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A PGD-based Method for Robot Global Path Planning: A Primer
N. Mont
´
es
1 a
, F. Chinesta
2
, A. Falc
´
o
1 b
, M. C. Mora
3 c
, L. Hilario
1 d
and J. L. Duval
4
1
Department of Physics, Mathematics and Technological Sciences, University CEU Cardenal Herrera,
46115, Alfara del Patriarca, Spain
2
PIMM, ENSAM ParisTech ESI GROUP Chair on Advanced Modeling and Simulation of Manufacturing Processes,
Paris, France
3
Department of Mechanical Engineering and Construction, Universitat Jaume I, Castell
´
on, Spain
4
ESI Group, RUNGIS CEDEX, France
Keywords:
Model Order Reduction Techniques, PGD, Path Planning, Potential Field Methods, Laplace Equation.
Abstract:
The present paper shows, for the first time, the technique known as PGD-Vademecum as a global path planner
for mobile robots. The main idea of this method is to obtain a Vademecum containing all the possible paths
from any start and goal positions derived from a harmonic potential field in a predefined map. The PGD is a
numerical technique with three main advantages. The first one is the ability to bring together all the possible
Poisson equation solutions for all start and goal combinations in a map, guaranteeing that the resulting potential
field does not have deadlocks. The second one is that the PGD-Vademecum is expressed as a sum of uncoupled
multiplied terms: the geometric map and the start and goal configurations. Therefore, the harmonic potential
field for any start and goal positions can be reconstructed extremely fast, in a nearly negligible computational
time, allowing real-time path planning. The third one is that only a few uncoupled parameters are required
to reconstruct the potential field with a low discretization error. Simulation results are shown to validate the
abilities of this technique.
1 INTRODUCTION
An essential in robotics is to guide the robot safely
from a start to a goal position among a set of ob-
stacles. For this purpose, a collision-free path must
be generated, which implies a computationally hard
geometric path planning unfeasible in real-time (RT)
applications (Reif, 1979). This problem is known
in the literature as motion planning or the piano
mover’s problem and its complexity has motivated
a lot of research works in the field of robot path
planning. Some works have studied subproblems of
the general approach (Kavraki and LaValle, 2008).
Other researchers have considered alternative plan-
ning paradigms under simplified but realistic assump-
tions such as, for instance, sampling-based planners,
grid-based searches, interval-based searches, geomet-
ric algorithms, etc (Kavraki and LaValle, 2008).
a
https://orcid.org/0000-0002-0661-3479
b
https://orcid.org/0000-0001-6225-0935
c
https://orcid.org/0000-0003-0627-6764
d
https://orcid.org/0000-0003-0729-6628
One of the most used algorithm is the Artificial
Potential Field method (APF), (Khatib, 1986). This
technique defines an artificial potential field in the
configuration space (C-space) that generates a path
from a start to a goal position. This method is very
fast for RT applications. However, the robot could get
stuck in a local minimum of the potential function.
This problem can be solved using harmonic functions
in the generation of the potential field (Canny, 1998),
which satisfy the Laplace equation in the C-space
and completely eliminate local minima as they sat-
isfy the Min-Max principle (Rimon and Koditschek,
1992). These functions were initially proposed in
(Zhachmanoglou and Thoe, 1986) and used for robot
path planning in (Connolly et al., 1990; Kim and
Khosla, 1992; Akishita et al., 1993; Guldner et al.,
1997; Waydo and Murray, 2003; Rosell and Iniguez,
2002; Saudi and Sulaiman, 2012). The main prob-
lem of this technique, addressed in (Waydo and Mur-
ray, 2003), is that the solution must be numerically
computed in a discrete mesh and, therefore, the com-
putational cost increases exponentially with the mesh
resolution. In (Gingras et al., 2010), one of the last
Montés, N., Chinesta, F., Falcó, A., Mora, M., Hilario, L. and Duval, J.
A PGD-based Method for Robot Global Path Planning: A Primer.
DOI: 10.5220/0007809000310039
In Proceedings of the 16th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2019), pages 31-39
ISBN: 978-989-758-380-3
Copyright
c
2019 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
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