CALCULATION OF OPTIMAL PATHS IN THE

CONFIGURATION SPACE USING ARTIFICIAL POTENTIAL

FIELDS AND A* AND D* ALGORITHMS FOR AN

ARTICULATED ROBOT. COMPARISON OF TECHNIQUES

Miguel A. Gutiérrez, Alejandro Morales

Depto. Informática y Ciencias Computación, Universidad Católica de Ávila, Ávila, Spain

Vidal Moreno, Belén Curto

Depto. Informática y Automática, Universidad de Salamanca, Salamanca, Spain

Keywords: Configuration space, Articulated robot, Potential f

ields, Path planning.

Abstract: In this paper, we use a calculation path technique based on the configuration space in the case of an

articulated robot of two degrees of freedom. We propose the use of artificial potential fields to represent the

configuration space and the use of techniques of artificial intelligence like A* and D* to search a free

collision path into the configuration space. This combination of techniques can be used in static and

dynamic environments with more than three dimensions without considering the geometry of the obstacles.

The results for this combination of techniques are presented, choosing in each case the best option for each

one of the techniques for the combination.

1 INTRODUCTION

In the design of autonomous robots, one of the tasks

on which more studies have been made is the path

planning. Different techniques like potential fields or

artificial intelligence algorithms like A* or D* have

been used.

We can use the potential field method to

calcu

late the path from the initial configuration to

the final one. One of the first works presented on the

use of potential fields in the path planning was

(Hwang 1988), where a survey of the possibilities of

this method was made. In (Melchior 2003) the

potential field method is used to represent the

environment of the robot in 2D. A risk coefficient is

assigned to each obstacle to increase the repulsion

field in the surroundings of some obstacles. The

algorithm used to search a path between two

positions of the map was A*. Also in 2D spaces, the

potential field method is used in (Prestes 2004) to

represent the surrounding of a Nomad2000 robot in

order to make easy to find paths towards the

unexplored zones of the map. In (Ge 2002)

(Kacandes 1989) (Lee 1995) the potential fields

method is proposed to find a path in dynamic

environments. For example in (Lee 1995), the

surrounding of the robot was observed in regular

intervals of time, and the space occupied by the

obstacles at the end of the interval was calculated at

the beginning of the same one. After this, a repulsion

field is generated around the space that is going to

be occupied by the obstacles and an attraction one

towards the goal. In this work, the robot is assumed

to be precise and the obstacles are supposed to be

circular. A similar exposition was made in (Ge

2002), where the speed and acceleration of the

obstacles were well-known with respect to the robot.

All these methods are proposed in 2D work spaces.

Other methods calculate potential fields without

local minima. In (Latombe 1991) (Wong 2000) the

potential field map is calculated by a road map

previously calculated by a front of waves. Potential

field method has been used in 3D spaces in (Elnagar

2002) (Yachida 1995) (Zhang 1997), for example in

the path planning for flying robots.

Another option for path planning is the use of

artificial in

telligence algorithms. In (Trovato 1996),

Karen I. Trovato makes a study of the algorithms A*

and D* in the case of a robotic arm and an

A. Gutiérrez M., Morales A., Moreno V. and Curto B. (2005).

CALCULATION OF OPTIMAL PATHS IN THE CONFIGURATION SPACE USING ARTIFICIAL POTENTIAL FIELDS AND A* AND D* ALGORITHMS

FOR AN ARTICULATED ROBOT. COMPARISON OF TECHNIQUES.

In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics - Robotics and Automation, pages 11-16

DOI: 10.5220/0001165800110016

 SciTePress

automobile. In (Gilliard 2002) the A* algorithm is

used on a search space represented using BSP trees.

This representation depends on the geometry of the

obstacles. This makes that this technique of path

planning very difficult to use in environments with

three or more dimensions. In (Adi 2004) the A*

algorithm has been used on hierarchic maps,

dividing the search in several levels. The main map

is divided in several smaller submaps, that can as

well be divided in smaller ones, etc. The search is

made on each one of the maps separately and the

obtained paths are united to get an only path.

All the mentioned path planning techniques

work in the work space. Other works use these

techniques in the configuration space.

The idea of reducing the robot to a point in an

appropriate space was introduced by Udupa (Udupa

1997), although the term configuration space had not

been used yet. Later, Lozano-Pérez (Lozano-Pérez

1983) adopted the notion of Cspace from the

mechanics and popularised it in path planning. In

some works the Cspace is calculated by means of

geometric methods as much for obstacles as for

robots with polygonal or polyhedral convex forms

(Lozano-Pérez 1983) (Seidel 1986). Other ones have

tried to make a representation in bitmap in such a

way that “elemental blocks” can be identified and

easily transformed into the Cspace, so that combined

form the Cspace for more complicated forms

(Branicky 1991). More recent works set out to

calculate bitmap of the Cspace like the convolution

of bitmap of the workspace with the one of the robot

(Kavraki 1995) (Moreno 1997) (Blanco 1997). In

these, the sufficiently small CPU time is used in the

calculation of the Cspace to use it in the planning

task without this supposes a boosting charge. In

(Blanco 1997) the potential field without minima is

used in the configuration space. In other works, the

geometric properties of the obstacles for the

construction of the Cspace are used to reduce the

time of calculation (Helgason 2001) (Williams

2001) (Chen 1996). In (Fox 1993), Maciejewski

proposes a path planning technique based on the

geometric characteristics of the obstacles in the

configuration space. Those techniques that use the

geometric properties of the obstacles are very

difficult to be applied in environments with more

than three dimensions.

In this work, we will use a combination of these

techniques. We will represent the configuration

space of an articulated robot using the potential field

technique and we will use path search algorithms

like A* and D* for searching collision-free paths.

We will use a bitmap to represent the configuration

space. This representation allows us to use the

technique independently of the geometry of the

robot or the obstacles. The selected search

algorithms work in static and dynamic

environments, so this technique can be used in both

type of environments. In addition, the measurement

of the performance of the algorithms implemented

will be presented.

The remainder of this paper is organized as

follows: in section 2 we present the problem

statement. The different techniques used to calculate

the configuration space of the robot and its

representation with potential fields are commented.

In section 3, we present a software platform

developed to validate the path planning process.

Finally, in section 4 we present the performance of

the technique in different environments and in

section 5 the main conclusions are presented.

2 PROBLEM STATEMENT

The path calculation technique used in this paper

works in the configuration space. First, the

configuration space corresponding to the work space

of the robot is calculated. The representation of this

configuration space is made by a bitmap. This

representation allows us to work independently of

the geometry of the obstacles and the robot. The

configuration space is represented by potential

fields: attraction to the goal and repulsion from

obstacles. Using this bitmap representation, the

repulsion field of the obstacles is calculated. In order

to obtain the path from the initial configuration to

the final configuration, we will use the A* in static

environments and the D* algorithm in dynamic

environments. Both algorithms guide their search by

an heuristic function. We use the potential field to

guide the search algorithms.

The result of the search algorithm is a series of

configurations that, if they are adopted by the robot,

it will move in the workspace without colliding with

obstacles.

2.1 Calculating the Configuration

Space

To calculate the configuration space we use the

method proposed in (Moreno 1997). In this method

the Cspace is calculated like the convolution of the

function that represents the robot and other function

that represents the workspace. In the particular case

of an articulated robot with two degrees of freedom

the final expressions to calculate the obstacles in the

Cspace are:

ICINCO 2005 - ROBOTICS AND AUTOMATION

Figure 1: Application develope

ϕϕθϕθ

drdrBrACB ),(),()( 1)0(111 ⋅−=

∫

ϕϕθϕθθ

drdrBrACB

∫

⋅−= ),()1,(),( )2,0(2212

where A

and A

are the subsets of the workspace

that represent the two links that form the robot and B

is the subset of the workspace occupied by the

obstacles.

Using this bitmap representation, the potential

field is calculated. To calculate the attraction force

of the potential field in one point, we use the

distance to the goal configuration. Possible functions

are: Euclidean, Manhattan, eight neighbours, etc.

This function is easy to calculate and valid in spaces

with more than three dimensions. To calculate the

repulsive force of the obstacles, we use a front of

waves that calculate the repulsive force in each point

in the surroundings of the obstacle. This force is

smaller as we move away from the obstacle.

With this method we can calculate the Cspace of

the robot in a short time, and it is easy to modify if

the robot is in a dynamic environment.

2.2 Using Potential Fields like

Heuristic Function

The A* and D* are heuristic guided algorithms. We

use the potential fields representation of the

configuration space like heuristic function to guide

the collision-free path search. The definition and

pseudocode of the algorithms can be found in

(Feigenbaum 1986) (Stentz 1994) and (Stentz 1995).

To do this, the function that chooses the best

neighbour in the A* algorithm must be modified.

The original function consists of two terms: cost

from the starting node (g) and the heuristic that

guides the search. This last term is divided in two

terms: the first is the attraction force in this point

) and the second is the repulsive force in this

point (h

). The resulting function is the following

one:

f = g + h where h = h

+ h

In the same function in the D* algorithm, the

resulting function consists of only two terms: the

first is the attraction force in this point (h

) and the

second is the repulsive force in this point (h

f = h

+ h

Both functions can be quick and easily

calculated in any dimension spaces.

3 PROCEDURE VALIDATION

We have developed a software platform to validate

the path calculation. This application presents the

user the work space and its corresponding

configuration space. In both representations we can

see the robot, the obstacles, the calculated path, the

explored nodes and the repulsion field around the

obstacles. When the user modifies the work space,

the corresponding configuration space is

automatically recalculated and showed to the user.

We can see two different path calculations in figure

3.1 Pseudocode

The pseudocode of the algorithm used may be the

following:

env = robot_environment();

cspace = cspace_from(env);

pot_cspace = repulsive_field(cspace);

result = search_alg(pot_cspace,

dist_function,data_struct);

wresult = to_workspace(result);

apply_robot(wresult);

CALCULATION OF OPTIMAL PATHS IN THE CONFIGURATION SPACE USING ARTIFICIAL POTENTIAL

FIELDS AND A* AND D* ALGORITHMS FOR AN ARTICULATED ROBOT. COMPARISON OF TECHNIQUES

The

robot_environment function generates a

bitmap representation of the environment of the

robot. This bitmap represents the workspace of the

robot and the calculation path technique used works

in the configuration space. The

cspace_from

function generates the Cspace. Then, the

repulsive_field function calculates the

repulsive fields of the obstacles in the configuration

space. At this time, the map for the path calculation

is ready. We use one search algorithm like A* or D*

to search a free-collision path using like heuristic

function the potential field generated before. The

path generated must be transformed for its use in the

workspace using the

to_workspace function.

Finally,

apply_robot acts in the robot to follow

the path.

4 PERFORMANCE OF THE

ALGORITHM

In this chapter, the performance of this technique is

presented. First we present the CPU time results

obtained using both algorithms in different

environments. The location and number of obstacles

and the initial and goal positions of the robot have

been generated of random way.

In the particular case of an articulated robot with

two degrees of freedom, two different free-collision

paths could be found: elbow-up and elbow-down

configurations. Those searches can be made in

parallel, so that we can choose the best solution, the

faster solution, etc. The CPU times choosing the best

solution are listed at figure 2.

Figure 2: Different CPU time results in different environments using both search algorithms

To calculate the attraction forces in the potential

field we use the distance to the goal configuration.

We can see the results using the Euclidean,

Manhattan and 8-distance functions in the following

table.

Path length Explored nodes Time (s)

Euclidean 100,774 61894,857 0,659

Manhattan 94,833 38947,389 0,421

8-distance 105,05 45177,01 0,573

As we can observe in this table, the number of

explored nodes is quite bigger in the case of using

Euclidean distance.

The most important data structure in both

algorithms is the open list, mainly because a good

selection of the data structure has an important

repercussion in the time of calculation and in the

memory usage. We have tested the algorithms with

two different data structures: an ordered list and a

binary heap. The results obtained are listed in figure

ICINCO 2005 - ROBOTICS AND AUTOMATION

3. The results show that the use of the binary head

data structure reduces the time of computation.

A parameter to be considered when representing

the work space with potential fields is the size of the

repulsion field that separates the robot from the

obstacles. Numerous tests demonstrate that, from

certain size, to increase the size of the repulsion field

makes the number of explored nodes to find the path

grow, increasing the time used to calculate it. This

tests show that a repulsion field size over ten points

causes the increase of the number of explored nodes.

Nevertheless, in the cases of a repulsion field size

below ten points (even the no existence of repulsion

field) the number of explored nodes is very similar.

This proves the importance of using a size of

repulsion field not excessively big, since we forced

the search algorithms to give a roundup to avoid the

obstacle. An example of this can be showed in the

figure 4.

5 CONCLUSIONS

The main target of this work has been the

representation of the environment of an articulated

robot using the potential field technique and the

implementation of path search algorithms in these

environments and the measurement of its

performance.

The results demonstrate that, in static

environments, there are not important differences of

performance between search algorithms.

The search algorithms have a higher

performance if the Manhattan distance is used

instead of the Euclidean distance or 8-distance in the

calculation of the distance to the goal configuration.

It is the same if a binary heap is used to store the

open list instead of a ordered list.

The size of the repulsion field that moves away

the robot from the obstacles must not be excessively

great, since the yield of the algorithms diminishes as

this field of repulsion grows. The results show that,

although the information provided by the potential

field is not much, the implemented algorithms

search allow us to obtain the optimal way between

two configurations in a fast way.

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