Algorithm for solving the Discrete-Continuous Inspection Problem

Abstract

The article introduces an innovative approach for the inspection challenge that represents a generalization of the classical Traveling Salesman Problem. Its principle idea is to visit continuous areas (circles) in a way, that minimizes traveled distance. In practice, the problem can be defined as an issue of scheduling unmanned aerial vehicle which has discrete-continuous nature. In order to solve this problem the use of local search algorithms is proposed.

Full text

10.24425/acs.2020.135845
Archives of Control Sciences
Volume 30(LXVI), 2020
No. 4, pages 653–666
Algorithm for solving the Discrete-Continuous
Inspection Problem
R. GRYMIN, W. BOŻEJKO, Z. CHACZKO, J. PEMPERA and M. WODECKI
The article introduces an innovative approch for the inspection challenge that represents a
generalization of the classical Traveling Salesman Problem. Its priciple idea is to visit continuous
areas (circles) in a way, that minimizes travelled distance. In practice, the problem can be defined
as an issue of scheduling unmanned aerial vehicle which has discrete-continuous nature. In order
to solve this problem the use of local search algorithms is proposed.
Key words: discrete-continuous optimization, UAV, VTOL, autonomous drone, zero emis-
sion
1. Introduction
In recent years, unmanned aerial vehicles (UAV) have become very popular in
many areas. Inspection is one of the applications. Their advantage over terrestrial
vehicles is independency of a finite number of routes, due to the fact they do
not participate in traffic jams and can perform inspections without having to
continously interact with humans. Manual control of an unmanned aerial vehicle
is associated with the danger of possible operator errors – in order to eliminate
these errors, autonomous inspection algorithms are used.
Copyright ©2020. The Author(s). This is an open-access article distributed under the terms of the Creative Com-
mons Attribution-NonCommercial-NoDerivatives License (CC BY-NC-ND 4.0 https://creativecommons.org/licenses/
by-nc-nd/4.0/), which permits use, distribution, and reproduction in any medium, provided that the article is properly
cited, the use is non-commercial, and no modifications or adaptations are made
R. Grymin, W. Bożejko (corresponding author) and J. Pempera are with Department of Control Sys-
tems and Mechatronics, Wrocław University of Science and Technology, Wyb. Wyspiańskiego 27, 50-370
Wrocław, Poland, e-mails: {radoslaw.grymin,wojciech.bozejko,jaroslaw.pempera}@pwr.edu.pl.
Z. Chaczko is with Faculty of Engineering and Information Technology (FEIT), University of Technol-
ogy, Sydney(UTS), Australia, NSW, Sydney, e-mail: zenon.chaczko@uts.edu.au.
M. Wodecki is with Department of Telecommunications and Teleinformatics, Wrocław Univer-
sity of Science and Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland, e-mail: mieczys-
law.wodecki@pwr.edu.pl.
The paper was partially supported by the National Science Centre of Poland, grant OPUS no.
2017/25/B/ST7/02181 and statutory grant no. 0401/0023 /18 of Faculty of Electronics, Wrocław University
of Science and Technology.
Received 10.08.2019. Revised 10.05.2020.

654 R. GRYMIN, W. BOŻEJKO, Z. CHACZKO, J. PEMPERA, M. WODECKI
A serious challenge connected with using an electric drive is a relatively
short flight time compared to the time required to fully charge its UAV battery.
For example, one of the most modern UAVs DJI Matrice 200 (Fig. 1a), widely
used in inspection, can fly from 13 up to 38 minutes until its complete discharge
(depending on battery used, mounted equipment and speed). An alternative to
UAV in the future will be the use of small VTOL (Vertical Take-Off and Landing)
planes that allow its users not only for vertical takeoff but also for vertical landing.
An example of an autonomous VTOL is the Airbus A3 Vahana [16] shown in
Fig. 1b.
(a) DJI Matrice 200
Source: dji.com
(b) Airbus A3 Vahana.
Source: vahana.aero
Figure 1: Examples of aerial vehicles enabling inspections
The use of an autonomous flight system in a vehicle, in the future, makes it
available to people who do not have a professional helicopter pilot license (as on
31/12/2017 only 278 people had such a license in Poland [5]). The vehicle has
an electric drive, thanks to which it has zero emissions and is so quiet that it can
carry out inspections in the city without exceeding the noise standards. While
writing this article, Airbus A3 Vahana was still in the testing phase.
The use of UAV or VTOL is associated with the problem of long battery
charging. Therefore, the autonomous inspection requires the use of effective
flight planning algorithms. In this article an effective algorithm for solving the
problem of autonomous inspection is presented. In Section 2, the concept of
Inspection Problem is introduced as a challenge that can be solved by applying a
novel two-level optimization solution strategy. At the first level, the fast Powell’s
method is used to optimize the flight path, and at the second level, a higher
level optimization procedure based on the 2-Opt and the Simulated Annealing
algorithms is introduced. Section 3 discusses details of experimental research
conducted, whereas Section 3.1 covers a description of the algorithm that was
used for generation of the presented Inspection Problem instance. Section 3.2
contains a description of computational experiments that were conducted. Finally,
Section 4 summarizes the results of the conducted research work.

ALGORITHM FOR SOLVING THE DISCRETE-CONTINUOUS INSPECTION PROBLEM 655
2. Inspection problem
In the problem of data inspection there are oobjects from the set O=
{1,2,3, . . . , o}, for which the inspection must be conducted. For each object
i∈Othere is defined three-dimensional subspace, from which clear pictures
can be taken. This space will be brought closer with the use of a sphere. The
optimization problem is to find the shortest path of the aircraft, which passes
through all areas of visibility at least once (in order to take pictures). The fly-
ing object starts from a certain starting point and must return to it after the
inspection.
This problem is a generalization (continuous version) of Set TSP, also known
as: Generalized TSP (GTSP), International TSP,Group TSP, One-of-a-Set TSP,
Coverng Salesman problem or Multiple Choice TSP. The problem is strongly
NP-hard as it can be reduced to TSP when the areas are single points. GTSP
appears frequently in traffic planning problems (Imeson and Smith [9]; Mathew
et al. [11]; Wolff et al. [18]). For this class of problems, many heuristics and exact
algorithms were proposed in the literature, including evolutionary algorithms
(Snyder and Daskin [15]), also in the parallel version (Bożejko and Wodecki [4]).
Thus far, there is a limited, if any, research material or literature that covers the
continuous version of GTSP.
In further considerations it is assumed that the visibility space of the object
being inspected can be approximated by means of a hemisphere (for example in the
case of photographing objects and transmitter – radio receiver communication)
and that the unmanned aircraft flies at a fixed ceiling and has no kinematic
limitations, i.e. it can navigate any curves as it is a holonomic robot. Then the
inspection area for each inspection object can be described with the use of a
sphere. This geometrical object will be called the visibility circle, whilst the point
at which the inspection (e.g., photographing) is performed will be called the the
inspection point.
For each i∈Oobject, there is a visibility sphere (space) specified with
triangle (xi,yi,ri)given, where xiand yidenote the center coordinates, while
ri(ri>0) denotes the radius of the visibility sphere of the object. Before the
flight begins, the unmanned vehicle is at the point with coordinates (x0,y0). After
the inspection, the unmanned vehicle must return to this point. The purpose of
the optimization is not only to determine the inspection point for each object
but also to find the order of visiting inspection points for which the length of
the distance traveled by the unmanned aircraft is the shortest. To illustrate the
problem, a sample solution acceptable for a certain problem instance is presented
in Figures 2 and 3.
The object’s inspection point i∈Owill be denoted by a pair (Xi,Y
i), which
of course must belong to the visibility circle (xi,yi,ri). The order of visiting
inspection areas can be described using sequences S=(0,S1,S2, . . . , So,0),

656 R. GRYMIN, W. BOŻEJKO, Z. CHACZKO, J. PEMPERA, M. WODECKI
Figure 2: Illustration of disjoint visibility circles, starting point 0 (zero)
and path found by the algorithm
Figure 3: Illustration of the overlapping circles of visibility, starting
point 0 (zero) and the inspection plan found by the algorithm

ALGORITHM FOR SOLVING THE DISCRETE-CONTINUOUS INSPECTION PROBLEM 657
where Si∈Odenotes the inspection object, while 0 (zero) denotes the starting
point.
For the given inspection points of the (Xi,Y
i),i∈Oobjects, and the order
of visiting the areas S=(0,S1, ..., So,0)the length of the flight path can be
determined from the formula
L(S,X,Y)=
o+1
∑
s=1
d(XSs−1,YSs−1,XSs,YSs),(1)
where d(Xk,Yk,Xl,Yl)is the Euclidean distance between points (Xk,Yk)
and (Xl,Yl). In the optimization problem being considered there are two types of
decision variables:
1) S– sequence of visiting inspection areas (permutation, see e.g. [3]),
2) X,Y– inspection points in each area (real variables).
Determining the shortest flight path of a flying object requires designing al-
gorithms combining the characteristics of continuous and discrete optimization
algorithms.
Before introducing the proposed optimization algorithm, we will present a
certain property of the problem allowing to reduce the size of the problem (the
number of considered inspection areas) without losing the possibility of finding
an optimal solution for the primary problem.
Lemma 1 Let (xi,yi,ri)be a circle contained in the circle (xj,yj,rj)i.e. there is
d(xi,yi,xj,yj)+ri¬rj
then the circle (xj,yj,rj)can be omitted during the path determination.
Proof. Let us assume that ri¬rj, and that a circle with radius riis contained in
a circle with radius rj. Then, by visiting a certain point (Xi,Y
i)belonging to the
smaller circle (xi,yi,ri), we visit simultaneously the point which belongs to the
biggest circle (xj,yj,rj), which ends the proof. □
2.1. Optimization algorithm
In the optimization problem under consideration, there should be two com-
ponents determined: the order in which inspection areas are visited and in each
area – the inspection point. Both the selection of the order and position of the
inspection points affects the length of the flight path of the flying object (see
Eq. (1)). To determine the shortest path of a flying object, we propose a hierarchi-
cal algorithm in which the high level is responsible for determining the sequence
of visited areas, while the low level is responsible for finding the inspection points

658 R. GRYMIN, W. BOŻEJKO, Z. CHACZKO, J. PEMPERA, M. WODECKI
for a given sequence, so that the path length is the shortest possible. The aim of
the high level is to find a sequence for which the route length determined by the
low level algorithm is the shortest.
Before discussing the algorithm for designating inspection points for a given
sequence of Sof visiting areas, we must notice that
Lemma 2 The feasible flight path of a flying object must have at least one point
in common with the circumference of each visibility circle.
For proof of lemma 2, it is enough to note that the inspection point must be in
the circle of visibility, so if it is not on the circumference, then the flying object
first had to go through a point on the circumference and then leave the same or
another point on the circumference circuit.
On the other hand, each point on the perimeter of the circle can be clearly
described by the angle between the OX axis and a line connecting inspection
point with the location of the inspection object. Finally, the object’s inspection
point i∈Ois uniquely specified using the angle αi. Cartesian coordinates of a
point can be determined by means of equations:
Xi(αi)=xi+ricos(αi),(2)
Y
i(αi)=yi+risin(αi).(3)
For a given sequence S,L(S,X(α),Y(α)),α=(0, α1, ...αo,0),X(0)=X0,
Y(0)=Y0is non-linear function with ovariables. The minimum of the function
L(S,X(α),Y(α)) can be determined with the use of such algorithms as: Limited-
memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algortihm (Byrd, Lu and
Nocedal [6], Zhu et al. [19]), The Nelder-Mead method (Nelder and Mead [12]),
Sequential Quadratic Programming (SQP) method (Kraft [10], Nocedal [13]).
Unfortunately, these methods are time-consuming due to the large number of
variables (inspection objects). Therefore, to find the shortest path, the greedy
algorithm is proposed, in which in each iteration the greatest improvement is
obtained by changing the value of one variable (the angle describing the point of
visibility).
More precisely, let us consider three consecutive inspection points
(Xa(αa),Ya(αa) ),(Xb(αb),Yb(αb)),(Xc(αc),Yc(αc) ). The angle αbis lo-
cally optimal angle for object bin reference to points (Xa(αa),Ya(αa)) and
(Xc(αc),Yc(αc)) if
d(Xa(αa),Ya(αa),Xb(αb),Yb(αb)) +d(Xb(αb),Yb(αb),Xc(αc),Yc(αc)) (4)
takes the minimum value. The angle α∗
b, for which the expression (4) takes the
minimum value was designated with the Powell’s method (Powell [14]).

ALGORITHM FOR SOLVING THE DISCRETE-CONTINUOUS INSPECTION PROBLEM 659
We determine by
∆b=L(S,X(α),Y(α)) −L(S,X(α∗),Y(α∗)),(5)
where α∗=(0, α1, ..., α∗
b, ..., αo,0), improving (shortening) the length of the
shortest path by optimizing the angle αb.
In each iteration of the algorithm, an optimal angle is found locally for each
inspection object. Then an object is designated for which the angle change gener-
ates the greatest improvement. This angle becomes the base angle for this object
in the next iteration of the algorithm.
The algorithm stops when the improvement is not greater than the arbitrarily
selected value of ϵor after the execution of M axIter iterations. The scheme of
the proposed method is presented in the Algorithm 1. In order to find the best
sequence of visiting inspection areas, one can apply one of many algorithms
dedicated to scheduling problems available. However, for the start, it should be
noted that the investigated problem is very similar to a well-known Traveling
Salesman Problem which aims to find a solution for the optimal order of visiting
inspection areas. In the investigated problem, the positions of the inspection
points and the resulting distances between the points depend on the sequence
of visiting inspection areas whilst in the classical Traveling Salesman Problem
(TSP) the position of inspection points are fixed.
Algorithm 1: Designation of the shortest path for a given sequence S
Input : α0– initial angles vector;
ϵ– precision parameter;
Output : α=(α1, α2, . . . , αo);
1α←α0
2for iter ←1to M ax It er do
3for b←1to odo
4determine ∆b
5determine such kthat k=arg maxs∈O∆s
6if ∆k< ϵ then
7return α
8αk←α∗
k
Considering the similarity of the investigated scenario to the Traveling Sales-
man Problem, not only a 2-Opt algorithm was developed but also an algorithm
based on the simulated annealing method was designed, which is one of the most
effective methods of constructing algorithms for sequential problems.

660 R. GRYMIN, W. BOŻEJKO, Z. CHACZKO, J. PEMPERA, M. WODECKI
2-Opt Algorithm
2-Opt Algorithm (Croes [7]) is one of the simplest and highly effective algo-
rithms for the traveling salesman problem. It is an algorithm that determines the
local optimum for the 2-Opt neighborhood. The 2-Opt neighborhood consists of
sequences being a modification of the base sequence S=(0,S1, ...., So,0). Each
modification (also called a move) is described by a pair v=(a,b). As a result
of the modification of the base sequence described by v=(a,b)is a new route
which takes the form
Sv=(0,S1, . . . , Sa−1,Sb,Sb−1, . . . , Sa+1,Sa,Sb, . . . , So,0).(6)
In each iteration of the neighborhood algorithm, the best sequence is selected
which becomes the base sequence in the next iteration. The algorithm terminates
when no better solution is located in the vicinity of the base solution, i.e. the base
solution is locally optimal.
Simulated Annealing Algorithm
The simulated annealing algorithm (SA, see e.g. [2,17]) refers to the thermo-
dynamic process of metal annealing, where the metal sample is subjected to a
cooling process to achieve the desired properties (hardness, elasticity, etc.). The
idea of operating of SA boils down to generating a trajectory of search (where
each successive solution is chosen randomly from the neighborhood of the cur-
rent solution) based on random moves in the neighborhood. The drawn solution
is accepted (it replaces the current solution in the next iteration) unconditionally
when it is not worse than the current one. It is also possible to accept worse
solutions with a probability depending, for example, on the temperature and the
difference in the value of the objective function (the so-called acceptance func-
tion). A detailed description of the method can be found in [1]. In the algorithm
implemented to solve the problem under consideration, the 2-Opt neighborhood
was used. The algorithm performed M axIter =2000 iterations. As a result, the
trajectory of searching for the SA algorithm is carried out in ’statistically good’
direction.
Finally, three algorithms were implemented: 2-Opt (C), 2-Opt and SA. In the
2-Opt(C) algorithm, the route length was determined based on the location of the
inspection objects, i.e.
L(S,x,y)=d(x0,y0,xS1,yS1)+
o−1
∑
s=1
d(xSs,ySs,xSs+1,ySs+1)+
+d(xSo,ySo,x0,y0).(7)

ALGORITHM FOR SOLVING THE DISCRETE-CONTINUOUS INSPECTION PROBLEM 661
3. Experimental research
The results of the research are strongly influenced by the selection of appro-
priate test instances and the selection of appropriate metrics. The aim of this
research was to check the efficiency of the algorithm for a problem that has not
been described yet, so there are no test results for other algorithms to compare its
efficiency.
3.1. Data generating
The algorithm should be used for any instances of the inspection problem, it
is not dedicated to specific instances of the problem (e.g., in which the inspection
areas are arranged schematically, according to some rule), hence the test instances
were generated in a random manner.
There were 6instance groups generated with the same number of objects to
visit n∈ {5,10,15,20,25,30}. Due to the fact that the instances were generated
randomly, each instance group contained 10 instances of the problem. Such a
procedure was designed to allow us to investigate what the average expected per-
formance of the algorithm for the ninstance would be. Each instance is described
by the means of centers of circles, the length of their radiuses and coordinates of
the base. The instance generating algorithm randomly placed the centers of circles
on a square of 1000 side, and the radiuses randomized with the uniform distri-
bution of U(a,b)with the distribution parameters characteristic for each group,
(a,b)∈ {(100,200),(50,150),(40,120),(30,100),(30,100),(30,100)}. Higher
size instances had to contain smaller circles to get some overlapping circles and
some non-overlapping circles. The way of generating data: non-overlapping and
overlapping circles is presented in Algorithms 2 and 3, respectively. An algo-
rithm that generates circles that are not overlapping, used to illustrate the case of
non-overlapping circles (Figure 2), works on the principle of adding successive
circles. It starts its operation from drawing the base coordinates basex,basey
(each coordinate is drawn with the uniform distribution U(0,1000)). Then, in a
loop, he adds more circles in turn. The addition of a new circle consists of two
stages: the drawing of the (center) and the radius. The center of the circle is
drawn by the computeCenter function until it is possible to add a circle that
has a radius at least equal to minRadius and is distant from the other circles
and base by at least minSpaceBetweenCircles.
When such a measure is found, the function maximalRadiusForCircle
sets the maximum possible radius of the circle maximalRadiusForCircle,
so that the distance from the remaining circles and base is not less than
minSpaceBetweenCircles. At the very end a radius draw is made with
the distribution of U(minRadius,maximalRadiusForCircle).
The algorithm that generates overlapping circles (Algorithm 2) at the begin-
ning draws coordinates of the base (each coordinate is drawn with the distribution

662 R. GRYMIN, W. BOŻEJKO, Z. CHACZKO, J. PEMPERA, M. WODECKI
Algorithm 2: The algorithm of generation of overlapping circles
1base_x ←rand() ·1000;
2base_y ←rand() ·1000;
3base ←(base_x, base_y);
4circles ←[ ];
5for iter =1to ndo
6x←rand() ·1000;
7y←rand() ·1000;
8r←rand( ) ·(b−a) +a;
9{append 3-tuple to the end of circles vector}
10 circles ←(circles |[(x, y, r)]);
11 return base, circles;
Algorithm 3: The algorithm generating non-overlapping circles
1centers ←[ ];
2radii ←[ ];
3base_x ←rand() ·1000;
4base_y ←rand() ·1000;
5base ←(base_x, base_y);
6for iter =1to ndo
7center ←computeCenter(centers, radii, base, minRadius, areaWidth,
areaHeight, minSpaceBetweenCircles);
8maximalRadiusForCircle ←maximalRadiusForCircle(center, base, centers,
radii, maxRadius, minSpaceBetweenCircles);
9r←rand( ) ·(maximalRadiusForCircle−a) +a;
10 radii ←(radii |r);
11 centers ←(centers |[center]);
12 return centers, radii, base;
of 1000). Next, further circles are generated in the loop. Coordinates of centers
are generated randomly with the distribution of U(0,1000), whereas the radius
is generated with the distribution of U(a,b). The tests used instances in which
some circles overlapped.
3.2. Computational experiments
2-Opt (C), 2-Opt and SA metaheuristics were programmed in C++ in Visual
Studio 10 and run on a single processor core Intel Core i7 3.5 GHz, with 8 GB
RAM running under the operating system Windows 7. The SA algorithm was
designed in the version with auto-tuning of the initial parameters (see [1]). In order

ALGORITHM FOR SOLVING THE DISCRETE-CONTINUOUS INSPECTION PROBLEM 663
to compare the effectiveness of individual optimization stages, the (percentage
relative deviation, PRD) was examined between the length of the path SAreturned
by the tested algorithm A∈ {2-Opt(C), 2-Opt, SA}and the length of the best
found path L∗.
This measurement is defined by the equation:
PRD(A)=L(SA)−L∗
L∗·100%.(8)
Figure 4 shows the order in which algorithms are run to find an optimized path.
The 2-Opt(C) algorithm generates in a very short time the path that is improved
by the 2-Opt algorithm and three times by the SA algorithm. After completing
each calculation phase, the length of the shortest path is memorized. During the
next phase of calculations, the best path from the previous phase becomes the
initial path for the next phase.
Figure 4: The order of running optimization algorithms
Table 1 presents the percentage relative error PRD of solutions returned by
the subsequent phases of the algorithm. This error was determined in reference
to the shortest path length L∗obtained in the last run of the SA algorithm (SA-3).
It can be seen that the length of the path when the aerial vehicle is approaching
the object is much longer than the length of the path returned using the proposed
method. It is longer on average by 39.14%, whereas the average value varies
between 32.56% and 46.03%. These observations confirm the high effectiveness
of the proposed methods of path determination for the aerial vehicle.
It is worth noting that the application of the proposed method to the sequence
generated by 2-Opt(C) (see column 2-Opt (C) L(S)) significantly reduces the
length of the route. The average PRD value is only 4.19% and ranges from

664 R. GRYMIN, W. BOŻEJKO, Z. CHACZKO, J. PEMPERA, M. WODECKI
Table 1: Average relative error of the algorithms
n2-Opt(C) LC (S)2-Opt(C) L(S)2-Opt L(S)SA-1 L(S)SA-2 L(S)
5 46.03 1.35 0.00 0.00 0.00
10 37.74 4.09 0.55 0.00 0.00
15 32.56 1.79 0.65 0.43 0.00
20 33.25 4.12 3.15 0.10 0.00
25 43.57 6.71 3.29 1.99 0.03
30 41.69 7.08 3.53 0.96 0.32
Average 39.14 4.19 1.86 0.58 0.06
1.35% to 7.08%. Unfortunately, its efficiency decreases as the number of objects
increases.
In the next phases, the algorithms operate on routes in which inspection points
are determined by the bottom-level method. In the case of the 2-Opt algorithm,
the average error is 1.86% and ranges from 0% to 3.53%. This error increases as
the number of objects increases.
4. Summary
The presented study considers a discrete-continuous problem of routing which
represents a generalization of the classical TSP problem concerning the deter-
mination of the unmanned vehicle routing. The test scenarios involved instances
where some of the path circles overlap with each other. The proposed cascade,
two-level optimization method allowed for a significant improvement in results
compared with the cases where the aerial vehicle was flying directly towards the
object being photographed, i.e. to the center of the target inspection area.
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