Performance Evaluation of Path Planning …1052215/FULLTEXT02.pdfPerformance Evaluation of Path Planning Techniques for Unmanned Aerial Vehicles AcomparativeanalysisofA-staralgorithmand

Thesis no: MSCS-2016-20

Performance Evaluation of PathPlanning Techniques for Unmanned

Aerial VehiclesA comparative analysis of A-star algorithm and

Mixed Integer Linear Programming

Apuroop Paleti

Dept. Computer Science & EngineeringBlekinge Institute of TechnologySE–371 79 Karlskrona, Sweden

This thesis is submitted to the Department of Computer Science & Engineering at BlekingeInstitute of Technology in partial fulfilment of the requirements for the degree of Master ofScience in Computer Science. The thesis is equivalent to 20 weeks of full-time studies.

Contact Information:Author(s):Apuroop PaletiE-mail: [email protected]

University Advisor:Prof. Lawrence HeneseyDept. Computer Science & Engineering

Dept. Computer Science & EngineeringBlekinge Institute of TechnologySE–371 79 Karlskrona, Sweden

Internet : www.bth.se Phone : +46 455 38 50 00 Fax : +46 455 38 50 57

Abstract

Context: Unmanned Aerial Vehicles are being widely being used forvarious scientific and non-scientific purposes. This increases the needfor effective and efficient path planning of Unmanned Aerial Vehicles.Two of the most commonly used methods are the A-star algorithmand Mixed Integer Linear Programming.Objectives: Conduct a simulation experiment to determine the per-formance of A-star algorithm and Mixed Integer Linear Programmingfor path planning of Unmanned Aerial Vehicle in a simulated environ-ment. Further, evaluate A-star algorithm and Mixed Integer LinearProgramming based computational time and computational space tofind out the efficiency. Finally, perform a comparative analysis of A-star algorithm and Mixed Integer Linear Programming and analysethe results.Methods: To achieve the objectives, both the methods are studiedextensively, and test scenarios were generated for simulation of thesemethods. These methods are then implemented on these test scenar-ios and the computational times for both the scenarios were observed.A hypothesis is proposed to analyse the results. A performance eval-uation of these methods is done and they are compared for a betterperformance in the generated environment.Results: It is observed that the efficiency of A-star algorithm andMILP algorithm when no obstacles are considered is 3.005 and 12.03functions per second and when obstacles are encountered is 1.56 and10.59 functions per seconds. The results are statistically tested usinghypothesis testing resulting in the inference that there is a signifi-cant difference between the computation time of A-star algorithm andMILP. Performance evaluation is done, using these results and the ef-ficiency of algorithms in the generated environment is obtained.Conclusions: The experimental results are analysed, and the efficien-cies of A-star algorithm and Mixed Integer Linear Programming for aparticular environment is measured. The performance analysis of thealgorithms provides us with a clear view as to which algorithm is betterwhen used in a real-time scenario. It is observed that Mixed IntegerLinear Programming is significantly better than A-star algorithm.

Keywords: A-star algorithm, Mixed Integer Linear Programming,performance evaluation.

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Acknowledgments

It is a genuine pleasure to express my heartfelt gratitude to my thesis supervisorDr. Lawrence Henesey for giving me the opportunity to pursue my thesis in atopic of my choice and interest. His advice and support were a great encourage-ment to me through my thesis.

I would also like to thank all my friends and peers for their valuable viewsand constant motivation and for all their support throughout my thesis.

I would like to thank my parents Bhanumurthy Paleti and Rajani Paleti fortheir unending support and love and for encouraging me throughout my Masters.

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Contents

Abstract i

Acknowledgments ii

1 Introduction 11.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Research Aim and Objectives . . . . . . . . . . . . . . . . . . . . 11.3 Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Contribution of the study . . . . . . . . . . . . . . . . . . . . . . 21.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Background and Related Work 42.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Path Planning Problem . . . . . . . . . . . . . . . . . . . . 42.1.2 Framework for automated path planning . . . . . . . . . . 52.1.3 A-star Algorithm . . . . . . . . . . . . . . . . . . . . . . . 52.1.4 Mixed Integer Linear Programming . . . . . . . . . . . . . 62.1.5 Efficiency of an algorithm . . . . . . . . . . . . . . . . . . 7

2.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Method 93.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.1 Inclusion and Exclusion Criteria . . . . . . . . . . . . . . . 103.2 Basic Evaluation of Path Planning Methods . . . . . . . . . . . . 113.3 Simulation Experiment . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3.2 Test Environment . . . . . . . . . . . . . . . . . . . . . . . 143.3.3 Test Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . 143.3.4 Implementation: . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 Comparative Analysis: . . . . . . . . . . . . . . . . . . . . . . . . 223.5 Formulation of Hypothesis . . . . . . . . . . . . . . . . . . . . . . 22

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4 Results 234.1 Experimental Results: . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1.1 A-star Algorithm: . . . . . . . . . . . . . . . . . . . . . . . 234.1.2 MILP: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Performance Evaluation and Comparative Analysis . . . . . . . . 304.2.1 A-star Algorithm . . . . . . . . . . . . . . . . . . . . . . . 314.2.2 MILP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2.3 Comparative analysis . . . . . . . . . . . . . . . . . . . . . 34

5 Analysis 365.1 Statistical Analysis of the Results . . . . . . . . . . . . . . . . . . 36

5.1.1 Sample Characteristics . . . . . . . . . . . . . . . . . . . . 365.1.2 Independent Samples T-test . . . . . . . . . . . . . . . . . 39

5.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2.1 Validation through Statistical Analysis . . . . . . . . . . . 425.2.2 Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2.3 Theoretical Validation . . . . . . . . . . . . . . . . . . . . 42

5.3 Computational Differences of A-star Algorithm and MILP . . . . 43

6 Discussion and Limitations 456.1 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.2 Threats to Validity . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.2.1 Internal Validity . . . . . . . . . . . . . . . . . . . . . . . 476.2.2 External Validity . . . . . . . . . . . . . . . . . . . . . . . 48

7 Conclusions and Future Work 497.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

References 51

Appendices 55

A Code for A-star algorithm 56

B Code for MILP 59

C Code for AMPL model 61

D Statistical Analysis 64

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List of Figures

2.1 Block diagram to show the framework of automated path planning 52.2 Pseudo code for A-star algorithm . . . . . . . . . . . . . . . . . . 6

3.1 The optimal path generated by A-star algorithm for Scenario 1 . . 173.2 The optimal path generated by A-star algorithm for Scenario 2 . . 183.3 A snapshot running the AMPL model for generating the environ-

ment variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4 The optimal path generated by MILP for Scenario 2 . . . . . . . . 21

4.1 Computation time vs Number of Functions for A-star algorithm inScenario 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 Computation time vs Number of Functions for A-star algorithm inScenario 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3 Computation time vs Number of Functions for MILP in Scenario 1 334.4 Computation time vs Number of Functions for MILP in Scenario 2 344.5 Performance of A-star algorithm and MILP in Scenario 1 . . . . . 344.6 Performance of A-star algorithm and MILP in Scenario 2 . . . . . 35

5.1 Descriptive Statistics of Results . . . . . . . . . . . . . . . . . . . 375.2 Results for F-test in Scenario 1 . . . . . . . . . . . . . . . . . . . 385.3 Results for F-test in Scenario 2 . . . . . . . . . . . . . . . . . . . 385.4 Group Statics for Independent Samples T-test in Scenario 1 . . . 395.5 Results for Independent Samples T-test in Scenario 1 . . . . . . . 405.6 Group Statics for Independent Samples T-test in Scenario 2 . . . 405.7 Results for Independent Samples T-test in Scenario 2 . . . . . . . 41

A.1 Creation of environment by assigning values to variables . . . . . 56A.2 Code for A-star algorithm . . . . . . . . . . . . . . . . . . . . . . 57A.3 Code for calculating the distance between the nodes . . . . . . . . 57A.4 Code for obtaining the optimal path . . . . . . . . . . . . . . . . 58A.5 Code for plotting the path of the UAV . . . . . . . . . . . . . . . 58

B.1 Code for initializing the environment of MILP . . . . . . . . . . . 59B.2 Code for solving MILP . . . . . . . . . . . . . . . . . . . . . . . . 59B.3 Code for plotting the path obtained by MILP . . . . . . . . . . . 60

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C.1 Code for environment in the model created for MILP . . . . . . . 61C.2 Code for setting up constraints for AMPL model . . . . . . . . . . 62C.3 Code for calculating the cost of paths in AMPL model . . . . . . 63C.4 Code for loading and running the AMPL model . . . . . . . . . . 63

D.1 Histogram Depiction of the Results . . . . . . . . . . . . . . . . . 64D.2 Normal Q-Q Plot Depiction of the Results . . . . . . . . . . . . . 65

vi

List of Tables

3.1 Results Obtained during Literature Review . . . . . . . . . . . . . 10

4.1 Computation time of A-star algorithm for Scenario 1 . . . . . . . 254.2 Computation time of A-star algorithm for Scenario 2 . . . . . . . 264.3 Computation of MILP for Scenario 1 . . . . . . . . . . . . . . . . 284.4 Computation of MILP for Scenario 2 . . . . . . . . . . . . . . . . 294.5 Results obtained from the experiment . . . . . . . . . . . . . . . . 31

5.1 Description of Variables used for Statistical Analysis . . . . . . . 38

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List of Abbreviations

UAV Unmanned Aerial Vehicle

MILP Mixed Integer Linear Programming

AMPL A Mathematical Programming Language

CPLEX Simplex method implemented in C programming

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Chapter 1Introduction

1.1 Problem StatementNowadays, Unmanned Aerial Vehicles (UAV) are being frequently used for mil-itary, scientific fields and also for commercial and non-commercial purposes likedisaster area monitoring, surveillance, and land surveying [1] [2]. As there is noneed of a person to be present inside these vehicles, they are less complex innature and provide a better chance of performing more dangerous missions witha minimum risk [2]. A lot of these applications requires UAVs to fly at altitudeswhere they may encounter obstacles. Also, UAVs flying at high altitudes shouldbe monitored to avoid any unwanted results [3]. Therefore path planning playsan important role for the safe operation of UAVs.

For the present scenario of the unmanned aerial vehicle system, people onthe ground operate the UAVs. However, with the increase in usage of UAVs inthe near future, there will be a need to automate the vehicles [2]. This increasesthe need for an effective and efficient path planning of the UAVs. Path planningfor UAVs is nothing but determining the path from the start point to the endingpoint such that the vehicle does not encounter any obstacles in its course [4]. Pathplanning is generally associated with a number of terms like trajectory planning,navigation planning, local navigation, global path planning, and motion planning[4]. Various path planning methods for UAVs are proposed mainly to providewith a safe and an optimal route for the UAVs from their starting point to theending point in a hostile or unknown environment.

1.2 Research Aim and ObjectivesThe main aim of this thesis is to perform a comparative analysis for two of thefrequently used methods for path planning of UAVs namely A-star algorithmand Mixed Integer Linear Programming (MILP). The two methods are comparedbased on two crucial factors, computational time and computational space [4].Finally, the results are analysed, and simulation is done by considering the prob-lem instance in terms of environment and applying the algorithm to the problem

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Chapter 1. Introduction 2

instance.

The primary objectives of this thesis are:

• Conduct a performance evaluation of A-star algorithm and MILP based ontwo important factors, computational space and computational time to findout the efficiency.

• Analyse the performance evaluation and simulate the methods to the prob-lem instance.

• Perform a comparative analysis of A-star algorithm and MILP based on theliterature and the experimental results.

1.3 Research QuestionsRQ 1. What is the efficiency of A-star algorithm and MILP with respect tocomputational space and computational time?

Motivation: These methods are both proposed for path planning of UAVs.It is better to know the efficiency of these methods to know which performs betterin a given environment. So performance evaluation is done to find out efficiencybased on evaluation metrics which are computational space and computationaltime.

RQ 2. What are the most significant computational differences between A-staralgorithm and MILP?

Motivation: The methods mentioned above are most commonly used meth-ods for path planning of UAVs. It is, therefore, useful to know computationaldifferences between these two methods.

RQ 3. What is the performance of A-star algorithm and MILP in a simulatedenvironment?Motivation: Simulation of algorithms is done to solve the optimisation problem.Here the performance of A-star algorithm and MILP are measured and comparedwith each other to analyse the algorithms in similar environments.

1.4 Contribution of the studyThe aims and objectives of this research are achieved on the completion of thisstudy. All the research questions that were proposed were answered. The maincontributions of the research are:

Chapter 1. Introduction 3

• The experiment facilitates us to identify the efficiency of A-star algorithmand MILP theoretically and also for the simulated environment. This effi-ciency helps us to identify the better algorithm among the two methods. Itis observed that MILP has a significantly lower computation time than A-star algorithm. Though MILP has a higher number of functions, it will bebetter for implementation in real-time for path planning and generation ofan optimal route as the space required for the function is significantly lowerwhen compared to the trade-off in time. Therefore the efficient algorithmfor path planning of UAV in the simulated environment is identified.

• Performance analysis of A-star and MILP provides us with a better under-standing of these two path planning methods in a general sense. Further,the complexity of the methods theoretically helps us to analyse in whatsituation is the algorithm better when implemented in a real-time scenario.

1.5 Thesis OutlineThe thesis report consists of 7 chapters. The chapters are as follows. Chapter 1is the introduction for the thesis where the motivation for the research is brieflydiscussed. Chapter 2 contains the background and related work that pertains tothis thesis. The algorithms that were considered were discussed in a generalisedsense. Chapter 3 deals with the research methods and methodology that is usedin this thesis. Chapter 4 provides the results observed and the findings of thisthesis. Chapter 5 deals with the analysis of the results observed in Chapter 4.Validation of the results is also dealt with in this chapter. Also, computationaldifferences for the proposed methods are also discussed in this chapter. Chapter6 deals with discussions and limitations regarding this study. Threats to validityare also addressed in this chapter. Finally, Chapter 7 concludes the study byproviding the answers to the research questions that provide motivation for thisstudy. Also, future work that could be done for this study is discussed.

Chapter 2Background and Related Work

2.1 BackgroundThis chapter provides a basic idea of the path planning problem and puts it in thecontext of Unmanned Aerial Vehicles and gives an overview of A-star algorithmand Mixed Integer Linear Programming. Also, a brief introduction about pathplanning problem and efficiency of the algorithm is presented in this section.

2.1.1 Path Planning Problem

Path planning is an area of interest in several fields of research [5]. The mainfunctionality of a path planner is to devise a path for a machine from a startingpoint to the ending point by avoiding obstacles. Path planning is an importantaspect in the area of flight navigation and UAV navigation to avoid problems withthe weather and other such aspects [5]. Technically path planning problem of anunmanned aerial vehicle, in general, can be defined as determining the path ofa vehicle in a well-defined environment from a starting point to an ending pointwhere the vehicle is free from collisions with the surrounding objects [4]. Oftenthese path planning problems for UAV navigation with a diversion from obstaclesturn out to be NP-hard problems [6]. Such problems are often simplified by di-viding one problem into sub-problems and solving each subproblem individually[6].

Path planning problems have been thoroughly researched and reviewed. How-ever when it comes to path planning problems a few criteria that are often takeninto account are computational time, completeness and path length [4]. A pathplanning algorithm is said to be more efficient if it has less computational time[4]. There however might be trade-offs among these criteria to achieve a perfectsolution [4].

Path planning problem is a form of an optimisation problem which consistsof recurring parameters that are to be optimised on the same path [7]. Pathplanning is one of the important issues that is to be addressed when it comes to

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Chapter 2. Background and Related Work 5

the topic of development of UAVs. A path planning algorithm formulates a routefor the UAV from its starting location to the desired ending location [8].

2.1.2 Framework for automated path planning

The framework consists of two important parts namely external information andanalytical process [9]. External information mainly consists of the environmentthat the vehicle is going to travel in along with variables and constraints. All suchfactors must be considered to set a goal [9]. The analytical part consists of anintegration module and a path generation module [9]. In the integration module,the costs and goals are computed from the external information and finally apath is generated based on the analysis done. The framework is represented inthe form of a block diagram as shown in Figure 2.1.

Figure 2.1: Block diagram to show the framework of automated path planning

2.1.3 A-star Algorithm

One of the most widely used path planning algorithms for finding the shortestpath is the A-star algorithm. The A-star algorithm uses a heuristic function h(n)to find the cost of the best path from start node to the end node [10]. A-staralgorithm is a best first algorithm and can be evaluated with the function f(n) as

f(n) = g(n) + h(n)

where g(n) is the actual length of the evaluated path from the start to the finishnode and h(n) is a heuristic estimate of the remaining distance to the goal state[11]. A-star algorithm is only admissible if the heuristic function always gives theexact distance to the goal. However, this may be practically impossible [12].

A-star algorithm might be admissible in some cases as it considers only a fewnodes. This is because A-star algorithm uses an optimistic estimate of the cost ofa path [10]. In a best case scenario, A-star algorithm is as fast as the Best First


Search algorithm. In a worst case scenario, as Best First Search algorithm shouldconsider all paths to all the possible nodes, its time complexity is increased [13].We can say that one of the best-suited algorithms for finding the optimal pathis A-star algorithm [13]. The pseudo code for A-star algorithm is presented inFigure 2.2

Figure 2.2: Pseudo code for A-star algorithm

2.1.4 Mixed Integer Linear Programming

Linear programming models are commonly used to solve mathematical problemsfor optimisation and try to arrive at an optimal solution [14]. Linear program-ming takes in the problem statement and represents it in the form of mathematicalequations [14]. An optimal solution is a point where the maximum or minimumvalue is taken by the objective function [14].

Linear programming has a special case called integer programming which re-quires the variables to be integers [14]. For such cases, mixed integer linearprogramming is used [14]. Mixed Integer Linear Programming is one of the bestoptimisation methods which is a combination of both continuous and discretevariables [15]. Therefore MILP can solve problems involving both continuousand discrete variables [15]. Formally Mixed Integer Linear Program is a problemhaving


• A linear objective function fTx, where x is a variable column vector and fis the column vector of constants.

• Linear constraints, a lower bound, and an upper bound.

It is mathematically represented as:

minx,z fT1 x+ fT

2 z

subject to A1x+ A2z ≤ b

MILP is a recent approach that has been developed in the filed of path plan-ning and has been proven to be an essential and a suitable way to avoid obstaclesin a dynamic environment [16]. The main advantage of MILP is that it can ef-ficiently solve optimization problems that commonly occur in obstacle avoidanceproblems [16]. MILP differs form a regular linear programming problem as onlya few variables can be integers. Therefore the solution space of linear program-ming problem and mixed integer linear programming problem are similar [17].The MILP problem is generally NP-complete but as the problem size increases itbecomes NP-hard [15].

2.1.5 Efficiency of an algorithm

The efficiency of an algorithm is related to the amount of resources used by thealgorithm [18]. For greater efficiency of the algorithm, we aim at minimising theresources used by the algorithm. Resources are mainly measured by two factors:

• Time: The total time taken for the complete execution of the algorithm.

• Space: The amount of memory needed by the algorithm to function prop-erly. This may be the memory required for the code and also the amountof memory required for the data using which the code operates.

Usually, time complexity is analysed using Big O notation to get a near esti-mate of the running time of the algorithm for the given data. This is useful forcomparison of algorithms when there is a large input data.

Space complexity is an analytical estimate of the amount of run time memoryneeded as a function of the size of input data. Big O notation is used to expressspace complexity. Space complexity considers the amount of memory needed bythe code of the algorithm, input data, output data, and also as the workspaceduring execution of the code.


2.2 Related WorkPreviously, many different path planning algorithms have been proposed. Thesealgorithms aim at providing an optimal route for the UAV in a dynamic envi-ronment [19]. The efficiency of these algorithms is generally measured with thehelp of criteria like computational cost and time [4]. Path planning algorithmscan be broadly classified into potential field algorithms, probabilistic algorithms,and graph search algorithms as mentioned in [7]. Heuristic approaches like tabusearch, ant colony optimisation, simulated annealing are more commonly usedmethods for path planning. Algorithms like mixed integer linear programming(MILP), A-star algorithm, market-based algorithm, and Voronoi partitioning arediscussed in detail in [6] [7].

In [7] the author briefly describes the algorithms and also mentions about asoftware tool called PCube which is used to generate waypoint sequences. In [3]the author aims to study about sampling based path planning algorithms. In[20], the author emphasises on path planning for ground vehicular systems andfeasibility with the environment. [21] deals with adaptive path planning problemand simulation of a real-time path planner. In [9] the authors aim to understandhow humans conduct path planning under time pressure by experimenting withfour cost functions in a domain. In [11] the author aims to develop an algorithmfor path planning to avoid obstacles and obtain an optimum trajectory with min-imum arrival time to destination.

In [22] the author proposes a simulation experiment using MILP for path plan-ning of UAVs using Archimedes spiral thereby solving the path planning problemfor UAVs within a dynamic environment.A-star algorithm and its importance inpath planning of UAVs is discussed in [23]. Also, the important features of graphsearch algorithms are discussed in [7]. In [19] the author talks about the impor-tance of path planning algorithms and compares A-star algorithm with Dijkstra’salgorithm. The author of [14] discusses in detail about task assignment and mis-sion scheduling using Mixed Integer Linear Programming. [24] clearly discussesthe efficiency of algorithms.

Chapter 3Method

This chapter describes the type of method used for implementing the study andthereby answering the research questions. Initially, both, A-star algorithm andMixed Integer Linear Programming are thoroughly studied. An understandingof what the algorithms should be evaluated on is acquired. In this experimentalscenario, computational time and computational space are considered as perfor-mance metrics.

3.1 Literature ReviewBased on the related work that has been done in the area of path planning forUAVs, a literature review is conducted. Articles related to path planning meth-ods and techniques along with various algorithms used for path planning wereconsidered and studied. The research papers, journal articles, conference papersand thesis reports were obtained form various databases.

The search was performed in five stages. The key words used were modifiedeach time such that initially all the research related to path planning algorithmsis considered. Later the search was narrowed to path planning methods and tech-niques that consist of obstacle avoidance and also most efficient path planningalgorithms.

A total of 543 articles were found in the initial search. These articles includejournal articles, conference papers, thesis documentations and research papers.These articles were filtered based on the availability of full text in the database.This criteria reduced the search result to 399 articles. Further, the articles wereconsidered only if they are from the last 20 years, which resulted in the resultsto consist 260 articles. The abstracts for these 260 articles were studied and ofthese 53 articles were found to be most relevant to the topic at hand.

The search strings are further modified thereby deepening the search usingsnowballing technique [25]. The relevant articles are thoroughly studied.

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Chapter 3. Method 10

3.1.1 Inclusion and Exclusion Criteria

Inclusion criteria

• Articles that discuss the key features of path planning methods were con-sidered.

• Articles that were published in the last 20 years were considered.

Exclusion Criteria

• Articles that were published in English were considered.

• Articles for which full text was not available or the articles that need to bepurchased were excluded.

• Articles without proper citations were excluded.

Keywords Number of arti-cles

Number of Ar-ticles after ex-clusion

"Path Planning Algo-rithms" 153 6

"Path Planning and"Optimization" 147 15

"Path Planning" and"Obstacle Avoidance" 92 13

"Path Planning" and"Efficiency" 140 15

"Path Planning" and"Complexity" 16 4

Table 3.1: Results Obtained during Literature Review

Motivation for Selection of A-star and MILP:From the Literature review that has been conducted, it has been observed thatthe most widely used path planning techniques that are being used for UAVsare A-star algorithm and MILP. A-star algorithm is being used with obstacleavoidance to get the efficient path of the algorithm [6]. Also A-star algorithm isthe most widely used algorithm among all the heuristic algorithms [6]. It is alsofound that MILP is an effective and an efficient method for path planning usingobstacle avoidance [16]. Further it is observed that MILP can solve optimizationproblems for path planning along with obstacle avoidance [17]. Since these twomethods are found out to be widely used and are considered to be efficient intheir respective areas, they are chosen for a comparative analysis in this study.


3.2 Basic Evaluation of Path Planning MethodsEvaluation of path planning methods means to determine the number of resources,generally space and time, required by the methods to execute [26]. Most of thealgorithms and methods are created in such a way that they can work with a setof arbitrary inputs. Consistently efficiency of an algorithm is measured as timecomplexity or space complexity [27].

Algorithm analysis is an important area as it provides a theoretical estima-tion of the required resources needed by an algorithm for computing and solvinga problem [28]. This estimation can potentially provide insight on appropriateways to find efficient algorithms.

Theoretically, in the analysis of algorithms and methods, the complexity ofalgorithms is estimated, as it is relatively easier to estimate the complexity fora large problem size [28]. Most commonly, Big O notation, Big-theta notationand Big-omega notation are used for measuring the complexity [29]. As there isa possibility for a difference in efficiency for different algorithms and methods fordifferent implementations with different inputs, asymptotic estimates are used tothis end. However, certain implementations of algorithms usually require certainassumptions for computing exact efficiency of the algorithm or methods.

Measuring the performance is the most common way to arrive at a definitiveconclusion regarding the efficiency of the algorithm. Analysis and evaluation ofalgorithms is an important aspect, as the unintended use of an inefficient algo-rithm can impact the performance of the system significantly [18]. In the case oftime-sensitive applications or programs, an algorithm that takes a lot of time forexecution can produce results that are useless and obsolete [18]. This can lead toserious problems in practical situations. Also, an algorithm which is inefficientcan wind up requiring additional resources and use them uneconomically to ren-der it useless [18].

For path planning algorithms in the case of UAVs, algorithmic efficiency playsa pivotal role. Evaluation of algorithms for finding the best-suited algorithm fora particular scenario is very important. The algorithms that were considered forperformance evaluation in this study were A-star algorithm and Mixed IntegerLinear Programming (MILP). These are the most commonly used methods forpath planning of UAVs.

The criteria on which these path planning algorithms were evaluated are com-putational time and computational space. Big O notation is used for evaluatingthe algorithms. This is because the exact counting of all the operations is quitedifficult. Big O notation allows us to estimate a cost function that facilitates us


to approximate the growth rate of the algorithm. An approximation is good whenthe problem size gets too large. Big O notation is commonly used as it does notgive an explicit interpretation of the cost function for a distinct problem size [15].

The general idea is that the behaviour of the algorithm is expressed as theproblem size grows large enough that it ignores constants and lower order terms.The results of this evaluation of A-star algorithm and MILP are presented clearlyin Chapter 4.

3.3 Simulation ExperimentSimulations are programs based on some systems or processes in a computer [30].The main motivation for using a simulated experiment is that they are simplifiedversions of real-world scenarios with a controlled environment [30]. This controlover the environment is implemented to focus on the main topic at hand therebyachieving the desired results with a greater degree of reliability [31].

A simulation program can be developed for an algorithm and can be executedto get results that are satisfactory for the algorithm in the created environment[31]. Every algorithm has its own advantages and disadvantages [31]. For simu-lating an algorithm the following steps are implemented.

• Generate a program that can mimic the functionality of the algorithm inreal-time.

• Implement the program containing the algorithm.

• Conduct experiments on this system and analyse the results.

A simulation experiment is chosen for this research because the environmentthat is generated is controlled. The execution of the algorithms in a real-time en-vironment is quite difficult because the problem size would be large. This will beinefficient as the results observed might be flawed or hard to analyse. Therefore,a simple environment is generated and, the algorithms are simulated accordingly.The simulation model that best suits this experiment is continuous simulationas the variables are changing continuously [32]. Discrete simulation model doesnot fit very well in this case as the variables do not change in discrete times andsteps for the simulated environment [32]. Also, discrete model is not consideredas the simulation becomes complex and the environment variables will be harderto simulate on the instance.

The two most widely used methods for path planning of UAVs are the A-staralgorithm and MILP [4]. So these two methods are taken for an experimentalanalysis wherein the time taken for each of the method to compute a given test


scenario is measured with respect to the number of functions that algorithm needsto compute the scenario.

This experimental analysis is done using Matlab software. First, both themethods were carefully studied and pseudo code for A-star algorithm and MILPwere analysed. Test scenarios were generated and a code was developed accord-ingly using Matlab. The simulation is expected to result in an optimal path foran unmanned aerial vehicle with collision avoidance using both A-Star algorithmand MILP. A general result from the simulations is a trajectory from the startingpoint to ending point where collision avoidance is enforced to make the route ofthe UAV optimal and more practical. This simulation for both the algorithmswas done using common scenarios.

3.3.1 Assumptions

Usually, in simulation experiments, assumptions are made to keep the systemstable and to avoid any external errors that may occur [33]. A few assumptionsare made for the experiment performed in this study are as follows:

• The UAV can only travel in X and Y coordinates which means that theUAV should travel around an obstacle and cannot fly over or under theobstacle.

• The environment created in the experiment is static, i.e., the obstacles arefixed and are not moving. In a real world setting these static objects canbe buildings, structures or trees that might occur in the path of a UAV.

• Moving obstacles (which can be other UAVs) are not considered so thatthe main emphasis of the results is on the performance of the path planningtechniques. This allows us to measure the performance of the path planningtechniques in a simple environment.

• The starting point and the ending points of the UAV are predetermined tomaintain optimality.

• The external factors such as wind and fuel consumed do not affect theperformance of the algorithms. This is because only a simplified model ofa UAV is considered

• The optimal path is obtained by calculating the shortest distance betweenthe starting point and the ending point of the UAV. Also, the path calcu-lated by the algorithms is always optimised for better results.


3.3.2 Test Environment

The simulation experiment was performed on a computer with the following spec-ifications:

• Operating system: Windows 10 Home

• Processor: Intel® Core ™ i3-4030U CPU @ 1.90 GHz

• Installed memory (RAM): 4GB

• System type: 64-bit Operating System, x64-based processor

3.3.3 Test Scenarios

An environment for the path planning is carefully designed for both A-star algo-rithm and MILP. Initially, a grid is taken as a 2-dimensional array map with sizeon the X axis as 50 units and size on the Y axis as 25 units. This array storesthe coordinates of the map and objects in each coordinate. The input values forthe test cases are the number of obstacles, starting point of the UAV and finalposition of UAV. These test scenarios are developed using Matlab software. Theposition of the obstacles is randomly generated. If there are more obstacles, thetime taken for computation of the path is also increased. Therefore we can saythat the number of obstacles encountered by the UAV is directly proportional tothe time taken for computing an optimal path. However, to maintain optimalitywe consider equal number of obstacles for A-star algorithm and MILP.

Number ofobstacles ∝ Computation time

A-star Algorithm:A test environment for A-star algorithm is created in such a way that there arestarting and ending points for the UAV. The number of obstacles is taken as 5.The main motivation for considering only 5 obstacles is that the path of the UAVwhen affected by the positions of 5 obstacles is observed to be similar when moreobstacles are considered i.e, the computation time is observed to be similar tothat when 5 obstacles were considered. Therefore, to maintain optimality andsimplicity and to obtain more accurate results only 5 obstacles were considered.The obstacles are randomly placed across the path of the UAV. The algorithmgenerates a path for the UAV in such a way that the obstacles are avoided, andthe UAV reaches the goal node without any collisions. The final route that is for-mulated will be the optimal route which gives the shortest path from the startingpoint to the ending point avoiding collision with the obstacles.


Two cases are considered for the environment for the path generation of the UAVusing A-star algorithm.Scenario 1: No obstacles are encountered by the UAV.Scenario 2: Obstacles are encountered by the UAV.

MILP:Test environment for MILP is generated in such a way that a starting point forthe UAV is predetermined, ending point of UAV is predetermined, and the ob-stacles are randomly placed. The number of obstacles for MILP is taken as 5.The UAV starts from the starting node and reaches the goal node in an optimalroute avoiding the obstacles in its path. The final route that is formulated by thealgorithm has an optimal time which gives the shortest path from the startingpoint to the ending point without any collisions.

Two cases are considered for the environment for the travel of the UAV usingMILP.Scenario 1: No obstacles encountered by the UAV.Scenario 2: Obstacles are encountered by the UAV.

For Scenario 1, 5 obstacles are randomly placed such that the path of theUAV is unobstructed and for Scenario 2, the 5 obstacles are placed in such a waythat the path of the UAV is obstructed and the collision avoidance function isused to generate the path.

3.3.4 Implementation:

Both the path planning methods are implemented in both the scenarios. Imple-mentation for each of the method is explained individually.

A-star Algorithm:A-star algorithm is used to calculate the shortest path from a starting point to anending point. Initially, an interface is created where a starting point and endingpoint of the UAV are generated. Obstacles are placed for each scenario such thatthe test cases are satisfied. This algorithm generates the shortest path betweenthe starting point of the UAV and the ending point. After the environment iscreated as a 2-dimensional map array and the starting point and ending pointsof the UAV are fixed, positions of the obstacles are fixed.

The obstacles are generated randomly. For one scenario, a path where noobstacles are encountered and another scenario where obstacles are encounteredby the UAV are considered. This may increase the computation time of thealgorithm as the collision avoidance function should be processed each time anobstacle is encountered.


Cost Function: The distance between any two nodes can be obtained by sim-ply calculating the straight distance between the nodes. This might not alwaysbe true practically as there may be obstacles present in the path. As long as thecost is not overestimated, the algorithm can generate an optimal path.

The implementation of A-Star algorithm is as follows:

1. Initially, set the start node on OPEN list and proceed to calculate thecost function f(n) where the heuristic function is 0 and g(n) is the distancebetween the start position of the UAV and the goal node.

2. Remove the node with the smallest cost function from the OPEN list andput it in the CLOSED list. Let this be node n.

3. If n is found to be the goal node, then terminate the algorithm and obtainthe solution path using pointers. Else continue to step 4.

4. Compute the cost function for each successor node on the CLOSED list.

5. Associate each successor node on OPEN list or CLOSED list with the cal-culated cost and put them on the OPEN list.

6. Associate with successors on the OPEN list with cost values smaller thanthe previous node.

7. Go to step 2.

The Matlab Program for A-star algorithm consists of

• A main file that needs to be executed to run the program.

• A distance function to calculate the distance between two nodes.

• A function to take in the list of successor nodes and calculate the costfunction.

• A function to generate the OPEN list with values.

• A function that takes values from the OPEN list and returns the node withthe least cost function.

• Also a function to index the location of the node in the OPEN list.

The path traversed by the UAV is displayed as a trajectory from the startingpoint to the ending point. Whenever an obstacle is encountered, the path of theUAV deviates. The trajectories for the two test cases are shown in Figure 3.1 andFigure 3.2.


Figure 3.1: The optimal path generated by A-star algorithm for Scenario 1

Figure 3.1 shows the path for a UAV for Scenario 1 when no obstacle is en-countered in its path from starting point to the ending point. The starting andending point are placed far apart from each other and are blue in colour. Theblue line represents the path travelled by the UAV. The obstacles are present atrandom positions and are represented by red circles.


Figure 3.2: The optimal path generated by A-star algorithm for Scenario 2

Figure 3.2 shows the path for a UAV for scenario 2 when obstacles are encoun-tered in its path. The starting and ending point are placed far apart from eachother and are blue in colour. The path of the UAV is represented as a blue line.The obstacles are present at random positions and are represented as red circles.When the UAV detects an obstacle, the path deviates. The A-star algorithmgenerates the shortest route to the final point.

The time taken for execution of the algorithm in these two scenarios is takenalong with the number of functions the algorithm needs to execute in order toget the optimal route.This done using the profiler in Matlab. The results arepresented in Chapter 4.

MILP:Mixed integer linear programming is used to find an optimal route that takes theleast amount of time to reach the goal node from the starting node. A set of way-


points determines the path travelled by the UAV. These waypoints are generatedin such a way that a continuous path from the waypoints gives the route of theUAV [34]. Time taken for the UAV to traverse is calculated for each scenario.Collision avoidance function is implemented so that the algorithm implementa-tion is practical. An optimal route is calculated using the IBM ILOG CPLEXoptimisation tool and A Mathematical Programing Language (AMPL) which isa modelling language used to solve complex problems [35].

AMPL is used to create and translate models [36]. It helps in indexing andsorting out of variables. Many solver codes are used in conjunction with AMPL.CPLEX optimisation tool is a commercial optimizer which efficiently generatesthe shortest route from the starting point to the ending point [35]. Matlab, AMPLand CPLEX optimizer are used together to get the best possible scenario usingMILP for generating the shortest path.

The implementation of MILP is as follows:

1. Initially, a default set of waypoints are generated for plotting the path ofthe UAV. These waypoints can be generated according to the test cases.

2. A continuous path is generated connecting these waypoints.

3. All the data required for a particular scenario is generated using a Matlabfunction writeAmplModel.

4. A .dat (data) file is generated which consists of all the data regarding theUAV and the environment.

5. AMPL is now used to load the model and implement the respective datafile for generation of the path.

6. Time taken by the UAV from the starting point to the ending point is nowcalculated. This is done from a function getTimeFromPath.

7. Different sets of waypoints are generated where for each set of waypointsthe UAV has its own time to travel from the starting point to the end point.

8. The solution for each path generated is recorded.

9. This solution is plotted to get the optimal path from start point of the UAVto the end point.

AMPL modelling language with the commands needed to run the model thatdefines the environment for MILP is presented in Figure 3.3. After assigningthe values to the variables for the environment, a data file is generated with allthe environment variables as shown in Appendix C.1. CPLEX optimisation tool


is used to optimise the cost for travelling from the starting point to the endingpoint.

Figure 3.3: A snapshot running the AMPL model for generating the environmentvariables

Matlab program for MILP consists of

• A main file that needs to be executed to run the program and set the datato the variables.

• A function to generate waypoints.

• AMPL and IBM ILOG CPLEX optimisation tool to generate a model andoptimise the solution.

• A function to calculate the time taken by the UAV from the starting pointto reach the goal.

• A function to optimise the path of the UAV using optimisation toolbox ofMatlab.


Figure 3.4: The optimal path generated by MILP for Scenario 2

Figure 3.4 shows the trajectory of the UAV using MILP when no obstacles areencountered. The starting point and the ending point are selected far apart fromeach other and the way points generated are in such a way that the shortest pathis reached from the start point to the end point. The obstacles are representedas boxes.

Similarly a path for UAV is optimised when obstacles are encountered. This isdone using the AMPL modelling where all the environment variables are assignedvalues. CPLEX optimisation tool is used to optimise the path.

Finally, the time taken for the program to execute is taken for both the sce-narios along with the functions needed by the program to execute. The resultsare presented in Chapter 4.


3.4 Comparative Analysis:A comparative analysis for A-star algorithm and MILP is done to determine theperformance of these algorithms and compare the performance for the test cases.The performance of each algorithm is measured with the time taken for executionof the program for that algorithm with the number of functions used to get anoptimal route. This measure of performance for both the algorithms is comparedand the algorithm with best execution time and the algorithm with least numberof functions is presented. These factors help us determine which algorithm isbetter for computation of an optimal route of UAV in the given environment.Thecomparative analysis for A-star algorithm and MILP is discussed in Chapter 4.

3.5 Formulation of HypothesisThe results obtained form the experiment are compared and analysed statisticallyto understand which method among A-star algorithm and MILP performs betterin Scenario 1 and Scenario 2. For this purpose we need to analyse the data andstatistically verify if there is actually a significant difference in the time observedfor A-star algorithm and MILP in each scenario. the hypothesis proposed is asfollows.

H0 (Null Hypothesis): There is no significant difference in the computationtime for A-star algorithm and MILP.H1 (Alternate Hypothesis): There is a significant difference in the computa-tion time for A-star and algorithm and MILP.

This hypothesis is tested statistically to verify the findings of this study andto validate the results that are observed through the experiment. The statisticaltesting method is elaborated further in Section 5.1.

Chapter 4Results

The following section presents the results of the experiment as per the researchmethod proposed in Chapter 3.

4.1 Experimental Results:The experiment for generating an optimal route for the UAV using A-star algo-rithm and MILP is performed using Matlab software. A program utilising theA-star algorithm is developed as presented in Appendix A to generate an optimalroute which is the shortest route from the starting point to the ending point. Alsoanother program using the similar environmental conditions is developed whichutilises MILP. This program utilises AMPL to create a model that generates theenvironment. CPLEX optimizer is used to optimise the path.

The experimental results utilising both the algorithms are presented individ-ually. Each scenario for A-star algorithm and MILP is implemented 10 times.A total of 10 iterations were considered as it was observed that the difference inthe mean for more iterations is insignificant and the standard deviation is verysmall. Therefore to obtain the results that are optimal, the number of iterationswas stopped at 10 as 10 data points for each algorithm in each scenario providesthe required results.

4.1.1 A-star Algorithm:

The program using A-star algorithm is executed in Matlab. This generates theshortest route from the starting point to the ending point. Two scenarios wereconsidered where Scenario 1 has an environment where no obstacles are encoun-tered by the UAV and Scenario 2 has an environment where obstacles are en-countered by the UAV. The program is executed for both the scenarios and isrepeated thrice. The execution time of the program utilising A-star algorithm ismeasured by taking the average of the three values for execution times in both

23

Chapter 4. Results 24

the cases.

The efficiency of the program utilising the algorithm is obtained by taking thenumber of functions used by the program to get an optimal route along with thecomputation time of the program. This process is done for the scenario whereobstacles are not encountered by the UAV and a scenario where the UAV encoun-ters obstacles. Efficiency, in this case, is determined as the time taken to executethe program utilising the functions to obtain an optimal shortest route from thestarting point to the ending point.

The experiment is repeated 10 times for both the scenarios by executing theprogram. The average of the 10 repetitions is taken and is considered as thecomputation time of A-star algorithm. Also the standard deviation of the 10iterations is calculated for both the scenarios to make sure the change in time isminimal.The standard deviation (s) is calculated as the square root of variance(s2) and can be represented by the formula

s =

√∑(X −M)2

n− 1(4.1)

where X is the time in each repetition, M is the mean, and n is the number ofrepetitions. The results for the computation time of the program utilising the A-star algorithm are presented for Scenario 1 where the UAV does not encountersobstacles and Scenario 2 where the UAV encounters obstacles along with theiraverage computation times in Table 4.1 and Table 4.2 respectively.


Repetition ComputationTime (seconds)

Repetition 1 37

Repetition 2 35

Repetition 3 31

Repetition 4 33

Repetition 5 34

Repetition 6 37

Repetition 7 33

Repetition 8 35

Repetition 9 36

Repetition 10 35

Mean (Average) 34.6

Standard Deviation 1.897

Table 4.1: Computation time of A-star algorithm for Scenario 1



Repetition 1 68

Repetition 2 65

Repetition 3 72

Repetition 4 69

Repetition 5 66

Repetition 6 60

Repetition 7 68

Repetition 8 65

Repetition 3 66

Repetition 10 65

Mean (Average) 66.4


Table 4.2: Computation time of A-star algorithm for Scenario 2

The functions utilised by the program is constantly observed to be 104 forthe program executing the A-star algorithm for both the cases. Therefore theefficiency of the program utilising A-star algorithm can be computed as follows

Scenario 1: The average computation time for the scenario where UAV doesnot encounter obstacles is observed to be 34.3 seconds for 104 functions. Thestandard deviation calculated is 1.897. Therefore, if we consider efficiency as afunction of computation time and the number of functions executed, we can saythat

Efficiency =Number of functions

Computation time=

104

34.6= 3.005

Therefore the efficiency of A-star algorithm in the given scenario is said to be3 functions per second.

Scenario 2: The average computation time for the scenario where the UAVencounters obstacles is observed to be 66.4 seconds and the number of functionsexecuted are observed to be 104. The standard deviation calculated is 3.169.Therefore the efficiency being the function of number of functions and the com-


putation time, we can say that,


Computation time=

104

66.4= 1.56

Therefore the efficiency of A-star algorithm for the given scenario is observedto be 1.56 functions per second.

4.1.2 MILP:

The program using MILP is executed in Matlab using the model created byAMPL and optimized using CPLEX optimizer. This program generates the op-timal route from the starting point to the ending point for both the scenarios.Scenario 1 where no obstacles are encountered by the UAV and Scenario 2 whereobstacles are encountered by the UAV. The program using MILP is executed andrepeated thrice for both the scenarios. The execution time of the program util-ising MILP is determined by taking the average of the three execution times inboth the scenarios.

The efficiency of the program utilising MILP is determined by considering thenumber of functions utilised by the program to generate an optimal route fromthe starting point to the ending point. This is done by counting the number offunctions utilised by the program for both the cases where obstacles are not en-countered by the UAV and where obstacles are encountered by the UAV. In thiscase, the efficiency is determined as a function of time taken to compute the func-tions utilised by the program to generate an optimal path from the starting pointof the UAV to the ending point and the number of functions used by the program.

The experiment is repeated 10 times for both the scenarios and the averagecomputation time is calculated.The results for computation time of the programutilising MILP is tabulated and presented for Scenario 1 and Scenario 2. Theaverage time is considered as the total computation time of the program for eachscenario. The computation times of the program for Scenario 1 and Scenario 2are presented in Table 4.3 and Table 4.4 respectively.



Repetition 1 14

Repetition 2 19

Repetition 3 17

Repetition 4 14

Repetition 5 15

Repetition 6 12

Repetition 7 16

Repetition 8 19

Repetition 9 17

Repetition 10 13

Mean (Average) 15.60


Table 4.3: Computation of MILP for Scenario 1



Repetition 1 17

Repetition 2 19

Repetition 3 21

Repetition 4 20

Repetition 5 18

Repetition 6 19

Repetition 7 15

Repetition 8 16

Repetition 9 19

Repetition 10 20

Mean (Average) 18.3


Table 4.4: Computation of MILP for Scenario 2

The functions utilised by the program is constantly observed to be 195 for theprogram using MILP in both the cases. Therefore the efficiency of the programfor MILP can be calculated as follows.

Scenario 1: The average computation time of MILP for the scenario where UAVdoes not encounter obstacles is observed to be 15.6 seconds for 195 functions. Thestandard deviation calculated is 2.431. Therefore, if we consider efficiency as afunction of computation time and the number of functions executed, we can saythat


Computation time=

195

15.6= 12.5

Therefore the efficiency of MILP in the given scenario is said to be 12 func-tions per second.

Scenario 2: The average computation time of MILP for the scenario whereUAV encounters obstacles is observed to be 18.3 seconds for 195 functions. Thestandard deviation calculated is 1.889. Therefore, if we consider efficiency as afunction of computation time and the number of functions executed, we can saythat


Computation time=

195

18.3= 10.6


Therefore the efficiency of MILP in the given scenario is said to be 10.6 func-tions per second.

4.2 Performance Evaluation and Comparative Anal-ysis

The performance of an algorithm is the amount of work done by that particularalgorithm in a specific time [29]. This is generally measured in terms of computa-tion time of the algorithm and space required for the execution of the algorithm[18]. Hence we can say that the two major aspects of algorithmic performanceare:

• Time

– Time taken for execution of instructions and functions.

– The total computation time of the algorithm.

• Space

– Functions and data structures used.

– Type of functions used.

However, the run time or computation time calculated is system dependent.This means that the performance of the algorithm might vary from system tosystem [29] [18].

The computation time which is determined experimentally in a contained en-vironment for both A-star algorithm and MILP are presented in Table 4.5. Alsothe number of functions that the algorithm utilises to execute the algorithmsalong with the efficiency are presented in Table 4.5.


ConstraintsA-starScenario1

A-starScenario2

MILPScenario1

MILPScenario2

Average Computationtime (seconds) 34.6 66.4 15.6 18.3

Standard Deviation 1.897 3.169 2.413 1.889

Number of functions 104 104 195 195

Efficiency (Functionsper second) 3.005 1.56 12.5 10.6

Table 4.5: Results obtained from the experiment

4.2.1 A-star Algorithm

For A-star algorithm, the computation time and the number of functions executedby the program in both the scenarios are taken and is represented as a graph.The time taken for computation of the program using the algorithm is shown onY-axis and the number of functions utilised by the program is shown on X-axis.

For Scenario 1, where no obstacles are encountered by the UAV, it is observedthat the number of functions executed by the program is 104 as presented inTable 4.5. The average computation time is observed to be 34.6 seconds as pre-sented in Table 4.5. The time-function graph for Scenario 1 for A-star algorithmis presented in Figure 4.1.

For Scenario 2, where obstacles are encountered by the UAV, it is observedthat the number of functions executed by the program is 104 as presented in Table4.5. The average computation time is observed to be 66.4 seconds as presentedin Table 4.5. The time-function graph is for Scenario 2 for A-star algorithm ispresented in Figure 4.2.


Figure 4.1: Computation time vs Number of Functions for A-star algorithm inScenario 1

Figure 4.2: Computation time vs Number of Functions for A-star algorithm inScenario 2

4.2.2 MILP

For MILP, the computation time and the number of nodes traversed by the UAVin both the cases is observed and recorded and is graphically represented. Thetime taken by the program using MILP is shown on Y-axis and the number of


functions executed by the program to implement the algorithm is taken on X-axis.

For Scenario 1, where no obstacles are encountered by the UAV, it is observedthat the number of functions executed by the program is 195 as observed in Table4.5. The average time taken by the program using MILP to execute is observedto be 16.2 seconds as observed in Table 4.5. The time graph for Scenario 1 forMILP is presented in Figure 4.3.

Figure 4.3: Computation time vs Number of Functions for MILP in Scenario 1

For Scenario 2, where no obstacles are encountered by the UAV, 195 as ob-served in Table 4.5. The average time taken by the program using MILP toexecute is observed to be 18.4 seconds as observed in Table 4.5. The time graphfor Scenario 2 for MILP is represented in Figure 4.4.

These time graphs help us analyse the efficiency of A-star and MILP for eachindividual case where the obstacles are encountered by the UAV and where noobstacles are encountered by the UAV. Performance of each of the algorithm isnow evaluated experimentally.


Figure 4.4: Computation time vs Number of Functions for MILP in Scenario 2

4.2.3 Comparative analysis

Comparison for A-star algorithm and MILP is done to evaluate the efficiencyof these algorithms for the scenarios considered in the generated environment.Graphs are plotted to compare the two algorithms in Scenario 1 and Scenario 2.

Figure 4.5: Performance of A-star algorithm and MILP in Scenario 1

From figure 4.5, we observe that in Scenario 1 the computation time of MILP


is significant less though the number of functions computed are more. So we caninfer from the graph in Figure 4.5 that the performance of MILP is better in thissimulated environment.

Figure 4.6: Performance of A-star algorithm and MILP in Scenario 2

From Figure 4.6, we observe that in Scenario 2 the computation for MILP isvery less although the number of functions computed is more. So we can inferfrom graph in Figure 4.6 that the performance of MILP is significantly betterthan A-star algorithm for Scenario 2 in this experiment.

Chapter 5Analysis

The results obtained from the experiment to evaluate the efficiency of A-starand MILP are presented in Chapter 4 in detail. These results explicitly answerresearch question RQ3. In this section, the results obtained are carefully analysedto answer research questions RQ1 and RQ2 explicitly.

5.1 Statistical Analysis of the ResultsThe efficiency of algorithms is dependent on two of the most important factors,time and space. The efficiency of an algorithm can be generalised as time effi-ciency or space efficiency. The performance of an algorithm is generally dependenton the problem size [29]. The measure of the efficiency of an algorithm is doneby measuring the time complexity or space complexity of the algorithm over aproblem. This can be done in different ways of which the most common wayis to measure the execution time of the program that implements the algorithm[29]. Also, the space requirement of an algorithm is often measured by manu-ally counting the number of operations the algorithm has to perform or profilingthe program that executes the algorithm. Profiling of an algorithm gives us thenumber of functions used by the program executing the algorithm along with thenumber of times each function is called [29] [18].

5.1.1 Sample Characteristics

Initially, it is necessary to check for normality of the data, i.e., if the results ob-tained are normally distributed. For this, we propose a hypothesis.

H0 (Null Hypothesis): The results observed are normally distributed.H1 (Alternate Hypothesis): The results observed are not normally distributed.

A Shapiro-Wilk’s test (p>0.05) [37] and a visual inspection of their histogramsand normal Q-Q plots (Appendix D) show that the time values obtained for the10 repetitions for A-star algorithm and MILP in Scenario 1 and Scenario 2 were

36

Chapter 5. Analysis 37

approximately normally distributed, with skewness and kurtosis as presented inFigure 5.1 [38][39][40].

Figure 5.1: Descriptive Statistics of Results

Further, we purpose we propose a hypothesis to verify if there is a statisticallysignificant difference in the variance of the two samples that are being consideredfor each scenario.

H0 (Null Hypothesis): There is no statistically significant difference betweenthe variance of the two samples.H1 (Alternate Hypothesis): There is a statistically significant difference be-tween the variance of the two samples.


This hypothesis is validated using an F-test using Levene statistic, performedusing SPSS tool [41]. The results of the F-test are presented in Figure 5.2 andFigure 5.3.

Figure 5.2: Results for F-test in Scenario 1

Figure 5.3: Results for F-test in Scenario 2

From Figure 5.2 and Figure 5.3 we observe that the significance in both thescenarios obtained from F-test is observed to be more than 0.05. Therefore weaccept the null hypothesis and reject the alternate hypothesis. Therefore we ob-serve that there is no statistically significant difference between the variance ofthe two samples.

Variables Groups Scenarios Description

Independent

A-star1

A-star algorithm and MILP when

no obstacles are encountered by the UAV.MILP

A-star2

A-star algorithm and MILP when

obstacles are encountered by the UAV.MILP

Dependent Time -Computation time for each

iteration for execution of

A-star algorithm and MILP.

Table 5.1: Description of Variables used for Statistical Analysis


The results obtained form the experiment are observed to be unpaired sam-ples, as clearly there is no effect of one sample on the other. Also the varianceof the samples as observed for Table 5.1, are statistically similar. Therefore thestatistical test Independent Samples T-test is used to test our hypothesis [42].

5.1.2 Independent Samples T-test

From the observations made in Section 5.1.1 and Table 5.1, a statistical test suit-able for the results was considered. Since the results are normally distributedwith no statistically significant difference between the variance of the two sam-ples, so Independent Samples T-test is found to be the most effective [42]. Thisstatistical test aims at verifying the hypothesis presented in Section 3.5.

The Independent Samples T-test is applied for both the scenarios where A-star algorithm and MILP are statistically analysed for one scenario at a time.This statistical analysis is carried out with SPSS tool [41].

Scenario 1 (When no obstacles are encountered by the UAV):

The result of the statistical analysis is as follows:

Figure 5.4: Group Statics for Independent Samples T-test in Scenario 1


Figure 5.5: Results for Independent Samples T-test in Scenario 1

From the above Figure 5.5 we can observe that the significance at 95 percentconfidence is observed to be less than 0.05. Therefore we ignore the null hypothesisand consider the alternate hypothesis. Also, here the t-value which signifies thesize of the difference relative to the variation of the sample data is observed tobe 19.574. Therefore we can conclude that when A-star algorithm and MILP areimplemented in Scenario 1, there is a significant difference in the computationtime.Scenario 2 (When obstacles are encountered by the UAV):

The result of the statistical analysis is as follows:

Figure 5.6: Group Statics for Independent Samples T-test in Scenario 2


Figure 5.7: Results for Independent Samples T-test in Scenario 2

From the above Figure 5.7 we can observe that the significance at 95 percentconfidence is observed to be less than 0.05. Therefore we ignore the null hypothesisand consider the alternate hypothesis. Also, here the t-value which signifies thesize of the difference relative to the variation of the sample data is observed tobe 31.094. Therefore we can conclude that when A-star algorithm and MILP areimplemented in Scenario 2, there is a significant difference in the computationtime.

5.2 ValidationSimulation is a concept related to experimental methods performed in a controlledenvironment. We can evaluate the performance of a technology by executing iton a model similar to the real world environment [22]. Sometimes, it is neededthat we predict or make a few assumptions about how the real world will reactto the technology. Simulations are widely preferred as it is easier and cheaper toexecute than running the entire system in the real world. Often the results of asimulation experiment are validated by comparing them to the theoretical resultsobtained [22]. In this case, the experimental results observed in Chapter 4 arevalidated with statistical analysis aswell as theoretical analysis.


5.2.1 Validation through Statistical Analysis

The results that were presented in Chapter 4 are statistically analysed using In-dependent Samples T-Test. Initially, the results are considered where the typeof method used i.e., A-star algorithm or MILP is taken as the independent vari-able and the computation time of the algorithm is considered as the dependantvariable. The results are now tabulated and are tested for normality, that is, ifthe results observed are normally distributed. From the observations presentedin Figure 5.1, we conclude that the results obtained are normally distributed.

Further, the results are tested for Homogeneity of Variance. This test is donewith Levenes Static. The test helps us determine if the variance of the samplesare statistically similar. From the observations made in Figure 5.2 and Figure5.3, we conclude that, in both the scenarios the variances are homogeneous.

5.2.2 Hypothesis

The results are subjected to Independent Samples T-Test to validate the hypoth-esis that is proposed in Section 3.5. From the results presented in Figure 5.5 andFigure 5.7, the null hypothesis is rejected. This conclude that there is indeed asignificant difference in the values of computation time for A-star algorithm andMILP in Scenario 1 aswell as Scenario 2. By comparison of means as presentedin Figure 5.4 and Figure 5.6, we can therefore conclude that MILP performs sig-nificantly better than A-star algorithm.

5.2.3 Theoretical Validation

A-star AlgorithmThe A-star algorithm computes the shortest distance between two nodes tra-versed by the algorithm. In this case, the shortest distance travelled by the UAVis calculated. The shortest path is calculated for two scenarios, when obstacles arenot encountered by the UAV and when the obstacles are encountered by the UAV.

As the time complexity of A-star algorithm is theoretically observed to beexponential, it means that there is a significant increase in the time taken by thealgorithm when the algorithm has to processes a new node. The results fromChapter 4 indicates that when no obstacles are encountered by the UAV, theaverage execution time observed is 34.3 seconds as seen in table 4.1. When anobstacle is encountered by the UAV in its path, the time taken for calculatingthe route is observed to be 68.3 seconds as seen in Table 4.2. This means thatthe time taken for calculating the route when obstacles are encountered is almost


twice as much as the time taken for generating a route without obstacles. Weobserve a significant increase in the computation time. This validates the resultsobtained for A-star algorithm.

To obtain the average time for generating the optimal route for the UAV fromstarting point to the ending point, the program using the algorithm is executedthree times and the average of the three repetitions is considered.

MILPMILP problems are generally NP-compete and sometimes may transform into NP-hard problems [15]. This means the estimation of computation time for MILPtheoretically is not possible. Therefore optimisation solvers are used along withmodelling languages for generation of the environment and solving the MILPproblem.

MILP is used in combination with AMPL and CPLEX optimisation tool. Itcomputes the shortest route from the starting point to the ending point. AMPL isgood for modelling environments for simulations along with CPLEX optimisationtool [36]. The program then proceeds to find the best path from starting point tothe final node by avoiding obstacles generated in the environment. The CPLEXoptimizer is used to generate the optimal result. The program execution is re-peated three times for each scenario to get consistent values. The computationtime for the algorithm in the case where no obstacles are encountered and thecase where obstacles are encountered by the UAV is taken for the three repetitionsand the results are averaged.

5.3 Computational Differences of A-star Algorithmand MILP

The A-star algorithm and MILP were simulated in a controlled environment andwere given test scenarios to compute. Both the algorithms generate an optimalpath for a UAV in two scenarios, where obstacles are not encountered and whereobstacles are encountered by the UAV. The computation times were measuredalong with the number of functions that the algorithms need to execute. Theefficiency of these algorithms was calculated based on these results.

Although both the algorithms produced an optimal route for the UAV, therewere a few computational differences between the two algorithms. These differ-ences affect the efficiency of the algorithm when a comparison is made againsteach other. These computational differences observed are quite significant forexperimentation with these algorithms. The factors that have a direct effect on


the performance of the algorithms are computational time and problem size.

Effect on computation time: The A-star algorithm, when executed, tends tofind out the best route from the starting point to the ending point. It calculatesthe cost of reaching the goal as a function of distance. The time taken to com-pute the optimal route using the A-star algorithm is, therefore, dependent on thedistance between the starting point of the UAV and the ending point of the UAVand the position of the obstacles. Whereas in the case of MILP, the computationtime depends on the performance of the computer and solver that is being usedto compute the problem. The computation time can be increased but often themethods used are problem specific.

Effect on problem size: For A-star algorithm, the problem size increases withthe computation of extra nodes. This might be the case when there are more ob-stacles. The computation cost increases exponentially when an additional nodehas to be parsed because the node should be parsed all the way down till itsleaf nodes. This increases the cost significantly. Therefore we can say that theproblem size has a direct effect on the efficiency of A-star algorithm. For MILP,however, the computation cost is dependent on the system and is problem spe-cific. With the increase in the size of the problem, the problem instance turns intoan NP-hard problem. This means that computation of a larger problem instancemay be slow.

For the generated environment A-star algorithm and MILP give an efficient result.But A-star algorithm is observed to have lower efficiency in both the scenarioswhen compared to MILP. However, the MILP uses AMPL modelling and CPLEXoptimizer for optimizing the path and generating the optimal results which makeit faster. Therefore we can say that in a case where the problem is a simple routeplanning problem A-star algorithm is easy to compute but for a complex problemMILP is more effective.

Chapter 6Discussion and Limitations

Unmanned Aerial Vehicles are those which can operate with a minimum humaninteraction [43]. The path for UAVs is generally generated using path planningalgorithms. Waypoints are generated and the UAV follows the path generated bythe algorithm [43].

The usage of UAVs has now increased in real time and are being used formany purposes effectively [43]. Complex tasks such as surveillance and monitor-ing, weather prediction and other such tasks have been made easier and risk-freewith the advent of UAVs. Remotely monitored UAVs, however, need line of sight.Remotely monitored air vehicles are currently used by the military for defencepurposes [43]. There will be a need to automate the UAVs. Therefore to makethe usage of UAVs efficient, a good path planning algorithm is needed.

Path planning is one of the areas that is quite commonly addressed in theplanning and development of UAVs [8]. A path planning algorithm computes thepath of a UAV from a desired starting location to a future location [8]. A pathplanning algorithm is said to be efficient if it optimizes several attributes [8]. Twoof the most commonly used path planning methods are the A-star algorithm andMILP. Although these methods have been previously proposed along with a fewvariations and improvements, a comparative study is lacking between these twomethods where the measure of performance is estimated in terms of computationtime and space.

The two methods were extensively researched upon and the pseudo codes wereanalysed. Both A-star algorithm and MILP result in an optimal path from thestarting point of the UAV to the final position. A-star algorithm and MILPwere simulated in Matlab where a program was written for A-star algorithm andMILP. The environment for both the methods was created so that the startingpoint and ending point of UAV is determined and also the position of obstacles isdetermined. Two scenarios were considered for the generation of the path of theUAV, one where no obstacles are encountered by the UAV and the other whereobstacles are encountered by the UAV. The optimal paths are generated for both

45

Chapter 6. Discussion and Limitations 46

the scenarios for the two methods.

The A-star algorithm and MILP were executed on Matlab software and thecomputation time is observed using the profiler. This provides us with the timefor the entire computation as well as the time taken for execution of each function.The execution times for both the methods are noted along with the number offunctions used by the program to execute the algorithm. The computation timesand the number of functions used is observed for both the methods when an ob-stacle is encountered and when no obstacle is encountered in the path of the UAV.

The performance of an algorithm is generally determined by the efficiency ofthe algorithm. The efficiency of an algorithm is the amount of resources used bythe algorithm for proper execution of the algorithm to produce the best possi-ble result for a given scenario. The efficiency of A-star algorithm and MILP arecomputed as a function of computation time and the number of functions used.This provides us with efficiencies of both the methods in the scenario where anobstacle is encountered and where no obstacle is encountered by the UAV. Theseefficiencies are compared against each other to get a clear view as to which algo-rithm performs better and in what conditions.

This experiment helps us to identify the efficiency of the A-star algorithm andMILP in conditions where obstacles are encountered and where obstacles are notencountered. The efficiencies are further compared with each other to observewhich algorithm performs better in the given scenario. From this research, wecan conclude that the performance of MILP is significantly better than the per-formance of A-star algorithm. This observation can help real-time path planningprograms to use MILP instead of A-star algorithm to achieve a better perfor-mance. The efficiency factor is useful in path planning for UAVs because in areal scenario there may be cases where the problem size is huge and difficult tocompute [15]. MILP proves to be useful in such a scenario [15].

A-star algorithm is useful for easy and simple computation of paths when theproblem size is limited according to [19] [44]. Also, the author of [12] presents thatthe computation time of A-star algorithm greatly increases when it has to parsea new node. This proved to be right in the case of this study as the number offunctions used for implementing A-star algorithm is significantly less. Also whenthe UAV encounters an obstacle, the computation time is increased. Thereforewe can infer from the analysis presented in [19] and [44] that the A-star algo-rithm is effective when the problem size is limited. The increase in computationtime is observed when the obstacles are encountered which validates the analysispresented in [12].

The results and analysis from this study justify that performance of MILP is


better when compared to A-star algorithm. This analysis can be used in real-time path planning for UAVs where the problem size is significantly large and thenumber of variables is higher. MILP yields better results in terms of efficiencywhen the constraints are higher and the problem size is large [15]. For a real-timepath planning program with a large and complex scenario, MILP is better thanA-star algorithm.

6.1 LimitationsThe performance evaluation of algorithms is always system dependent. Thismeans there might be a slight change in the values observed with the computa-tion time of the algorithms when executed on a different system with a differentprocessor. Also, for a simulation experiment, the values of the environment onwhich the algorithm is executed will be valid only for that algorithm on thatsystem. When the algorithm is applied to UAV in real-time considering a largeinput, the values may differ accordingly but the change in the values is relative.This is the major limitation for the proposed simulation experiment.

6.2 Threats to ValidityAn experiment should always be validated to maintain the quality of the resultsobtained. Therefore an emphasis on the threats to validity is essential for anexperiment. Generally, for an experimental research as done in this study, thereare two major threats to validity which includes internal validity and externalvalidity [45]. Each of these types of threats is individually explained and ways tomitigate them are discussed in this section.

6.2.1 Internal Validity

Internal validity deals with the degree of control over the design of the study athand [46]. The level of internal validity depends on the degree of control appliedon the variables [46]. In the experiment conducted for this study, a number ofvariables are considered for defining the environment. These variables have adirect effect on the results. Therefore there is a threat to internal validity wherethe values for the variables are huge thereby increasing the total problem size.Also, the number of iterations that were performed were 10 for each scenario forA-star algorithm and MILP which may result in a change in the mean of thecomputation times.


This threat can be mitigated by carefully checking the values that are beingassigned to the variables. These environment variables define the path generatedby the algorithm for the traversal of UAV from the starting point to the endingpoint. Also, the execution of the program is repeated so that the computationis done properly and the results obtained are properly validated. The numberof iterations is taken as 10 because the change in the means is observed to beinsignificant as the standard deviation observed is very less.

6.2.2 External Validity

External validity can be described as the extent to which the results that are ob-served in a study can be generalised [46]. This observation may affect the observedresults for other situations [46]. In the experiment conducted for this thesis, gen-eralisation in consideration of the obstacles that the UAV may encounter in itspath applies to this experiment. This might affect the computation time of the al-gorithm thereby affecting the performance of the algorithm for different scenarios.

This threat can be mitigated by randomizing the distribution of the obstaclesfor each repetition of the experiment. This is done in the simulation because, inreal-time, the occurrence of obstacles is random. Though this has an affect on theperformance of the algorithm, the risks of a drastic change in time are minimal.Therefore we can say that randomization of objects leads to results that are moregeneralised and help in computing the execution time of the algorithms close toreal-time.

Chapter 7Conclusions and Future Work

Based on the extensive research and experiment performed as a part of this re-search study, the answers to research questions that were formulated are as fol-lows:

RQ 1. What is the efficiency of A-star algorithm and MILP withrespect to computational space and computational time?

The efficiency of A-star algorithm with respect to computational space andcomputational time is exponential. It can be represented using Big-O notationas O(2n). This means that the computational time and computational spaceincreases drastically when a new node is to be traversed by the algorithm. Theefficiency of MILP depends on the environment and problem variables. The exacttime complexity and space complexity cannot be evaluated theoretically. This isbecause the problem is generally NP-hard. Therefore the efficiency of algorithmis usually computed based on simulations using optimizers for optimal route gen-eration. This means that the efficiency of MILP is dependent completely on theproblem and the system that is being used for computation.

RQ 2. What are the most significant computational differences be-tween A-star algorithm and MILP?

There are many computational differences between A-star algorithm and MILP.The few significant ones being that A-star algorithm aims at solving the pathplanning problem to obtain the best possible route by checking for an optimalsolution, whereas MILP is completely dependant on the environment variablesand depends on optimisation solvers for generating an optimal route based onthe cost of the path. This makes MILP more complex but effective in many casesin a real-time environment where the occurrence of the objects is dynamic. Theremaining computational differences are discussed in Section 5.3 of Chapter 5.

RQ 3. What are the performance of A-star algorithm and MILP pathplanning algorithm in a simulated environment?

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Chapter 7. Conclusions and Future Work 50

The performance of an algorithm is the extent to which the resources areused for execution of the algorithm [4]. The performance of A-star algorithm ina simulated environment can be generalized and said to be exponential. Thismeans the performance decreases with the increase in the problem size. Howeverthe performance of MILP is observed to be optimal and completely dependent onthe variables that affect the environment and the objects involved.

7.1 Future WorkFuture work in the field of path planning algorithms for UAVs is quite signifi-cantly large as there are a lot of features that can be studied in this field. Thefuture work that can be done for this is study involves the development of aneffective and an efficient way to implement the algorithm in a dynamic environ-ment where the position of objects in the path of UAV are changing from time totime. This makes the computation of performance more likely to be applied in areal-time scenario. Potential field algorithms can also be used for better execu-tion of collision detection and avoidance of the UAVs. Also, the existing A-staralgorithm and MILP could be worked upon to formulate new and more efficientpath planning strategies using these algorithms. Further, a few elements of thealgorithms can be reworked upon to generate more efficient algorithms therebyimproving the productivity of the algorithms.

D = log2(1 +1

D0

K∑k=1

|d̂k(pk||qk)|) (7.1)

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Appendices

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Appendix ACode for A-star algorithm

Figure A.1: Creation of environment by assigning values to variables

56

Figure A.2: Code for A-star algorithm

Figure A.3: Code for calculating the distance between the nodes

57

Figure A.4: Code for obtaining the optimal path

Figure A.5: Code for plotting the path of the UAV

58

Appendix BCode for MILP

Figure B.1: Code for initializing the environment of MILP

Figure B.2: Code for solving MILP

59

Figure B.3: Code for plotting the path obtained by MILP

60

Appendix CCode for AMPL model

Figure C.1: Code for environment in the model created for MILP

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Figure C.2: Code for setting up constraints for AMPL model

62

Figure C.3: Code for calculating the cost of paths in AMPL model

Figure C.4: Code for loading and running the AMPL model

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Appendix DStatistical Analysis

Figure D.1: Histogram Depiction of the Results

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Figure D.2: Normal Q-Q Plot Depiction of the Results

65

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