Optimal Path Planning using Equilateral Spaces Oriented Visibility Graph Method

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 7, No. 6, December 2017, pp. 3046~3051
ISSN: 2088-8708, DOI: 10.11591/ijece.v7i6.pp3046-3051  3046
Journal homepage: http://iaesjournal.com/online/index.php/IJECE
Optimal Path Planning using Equilateral Spaces Oriented
Visibility Graph Method
Nor Badariyah Abdul Latip, Rosli Omar, Sanjoy Kumar Debnath
Universiti Tun Hussein Onn Malaysia, Faculty of Electrical and Electronic Engineering, 86400 Parit Raja,
Batu Pahat, Johor, Malaysia
Article Info ABSTRACT
Article history:
Received Aug 4, 2017
Revised Oct 2, 2017
Accepted Oct 16, 2017
Path planning has been an important aspect in the development of
autonomous cars in which path planning is used to find a collision-free path
for the car to traverse from a starting point Sp to a target point Tp. The main
criteria for a good path planning algorithm include the capability of
producing the shortest path with a low computation time. Low computation
time makes the autonomous car able to re-plan a new collision-free path to
avoid accident. However, the main problem with most path planning
methods is their computation time increases as the number of obstacles in the
environment increases. In this paper, an algorithm based on visibility graph
(VG) is proposed. In the proposed algorithm, which is called Equilateral
Space Oriented Visibility Graph (ESOVG), the number of obstacles
considered for path planning is reduced by introducing a space in which the
obstacles lie. This means the obstacles located outside the space are ignored
for path planning. From simulation, the proposed algorithm has an
improvement rate of up to 90% when compared to VG. This makes the
algorithm is suitable to be applied in real-time and will greatly accelerate the
development of autonomous cars in the near future.
Keyword:
Optimal path planning
Path planning
Visibility graph
Copyright © 2017 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Rosli Omar,
Faculty of Electrical & Electronic Engineering,
Universiti Tun Hussein Onn Malaysia,
86400 Parit Raja, Batu Pahat, Johor, Malaysia.
Email: roslio@uthm.edu.my
1. INTRODUCTION
Path planning has been an emerging trend research nowadays to cater the needs of autonomous
systems. Without path planning, autonomous cars could not be materialized. Currently, there are many path
planning algorithms that have been developed based on methods such as Voronoi diagram (VD) [1], [2], cell
decomposition (CD) [3], [4], genetic algorithm (GA) [5], [6] and visibility graph (VG), to name a few. A
path planning algorithm’s performance is normally measured based on three criteria which include the
computation time, path optimality and completeness.
Basically, the main constraint that affects the computation time of path planning is the number of
obstacles contained in a configuration space (C-space). In C-space size of the autonomous car is reduced to a
point and the size of obstacle is enlarge based on the size of autonomous car [7]. The higher the number of
obstacles in C-space, the higher the computation time to find a collision-free path.
An optimal path can be defined as the shortest path generated by a path planning algorithm among
all the produced path commencing from the starting point Sp to the end point Tp [2], [8-11]. A path planning
algorithm holds the completeness criterion if it is able to produce a path if one exists.
There are several types of path planning such as combinatorial and bio-inspired. Each of them has
their benefits and drawbacks in satisfying the above-mentioned criteria.

IJECE ISSN: 2088-8708 
Optimal Path Planning using Equilateral Spaces Oriented Visibility …. (Nor Badariyah Abdul Latip)
3047
VG is a complete algorithm and capable of producing a path with the shortest distance but its
drawback is the computation time increases when the number of obstacles in C-spaces increases [9], [10],
[12]. VD, on the other hand, uses equidistant techniques to discover a path for which an optimal path cannot
be accomplished although it has relatively lower computation time and the algorithm is complete [2], [13].
CD divides the C-spaces into cells which are discrete and non-overlapping. A path is then produced
through a cell that is not occupied by the obstacle. The drawback of this method is that it cannot produce
optimal path albeit it has a lower computation time. CD is complete where it finds a path if one exists.
The bio-inspired type gets motivation from nature. An application of operators is used in the Genetic
algorithm (GA) to emulate natural selection process. The disadvantage of GA is that there are possibilities of
not finding the solution in narrow environments as the local minima conditions may occur. Conversely, GA
works in parallel and thus, uses less computation time. It is meta-heuristics and hence, does not guarantee the
shortest distance[5], [6], [14].
As VG is capable of producing the shortest path and complete, in this paper, an algorithm based on
VG is proposed and the simulation is performed using MATLAB to evaluate the performance of the
proposed algorithm.
2. PATH PLANNING ALGORITHM
The proposed algorithm, called Equilateral Space Oriented Visibility Graph (ESOVG), is depicted
in Figure 1. The idea of the algorithm is to reduce the number of obstacles used when planning a path, which
will in turn, lessen the computation time.
The algorithm starts with creating a base line, which connects the starting point Sp and target point
Tp. The opening angle  is then set to nominal value of 20º. After that, a mid-point between Sp and Tp is
identified. This is followed by creating a mid-line which is perpendicular to the base line and intersecting the
mid-point. Then, a pair of imaginary lines, which emerge from Sp towards the mid-line with an opening
angle of  are created. Similarly, another pair of imaginary lines emerging from Tp towards the mid-line are
drawn. Both pairs of lines should intersect the points denoted by C1 and C2. The equilateral space, shown in
darker colour, which is formed by four imaginary lines is illustrated in Figure 2.
Algorithm: ESOVG
1: Create a base line connecting starting point Sp and target point Tp
2: Set the nominal opening angle  to 20º
3: Identify a mid-point on the base line between Sp and Tp
4: Create a mid-line passing through the mid-point and perpendicular to the base line
5: From each Sp and Tp, create a pair of imaginary lines with an opening angle of ρ towards the mid line.
6: Create an equilateral space from the enclosed area by the four imaginary lines drawn in step 5
7: Identify the obstacles, O located in the equilateral space
8: Identify nodes list of the obstacle in step 7
9: Construct a cost matrix based on the nodes list
10: Find the shortest path from Sp to Tp using Dijkstra’s algorithm
11: If no path is found within the equilateral space, go to step 2 and increase  by 5º.
Figure 1. The proposed algorithm
Figure 2. The equilateral space

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In this algorithm, the size of the generated equilateral space depends on the opening angle . The
nominal angle is set at 20° so that the path planning can be performed with a low computation time as the
number of obstacles contained in the area is small. However, if a path is not found in the space, the angle is
then increased by 5º until one is found.
Figures 3(a)-Figure 3(d) show the simulation of ESOVG using  = 20º, 30º, 40º and 50°,
respectively. The figures show 100 obstacles present in the C-spaces but only a few obstacles (shown in
green) are being considered by ESOVG for path planning as they are fully or partially contained in the
equilateral space (enclosed by the solid lines). On the other hand, the obstacles shown in white are ignored as
they are totally outside the space. Notice that the smaller the opening angle , the lesser the number of
obstacles are taken into account. The computation time and path length for each  are shown in Table 1.
Figure 3. Simulation of path planning using ESOVG, (a) ρ=20°, (b) ρ=30°, (c) ρ=40°, (d) ρ=50°
Table 1. Computation time and path length using different values of 
 Computation time (s) Path length (unit)
20º 0.4389 645.675
30º 0.7145 645.675
40º 1.5478 645.675
50° 2.3856 645.675
The piecewise linear segments shown in dashed line are the resulting path of ESOVG. From
Table 1, it is observed that as  is enlarged, the computation time increases. This is due to the fact that the
number of obstacles contained in the equilateral space is increased by enlarging . However, the path lengths
are identical with different . Thus, it is important to decide the value of  that can produce an optimal path
while minimizing the computation time.
3. SIMULATION RESULTS
The efficiency of the algorithm can be observed through a simulation with different number of
obstacles, i.e. between 15 and 150. These numbers of obstacles represent different density of the obstacles in
a C-space. To see the effect of the opening angle , three different values are applied which are 20°, 30º and

IJECE ISSN: 2088-8708 
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40°. Table 2 shows the comparison of computation time to search the collision-free paths by ESOVG with
different number of obstacles and opening angles . When  is 20º and with 15 obstacles in the search space,
the computation time to find a path is 0.0679s. As  is enlarged to 30° and 40°, the computation times are
0.0955s and 0.1251s, respectively.
When the number of obstacles is increased to 120, the computation time at =20° is 0.6156s. With
the same number of obstacles, at =30° and =40°, the computation times are 1.0248s and 2.1233s
respectively. With the number of obstacles of 150 in the C-space, at  = 20°, 30° and 40°, the computation
times are 1.0334s, 1.6662s and 3.0275s, respectively. The simulation result shows that when ρ is small,
which results in small equilateral space and a low number of obstacles, the computation time is lower. The
trend of the computation times of the simulation is depicted in Figure 4.
The performance of the proposed algorithm is evaluated by comparing it with the conventional VG
method. Table 3 shows the comparison of computation time and path length between the conventional VG
and ESOVG with =20°.
Table 2. Comparison of computation times with different values of ρ and obstacles
Number of obstacles Computation time(s) at ρ=20° Computation time(s) at ρ=30° Computation time(s) at ρ=40°
15 0.0679 0.0955 0.1251
30 0.1311 0.1808 0.2753
45 0.1599 0.1992 0.3683
60 0.2439 0.2903 0.5371
75 0.3541 0.4602 0.8798
90 0.3880 0.5607 1.2418
105 0.4545 0.7324 1.5761
120 0.6156 1.0248 2.1233
150 1.0334 1.6662 3.0275
Figure 4. Simulation results of ESOVG with different  values and different numbers of obstacles
Table 3. Performance comparison between conventional VG and ESOVG
Conventional VG ESOVG with ρ=20°
No of
obstacles
Computation time (s) Path length (unit) Computation time (s) Path length
(unit)
Improvement rate
(%)
15 0.1862 520.1343 0.0679 520.1343 63.5
30 0.6523 520.1342 0.1311 520.1343 79.9
45 1.4568 627.4492 0.1599 627.4492 89.0
60 2.2357 631.0024 0.2439 631.0024 89.0
75 3.2087 645.6750 0.3541 645.6750 89.0
90 4.4143 645.6750 0.3880 645.6750 91.2
105 6.0465 645.6750 0.4545 645.6750 92.5
120 7.3873 750.2258 0.6156 750.2258 92.0
150 11.8671 910.8723 1.0334 910.8723 91.3

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When the number of obstacles is 15, the computation time for finding a path by conventional VG is
0.1862s, and by ESOVG is 0.0679s. ESOVG improves the computation time by up to 63.5 %. When the
number of the obstacle is increased to 75, the computation time of conventional VG is 3.2087s while the
ESOVG’s is 0.3541s, which reduces the VG computation time by 89 %. With 150 obstacles in the
environment, the computation time by conventional VG and ESOVG are 11.8671s and 1.0334s respectively.
The improvement recorded by ESOVG is 91.3 %.
Figure 5 clearly shows the trends of computation times of conventional VG and ESOVG. The trend
indicates that when the number of obstacles escalates, the computation time of conventional VG increases
exponentially, whereas ESOVG has computation time that rises almost linearly.
In terms of path length, from Table 3, it can be seen that both VG and ESOVG produce path with
identical lengths. This proves that while ESOVG reduces the computation time, it maintains the optimality of
the resulting path in terms of length.
Figure 5. Comparison VG and Equilateral spaces VG
4. CONCLUSION
Equilateral Spaces Oriented Visibility Graph (ESOVG) algorithm has been proposed in this paper to
overcome a high computation time in conventional VG. ESOVG creates an equilateral space using four
imaginary lines to reduce the considered number of obstacles when planning a collision-free path. In
ESOVG, the size of the equilateral space is determined by the opening angle . The nominal value of  is set
to 20º and if a collision-free path could not be found in the space, it will be increased gradually until a path is
found in the space. ESOVG has been compared with the conventional VG in terms of computation time and
path length. It was found that ESOVG was able to improve the computation time drastically in comparison
with the conventional VG while the optimality of the path length was maintained. In the future ESOVG could
be tested in a dynamic environment which has moving and pop-up obstacles. If ESOVG were success in
dynamic environment, it could be applied in autonomous car, in which a lower computation time is necessary
in order to be applied in real time.
ACKNOWLEDGEMENTS
This work is supported by Universiti Tun Hussein Onn Malaysia (UTHM) and funded by Vot 1489
of Fundamental Research Grant Scheme (FRGS).
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BIOGRAPHIES OF AUTHORS
Nor Badariyah Abdul Latip, received her bachelor in Electronic Engineering from Universiti
Tun Hussein Onn Malaysia (UTHM) in 2015. Her research interests are in robotic path planning,
control system and medical electronic. Upon graduation she pursues his master in Electrical
Engineering in UTHM. She is currently a Master degree student at UTHM.
Rosli Omar is a senior lecturer at Faculty of Electrical & Electronic Engineering, Universiti Tun
Hussein Onn Malaysia. He received his PhD in engineering from University of Leicester, United
Kingdom in 2012. His research interests are in robotic engineering, autonomous system and
system identification.
Sanjoy Kumar Debnath is a 2nd-year PhD student in the Department of Mechatronic and
Robotic engineering, Faculty of Electrical & Electronic Engineering in the Universiti Tun
Hussein Onn Malaysia (UTHM). He received his Masters of Engineering from Universiti
Teknologi Malaysia in 2014 and Bachelor of Engineering Degree from the Presidency
University Bangladesh in 2008. He joined a research on “Optimal Energy Efficient Path
Planning for an Unmanned Air Vehicle (UAV) in Obstacle-Rich Environment” in 2015 at
UTHM with research Grant under the Office for Research, Innovation, Commercialization, and
Consultancy Management (ORICC).

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