A REACTIVE MOTION PLANNER ARCHITECTURE FOR

GENERIC MOBILE ROBOTS

BASED ON MULTILAYERED CELLULAR AUTOMATA

Fabio M. Marchese

Universit

a degli Studi di Milano - Bicocca

Via Bicocca degli Arcimboldi 8, I-20126, Milan (Italy)

Keywords:

Mobile Robots, Motion Planning, Cellular Automata, Trajectories Generation.

Abstract:

The aim of this paper is to describe the architecture of a Path Planner for Mobile Robots based on the paradigm

of Cellular Automata. The environment representation is distributed, as the robot shape; both and the robot

kinematics are parameters for the planner. Hence, it results to be very ﬂexible, handling robots with quite

different kinematics (omnidirectional, car-like, asymmetrical, etc.), with generic shapes (even with concavities

and holes) and with generic cinematic center positions. Because of these characteristics, it is applicable for the

assembly planning in the manufacturing industry, as in the Piano Mover’s problems, or in vehicles trajectories

generation. It can be applied to ﬂat (Euclidean) Work Space and to natural variable terrains. Considering

robots moving with smoothed trajectories, the underlying algorithm is based on a Potential Fields Method,

using an anisotropic propagation of potentials on a non-Euclidean manifold. The collision-free trajectories

are found following the minimum valley of the potential hypersurface embedded in a 4D space. Thanks to

the Multilayered Cellular Automata architecture, it turns out to be very fast, complete and optimal, allowing

to react to the wold dynamics (reactive planning), generating new optimal solutions every time the obstacles

positions changes.

1 INTRODUCTION

The work presented in this paper concerns Mobile

Robots Path-Planning exploiting Multilayered Cellu-

lar Automata (MCA). The Path-planning problem is

very important to drive mobile robots in environments

avoiding collision with obstacles. In our work, we

consider robots with different types of kinematics: for

example, robots moving thanks to differential drives

(moving forward and backward, and rotating in the

place), robots moving as cars (car-like kinematics or

non-holonomic), robots also translating in any direc-

tions and rotating (sphere-like kinematics or omnidi-

rectional), etc. To realize a path-planner able to face

with different types of motion is quite problematic.

More over, we added other constrains to this prob-

lem. When a robot moves between obstacles, its real

shape and size must be considered to avoid collisions.

It is more difﬁcult to handle robot with asymmetrical

shapes, even with concavity, holes, etc. Most of the

works in literature treat this problem substituting the

real shape with an equivalent cylinder with the same

radius, enlarging the space occupancy. This solution

is feasible when the robot moves in wide spaces, but it

is not when considering narrow spaces, cluttered en-

vironments, etc. A further complication derives from

the geometry of the world in which the robot is sit-

uated. In a ﬂat world, as ofﬁce-like structured envi-

ronments, the geometry is quite simple: it is a planar

surface (Euclidean 2D Space) on which the robot nav-

igates. We want to face different situations, as motion

on natural terrains, where the Euclidean metric is no

more applicable. In this work we introduce the archi-

tecture of a complete optimal path-planner applicable

on robots with different shapes and kinematics op-

erating in a natural world. Because the use of the

paradigm of Cellular Automata, the approach is dis-

tributed and very fast, even on single-processor com-

puters. During the last twenty years, many authors

have proposed different solutions, based on geomet-

rical descriptions of the environment (e.g. (Lozano-

erez and Wesley, 1979; Lozano-P

erez, 1983)). In

the eighties, Khatib in (Kathib, 1985) ﬁrst introduced

a new method for the collision avoidance problem in

a continuous space for a 6 DOF manipulator robot.

This alternative approach is less precise, but more ef-

ﬁcient: the Artiﬁcial Potential Fields Methods. Ja-

hanbin and Fallside introduced a Wave Propagation

M. Marchese F. (2005).

A REACTIVE MOTION PLANNER ARCHITECTURE FOR GENERIC MOBILE ROBOTS BASED ON MULTILAYERED CELLULAR AUTOMATA.

In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics - Robotics and Automation, pages 82-87

DOI: 10.5220/0001184500820087

 SciTePress

algorithm in the discretized Conﬁguration Space (C-

Space) (Distance Transform (Jahanbin and Fallside,

1988)). Barraquand et al. in 1992 (Barraquand et al.,

1992) used the Numerical Potential Field technique

on the C-Space to build a generalized Voronoi Dia-

gram. Zelinsky extended the Distance Transform to

the Path Transform (Zelinsky, 1994). Tzionas et al.

in (Tzionas et al., 1997) described an algorithm for

a diamond-shaped holonomic robot in a static envi-

ronment, where they let a CA to build a Voronoi Di-

agram. Pai and Reissel in (Pai and Reissell, 1998)

introduced a Motion Planning algorithm on multires-

olution representation of terrains using wavelets. The

path ﬁnding is based on a variation of Dijkstra’s

algorithm on a regular 2D lattice. In (Kobilarov

and Sukhatme, 2004), the authors presented a Path-

Planner for outdoor terrain based on the Control The-

ory and the technique of random trees (RRT). We used

CA as a formalism for merging the Grid Model of

the world (Occupancy Grid) with the C-Space of a ro-

bot and Numerical (Artiﬁcial) Potential Field meth-

ods, with the task to ﬁnd a simple and fast solution

for the path-planning problem for mobile robots with

different kinematics. This method uses a directional

(anisotropic) propagation of distance values between

adjacent automata to build a potential hypersurface

embedded in 4D space. Using a constrained version

of the descending gradient on the hypersurface, it is

possible to ﬁnd out all the admissible, equivalent and

shortest (for a given metric of the discretized space)

trajectories connecting two conﬁgurations of the ro-

bot C-Space.

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 (Lozano-P

erez, 1983). The

C-Space C of a rigid body is the set of all its conﬁgu-

rations q (i.e. poses). If the robot can freely translate

and rotate on a 2D surface, the C-Space is a 3D mani-

fold (R

×SO(2) ≡ SE(2)). It can be modelled using

Figure 1: Mobile Robot Path-Planner Architecture

a 3D Bitmap GC (C-Space Binary Bitmap), a regular

decomposition in cells of the C-Space, represented by

the application GC : C → {0, 1}, where 0s represent

non admissible conﬁgurations. The C-Potential is a

function U(q) deﬁned over the C-Space 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 (non-holonomic vehicle). The

latter assumption introduces important restrictions on

the types of trajectories to be found. In the following

section, we will describe each layer and its properties.

3 THE PATH-PLANNER

ARCHITECTURE

The planner architecture is organized in four main

packages (Fig. 1). The main package is the Path-

Planner itself: it coordinates the two lateral packages

(Environment and Robot Models). It is worth not-

ing that the World Model, the Robot Model and the

Path-Planner use the same structure based on Cellu-

lar Automata, i.e. there is an isomorphism based on a

regular decomposition structure.

3.1 CA Kernel

The CA Kernel realizes the Cellular Automata para-

digm (Fig. 2.b). The architecture is organized in lay-

ers of cells depending on the underlying topological

space. There are bi-dimensional spaces, such as the

Work-Space, and 3D spaces (C-Space, Attraction Po-

tential, etc.); some of them are active and are used to

make calculations, others are static and are used only

to represent speciﬁc information. The package makes

available the basic structures to represent the informa-

tion of both types of spaces and the related calculation

kernel.

3.2 Environment CA Model

The Environment Model (Fig. 3.a) is subdivided in

two parts: the C-Space and the Terrain Elevation Map.

The ﬁrst is a 3D space (SE(2), i.e. position and orien-

tation) in which are represented both the C-Obstacles

and the C

free

-Space. The second is a regular 2D man-

ifold representing the elevation map (z = f(x, y)) of

the terrain on which the robot navigates. In a struc-

tured world (e.g. an ofﬁce-like environment), it is

simply as a ﬂat surface (z = 0). There is space metric

associated to the C-Space. Formally, in a continuous

space, it is a set of parameters used to deﬁne a matrix

(fundamental tensor) necessary to evaluate the inﬁni-

tesimal distance between two neighboring points. In

A REACTIVE MOTION PLANNER ARCHITECTURE FOR GENERIC MOBILE ROBOTS BASED ON

MULTILAYERED CELLULAR AUTOMATA

(a) (b)

Figure 2: Packages inside views (A)

(a) (b)

Figure 3: Packages inside views (B)

our case, the space is discretized in cells (CA), and

the set of parameters are slightly different, but the has

the same role: the evaluation of the linear distance

between two adjacent cells in the 3D C-Space.

Figure 4: An example of Environment Model as a combi-

nation of Terrain and Obstacles Layers

3.3 Robot CA Model

This package is used to model the robot (Fig. 3.b).

Two major components are necessary to model the

robot: its shape and its kinematics. The shape is de-

ﬁned using a 2D CA (Fig. 5.a): it is a small occupancy

map centered on the robot cinematic center. The ro-

bot kinematics is deﬁned as a set of available moves.

Each move allows the robot to rototranslate from the

current cell to an adjacent one. The kinematics de-

ﬁnition is completed with a robot metric. This is a

set of costs associated to each robot move, to evalu-

ate the cost (in term of energy or time or path length)

the robot has to spend to reach a new pose. From this

two structures is derived a secondary structure con-

taining the Motion Silhouettes. In Regular Decom-

position world models, the obstacles are decomposed

in full or empty cells (Occupancy Grids) and the ro-

bot is shrunk to a point (the robot cinematic center).

The well-known technique of enlarging the obstacles

by the robot maximum radius is then used to take into

account of its real extension (Lozano-P

erez and Wes-

ICINCO 2005 - ROBOTICS AND AUTOMATION

ley, 1979). An isotropic enlargement with a constant

radius would give the same results as using an equiva-

lent cylindrical robot. The consequence is a great loss

(a) (b)

Figure 5: An example of CA silhouette for a L-shaped ro-

bot (red cross: cinematic center): a) basic silhouette (robot

shape); b) a motion silhouette obtained sweeping the robot

silhouette during a single movement (rotation)

of space around the obstacles, and many trajectories

are lost. An anisotropic enlargement (Lozano-P

erez

and Wesley, 1979), i.e. a different obstacles enlarge-

ment for each robot orientation, would solve only par-

tially the problem for asymmetric robots. Counter-

examples can be shown where the robot still collides

with obstacles between two consecutive poses. We

adopted a different approach to address this problem,

by deﬁning the Motion Silhouettes (e.g. Fig. 5.b)

and evaluating cell by cell the set of admissible ro-

bot movements that avoid collisions (see Repulsive

Layer in 3.4.1). The Motion Silhouette is generated

by sweeping the basic robot shape during a single

move and marking the cells covered. To avoid col-

lisions, there must not be any obstacles cells in the

marked cells. The Motion Silhouettes are calculated

off-line once for all for any robot orientation, combin-

ing the set of movements (robot kinematics) and the

basic silhouette (robot shape).

3.4 CA Path-Planner

The Path-Planner of Fig. 2.a is a sort of ”bridge” be-

tween the Environment Model and the Robot Model.

It combines the information from both to generate a

set of optimal trajectories. It is organized in two ma-

jor subsystems: an Input Subsystem and an Output

Subsystem (Fig. 6). They are subdivided in ﬁve sub-

layers (3 + 2), some of them are statical and the others

evolve during the planning time. The Input Subsys-

tem is an interface toward the outstanding environ-

ment. Its layers have to react as fast as possible to

the external changes: the robot starting pose, the ro-

bot goal pose, the elevation map of the terrain and,

the most important, the changes of the environment,

i.e. the obstacles movements in a dynamic world.

The ﬁrst two layers (Starting Pose L. and Goal Pose

Figure 6: Path-Planner Layers Structure

L. are considered quasi-statical, because they are up-

dated from the outside at a very low frequency (much

lower than the internal updating frequency); the Re-

pulsive L. is updated externally (by means of a per-

ception system) and it evolves testing the robot move-

ments admissibility to avoid collisions. The Output

Subsystem returns the results of the planning, that is

the set of complete trajectories from the Path Extrac-

tion L., or a single motion step from the Attraction L.

3.4.1 Repulsive Layer

The Repulsive Layer is the dynamical version of the

C-Space. It is ﬁrst initialized with the C-Obstacles in

the Environment CA Model, then it starts evolving to

ﬁnd out cell by cell any admissible move (Collision-

free Moves) exploiting the Motion Silhouettes previ-

ously determinated. The admissibility of a move is

inﬂuenced: 1) globally, by the speciﬁc robot kinemat-

ics; 2) locally, by the vicinity of an obstacle and by

the robot shape. This layer is decomposed in sub-

layers, one for each robot orientation and move: it is

itself a Multilayered CA on a 2D domain (the space

of positions). It is a particular type of Cellular Au-

tomata also for another reason: the cell neighborhood

does not have the standard square shape, but a non-

standard ﬁxed architecture (Sipper, 1997). Its par-

ticular shape reﬂects the robot Motion Silhouette as

shown in the example of Fig. 5.b. In Fig. 7 is shown

the result of the evolution of this layer for a rectangu-

lar robot using the terrain of Fig. 4.b. The grey level

is proportional to the number of admissible moves,

ranging from white (all moves admissible) to black

(no move).

A REACTIVE MOTION PLANNER ARCHITECTURE FOR GENERIC MOBILE ROBOTS BASED ON

MULTILAYERED CELLULAR AUTOMATA

(a) (b)

Figure 9: An example with a car-like robot: a) Attraction Potentials hypersurface (3D skeleton); b) Trajectories on the

elevation map;

Figure 7: An example of the Obstacles Layer for a rectan-

gular Robot moving in Fig. 4 world

3.4.2 Attraction Layer

This layer is the main part of the entire Path-Planning

Algorithm. The Attraction Layer is the digitalized

representation of the C-Potential function U(q) de-

ﬁned on the C-Space Bitmap: for any position and

orientation is calculated a potential value. The C-

Potential is a potential surface with a global minimum

in the goal cell. The potential represents the inte-

ger ”distance” of the cell c from the Goal cell when

the robot moves with an outgoing direction along a

collision-free path, or more simply it is the cost to

reach the goal from the cell c. To evaluate the path

cost, we have introduced the Robot Metric as a set

of costs for the basic robot movement: (forward, for-

ward

diagonal, direction change, stop, rotation, lat-

eral, backward, backward

diagonal). These basic

movements are combined in different ways to repre-

sent the different robots kinematics. The robot can

be subjected also to non-holonomic constraints, there-

fore not every movement can be done as an omnidi-

rectional robot does. To treat different kinematics, we

have introduced a subset of admissible moving direc-

tions (D

′

(c, d) ⊆ D) depending on the robot posi-

tion (cell c) and orientation d compiled off-line on the

base of the robot kinematics. Changing the metric,

we can realize the kinematics of different robots. For

example, the kinematics (2, 3, 1, 0, High, High, 2, 3)

emulates a common car-like kinematics (Reed’s and

Shepp’s vehicle), while A robot also translating in

any direction (omnidirectional) has a kinematics like:

(2, 3, 1, 0, 1, 2, 2, 3). To compute the path length, an

Environment Metric (3.2) is needed. On a ﬂat environ-

ment (Euclidean Space), the distance between cells

(the space metric) is invariant with respect to the po-

sition. For variable terrains, the cost value must de-

Figure 8: Manoeuvering example for a L-robot in a clut-

tered world

pend on the 3D distance between points on the sur-

face, on the difference of elevation between two po-

sitions, hence the space metric changes from point

to point. For this reason, it is necessary to include

the surface gradient in the cost function. The over-

ICINCO 2005 - ROBOTICS AND AUTOMATION

all path cost results in a combination of robot met-

ric, environment metric and surface gradient (a kind

of ”path-planner metric”). Two main properties can

be demonstrated: the termination of the propagation

of the potential values through the C-Space and the

absence of local minima. The later property ensures

to achieve the goal (in the global minimum) just fol-

lowing the negated gradient vector of the C-Potential

function without stalling in any local minima.

4 EXPERIMENTS AND RESULTS

We have generated some synthetic elevation maps and

introduced an obstacle distribution to observe the al-

gorithm behavior. An interesting property of this al-

gorithm is the simultaneous computation of trajecto-

ries from more than one starting position. We exploit

this property to show multiple problems in the same

environment. In the example of Fig. 9, the terrain has

a group of three ”hills” in the middle, and we con-

sider ﬁve different starting points of the robot and one

goal (bottom-left). From any position, the robot tries

to move around the hills, unless the shortest path is

to pass over them (in any case, at the minimum ele-

vation). The performance tests, carried out with an

Intel Pentium IV 2.26 GHz PC, gave the following

result (mean times over 500 repetitions ): 182 ms

(Fig. 9), 26.7 ms (Fig. 8). The complexity of a path-

planning algorithm is always strictly related to the ob-

stacles distribution. A good upper-bound estimate, in

the worst cases without obstacles enlargements, can

be done. Considering that the longest paths cover ap-

proximatively 1/2 of the total number of cells N of

the 2D Workspace Bitmap, and require nearly 2

cells updates to be computed, a realistic upper-bound

of the complexity is O(N

). If we take also into ac-

count of the obstacles enlargements, the result is even

better since the number of free cells is much lower,

especially in a cluttered world.

5 CONCLUSION

In this paper we have described an architecture solu-

tion for the Path-Planning Problem for mobile robots

with generic shapes (user deﬁned) and with generic

kinematics on variable (regular) terrains based on

(Multilayered) Cellular Automata. Another impor-

tant property of this algorithm is related to the con-

sistency of the solution found. For a given terrain sur-

face, the solution found (if it exists) is the set of all

shortest paths (for the given metric) that connect the

starting cell to the goal cell. The CA evolution can be

seen as a motion from one point to another point of a

global state space until an optimal solution is reached.

This is a convergence point for the given problem or

a steady global state. If we make some perturbations,

such as changing the environment (e.g. adding, delet-

ing or moving one or more obstacles), then the point

becomes unstable and the CA starts to evolve again

towards a new steady state, ﬁnding a new set of opti-

mal trajectories (Incremental Updating).

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A REACTIVE MOTION PLANNER ARCHITECTURE FOR GENERIC MOBILE ROBOTS BASED ON

MULTILAYERED CELLULAR AUTOMATA