Preprints, 10th IFAC Conference on Control Applications in Marine Systems
September 13-16, 2016. Trondheim, Norway
Integration of risk in hierarchical path
planning of underwater vehicles
Nicolas Lefebvre ∗ Ingrid Schjølberg ∗ Ingrid B. Utne ∗
∗
Department of Marine Technology, Norwegian University of Science
and Technology (NTNU), NO 7491 Trondheim, Norway
(e-mail: [email protected], [email protected],
[email protected])
Abstract: This paper focuses on path planning problems integrating the risk of collision. This
challenge is crucial regarding the use of autonomous underwater vehicles (AUVs) for inspection,
maintenance and repair operations. The solving of this problem in a reasonable amount of time
enables integration of path planning in the AUV control architecture and, by this way, enhances
its autonomy capability. Classical approaches rely on the A-Star algorithm to solve this problem,
but the heuristic associated to the collision risk appears to be inefficient in some cases. Based on
the hierarchical technique, i.e., HPA-Star, another approach is proposed. It leads to paths close
to the optimal ones calculated in a faster way. The performances are illustrated in the context
of a multi criteria optimization: minimal length and minimal risk path planning.
Keywords: Path planning, Probabilistic risk assessment, Multiobjective optimisations,
Autonomous vehicles, Hierarchical decision making
1. INTRODUCTION
integration of path planning in the AUV control architecture. This is obtained through the hierarchical path
planning methodology.
In underwater inspection, maintenance and repair (IMR)
operations, the use of autonomous underwater vehicles
(AUVs) as an alternative to remotely operated vehicles
(ROVs) constitutes a significant potential for saving costs.
The benefits are particularly related to the ability of
autonomous systems to operate without the support of
a surface vessel, see Winnefeld and Kendall (2011).
AUVs are exposed to random environmental conditions
and operational hazards challenge risk management of the
IMR operations. In this context, a direct approach to
risk control is path planning adaptation, specially related
to AUV inspection tasks where collision is one of the
major undesired events. This gives significant advantages
compared to existing systems which mainly optimize the
path planning on minimal cost or operational time.
Bae et al. (2015) consider the problem of finding a riskconstrained shortest path for an unmanned combat vehicle. The problem is solved by the use of a dynamic programming approach. The problem of computational time
is tackled by a straightforward limitation of the explored
area. Pereira et al. (2013), as Greytak and Hover (2009),
use the A-Star algorithm to find the minimum risk path for
an AUV or underactuated surface vessel. Path length and
risk is discussed, but no solution is suggested to balance
out the inefficiency of the considered heuristic in A-Star.
De Filippis et al. (2011) discuss the use of A-Star and
genetic algorithm to find minimal risk path.
The objective of this paper is to present an approach
for risk-based path planning. Its main contribution is to
propose a trade-off between computational requirements
and the optimality of the found path to enable embedded
Copyright © 2016 IFAC
226
The remainder of this paper is organized as follows: section 2 gives an overview of risk and general considerations
on how risk can be associated with the path planning
task. Section 3 focuses on path planning algorithms and
definition of heuristics. Section 4 presents the case study.
Section 5 provides computational experiments to illustrate
the developed strategy. Finally, the article ends with concluding remarks and perspectives.
2. RISK IN PATH PLANNING
The aim of this section is, firstly, to provide a brief introduction to risk. Secondly, to discuss the interactions
between risk and AUV control and especially the path
planning. Moreover, risk assessment along a path is presented.
2.1 Risk
According to Kaplan and Garrick (1981) and Rausand (2011), risk analysis answers three main questions:
1. What can go wrong? 2. What is the likelihood of that
happening? and 3. What are the consequences? Let risk
nR
be defined as a set R = {(Ei , Pr {Ei } , Ci )}i=1
of nR
triplets where Ei denotes the hazardous event i, Pr {Ei }
the probability of occurrence of this event and Ci the
cost associated to its consequences. The risk management
framework, proposed in ISO 31000 (2009), describes the
steps to identify the set R. If all hazardous events have
been considered and included, the set is considered to
represent risk.
2016 IFAC CAMS
Sept 13-16, 2016. Trondheim, Norway
PnR
Let rT =
i=1 Ci I{Ei } be the random variable which
characterises the cost associated with the occurrence of
the hazardous events. The total risk RT is defined as the
mean value of rT :
nR
nR
X
X
Ci Pr {Ei }
(1)
Ci E I{Ei } =
RT = E [rT ] =
i=1
i=1
Part of the hazard identification process may focus on
deriving risk influencing factors (RIFs). RIFs can be
defined as an aspect (event/condition) of a system or an
activity that affects the risk level of this system or activity,
see Oien (2001). For AUVs, RIFs and hazardous events are
related to the operation of the vehicle, i.e., the technical
condition of the AUV, its mission and operating context.
Collision is one important hazardous event to mitigate for
path-planning. Hence, to limit the extent of the paper,
the path planning approach is exemplified by focusing on
collision risk.
2.2 Integration of risk in path planning
Referring to National Research Council (US) (2005), an
AUV is a vehicle that possesses self-governing capabilities
allowing it to carry out tasks without human intervention.
approaches, combinatorial (deterministic) approaches. To
avoid the inherent problem of potential fields to get
trapped in local minima and the intrinsic weaker of
random approaches (near optimal solution if a solution
is found), the combinatorial approaches and especially
heuristic based algorithm are considered in the following. Therefore, it is assumed that the continuous vehicle
environment (i.e., the search space) can be modelled as
a graph G = (S, T ) where S = {s1 , s2 , . . . , sn } is the
set of possible state (e.g., locations of the vehicle) and
T = {(si , sj ) : si , sj ∈ S, si 6= sj } the set of possible transitions between states S. All transitions in T can be weighted
with several non negative values (e.g., length, travel time,
risk).
Path planning corresponds to the search in G of a path
from a starting state ss to a goal state sg (ss , sg ∈ S). A
path pss ,sk = [ss , si , . . . , sj , sk ] defines the successive states
that have to be reached to go from the starting state ss to
the state sk . In the following, path pss ,sk is noted psk if the
starting state is ss ; psk (i) denotes the ith states reached
and |psk | is the number of states in the path. Path planning
can be formulated as an optimisation problem
(2)
p∗sg = arg min g psg
psg ∈Psg
p∗sg
According to Campbell et al. (2012), a simple way to
integrate risk is at the supervision level, through the
adaptation of the vehicle path to the safety requirements.
Basically, this task is achieved off-line on the basis of
a mapped environment with known static obstacles, but
it can also be considered on-line to enhance the AUV’s
autonomous capabilities. Therefore, one of the challenges
is to carry out this new capability in combination with
standard reflexive avoidance strategies. Another one is to
provide path (re)planning solutions in accordance with the
computational capabilities and time constraints intrinsic
to the embedded system in AUV applications.
denotes the path, amongst the set Psg of possible
where
paths from ss to sg , that minimizes the cost function g.
(i)
Constraints may be added to (2), e.g., f (i) (psg ) − fth < 0
where f (i) , i = 1, . . . , nc is a set of cost functions and
(i)
fth a set of specified maximal threshold. Through this
formulation the path planning problem may be formulated
as the search of i) the minimal length path, ii) the minimal
risk path, iii) the minimal length (respectively risk) path
subject to a constraint of a maximal level of risk (resp.
length) in the set Psg .
The aim of the proposed approach is to provide a solution for path planning, with risk integration, compatible
with an embedded use, i.e., with limited computational
requirements. A mapped environment with known static
obstacles is assumed. The start and goal position, as well
as weights put on the different criteria considered for path
planning might be provided by the AUV itself, increasing
by this way its autonomy.
2.4 Cost assessment along a path
In the next section, the path planning problem and the
integration of risk is formalized.
Several cost functions g, depending on the criteria considered for path planning (length and risk), are defined in the
following.
Path length is defined by (3) where the distance associated
with the transition from state psg (i − 1) to state psg (i) is
denoted by d psg (i − 1), psg (i) .
|psg |−1
glength (psg ) =
X
d psg (i), psg (i + 1)
(3)
i=1
2.3 Problem formalization
Path (or motion) planning is the subject of an extensive
research. For an overview, see, for instance, LaValle (2006)
or Paull et al. (2013). The general problem statement is to
find a path between the given start and goal states (i.e.,
positions). Generally, this problem is associated with the
optimisation of a performance criterion (e.g., path length,
travel time) and is potentially subject to constraints (e.g.,
path smoothness). So, risk integration in path planning
can be seen as adaptation of the performance criterion or
of the constraints.
There are several approaches to achieve the path planning: e.g., potential fields, sampling-based (randomized)
227
Regarding risk, as mentioned before, we focus only on the
risk of collision. Moreover it is assumed that whatever is
the collision consequence, it leads to a constant cost CCo
and aborting the mission. Therefore, only the first collision
is of interest.
Consider the function Co (si , sj ) which equals 1 if there
is collision in the transition from the states si to sj with
(si , sj ) ∈ T and equals 0 if not. Let Pr {Co (si , sj ) = 1}
be the probability that a collision occurs during this
transition. Such quantity is associated with all transitions
in the set T of G. Due to the sequencing of displacements,
the risk associated with the path psg corresponds to the
product of the cost CCo with the probability that:
2016 IFAC CAMS
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• the collision occurs during the transition from the first
state psg (1) = ss to the second one psg (2) of the path.
• or, that the collision occurs during the transition from
state psg (i) to psg (i + 1) with 2 ≤ i < |psg | − 1 and
has not occurred during the previous transitions.
Hence:
grisk (psg ) = CCo Pr Co (psg (1), psg (2)) = 1
|psg |−1
+
X
Pr Co (psg (i), psg (i + 1)) = 1
(4)
i=2
i−1
Y
1 − Pr Co (psg (j), psg (j + 1)) = 1
j=1
hrisk (sk , sg |psk ) = CCo Prmin {Co = 1}
N +1
1 − (1 − Prmin {Co = 1})
α(psk )
1 − (1 − Prmin {Co = 1})
(7)
where:
• N is the minimum number of transitions between the
state sk and the goal one:
de (sk , sg )
c
(8)
N =b
dmax
with dmax the maximum euclidean distance between
adjacent states,
• Prmin {Co = 1} is the minimum collision probability
affected to a transition in the graph G.
Prmin {Co = 1} = min Pr {Co (si , sj ) = 1} (9)
(si ,sj )∈T
Note that a conservative approach would be to consider
equation (5) for risk assessment along the path. This more
simple formulation leads to equivalent solutions when the
minimal risk path is targeted, but is less relevant, although
conservative, when risk is formulated as a constraint of (2)
since risk is overestimated.
0
grisk
(psg ) = CCo
|psg |−1
X
(5)
Pr Co (psg (i), psg (i + 1)) = 1
i=1
3. PATH PLANNING ALGORITHMS
In this section, we briefly present A-Star and Disjkstra’s
algorithms and propose heuristics in accordance with the
different cost functions developed in section 2.4. The
hierarchical approach for path planning that can exploit
one or the other of the two mentioned algorithms, is
presented. Finally, constrained path planning is discussed.
3.1 A-Star, its heuristics, and Dijkstra’s algorithms
A-Star and Disjkstra’s algorithms are very common solutions to solve minimal cost path planning, see for instance
Ferguson et al. (2005). These two algorithms operate in a
similar way but A-Star relies on an estimate, a heuristic
h, to drive the graph exploration to the most favourable
areas. By this way, A-Star may be much less computationally expensive than the Dijkstra’s algorithm and leads, as
the Dijkstra’s algorithm, to an optimal solution provided
that the considered heuristic is admissible.
Let h(sk , sg ), with sk , sg ∈ S, be an estimate of the cost
to reach the state sg from the state sk , without presuming
a specific path. This heuristic is admissible if it does not
overestimate the effective minimal cost h∗ to reach the
goal state (i.e., h(sk , sg ) ≤ h∗ (sk , sg ) ∀sk ∈ S). Dijkstra’s
algorithm can be seen as a special case of A-Star when
h(sk , sg ) = 0 ∀sk ∈ S which constitutes an admissible
heuristic, but not an efficient one.
When length cost is considered, as in (3), an admissible
heuristic is
hlength (sk , sg ) = de (sk , sg )
(6)
where de denotes the euclidean distance between the two
states.
When risk cost is considered, refer to (4), an admissible
heuristic is
228
• α(psk ) is a coefficient enabling to take into account
the path followed from ss to sk

 1, if |psk | = 1, else,

|psk |−1
Y
α(psk ) =

1 − Pr {Co (psk (i), psk (i + 1)) = 1}

i=1
The last term of (7) corresponds to the sum of the terms of
the geometrical suite generated by the N transitions with
a probability of collision equal to Prmin {Co = 1}.
If the cost is expressed by (5), an admissible heuristic is
(10)
hrisk (sk , sg |psk ) = N CCo Prmin {Co = 1}
where N and Prmin {Co = 1} are respectively expressed
by (8) and (9).
3.2 Hierarchical approaches
Hierarchical approaches, like HPA-Star as described in
Botea et al. (2004), can reduce problem complexity and
provide a solution in less time than A-Star. Even if the
optimality of the obtained solution is not guaranteed, this
approach constitutes an interesting alternative considering
the following argument: it is not necessarily relevant to
spend time on finding an optimal path for a mission that
is unique (specific environmental conditions, start and/or
goal positions, etc.).
The principle of the hierarchical approach (with 2 levels)
is to consider an abstraction of the graph associated with
the search space. This abstraction clusters adjacent states.
These clusters are connected through gates, that are
specific states at the periphery of clusters. The resulting
abstraction constitutes a smaller graph whose processing
can be summarized as follows:
(1) for the start and goal states, within each cluster
that encompasses the considered state, find a path to
connect this state to one of the gates of the cluster.
This gate is hereafter called the start gate (the goal
gate, respectively).
(2) using the abstract graph, find a path between the
start gate and the goal gate.
(3) refine each step of the abstract path, i.e, find a path
inside each considered cluster between gates that
constitutes the path.
In this way, the initial path planning problem has been
divided into a set of small path planning problems, faster
to solve with classical A-star or Dijkstra’s algorithms.
2016 IFAC CAMS
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3.3 Constrained optimization
Path planning formulated in (2) can be extended to a
constrained problem. Then, according to Handler and
Zang (1980), the problem is at least NP-complete while
the initial one can be solved in a polynomial time by
the Dijkstra’s algorithm. Different approaches (e.g., kshortest paths) have been proposed to solve this problem.
One of them relies on the Lagrangian relaxation method
and introduces the constraint into the objective function.
Hence, a constrained formulation of (2) might be given by
p∗sg = arg min g psg + λ f (psg − fth )
(11)
psg ∈Psg
where λ is a positive weight balancing cost and constraint
functions. Typically, functions g and f are combination of
those proposed in section 2.4.
The determination of λ is then challenging. Different
solutions have been proposed, see latter reference and
Santos et al. (2007).
4. CASE STUDY
si ∈ S is associated with a collision probability denoted
Pr {Co (si ) = 1} where Co (si ) is a function which equals 1
if there is collision in si and 0 if not. The probability of
collision during a transition between si and sj , (si , sj ) ∈ T ,
is defined as the average of probabilities of collision in cells
associated with states si and sj
Pr {Co (si , sj ) = 1} =
1
(Pr {Co (si ) = 1} + Pr {Co (sj ) = 1})
2
(12)
The probability Pr {Co (si ) = 1}, ∀si ∈ S is the result of
the aggregation, through an ”or” rule of collision probabilities related to each obstacle j for j = 1 to nO present
in the environment
nO n
o
Y
1 − Pr Co(j) (si ) = 1
(13)
Pr {Co (si ) = 1} = 1 −
j=1
(j)
where Pr{Co (si ) = 1} is the collision probability of the
vehicle and the obstacle j when the reference state of the
vehicle is si .
(j)
4.1 Environment
Since 2012, the Applied Underwater Robotics Laboratory
(AUR-Lab) at NTNU uses an area of the Trondheim
harbor, not far from Munkholmen, as a test ground for
applications of underwater robots and sensors in marine
archaeology. This area is called The Reference Wreck due
to the presence of a shipwreck dated to late 17th century
at a water depth of 60 meters. In the following, we use the
map of this area to illustrate the proposed approach.
Obstacles inside this area have been defined based on a
simple threshold on the depth and in such a way that it is
forbidden for the vehicle to go outside the mapped area.
Additional obstacles have been randomly defined to make
the path planning task more challenging.
A basic cellular decomposition technique into non overlapping cells of simple predefined shapes is used to discretize
the search space. The entire cell is considered as an
obstacle and an invalid state, if it encompasses a part of
an obstacle. The cells dimension and connectivity directly
yields the size to the graph figuring the search space. In
the following, cells are a 0.125m square side, 8-connected
neighbourhood, leading to a graph composed by 18495
nodes and 138908 edges.
4.2 Map of collision probability
In the present application, apart from obviously the distance to the obstacle, other RIFs are assumed to be
constant and collision probabilities are homogeneously
distributed around single obstacle. The resulting map of
collision probabilities is illustrated on Fig. 1. Obstacles are
depicted in black. Free cells are coloured depending on the
collision probability according to the scale depicted above
the figure. Start and goal positions have been arbitrarily
set and are depicted with green and red dots respectively.
16
14
12
y−axis [m]
The considered use-case is in two dimensions, but can
be expanded to a higher dimension (e.g., vehicle depth,
vehicle speed). Only the risk of collision is considered. In
addition, the collision cost is assumed to be constant all
along the path. Therefore, for simplicity, risk is reduced to
the collision probability in the following (i.e., CCo = 1).
The function used to determine Pr{Co (si ) = 1} for
∀si ∈ S and all obstacles j is not detailed here. It depends on several RIFs, for example, the shortest distance
between the state si and the obstacle j, the sea current
strength and direction, visibility, vehicle dimensions, manoeuvring capabilities and the way the states si is crossed
(a pass with a diagonal way is longer than a pass with a
longitudinal or a lateral way). In practice, this function
might be tuned on the basis of experts judgement and/or
collected data.
10
8
6
4
2
0
0
5
10
15
20
25
1.5
2
2.5
x−axis [m]
According to section 2.4, it is assumed that all transitions (si , sj ) ∈ T , in addition to the distance between
state si and sj (de (si , sj )), are valuated with the probability Pr {Co (si , sj ) = 1}. To determine this probability, a ”map” is used where each state (i.e., each cell)
229
0.5
1
−3
x 10
Fig. 1. Map of collision probabilities
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5. SIMULATION RESULTS
5.2 Minimal collision risk path
In this section, we compare results obtained with 4 algorithms: Dijkstra, A-Star, hierarchical Dijkstra (i.e. Dijkstra is used in the hierarchical framework each time that
a path has to be determined), and hierarchical A-Star.
Hierarchical abstraction is obtained by the clustering of
block of 8 × 8 adjacent states (i.e., locations), including
each up to 8 gates. This leads to an abstract graph of 2223
nodes and 16556 edges. The preprocessing of the abstract
graph is achieved off-line, but paths within clusters are not
stored in cache and have to be determined on-line.
Minimal collision risk path is considered in this section.
Paths are illustrated Fig. 3 while table 2 gives the numerical results.
Table 2. Minimal collision risk paths.
Algorithm
Dijkstra
A-Star
Hier. Dijkstra
Hier. A-Star
Cost and heuristic functions are defined by (3) and (6),
respectively, when the path length criteria is considered,
and by (4) and (7) when considering path risk. If the
criteria is a linear combination of length and risk costs, the
linear combination of respective heuristics is considered.
Risk [probability]
2.2853e-03
2.2853e-03
2.4197e-03
2.4197e-03
Duration [%]
76.84%
79.00%
9.85%
10.76%
Dijkstra
A−Star
Hier. Dijkstra
Hier. A−Star
16
14
12
y−axis [m]
5.1 Minimal length path
Length [m]
28.3618
28.3618
29.5208
29.5208
10
8
6
In this section, the problem of determination of the minimal length path is considered. Obtained paths are illustrated in Fig. 2 while corresponding length and risk are
given in table 1. The table also includes the normalised
computational time to obtain the solution (reference time
is Disjkstra on minimal length path).
Table 1. Minimal length paths.
Algorithm
Dijkstra
A-Star
Hier. Dijkstra
Hier. A-Star
Length [m]
17.9905
17.9905
18.3566
18.3566
Risk [probability]
1.2999e-02
1.3716e-02
1.3632e-02
1.1855e-02
14
Duration [%]
100.00%
14.95%
10.16%
4.29%
y−axis [m]
10
8
6
4
2
10
15
20
0
5
10
15
20
25
x−axis [m]
Fig. 3. Minimal collision risk paths.
5.3 Pareto front
0
5
0
Lastly, an unexpected effect is the decrease of the computational time associated with the Dijkstra’s algorithm
(around -25%). It appears that the number of visited
states is reduced when the risk cost is considered instead
of the length cost (12602 vs. 16522). Actually, the exploration of some areas requires to go through narrow
passages between obstacles (this is especially the case for
the bottom left quarter of the map, see Fig. 1). The risk
associated with theses passages is high compared to the
risk of other paths. These passages are not exploited and
the corresponding areas are not explored here.
12
0
2
Paths obtained from standard approaches and hierarchical
ones are the same within each of these two categories. The
latter ones are suboptimal with a gap of around 6% but are
obtained in less time with a factor close to 8, showing that
hierarchical approaches are here still relevant. This is not
the case for A-Star (in the standard and the hierarchical
configuration). Indeed, due to the ”inefficiency” of the
heuristic used, it is not able to reduce the computational
time compared to Dijkstra, as seen in the previous section.
Dijkstra
A−Star
Hier. Dijkstra
Hier. A−Star
16
4
25
x−axis [m]
Fig. 2. Minimal length paths.
Dijkstra and A-Star algorithms do not lead to the same
paths, but distances are equal and are the minimal one
that can be expected. The hierarchical algorithms do not
find the same paths. The distance of the obtained paths are
around 2% higher than the optimal one. But, in return, the
computational time is reduced by a factor 10 in the case
of the Dijkstra’s algorithm and by a factor 3 for A-Star.
As these 4 paths are different, the associated risks are also
different.
230
In this section, to compare algorithms, Pareto fronts
between path length and path risk are considered (see
Fig. 4). These Pareto fronts are obtained by varying a
parameter w from 0 to 1 to weight the cost functions
associated with the path length and the path risk. At
w = 0, only the length cost is considered while at w = 1
it is the risk cost. In addition to the Pareto fronts, the
computational times for each algorithm depending on w
are given in Fig. 5.
As expected, the curves associated with Dijkstra and Astar are for the mostly overlapping due to the admissibility
2016 IFAC CAMS
Sept 13-16, 2016. Trondheim, Norway
to the associated heuristic which, although admissible, is
inefficient in the considered use case.
−3
14
x 10
Dijkstra
A−Star
Hier. Dijkstra
Hier. A−Star
w=0
Risk (Collision probability) [−]
12
An extension of this work is the consideration and inclusion of a dynamic environment. The risk of collision
can change during operation, due to changes in the RIFs
(e.g., current). In this case, re-planning algorithms based
on hierarchical approaches should be considered to limit
the computational load.
10
8
ACKNOWLEDGEMENTS
This work was supported by the Norwegian Research
Council (grants no. 234108).
6
REFERENCES
4
w=1
2
16
18
20
22
24
26
28
30
Length [m]
Fig. 4. Pareto fronts Risk/Length.
Dijkstra
Norm. Comp. Time [%]
120
A−Star
Hier. Dijkstra
Hier. A−Star
100
80
60
40
20
0
0
0.1
0.2
0.3
0.4
0.5
w
0.6
0.7
0.8
0.9
1
Fig. 5. Computational time for the Pareto fronts.
of the heuristic used with the A-Star algorithm. These
Pareto curves show that, in the considered use case, it is
possible to reduce the risk associated with a path without
degrading too much its optimality regarding the length
criteria (see the almost vertical part of curves).
The non optimality of hierarchical solutions is manifest,
but there is in return a drastic reduction of the computational time (see Fig. 5). This is even more clear when
w increases and when the weight on the risk cost is more
important. Indeed, the efficiency of the heuristic associated
with this cost function is weak and it has a strong impact
on the computational time of the A-Star algorithm. This
is also true, in a less extent, in the case of the hierarchical
A-Star. For Dijkstra’s algorithm, we can see the computational time decreases as w increases. This is a consequence,
as explained in section 5.2, of a more limited exploration
of the search space when risk is considered: it corresponds
to a particular property of the considered environment.
6. CONCLUSIONS AND FUTURE WORKS
This paper considers the problem of path planning with
integration of collision risk. First, the assessment of collision risk along a path and the definition of the associated
heuristic suitable to path planning have been closely presented. Then, a hierarchical approach has been considered
to solve path planning problems combining length and collision risk criteria. The obtained solutions are sub-optimal,
but are achieved -on a specific use case- up to 8 times faster
than with a conventional approach based on the A-Star
or Dijkstra’s algorithms. This is especially the case when
the weight on the collision risk criterion is important due
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