An Evolutionary Approach for the Dubins’ Traveling
Salesman Problem with Neighborhoods
Douglas G. Macharet
Armando Alves Neto
∗
Vilar F. da Camara Neto
Mario F. M. Campos
Computer Vision and Robotics Laboratory (VeRLab)
Department of Computer Science
Universidade Federal de Minas Gerais
Belo Horizonte - MG - Brazil
{doug,aaneto,neto,mario}@dcc.ufmg.br
ABSTRACT
General Terms
In this work we propose an efficient and simple three-stage
evolutionary algorithm to tackle the difficult problem of
planning shorter paths through regions of an environment
which are feasible for a nonholonomic vehicle with curvature constraints (e.g. Dubins’ vehicle). Our method is able
to efficiently solve both the combinatorial and the continuous steps of the problem in a combined manner. In the
first phase, the method varies the position of the waypoints
within the boundaries of each region, it then optimizes the
path orientation at each waypoint, and finally it chooses
the best actual sequence of visit. Numerous trials, under
different scenarios in a simulated environment, were executed providing a thorough evaluation and validation of the
methodology. The results show that a substantial improvement was obtained on the search for optimal paths in the
Dubins’ Traveling Salesman Problem with Neighborhoods
(DTSPN) over current works in the literature. Numerical
simulations also exhibit a significant performance improvement when compared with classical solutions that use the
Alternating Algorithm, and they also show that our method
outperforms a random sampling based technique. Our results present a reduction on the final path length of about
25% on average when compared to paths generated by the
aforementioned methods.
Algorithms, Performance, Experimentation
Keywords
TSP, Dubins’ path, Nonholonomic vehicles
1.
INTRODUCTION
Finding paths that are either length or time optimized has
been the goal of several research fields, and throughout the
years, several important results have been published. More
than ever, parsimonious use of energy is paramount, and the
search for energy efficient motion planning approaches are
fundamental. Broadly speaking, the well known Traveling
Salesman Problem (TSP) has been much investigated, and
several important results have been made available.
As it happens with several hard problems, the first solutions were concerned with finding the best possible route,
without taking into account the characteristics of the vehicle that would eventually perform it. In other words, several such approaches were designed for holonomic vehicles,
for which mobility restrictions are imaterial. However, when
motion constraints are present, due to the kinematic characteristic of the vehicle, the motion optimization tasks may
turn out to be very difficult or even impossible to solve. It
happens to be that the large majority of real world vehicles
are non-holonomic, and are based on platforms with several
motion constraints, in which case the classical TSP becomes
more than just a combinatorial problem.
Car-like vehicles, for example, are unable to instantaneously move orthogonally to the rolling plane of their wheels
due to kinematic constraints. Fixed-wing airplanes have limitations in their admissible curvature, climb angle and speed,
which may even prevent them from following straight line
paths produced by classical Euclidean TSP planners.
As far as the environment model is concerned, it is usually
assumed that the vehicle is required to reach an exact position (e.g. cities in the TSP formulation, or a given waypoint
for a robot). However, this is hardly the case in real world
applications, where such precision may be unattainable or
may not be even necessary. Therefore, when a vehicle does
not have to reach a pre-defined set of points, but a group of
regions instead, the difficulty of the problem becomes even
greater. This is the type of problem we deal with in this
paper.
Categories and Subject Descriptors
I.2.9 [Artificial Intelligence]: Robotics—autonomous vehicles; I.2.9 [Artificial Intelligence]: Robotics—kinematics and dynamics
∗Vilar F. da Camara Neto is also with the Fundação Centro de Análise, Pesquisa e Inovação Tecnológica (FUCAPI),
Manaus, AM, Brazil. E-mail: [email protected].
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GECCO’12, July 7-11, 2012, Philadelphia, Pennsylvania, USA.
Copyright 2012 ACM 978-1-4503-1177-9/12/07 ...$10.00.
377
This work describes our approach to tackle the problem
of planning shorter paths through regions of an environment
which are feasible for a non-holonomic vehicle with curvature constraints, based on an efficient and simple three-stage
evolutionary algorithm.
Our robot is modeled as Dubins’ vehicle, that is, a vehicle
whose constraints are satisfied by paths modeled as Dubins’
curves [6]. In that work, Dubins proved that optimal paths
for vehicles with limited curvature radius in R2 are composed by three subcurves, tangent to the initial and final
points with orientation angles previously specified.
Even though there exists a closed form solution to compute the length of a Dubins’ curve, the problem of calculating a circuit through a set of points turns out to be quite
difficult, since the orientation at each visited “city” must be
taken into account. Actually, this is the description of the
Dubins’ TSP (DTSP), a variant of the TSP in a manifold
space SE(2), which is one order greater than the conventional two-dimensional space.
Further extending the idea, we may also consider the use
of Dubins’ vehicles in even more complex scenarios, such as
the Traveling Salesman Problem with Neighborhoods (TSPN).
In the Dubins’ Traveling Salesman Problem with Neighborhoods (DTSPN) problem, there exist regions surrounding
the waypoints that the Dubins’ robots have to reached with
any orientation, and finding an optimal solution for this case
is quite expensive. Furthermore, the final path must be as
short as possible in order to increase the vehicle’s autonomy.
In this paper we present a solution to the problem of generating efficient (by efficient we mean short) paths for visiting regions in a known environment using a Dubins’ vehicle.
Combinatorial (referring to the sequence of visitation) and
continuous (as far as the values of position and orientation
at the waypoints is concerned) steps are treated in a combined manner using an Evolutionary Algorithm, in order to
minimize the overall length of the complete tour, thereby
reducing power consumption of the vehicle.
Our approach, which will be described later in the paper,
produces tour lengths that are statistically (≈ 25%, with
confidence interval of 95%) shorter than those provided by
state of the art algorithms in the literature.
The remainder of this paper is organized as follows: Section 2 discusses the works previously reported in the literature that have dealt with the related problems. Our methodology is detailed in Section 3 and validated by several experiments whose results are shown in Section 4. Finally,
Section 5 discusses the results and indicate future directions
for this investigation.
2.
RELATED WORK
The generation of paths for nonholonomic vehicles is a
topic of great relevance, especially in robotics. Numerous
techniques that address this problem have been proposed in
the literature.
Several shortest path algorithms for vehicles with no constraints have been reported. As an example, the Traveling Salesman Problem (TSP) is a fundamental optimization
problem and has been widely studied [3]. The fundamental
objective of the TSP is to determine the shortest path that
passes through a set of points (“cities”), starting from and
returning to any given point after having visited all other
points exactly once. Formally, the problem can be posed as
378
how to determine the shortest Hamiltonian cycle, which is
known to be NP-hard.
Several papers dealing with this problem, but considering
the constraints of the vehicle, are based of probabilistic algorithms. In [2] the authors propose a trajectory planning
technique for fixed-wing UAVs in environments with obstacles based on Rapidly-Exploring Random Tree (RRT). In
[12], path generation is obtained using Genetic Algorithms
and Bézier curves.
However, most of the above methods have as their main
focus the paths’ feasibility (e.g. which satisfies the kinematic
constraints of the vehicle), and usually neglect the length of
the generated path.
In its seminal work, Dubins showed a way to calculate the
shortest path between two points with assigned orientations
in the two-dimensional space, considering a vehicle with a
curvature constraint [6]. The resulting path is composed of
line segments (S) and arcs (C) that respect the curvature
radius of the vehicle. Figure 1 shows examples of Dubins’
curves connecting different configurations. There are basically two types of paths: the short case (CCC) and the
long case (CSC), or a subpath of a path of either of these
two types. The first one is a composition of three arcs with
the minimum curvature radius, while the other includes a
straight line between arcs.
ρ
ρ
(a)
(b)
Figure 1: Examples of Dubins’ curves connecting
configurations in two different cases: (a) the short
case and (b) the long case.
The problem of generating the minimum length circuit
visiting a set of points and considering the curvature constraint of the vehicle and making use of Dubins’ curves, was
initially introduced in [15]. This problem was named DTSP.
Figure 2 shows the difference between the results given by
the classical TSP and the DTSP. As it can be observed in
the figure, curvature restriction and distribution of points
in the environment may severely impact the length of the
generated path.
Another generalization of the TSP is known as TSPN, and
was presented in [4]. In this problem the salesman continues
to possess a circuit of visits to be made. However, it is assumed that each point has a given associated neighborhood.
So both the salesman and the buyer may meet in any place
within this neighborhood. The problem can then be defined
as how to generate the shortest path that intersects every
neighborhood at least once.
In [7], an algorithm is proposed for the TSPN with nonoverlapping regions. In [8] this restriction is lifted, and
neighborhoods are allowed to have intersecting regions. The
TSPN is often used to model problems related to the generation of tours to collect data from a wireless sensors network
[18, 5].
(a)
This representation encompasses a large class of nonholonomic vehicles that range from Ackerman steering cars to
fixed-wing airplanes.
As far as the underlying physics of the system is concerned, the curvature may be defined as a quantity that is
directly proportional to the lateral acceleration of the robot
in the plane. The value of κmax is inversely proportional
to the minimum curvature radius (ρ) of the curve that the
vehicle is capable to execute. This is also related to the
maximum lateral acceleration of the vehicle.
The DTSP is a TSP problem for which the path is composed by a Dubins’ curves. This implies two requirements:
(i) between any two points, the path is a Dubins’ curve and
(ii) orientations of two Dubins’ curves that meet at the same
point are equal.
Taking into account the formalization of the DTSPN problem, let H = {H1 , H2 , . . . , Hn } ⊂ R2 be a set of n closed regions representing waypoint areas to be reached by the vehicle. These regions are centered at coordinates ni = (xi , yi ),
which forms a set of n nodes. For simplicity, in this paper
we define the i-th neighborhood as a circular region centered
on the i-th node, or:
(b)
Figure 2: Comparison between the circuits obtained
by the (a) TSP and the (b) DTSP with orientations
given by the Alternating Algorithm [15].
A similar problem related to the TSPN is called Euclidean
Group Traveling Salesman Problem (EGTSP) [9]. In this
problem, the points of interest are grouped into clusters, and
the shortest path generated passing through at least one of
the points in each cluster must be calculated. This is an
important difference from the TSPN, in which any part of
the cluster can be visited (and not the exact position of a
point). Figure 3 shows this difference between EGTSP and
TSPN.
(a)
Hi = {p ∈ R2 : p − ni ≤ ri },
where p are positions in the two-dimensional space and ri
is the radius of each circular region.
In the kinematic of the Dubins’ vehicle, not only position
p, but also the orientation ψ of the robot is important to
the problem. We use a configuration vector qi = pi , ψi to represent the complete state of the vehicle in the SE(2)
domain. Then, the goal of the DTSPN is to find a configuration qi inside each Hi of the n regions and also a sequence
Σ of visiting configurations that minimize the total circuit
length Lρ composed of Dubins’ curves among those configurations. In other words, we may find a configuration set
QΣ composed of PΣ = p1 , . . . , pn (a set of positions) and
ΨΣ = ψ1 , . . . , ψn (a set of corresponding orientations) in a
sequence Σ = 1, . . . , n that generates the shortest Hamiltonian tour.
In this case, the following constraints hold:
(b)
Figure 3: Comparison between the circuits obtained
by the (a) EGTSP and the (b) TSPN [9].
Only recently, the union of the restrictions imposed by the
DTSP and TSPN problems began to be considered as part
of the same problem, which will be referred here as DTSPN.
In [13] a Genetic Algorithm is proposed, while in [14, 11] a
sampling based approach is used.
For both DTSP and TSPN, most of the techniques solve
the combinatorial step (visiting sequence) first, and uses the
sequence in subsequent calculations. It has been shown that
although this approach produces good results, the sequence
has great impact on the final result. Hence, we propose here
to jointly consider both steps. We show that the use of a
simple evolutionary algorithm to solve both combinatorial
and continuous steps of the problem in a combined manner
statistically produces better results.
3.
3.1
(1)
1. each neighborhood Hi contains at least one of the waypoints; and
2. a predefined cost function (e.g., the total traveled distance) is minimized.
With this approach, we can define the problem at hand as
finding the shortest Hamiltonian path through a set of waypoint regions for robots with Dubins’ kinematic constraints.
Formally:
Problem 3.1. Dubins’ Traveling Salesman Problem with
Neighborhoods: Let QΣ = PΣ , ΨΣ be some permutation Σ
of configurations q that reach n regions H. And let PΣ be the
position vector and ΨΣ the orientation vector of this permutation. Let also P : SE(2) → R2 be the projection function
that transforms the robot’s configuration into the workspace.
Then, we must optimize the Dubins’ circuit length accordingly to
METHODOLOGY
Theoretical formalization
As we stated before, we are dealing with the path planning
problem for Dubins’ vehicles in the context of the TSPN.
Then, we first define the vehicle’s nonholonomic constraint
before formally presenting our problem. In this paper, we
model the robot as a classical Dubins’ vehicle [6], which has a
maximum curvature (κmax ) as well as a velocity constraint.
arg min Lρ (QΣ )
P, Ψ, Σ
subject to P (qi ) ∈ Hi , i = 1, . . . , n,
379
(2)
Algorithm 1 Evolutionary-DTSPN(P)
with
Lρ (QΣ ) = Dρ (qn , q1 ) +
n−1
Dρ (qi , qi+1 )
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
16:
17:
18:
19:
20:
(3)
i=1
+
where Dρ : SE(2) × SE(2) → R represents the length of
the shortest path feasible by a Dubins’ vehicle with minimum
curvature radius ρ.
3.2
Evolutionary algorithm
Even though several methods to approach each individual subproblem of the larger problem posed here have been
reported in the literature, we have not identified a suitable
methodology that combines both the generation of the visiting sequence and the subsequent optimization steps. Clearly
the problem is quite untractable, and finding good methodologies is a quite challenging task. Among the solutions for
similar problems, evolutionary based ones have shown to be
a very good choice.
The field of evolutionary computing encompasses a large
literature describing techniques for solving problems in complex search spaces, based on simulation of processes ascribed
to the evolutionary theory. The fundamental goal of these
techniques is not simply an attempt to reproduce phenomena that occur in nature, but to use them as a model for the
development of algorithms which evolve in pursuit of better
solutions.
The methodology we propose in this paper consists of
three basic optimization steps that aims to solve both combinatorial and continuous phases of Problem 3.1 in a combined
manner. These steps will be further discussed in the next
sections, but an overview of each step is presented here:
Σ, Ψ ← AA (P)
L∗ρ ← Lρ (PΣ , ΨΣ )
while stopping criteria not met do
P̂ ← positionUpdate(P)
if Lρ (P̂Σ , ΨΣ ) < L∗ρ then
P ← P̂
L∗ρ ← Lρ (P̂Σ , ΨΣ )
end if
Ψ̂ ← orientationUpdate(Ψ)
if Lρ (PΣ , Ψ̂Σ ) < L∗ρ then
Ψ ← Ψ̂
L∗ρ ← Lρ (PΣ , Ψ̂Σ )
end if
Σ̂ ← ATSP(M)
if Lρ (PΣ̂ , ΨΣ̂ ) < L∗ρ then
Σ ← Σ̂
L∗ρ ← Lρ (PΣ̂ , ΨΣ̂ )
end if
end while
return P, Ψ, Σ
straight line segments. The remaining (non-straight line)
links are completed with Dubins’ curves based on the orientations previously assigned to the respective waypoints.
Algorithm 2 Alternating Algorithm(P) [15]
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
1. Position update: Moves the waypoint qi inside the
boundaries of a neighborhood Hi searching for the
placement that best helps minimize the cost function
of Equation 3;
2. Orientation update: Varies the orientation ψi assigned
to each waypoint qi in order to find a shorter Dubins’
curve connecting them;
3. Visiting sequence update: Checks if there is a better
visiting sequence Σ that produces a shorter Dubins’
circuit.
Our technique uses, as optimization process, one of the
simplest evolutionary algorithms found in the literature, the
(1+1)-Evolution Strategy [17]. The main characteristics of
this method are: (i) it mantains a population of size one
through generations, and (ii) a new candidate is generated
accordingly to a mutation process based on a Gaussian distribution. The optimization steps previously mentioned are
then grouped to form Algorithm 1.
Let P be a set of position waypoints, with each pi initially corresponding to the center ni of a region Hi , and
i = 1, . . . , n. With this as input, the algorithm starts by determining a initial sequence of visit Σ and orientation angles
Ψ at each waypoint to compute the corresponding Dubins’
circuit.
The sequence and the orientation are initially estimated
by using a classical technique called Alternating Algorithm
(AA) [15] (Algorithm 2), which is actually a straightforward heuristic. In short, it builds the circuit in such a way
that consecutive pairs of waypoints are linked together with
Σ ← ETSP(P)
ψ1 ← atan2(p1 , p2 )
for i = 2 to n − 1 do
if i is even then
ψi ← ψi−1
else
ψi ← atan2(pi , pi+1 )
end if
end for
if n is even then
ψn ← ψn−1
else
ψn ← atan2(pn , pi )
end if
return Σ, Ψ
The objective function being optimized here corresponds
to the length of the circuit Lρ composed by Dubins’ curves
and computed according to Equation 3.
While a stop criteria is not met, Algorithm 1 iterates executing the three steps previously described, optimizing the
sets containing the values of the position of the waypoints
(P), the values of orientation (Ψ) and the permutation of
the visit sequence (Σ). The variable L∗ρ represents the best
(shortest) path found so far.
It is important to note that our approach is different from
those in the literature, where a new candidate solution is
evaluated only at the end of a generation. In our case, we
evaluate each each new candidate after the execution of each
optimization step. Therefore, if the overall path length obtained by a candidate solution is shorter than the current
best solution, the modification made would be already assigned to the best solution. The reasoning behind this deci-
380
381
The first two steps of the methodology (position and orientation update) run in linear time O(n), where n is the
number of waypoints. The third step is the most costly one,
since it takes O(n2 ) to compute the cost matrix and O(n3 )
is the worst case time complexity for solving the ATSP [10].
Since the stop criterion of the algorithm is the number β of
generations (iterations), algorithm has a final time complexity of O(βn3 ) , or simply O(n3 ).
The length of the path generated by our methodology has
the same upper bound as the Alternating Algorithm, since
we use it in the initialization step, and the final result is
updated only if a better solution is found.
Theorem 3.4 from [16] demonstrates that the Dubins’ distance between two configurations is bounded by
Dρ (qi , qj ) ≤ pi − pj + τ ρπ,
250
200
150
100
y (m)
50
0
−50
−100
−150
(7)
−200
where τ ∈ [2.657, 2.658]. Then, one may directly observe
that a not tight upper bound for the length of the path is
n
τ ρπ,
(8)
Lρ (P, Ψ) ≤ ETSP(P) +
2
−250
−300
−200
−100
0
x (m)
100
200
300
(a) AA (2 335.2 m)
with n > 2 and ρ > 0, since we can consider that half of
the connecting links are lines and the other half are Dubins’
curves.
250
200
150
4.
EXPERIMENTS
100
y (m)
In this section, we describe our experiments and the corresponding statistical analysis, showing the improvement
(path reduction) obtained with the proposed methodology
when compared with other techniques in the literature. All
ETSP and ATSP problems instances were solved to optimality using the well-known TSP solver Concorde [1].
Initially, as depicted in Figure 7, we show an example of
possible results given by the techniques that will be used
in our comparison. In this first example we have an environment with dimensions 500 m × 500 m and 10 regions
uniformly randomly distributed, and a vehicle with minimum curvature radius of ρ = 50 m. The radius (area) of a
region was chosen randomly from the interval [25 m, 75 m].
Figure 7(a) presents a path obtained by the classical DTSP
solution (AA), which uses the Alternating Algorithm considering the exact center of each region (dashed blue line) and
Figure 7(b) shows the path computed by a random sampling
technique presented in [11] (Random), both techniques will
be used for comparison. Finally, Figure 7(c) is the result
produced by our methodology (Evolutionary).
One may clearly observe in Figure 7(c), the effect of each
step of our methodology on the final outcome. The visiting
positions have moved towards the center of the regions (red
triangle), while the orientation and visiting sequence have
also changed. The stop criterion used here was defined as
100 iterations, and we set σ = 1 (position update step) and
σ = 0.5 (orientation update step), with μ = 0 for both
steps. For this initial example, it was possible to obtain a
path reduction of ≈ 27% and ≈ 40% when compared to the
results of AA and Random, respectively.
As the proposed technique is a probabilistic method, in
order to perform a thorough statistical analysis, we present,
next, an overall analysis with a significant number of experiments. We run 200 experiments in an environment with
dimensions 500 m × 500 m containing a total of 25 regions
whose positions were uniformly randomly distributed. Each
region had a fixed radius chosen randomly from the inter-
50
0
−50
−100
−150
−200
−250
−300
−200
−100
0
x (m)
100
200
300
(b) Random (2 813 m)
250
200
150
100
y (m)
50
0
−50
−100
−150
−200
−300
−200
−100
0
x (m)
100
200
300
(c) Evolutionary (1 694.1 m)
Figure 7: Example solutions obtained by the compared techniques.
382
40
val [30 m, 60 m]. We also defined a simulated vehicle with a
minimum curvature radius of ρ = 50 m, which is adequate
when compared to the dimensions of the environment. For
each instance, all techniques (AA, Random and Evolutionary) were executed 15 times (the average path length is used
for comparison). The parameters of the proposed methodology were the same as described earlier.
Figure 8 presents a histogram of the path reduction obtained by the proposed methodology compared to the results given by the AA. We can see that the mean percentage
of length reduction is approximately 25%, with a standard
deviation of about 4.9%. The mean path reduction is significantly larger than zero, t(199) = 59.15, two-tail p = 0.
The average path lengths obtained for each technique were
5 078.05 m (AA) and 3 798.63 m (Evolutionary), a 95% C.I.
about path reduction is (1236.76, 1322.07).
Number of instances
35
30
25
20
15
10
5
0
0
10
20
30
Path reduction (%)
40
50
40
Figure 9: Path reduction histogram between Random versus Evolutionary.
30
5
10
AA
Random
Evolutionary
DTSP asymptotic lower bound
25
20
Average Tour Length
Number of instances
35
15
10
5
0
0
10
20
30
Path reduction (%)
40
50
Figure 8: Path reduction histogram between AA
versus Evolutionary.
4
10
3
10
2
10 0
10
Figure 9 presents a histogram of the path reduction obtained with our methodology compared to the results produced by the Random algorithm. The reduction reaches
a mean value of 28.3% and a standard deviation of 4.4%.
The mean path reduction is significantly greater than zero,
t(199) = 86.22, two-tail p = 0. The average path length
obtained by the Random algorithm was 5 302.40 m, a 95%
C.I. about path reduction is (1469.38, 1538.17).
We also deseigned an experiment with a varying number of
targets in the environment in order to analyze the impact of
this variable in the performance of the algorithm. The number of targets varied in the interval [5, 10, 15, ..., 100], and for
each specified value, 10 random instances were generated (all
techniques were executed 15 times for each instance). The
parameters of the algorithms were the same as those used
in the previous experiment.
Figure 10 presents the results comparing the actual performance of the three different algorithms. The DTSP asymptotic lower bound shown was taken from [16] and is given
by Cn2/3 for any ρ > 0, with
3 3
C=
3ρW H,
(9)
4
1
10
Number of Regions
2
10
Figure 10: Average tour length behavior accordingly
to the number of regions, plot on a log-log scale.
As it can be readily noted, the Evolutionary technique
produced better results than the others two algorithms for
all different number of regions used in the experiment. A
noteworthy result is that the AA method had better performance than the Random for low region densities, but as the
number of regions increase (≈ 40), the latter starts to show
a better performance.
Table 1 presents the empirical growth rates for the different algorithms obtained by a linear regression based on
the simulation results presented in Figure 10. The proposed
methodology presented a good asymptotic performance, significantly lower than the trivial solution (AA), and slightly
above the lower bound curve.
We also executed some experiments in order to verify the
gain obtained by increasing the number of iterations. Although it is evident that running for more iterations will
produce better results, we concluded that the improvement
wasn’t significant. However, it is up to the user to decide on
the trade off between computation time and solution quality
by adjusting the number of iterations.
where W and H represent the width and height of the environment, respectively. Considering the parameters used in
our experiment, we have C ≈ 251.
383
Table 1: Experimental growth rate of the average
tour length, with uniform randomly distributed regions in a 500x500 environment.
Algorithm
Average Tour Length
AA
5.7 × n0.88
Random
6.2 × n0.75
Evolutionary
5.9 × n0.71
Lower bound
5.5 × n0.67
[7]
[8]
5.
CONCLUSION AND FUTURE WORK
In this work we described a three-stage optmization process based on an evolutionary algorithm to solve the Dubins’
Traveling Salesman Problem with Neighborhoods (DTSPN).
The results were satisfactory and show a significant improvement when compared to the general solution using the classical DTSP approach (AA) and a technique based on a random sampling method.
Several issues will be addressed in the future to improve
the proposed methodology. As observed during the experiments, there are several overlapping areas among regions.
We intend to take advantage of this fact, by choosing waypoints that belong to several regions at the same time (intersection among areas).
Future directions also include the extension of the proposed methodology to multi-robot ensambles and to environments containing obstacles, which is a better representation of real world scenarios.
One of the main problems concerning the use of Dubins’
curves is the discontinuity on the curvature profile, which
leads to abrupt lateral accelerations resulting in paths that
are not feasible by a real robots. Therefore, we intend to
study possible techniques for generating smoother variations
of acceleration, for example, the use of other types of curves
such as clothoids.
6.
[9]
[10]
[11]
[12]
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