REPRESENTATION OF ODOMETRY ERRORS ON OCCUPANCY

GRIDS

Anderson A. S. Souza, Andre M. Santana, Ricardo S. Britto, Luiz M. G. Gonc¸alves

and Aderlardo A. D. Medeiros

Depatment of Computating Engineering and Automation, UFRN, Natal, Brazil

Keywords:

Mapping, Occupancy grid, Odometry.

Abstract:

In this work we propose an enhanced model for mapping from sonar sensors and odometry that allows a robot

to represent an environment map in a more suitable way to both the sonar sensory data and odometry system

of the robot. We use a stochastic modelling of the errors that brings up reliable information. As a contribution,

we obtain a ﬁnal map that is more coherent with the reality of the original data provided by the robotic system.

Practical experiments show the results obtained with the proposed modiﬁcation to be trustable in such a way

that this map can be used to provide previous knowledge to the mobile robot in order to perform its tasks in an

easier and accurate way. Moreover, the map can help the robot to support unexpected situations inside of the

environment.

1 INTRODUCTION

In order for a robotic system to be efﬁcient in the ac-

complishment of tasks, an important requirement is a

correct spatial description of its underlying space that

can be constructed from its own sensorial data. This

description, a trustable map, makes possible a coher-

ent interaction of the robot with its environment, so

that it can perform its tasks efﬁciently and also can

deal with unexpected situations like dynamical obsta-

cles appropriately. The process of construction of this

representation is generally named as mapping and the

result is a map of environment.

Two main approaches are used to represent an en-

vironment in a map: the topological and the metric. In

the topological approach, the environment map is rep-

resented by using graphs, in which the nodes are free

spaces and the edges have information about connec-

tion among the nodes, for example, distance. Topo-

logical mapping makes easier the accomplishment of

high level tasks, as navigation, with smaller computa-

tional cost.

In the metric approach, the geometry of the envi-

ronment is deﬁned in a more detailed way, presenting,

accurately, the position of the objects inside of the en-

vironment, for instance walls, chairs and desks.

A common method used in the construction of

metric maps is the occupancy grid. In this method,

the environment is represented by a matrix in which

each element represents a place in the environment

that can be empty or occupied, or can be a unknown

area.

In this work, we propose a mapping methodology

with representation using an occupancy grid with a

modiﬁcation proposed in the model of the sonar sen-

sor that embodies the uncertainty of the odometry sys-

tem.

2 RELATED WORKS

There are many works in the literature dealing with

occupancy grid. In 1987, Elfes (Elfes, 1987) pro-

posed the occupancy grid method that is better for-

malized in his Ph.D thesis (Elfes, 1989). His work is

implemented in two robots, Neptune and Terregator

and is part of a more complete system, that integrates

navigation and mapping based on sonars. This sys-

tem is named Dolphin. Our current work also deals

with sonars in the construction of the map in occu-

pancy grid, but we have improved the sonar model by

treating the noise of the system in a better way.

Moravec (Moravec, 1988) proposes a system, in

wich the occupancy grid map is based on sonar data

and stereo data. Informationfrom a sonar sensor array

and from a stereo vision system is combined to build

202

A. S. Souza A., M. Santana A., S. Britto R., M. G. Gonçalves L. and A. D. Medeiros A. (2008).

REPRESENTATION OF ODOMETRY ERRORS ON OCCUPANCY GRIDS.

In Proceedings of the Fifth International Conference on Informatics in Control, Automation and Robotics - RA, pages 202-206

DOI: 10.5220/0001500302020206

 SciTePress

the metric map of environment.

In a subsequent work, Moravec (Moravec, 1996)

introduces the idea of a map represented in a three

dimensional occupancy grid. During the mapping, a

sequence of stereo images is processed and the result

of the mapping stored in a three dimensional array

named evidence grid. The cells are initializes with

zero value, indicating that there is no occupation ev-

idence. After several sensor readings, the cells are

ﬁlled out so that blocks of negative cells indicate free

space, while positive blocks indicate obstacle. In his

work, Moravec has used the theory of evidence of

Dempster-Shafer. We use here the Bayesian approach

also used by Elfes (Elfes, 1987).

Konolige (Konolige, 1997) has presented a

method to treat the problems relative to the sonars

in a more efﬁcient way. Example of problems are

the specular reﬂection and redundancy of reading.

The proposed method is a mathematical reﬁnement

of the method introduced by Elfes (Elfes, 1987). The

method is named as MURIEL (MUltiple Representa-

tion Independent Evidence Log).

In the Elfes’ works, mapping is performed without

considering dependence among the cell and its neigh-

bours. This implicates in inconsistent maps when

the mapping is performed in cluttered environments.

In a more recent work, Thrun (Thrun, 2003) has in-

troduced an advanced sensor model to deal with the

problem of dependence among cells presented by the

standard algorithm of Elfes (Elfes, 1987). The model

introduced by Thrun (Thrun, 2003) veriﬁes the val-

ues of occupation of the neighboring cells and then

attributes the value of occupation to the current cell.

Thrun has based his work on the Bayesian theory.

Thrun et. al. (Thrun et al., 2005) afﬁrm that

the main usefulness of the mapping with occupancy

grid is in the post-processing, that is, in the map con-

structed. The map can be useful in several appli-

cations like, navigation, path planning, recognition

of landmarks, obstacle avoidance and localization.

Borenstein and Koren (Borenstein and Koren, 1991)

for example, have implemented a method of obsta-

cle avoidance in real time named VFF (Virtual Force

Field). This method uses an occupancy grid map, ob-

tained from sonar data, to deﬁne the localization of

the obstacles inside of the environment.

In the Dutra’s work (Dutra et al., 2003), a robot

provided with an array of 24 sonars builds an occu-

pancy grid map of its surroundings and store it in its

internal memory. Later, this map is used for naviga-

tion, but the results obtained by this work in both nav-

igation and mapping were quite inﬂuenced by accu-

mulated errors of the odometry system. In our work,

we model the sonars taking into account the intrinsic

errors also to the odometry system.

Other works in the literature exemplify the use of

occupancy grid maps. Kong et. al. (Kong et al.,

2006) has implemented a localization system based

on EKF (Extended Kalman Filter) in which features

inside the environment, as corners and ﬂat surfaces,

are detected. The information obtained about the fea-

tures is integrated with an occupancy grid map known

a priori to yield an accurate localization of the robot

in the environment.

3 THE STANDARD ALGORITHM

OF OCCUPANCY GRID

MAPPING

The standard algorithm formalized by Elfes (Elfes,

1989), builds the map from both the sensorial data and

the robot position (localization and orientation). The

mathematical formulation of the occupancy grid map-

ping is derived from Equation (1), which gives the

value of occupation of the whole map (Elfes, 1987;

Thrun et al., 2005; Thrun, 2003).

p(m|z

1:t

) (1)

In this equation m represents the obtained map, z

1:t

represents the set of measurements until the instant t.

The continuos space of the environment is converted

in a discrete space of cells, which form together an

approximation of the environment. Thus, the map is

deﬁned as a ﬁnite set of cells m

x,y

. Each cell posses a

value, among 0 and 1, associated which corresponds

to the state of the cell, occupied and empty, or can

represent an unknown state. The value 0 means empty

cell and 1 occupied cell. The notation p(m

x,y

) refers

the probability of a cell of index < x, y > to be occu-

pied.

The standard algorithm divides the problem of

construction of the map in a set of smaller problems

of estimate of the values of each cell m

x,y

separately.

p(m

x,y

1:t

) (2)

Due to reasons of numerical instabilities for probabili-

ties near 0 or 1, it is common to calculate the log-odds

of p(m

x,y

1:t

) instead of p(m

x,y

1:t

). The log-odds is

deﬁne for:

x,y

= log

p(m

x,y

1:t

)

1− p(m

x,y

1:t

)

(3)

The probabilities are easily recovered from the log-

odds ratio:

p(m

x,y

1:t

) = 1−

1+ e

x,y

(4)

REPRESENTATION OF ODOMETRY ERRORS ON OCCUPANCY GRIDS

203

The value of the log-odds can be estimated recur-

sively in any instant t by the rule of Bayes applied

on p(m

x,y

1:t

p(m

x,y

1:t

) =

p(z

1:t−1

, m

x,y

)p(m

x,y

1:t−1

)

p(z

1:t−1

)

(5)

Supposing that we are mapping static environment,

we can afﬁrm that the current measurements of the

sensors ones are independent of the last ones:

p(z

1:t−1

, m) = p(z

|m) (6)

Because the map is decomposed in cells, this suppo-

sition is also extended all. We assume, then, the inde-

pendence of each cell. Without take into account the

occupation of the neighboring cells, we have:

p(z

1:t−1

, m

x,y

) = p(z

x,y

) (7)

This allows us to simplify the Equation (5).

p(m

x,y

1:t

) =

p(z

x,y

)p(m

x,y

1:t−1

)

p(z

1:t−1

)

(8)

Applying the Total Probability Theorem to p(z

x,y

)

we obtain:

p(m

x,y

1:t

) =

p(z

x,y

)p(m

x,y

1:t−1

)

∑

x,y

p(z

x,y

)p(m

x,y

1:t−1

)

(9)

The Equation (9) provides the probability of a cell

x,y

to be occupied.

4 THE PROPOSED MODEL OF

THE SONAR

In our work, we consider that noise is intrinsic to both

data sources, sonar sensor and odometry system, in

the construction of the map. The sonar sensors have

internal and external features which arouse errors in

their measurements. Usually, these errors are pub-

lished by the manufacturers and made available to the

public. In Figure 1, for example, we can verify some

typical features of a sonar Polaroid series 6500, that

are used in our work. This sonar presents higher sen-

sibility in the area near to its main axis. Besides, it

presents an error of absolute measurement of +/- 1%.

Besides the typical errors of the sonar, we attempt

to the errors accumulated by the odometry during the

motion performed by the robot. The odometry cal-

culates the robot’s current pose in relation to the pre-

vious pose. With this, errors accumulate in an incre-

mental way.

If we neglect these two sources of error, the ﬁnal

map will not be kept in accordance with the data orig-

inated from the robot, committing others applications

like, navigation, path planning and others.

Figure 1: Features of the Polaroid sensor.

To deal whit those errors, we looked for a way of

include them in the occupancy grid map representa-

tion. Thus, we modiﬁed the sonar sensor model of the

standard algorithm, so that, in a probabilistic way, the

typical errors of the sonars and odometry are incorpo-

rated to the value of occupation of the cell in the grid

(Equation 10).

p(z

x,y

) =

2πσ

× e



−



x,y

−z

)

(θ

x,y

−θ)



(10)

Where:

represents the sensor reading in the instant t;

represents the orientation of the sensor;

represents the standard deviation regarding the er-

ror in the measured distance by the sensor;

represents the standard deviation regarding the er-

ror in the orientation of the sensor;

x,y

represents the Euclidean distance between the

sensor and the cell m

x,y

;

x,y

represents the angle of orientation of the cell m

x,y

The standard deviation regarding the error in the

measured distance depends the value of the odometry

error of a translational movement, that is,

= z

× k+ f (11)

Where:

k is the factor of errors intrinsic of the sonar (in our

case, +/- 1%);

f is a function which describes the odometric system-

atic error, when the robot moves linearly.

The standard deviation regarding the error in the

orientation angle of the sensor depends on the value

of the odometry error of a rotational movement, that

is,

+ g; (12)

Where:

β is the solid angle subtending the main lobe of the

area of sensitivity;

g is a function which describes the odometric system-

atic error, when the robot moves in a rotational way.

Function f in Equation 11 and function g in Equa-

tion 12 were experimentally deduced from several

data samples. Figure 2 shows a graph of the sensor

model deduced.

ICINCO 2008 - International Conference on Informatics in Control, Automation and Robotics

204

Figure 2: Graphics of the proposed model.

With this modiﬁcation, the value of the probabil-

ity of occupation of a cell passes to be weighed by

the uncertainties in the real robot position, due to the

odometry errors. Thus, the map is corrupted gradu-

ally, representing the real error.

5 EXPERIMENTS

To validate de proposed model, we have made some

experiments with the mobile robot named Galatia,

model Pioneer 3-AT of the ActivMedia Robotics, pro-

vided with two array sonar sensor (front and back)

and odometry system (Figure 3).

Figure 3: The robot Pioneer 3-AT.

The experiments are performed inside of the De-

partment of Computing Engineering and Automation

- UFRN, trying to map the corridors. The ﬁrst experi-

ment is performed using the standard algorithm of oc-

cupancy grid mapping without relying on the odome-

try errors. The result of this experiment can be seen

in Figure 4.

Figure 4: Map without representation of the errors.

Later, we perform the same experiment again

using the probabilistic sonar model proposed here,

to represent the systematic errors of the sonars and

odometry (Figure 5).

Figure 5: Corrupted map.

In the maps, the white cells represents free areas or

no obstacles, the black cells represents occupied areas

or obstacles and the greycells represents the unknown

areas. The sketched lines show the real contour of the

corridors.

6 CONCLUSIONS

In this work, we propose a modiﬁcation in the sensor

model used in the standard algorithms based on oc-

cupancy grid mapping, including in the probabilistic

model the intrinsic uncertainties of the robotic sys-

tem. In order to consider these uncertainties in the

mapping, we have found a way to include them in the

representation of the map of the environment. Thus,

we basically modify the sensor model so that, in a

probabilistic way, the typical errors from sonars and

odometry can be incorporated in the occupancy value

of a cell. The model was tested in practice with the

robot Pioneer 3-AT and demonstrated to be correct,

giving the right expectation for the error

Based on the presented results, we can see that the

proposed model supplies a more realistic way to rep-

resent a mapped environment using occupancy grid,

being known that the originated information from the

robot have errors, and what are really these errors.

That is, the errors corrupt the quality of the map,

showing in this way coherence whit the sensorial data.

As extension of this work, we will improve the

treatment of the incoherent readings from the sonars,

like speculate reﬂections and so. Furthermore, a way

for relocation of the robot, when the errors grow a lot,

will be studied and implemented.

ACKNOWLEDGEMENTS

We thanks CAPES, RNP and CNPq by the ﬁnancial

support.

REPRESENTATION OF ODOMETRY ERRORS ON OCCUPANCY GRIDS

205

REFERENCES

Borenstein, J. and Koren, Y. (1991). The vector ﬁeld his-

togram - fast obstacle avoidance for mobile robots.

IEEE Journal of Robotics and Automation, 7(3):278–

288.

Dutra, P. R. C., de Souza, M. M., Andriolli, G. F., lvares,

A. J., and Ferreira, J. C. E. (2003). Navmap: Um

sistema para navegao por mapeamento do rob mvel

nomad xr4000. In VI Simpsio Brasilieiro de Automao

Inteligente (SBAI), Bauru, SP.

Elfes, A. (1987). Sonar-based real-world mapping and nav-

igation. IEEE Journal of Robotics and Automation,

3(3):249–265.

Elfes, A. (1989). Occupancy Grid: A Probabilistic Frame-

work for Robot Perception and Navigation. PhD the-

sis, Carnegie Mellon University, Pittsburg, Pennsylva-

nia.

Kong, F., Chen, Y., Xie, J., Zhang, G., and Zhou, Z. (2006).

Mobile robot localization based on extended kalman

ﬁlter. In 6th World Congress on Inteligent Control and

Automation, Dalian, China.

Konolige, K. (1997). Improved occupancy grids for map

building. Autonomous Robots, (4):351–367.

Moore, R. and Lopes, J. (1999). Paper templates. In TEM-

PLATE’06, 1st International Conference on Template

Production. INSTICC Press.

Moravec, H. P. (1988). Sensor fusion in certainty grids for

mobile robots. AI Magazine, 9(2):61–74.

Moravec, H. P. (1996). Robot spatial perception by stereo-

scopic vision and 3d evidence grid. Technical Re-

port CMU-RI-TR-96-34, CMU Robotis Intitute, Pitts-

burgh, Pennsylvania.

Smith, J. (1998). The Book. The publishing company, Lon-

don, 2nd edition.

Thrun, S. (2003). Learning occupancy grid maps with for-

ward sensor models. Autonomous Robots, (15):111–

127.

Thrun, S., Burgard, W., and Fox, D. (2005). Probabilistic

Robotics. MIT Press, Cambrige, Massachusetts, USA.

ICINCO 2008 - International Conference on Informatics in Control, Automation and Robotics

206