persurface embedded in 5D space. Applying a con
strained version of the descending gradient on the hy
persurface, it is possible to ﬁnd out all the admissible,
equivalent and shortest (for a given metric of the dis
cretized space) trajectories connecting two positions
for each robot CSpaceTime.
2 PROBLEM STATEMENTS
A wide variety of world models can be used to de
scribe the interaction between an autonomous agent
and its environment. One of the most important is the
Conﬁguration Space (Latombe, 1991; LozanoP
´
erez,
1983). The CSpace C of a rigid body is the set of all
its conﬁgurations q (i.e. poses). If the robot can freely
translate and rotate on a 2D surface, the CSpace is a
3D manifold R
2
× SO(2). It can be modelled using
a 3D Bitmap GC (CSpace Binary Bitmap), a regular
decomposition in cells of the CSpace, represented by
the application GC : C → {0, 1}, where 0s represent
non admissible conﬁgurations. The CPotential is a
function U(q) deﬁned over the CSpace that ”drives”
the robot through the sequence of conﬁguration points
to reach the goal pose (Barraquand et al., 1992). Let
us introduce some other assumptions: 1) space topol
ogy is ﬁnite and planar; 2) the robot has a lower bound
on the steering radius (nonholonomic vehicle). The
latter assumption introduces important restrictions on
the types of trajectories to be found.
Cellular Automata are automata distributed on the
cells of a Cellular Space Z
n
(a regular lattice) with
transition functions invariant under translation (Goles
and Martinez, 1990): f
c
(·) = f (·), ∀c ∈ Z
n
, f (·) :
Q
A
0

→ Q, where c is the coordinate vector identi
fying a cell, Q is the set of states of an automaton and
A
0
is the set of arcs outgoing from a cell to the neigh
bors. The mapping between the Robot PathPlanning
Figure 1: MultiLayers Architecture.
Problem and CA is quite simple: every cell of the
CSpace Bitmap GC is an automaton of a CA. The
state of every cell contributes to build the CPotential
U(q) through a diffusion mechanism between neigh
bors. The trajectories are found following the mini
mum valley of the surface U(q). In this work, we use
a simple extension of the CA model: we associate a
vector of attributes (state vector) to every cell. Each
state vector depends on the state vectors of the cells
in the neighborhood. There is a second interpretation:
this is a Multilayered Cellular Automaton (Bandini
and Mauri, 1999), where each layer corresponds to
a subset of the state vector components. Each sub
set is evaluated in a single layer and depends on the
same attribute of the neighbor cells in the same layer
and depends also on the states of the corresponding
cell and its neighbors in other layers. In the follow
ing sections, we describe each layer and the transition
function implemented in its cells.
3 MULTILAYERED
ARCHITECTURE
In Fig. 1 is shown the layers structure and their de
pendencies. There are two main layers: Obstacles
Layer and the Attraction Layer. Each layer is sub
divided in more sublayers: the Obstacles L. has 3 di
mensions (2 for the workspace (X, Y ) and 1 more for
the time), while the Attraction L. has up to 5 dimen
sions (1 for the robots, 2 for the robots workspaces +
1 for their orientations (X, Y, Θ) and 1 for the time).
In the following subsections, we will brieﬂy describe
each layer and their roles. The Obstacles L. concep
tually depends on the outside environment. Its sub
layers have to react to the ”external” changes: the
changes of the environment, i.e. the movements of
the obstacles in a dynamical world. Through a sen
sorial system (not described here), these changes are
detected and the information is stored in Obstacles L.
permitting the planner to replan as needed.
3.1 The Obstacles Layer
The main role of the Obstacles Layer is to create a re
pulsive force in the obstacles to keep the robots away
from them. In the present work, only static obstacles
are considered (e.g. walls). For the single robot, the
other robots are seen as moving obstacles, with un
known and unpredictable trajectories. We are consid
ering a centralized planner/coordinator, that can de
cide (and, of course, it knows) the trajectories of all
the supervised robots. Thanks to this knowledge, the
planner considers the silhouette of the robot as an ob
stacle for the other robots. In this work, we introduce
a discretized version of the CSpaceTime as in (War
ren, 1990). With the introduction of the Time axis,
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