A probabilistic approach to Hough localization

A probabilistic approach to Hough Localization

Luca Iocchi, Domenico Mastrantuono, Daniele NardiDipartimento di Informatica e SistemisticaUniversita di Roma “La Sapienza”, Italy

{iocchi,mastrant,nardi}@dis.uniroma1.it

Abstract

Autonomous navigation for mobile robots performingcomplex tasks over long periods of time requires effec-tive and robust self-localization techniques. In thispaper we describe a probabilistic approach to self-localization that integrates Kalman filtering with mapmatching based on the Hough Transform. Several sys-tematic experiments for evaluating the approach havebeen performed both on a simulator and on soccerrobots embedded in the RoboCup environment.

1 Introduction

Self-localization is a crucial feature for autonomousnavigation of mobile robots performing complex tasksover long periods of time. Indeed several practical ap-plication domains require mobile robots to know theirposition within the environment, in order to effectivelyand reliably accomplish their tasks.

The use of a particular kind of sensor usually affectsthe design choices for the localization method. A typi-cal configuration for a mobile robot is having a relativepositioning system (e.g. motor encoders), which pro-vides an estimate of the displacement of the robot fromthe previous pose, and a range sensor (e.g. ultrasonicsonars, laser range finders, vision systems), which re-turns a set of 2D points, in the local coordinates of therobot, corresponding to the visible surfaces of objectsclose to it. This configuration is suitable for applyinglocalization methods that are based on model match-ing or map matching (see [2] for a survey).

Among the several existing methods for robot self-localization, map matching has been extensively stud-ied in the past years and the proposed approaches canbe divided into two groups depending on the repre-sentation of the reference map: 1) set of points (rawmap); 2) set of geometric features (geometric map).

The first group includes algorithms performing mapmatching by using all the points captured by the sensordevice, without any geometrical assumption on thesedata. A common feature for these approaches is that

matching tends to be computationally hard, and insome cases the proposed methods require heavy opti-mization to implement an effective real-time localiza-tion task on a mobile robot. In [5] a local search inthe robot’s pose space is performed in order to findthe best overlap between the current scan and the ref-erence map. In the Markov Localization described in[4] there is an explicit representation of the probabilitydistribution of the robot’s pose in the environment andevery sensor points is used for updating this distribu-tion according to the reference map. The second groupof methods make use of geometric features instead ofraw points: therefore they requires a preprocessing inorder to extract features (or natural landmarks) fromthe sensor data. Most of these methods deals withlines, segments, corners and the reference map is thusrepresented as a set of these features. The main draw-back of these kind of methods is that they rely on theavailability of features in the environment.

The probabilistic approach to self-localization [4] isbased on estimating the most likely pose of the robotgiven all the information on the environment com-ing from the sensor devices. The task is usually per-formed by matching range data against a given refer-ence model (a map) of the environment, in order todetermine the absolute pose of the robot in this map.Information coming from map matching are then inte-grated with odometric information in order to increasethe reliability and the precision of localization.

In this paper we describe an approach to self-localization in which map matching is performed in thedomain obtained by applying the Hough Transform torange data. This method, originally developed for theRoboCup games [6], applies to any environment thatcan be represented by a set of segments (polygonalenvironment) and it provides for a solution to the po-sition tracking problem, assuming that the robot hasat every time an initial guess of its pose. The workdescribed in [7] follows our approach using an omnidi-rectional camera installed on the robot. In this setting,since the robot is able to see at every time all the en-

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Proceedings of the 2001 IEEE International Conference on Robotics & Automation

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vironment in which it must act, the method providesfor a global localization.

The robotic soccer environment provided by theRoboCup organization [1] is an interesting setting fortesting solutions for self-localization, and in particularin the F-2000 League, where global positioning sensorsare not allowed and thus localization can be based onlyon sensors that are mounted on the robot. We havesuccessfully tested this method in the RoboCup envi-ronment within the ART team [8] during the officialcompetitions by making use of vision based line ex-traction procedures performing as a range data sen-sor. The main features of the method are robustnessand reliability in a very dynamic environment. Thecontribution of this paper is twofold: first we extendthe approach described in [6] by introducing a proba-bilistic viewpoint that allows us to address the prob-lem of integrating the Hough map matching processwith odometric data; second, besides the good resultsof the method demonstrated during the official games,we have developed a set of systematic experiments thatare described in details in section 4.

2 Hough Transform

In this section we present the Hough Transform andhighlight the properties that will be useful for devel-oping our localization method.

The Hough Transform is a robust technique for find-ing lines fitting a set of 2D points [3]. It is based ona transformation from the (x, y) plane (a Cartesianplane) to the (θ, ρ) plane (the Hough domain).

The transformation from (x, y) to (θ, ρ) is achievedby associating every point P (x, y) with the followingcurve in the Hough domain

ρ = x cosθ + y sinθ (1)

At the same time, a point in the Hough domain corre-sponds to a line in (x, y). Notice that this is a uniqueand complete representation for lines in (x, y) as longas 0 ≤ θ < π.

Given a set of sensor data S = {(xi, yi) | i = 1, .., n},let us define the following functions:

hSi (θ, ρ) =

{

1 if ρ = xicosθ + yisinθ0 otherwise

HTSc (θ, ρ) =

n∑

i=1

hSi (θ, ρ)

The function HTSc (θ, ρ) will be called the Hough

Transform of the sensor data S. In the following sec-tions, however, we will make use of a discrete represen-tation of this function that we denote with HTS(θ, ρ)

and that is obtained by generating a discrete grid ofthe (θ, ρ) plane (let δθ and δρ be the step units) andby defining HTS(θ, ρ) as the number of points (x, y)whose corresponding curve (1) lies within the interval[θ, θ + δθ]× [ρ, ρ + δρ].

Observe that it is possible to consider the discreteHough Transform of S as a voting space for points in(x, y), in which every point in (x, y) “votes” for a setof lines (represented as points in (θ, ρ)), that are allthe lines passing through that point.

The Hough Transform has a number of propertiesthat are useful for self-localization: 1) given a set of in-put points, a local maximum of HT (θ, ρ) correspondsto the best fitting line of these points; 2) in presenceof points originally belonging to several lines, no clus-tering is needed since local maxima of HT (θ, ρ) corre-spond to the best fitting lines for each subset of pointsrelative to each line; 3) the Hough Transform is veryrobust to noise produced by isolated points (since theirvotes do not affect the local maxima) and to occlusionsof the lines (since point distances are not relevant);4) measuring displacement of lines in the Cartesianplane corresponds to measuring distance of points inthe Hough domain.

An interesting property, that will be useful in the fol-lowing sections, is in the relation between the transfor-mations of the sensor readings when the robot moves.

Property 1. Given the Hough Transform of aset of sensor readings S, HTS(θ, ρ), and a rota-tion/translation (Tx, Ty, θR) of the robot (we assume| θR |≤ π), the Hough Transform of S with respectto the new pose of the robot will be HTS(θ′, ρ′) suchthat:if 0 ≤ θ + θR < π then

θ′ = θ + θR

ρ′ = ρ + Tx cos(θ + θR) + Ty sin(θ + θR)

if θ + θR ≥ π then

θ′ = θ + θR − π

ρ′ = −(ρ + Tx cos(θ + θR) + Ty sin(θ + θR))

if θ + θR < 0 then

θ′ = θ + θR + π

ρ′ = −(ρ + Tx cos(θ + θR) + Ty sin(θ + θR))

It is important to notice here that if θR = 0 (i.e.the robot does not rotate) then θ′ = θ (i.e. θ doesnot change), and conversely if Tx = Ty = 0 (i.e. therobot does not translate) then ρ′ = ρ (i.e. ρ does notchange). In other words, robot’s alignment can be di-vided in two separate steps: first determining the ori-entation with a null translation, and then determiningthe translation with a null rotation.

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The Hough Transform can be extended for detect-ing circles from a set of points by using the followingparametric curve:

(x− α)2 + (y − β)2 = r2

If we assume that r is known (and thus constant),we have to determine only two parameters α and βcorresponding to the center of the circle. The CircleHough Transform for the sensor data S will be denotedwith CHTS(α, β).

3 Hough Localization

In this section we introduce the Hough Localizationmethod, obtained by developing the framework pro-posed in [6]. Hough Localization is based on a match-ing between the Hough representation of a known mapof the environment and a local map built by the robot’ssensors.

The task of estimating the most likely pose of therobot in the environment can be addressed by eval-uating the probability that the robot is at a certainlocation, given all the sensor readings.

We assume that at every time-step t the followingdata are available to the robot: data from the rela-tive positioning system At and data from the rangesensor St. We also denote with p(l) the probabilitydistribution of the robot’s pose, with l = (x, y, θ) ∈<2 × [0, 2π).

Localization can be expressed as the task of com-puting the probability distribution p(l | At, St) fromthe previous distribution p(l | At−1, St−1), the currentsensor readings At and St, and a reference map M.

This task is usually performed in two steps:

1. Prediction. Predicting the new pose of the robotby dead reckoning from the previous position(that is computing p(l | At, St−1) from p(l |At−1, St−1) and At).

2. Update. Updating the robot pose with the re-sults of a map matching process between St andM (that is computing p(l | At, St) from p(l |At, St−1), St, and M).

Hough Localization is based on map matching be-tween sensor data and a reference map that, under theassumption that the environment can be representedby a set of segments, is performed in the Hough do-main.

The overall Hough Localization method consists inthe following steps:

1. extracting range information from the environ-ment in the form of a set of point S in the (x, y)plane,

2. generating the discrete Hough TransformHTS(θ, ρ) of such points,

3. determining the local maxima of HTS(θ, ρ) (forinstance by a threshold),

4. finding correspondences between local maximaand reference points,

5. measuring the displacement between local max-ima and the corresponding reference points in theHough domain (that corresponds to the displace-ment between the predicted and the actual poseof the robot),

6. integrating map displacement with odometric in-formation.

The critical step of this procedure is the fourth one,that is finding the correct correspondence between lo-cal maxima and reference points. Indeed errors in as-signing correspondences usually lead to large position-ing errors that are then difficult to recover.

In this article we focus the attention on the positiontracking problem, in which an initial guess of the po-sition of the robot is available before performing themap matching process. Position tracking is usuallyadopted when a bounded error assumption can be rea-sonably made. This assumption means that the posi-tioning error of the robot is within a certain threshold,that usually depends on the characteristics of the en-vironment. For example in the RoboCup environmentthese thresholds can be up to 50 cm and 45 degrees,and this allows the robot to deal with some of the col-lisions taking place during the games.

In the general case, in which it is not possible to relyon any information about the current position of therobot (global localization), a different technique mustbe integrated with the Hough Localization. In [6] wedescribe specific solutions for global localization in theRoboCup environment based on landmark recognitionand active localization.

3.1 Position tracking

Position tracking is usually addressed by representingthe probability distribution of the robot’s pose p(l) asa Gaussian, whose mean lk is the most likely positionof the robot and the covariance matrix Pk representsthe variance of this information.

Under the bounded error assumption, the line corre-spondence problem can be easily addressed by adopt-ing a closest matching approach. Given a refer-ence point (θM, ρM) and a local maximum of HTS

(θS , ρS), a match will be considered if and only if(θM − θS)2 + (ρM − ρS)2 < δ.

In other words, the HT grid can be partitioned in anumber of regions (one for each reference point in M),

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a ∆x

π2

π

b

∆θ

b’∆y

ρ

∆θ

a’

0 θ

b

a a’

b’

x

y

Figure 1: Map matching in the Hough domain

such that a matching will be considered only if a localmaximum of HTS is within the corresponding region.

Consider the example shown in Fig. 1, where therobot faces a corner. The solid segments a, b representthe map model and the set of points a’, b’ representdata coming from sensor device. The four segmentsare also displayed in the Hough domain: a, b (indi-cated by a circle) are the reference points, while a’,b’ (indicated by a cross) represent the local maximaof the Hough Transform applied to the set of inputpoints. Under the bounded error assumption, the cor-respondence problem is solved in this case by assigninga′ to a and b′ to b. For the Property 1 described insection 2, the displacement between the estimated andthe actual pose of the robot is determined by first com-puting the orientation ∆θ (with ρ constant) and thenthe translation ∆x, ∆y (with θ constant).

The computational complexity of the map matcingprocess is O(nk), where n is the number of points re-turned by the range sensor and k is the number ofsegments in the map M. Indeed, since a local searcharound each reference point is performed, HTS is com-puted only in a limited region around the referencepoints. This complexity bound makes the methodsuitable for real time implementation (typical compu-tation time is below 10 ms on a Pentium CPU).

3.2 Integrating map matching andodometry

The map matching method described above providesfor a correction of the estimated position of the robotthat must be integrated with odometric information.A standard technique for this integration (that is suit-able when the probability distribution of the pose ofthe robot is represented by a Gaussian) is using anExtended Kalman Filter [5].

We can describe the dynamics of the robot, withinternal state lk = (xk, yk, θk)T , input from odome-try uk = (δk, αk)T and output zk = (xk, yk, θk)T , like

follows

lk+1 = lk + Bkuk + Wkw

zk = lk + v

where

Bk = Wk =

cos θk 0sin θk 0

0 1

The vectors w = (wδ, wα)T and v = (vx, vy, vθ)T

are random variables representing respectively noisein odometric data and noise in the map matching pro-cess. For these random variables we assume a Gaus-sian white noise with zero mean and covariance matri-ces Qk and Rk.

Extended Kalman filtering is performed in twosteps:

1. Prediction. An estimated pose l−k+1 of the robotis computed from the previous pose and odometry andthe covariance matrix is updated.

l−k+1 = lk + Bkuk

P−k+1 = Pk + WkQkWkT

2. Correction. The pose of the robot is correctedby the result of the map matching process. Indeedzk+1 represent the new pose of the robot according tomap matching.

K = P−k+1(P−k+1 + Rk)−1

lk+1 = l−k+1 + K(zk+1 − l−k+1)

Pk+1 = (I −K)P−k+1

As we will see in section 4, extended Kalman filterprovides for an improvement in precision and stabi-lization of the robot’s pose.

4 Experiments

In this section we describe several experiments for eval-uating the precision and the robustness of the HoughLocalization: first experiments that make use of a sim-ulator in which we could test the effectiveness of theapproach under controlled conditions, and then exper-iments with real robots. The accuracy of a localizationmethod usually depends on the precision of the rangesensor. If we consider an ideal range sensor, the noiseintroduced by the Hough method is due only to thediscretizazion of the Hough grid. Therefore, the gridintervals δθ and δρ characterize the accuracy of theHough localization method itself and must be tunedaccording to the precision of the range sensor.

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Figure 2: Vision range simulator

4.1 Experiments with a simulator

We are using a simulator for mobile robots that in-cludes a mathematical model of several different er-rors that occurs during robot navigation and sensorperception. In particular, because of our applicationoriented to robotic soccer, we emulate a vision rangesensor that is able to extract points belonging to thelines in the RoboCup game field (see Fig. 2)1.

The following errors are considered in the simula-tor: 1) odometric error: the position of the robot isaffected by a random noise, such that the position er-ror increases over time; 2) sensor noise: sensor dataare affected by a random noise that increases with thespeed of the robot; 3) systematic error: sensor dataare affected by a systematic error that corresponds tousual errors in the calibration of the camera; 4) robotbumps: random movements of the robot; 5) false pos-itives and occlusions: points that do not belong to aline and occlusions due to objects (other robots) thatare in the field.

The models of the environments considered in ourexperiments are formed by sets of segments: in theRoboCup environment we have segments represent-ing the boards and the lines drawn in the field, andone circle drawn in the field; in an office environmentsegments represent walls of the corridors. These seg-ments are represented as points in the Hough domainfor lines, and the circles are represented as points inthe Hough domain for circles. Observe that the wallsare real obstacles for the robot, while the other linesand the circle are drawn in the field and do not corre-spond to obstacles. However, the vision range sensorthat we are using on our robots [6] is able to extractrange information from both of them and thus we con-sider them also in the simulator.

With the use of a simulator it is possible to know

1We are grateful to Kurt Konolige for his permission to ex-tend the Pioneer simulator.

exactly the actual pose of the robot at every time andto evaluate the position error as the difference be-tween the actual position and the estimated one. InFig. 3 we display two typical results of our experi-ments: in the first experiment (Fig. 3a) we have con-sidered only odometric errors, while in the second one(Fig. 3b) three bumps of the robot have been sim-ulated. The three lines on the graphs (Fig. 3a, 3b)represents the odometric error (red), the error withthe Hough Localization and without the Kalman filter(blue), and the error with the Hough Localization andthe Kalman filter (bold green). The temporal analysisshows that: 1) Hough Localization provides an upperbound to the localization error, while odometric er-ror generally increases over time; 2) odometric errorincreases smoothly, while Hough Localization updatesthe robot’s position sharply; 3) the use of a Kalman fil-ter provides for smoothing the robot position updates,while keeping the bound on the localization error.

4.2 Experiments with the robots

Hough Localization has been implemented in ourrobots by making use of a vision based range sen-sor (see [6] for details on this sensor) in two differentreal environments: during the official RoboCup soccercompetitions and within the corridors of our lab.

A first qualitative evaluation of the method has beenperformed during the RoboCup games, by using amonitor displaying the robot pose and by a visual in-spection of its position in the field and the estimatedposition. Furthermore, we have performed more sys-tematic experiments for evaluating the precision of theHough Localization.

The first kind of experiments follows a classical ap-proach [5]. We chose a number of reference positionsin the environment, then we drove the robot on a pathand we measured several times the distance betweenthe actual position of the robot and its internal esti-mation. The results in evaluating the position errorin the two operation fields are summarized as follows:average = 13cm, maximum = 29cm, variance = 8cm.

The above procedure attempted to measure an aver-age precision of the self-localization method, but suchvalues are not fully adequate to evaluate our methodsince they do not consider many sources of errors aris-ing in the actual operation (collisions with obstacles,occlusions) and they may depend on various experi-mental conditions: type of range sensor, noise in theenvironment, choice of the path, robot velocity, etc.Moreover, since the samples are acquired only in pre-defined positions, and when the robot is not moving, itis not possible to monitor the effect of the localizationmethod during robot navigation. For proving also the

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a) b) c)

Figure 3: Position error: a) simulator without bumps, b) simulator with bumps, c) real robot.

robustness of a localization method we need a more de-tailed analysis of the robot position error during theexecution of its tasks. We have thus implemented aglobal vision system, that makes use of a fixed camerapositioned outside the game field for measuring the ac-tual position of the robot. The images contain a globalview of the field and they are analyzed for recognizinga special marker put on the robot and for determiningits pose in the field.

In this setting, as in the experiments with the simu-lator, we are able to monitor the robot’s position errorduring navigation and thus evaluate the robustness ofthe self-localization under real conditions. In Fig. 3c)we show a representation of two trajectories computedduring a normal activity of one of our soccer robots.The green trajectory has been computed by the in-ternal localization method of the robot, while the redone has been computed by the tracking system of theglobal vision device. Note that this setting has notbeen used for evaluating the accuracy of the method,since it is affected by the errors of the global visionsystem in computing the position of the robot (theaverage position error of the global vision system isabout 16 cm). Instead it is very useful for evaluat-ing the robustness of our localization method, in factwe can show that the position error is always limitedwithin a certain threshold.

5 Conclusion

Hough Localization presented in this article is basedon a geometric representation of the reference map:lines and circles. This representation is suitable forthe RoboCup environment, but also in office-like en-vironments the availability of straight walls is usuallyguaranteed. With respect to other map matching tech-niques, the advantages of using the Hough Localizationare: 1) it is computationally efficient, since positiontracking is linear in the number of sensor points; 2) the

Hough Transform and thus the line extraction processis very robust to occlusions and false positives.

The probabilistic approach to self-localization hasbeen essential for an optimal integration of a robustmap matching process based on the Hough Transformand odometric information. The experiments we haveperformed have proven that the use of an ExtendedKalman Filter is relevant for both reducing the over-all localization error and for smoothing the positionupdates.

References

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[3] R. Duda and P. Hart. Use of the Hough Transfor-mation to detect lines and curves in the pictures.Comm. of the ACM, 15(1), 1972.

[4] D. Fox, W. Burgard, S. Thrun. Markov localizationfor mobile robots in dynamic environments. Jour-nal of Artificial Intelligence Research 11, 1999.

[5] J. S. Gutmann, W. Burgard, D. Fox, and K. Kono-lige. An experimental comparison of localizationmethods. In International Conference on Intelli-gent Robots and Systems, 1998.

[6] L. Iocchi and D. Nardi. Self-localization in theRoboCup environment. In RoboCup-99: RobotSoccer World Cup III, 1999.

[7] C. F. Marques and P. U. Lima. A localizationmethod for a soccer robot using a vision-basedomni-directional sensor. In Proc. of 4th Interna-tional Workshop on RoboCup, 2000.

[8] D. Nardi, et al. ART: Azzurra Robot Team. InRoboCup-99: Robot Soccer World Cup III, 1999.

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