameworkā€–, Technologies for Sustainable Development (ICTSD), 2015 International Conference on Mumbai, IEEE, PP 1-5, 2015 DOI 10.1109/ICTSD.2015.7095872 [7]. Changdon Kee et al., ā€•Centimeter-Accuracy Indoor Navigationā€–, GPS World, November 2001. ā€Ø [8]. Guerrero, L.A.; Vasquez, F.; Ochoa, S.F., ā€•An Indoor Navigation System for the Visually Impairedā€–, .Sensors, Vol.12, pp. 8236- 825, 2012. [9]. Kim H.,Chang S., ā€•High-Resolution Touch Floor System Using Particle Swarm Optimization Neural Networkā€–,Sensors Journal, IEEE, Vol. 13, pp. 2084-- 2093, June 2013. [10]. Sun M., Bruke L., Mao et al., ā€•eButton: A Wearable Computer for Health Monitoring and Personal Assistanceā€–, Proceeding DAC '14 Proceedings of the 51st Annual Design Automation Conference, , pp. 1-6, 2014. [11]. Zhang Y. ; Zhang R.; Miao W. ; Xia W., Jia Z. ; Shen L., ā€•An Indoor Positioning Algorithm for Mobile Objects Based on Track Smoothingā€–,Advanced Infocomm Technology (ICAIT), 2014 IEEE 7th International Conference , pp. 252, 14-16Nov. 2014. [12]. Abou Haidar G., Achkar R., Maalouf R., et al. ā€•SMART WALKERā€–,2014 International Conference on Future Internet of Things and Cloud, pp. 415, 2014. [13]. Carsten J., ā€•Global Path Planning on board the Mars Exploration Roversā€–, IEEE Aerospace Conference, 2007. [14]. Mustafa KilinƧarslan, ā€•Implementation Of A Path Finding Algorithm For the Navigation of visually Impaired Peopleā€–, Master thesis. CH 2, PP26, 2007 [15]. Reddy H, ā€•Path Finding ā€“Dijkstraā€™s and A* Algorithmā€™sā€–, International Journal in IT and Engineering, pp 1-15, 2013. Figure 10: The evaluation of path points of run 1 Figure 11: The evaluation of path points of run 2 0 5 10 15 20 1 2 3 4 5 6 7 8 9 X Y 5 10 15 20 4 6 8 10 12 X Y