Learning safe navigation in uncertain environments

Robotics and Autonomous Systems 31 (2000) 43–52

Learning safe navigation in uncertain environments

Maria João Rendas∗, Stefan RolfesLaboratoire d’Informatique, Signaux et Systèmes de Sophia Antipolis (I3S), 2000 route des Lucioles Batiment Algorithmes/Euclide, BP 121,

06903 Sophia Antipolis, France

Abstract

We present a strategy for the navigation of mobile robots in open sparse uncertain environments. Our work assumes thatthe robot has no access to external informations about its absolute position, and that the environment consists of a set ofobjects whose separation is large compared to the range of the perception sensors. The approach is based on: (i) maximizingthe probability of successful transition between any pair of known objects — by requiring if necessary that the robot usesother objects as intermediate landmarks — and (ii) guaranteeing — by controlling the extension of dead-reckoning motionsand resorting if necessary to active search for perceptual information — that a reliable path to the homing region alwaysexists. The paper presents experimental results that clearly demonstrate the increase in autonomy resulting from the proposednavigation strategy. © 2000 Elsevier Science B.V. All rights reserved.

Keywords:Safe navigation; Mobile robots; Open field robotics

1. Introduction

In this paper we present a navigation strategy formobile robots operating in open space regions, wherea set of isolated objects are present, the distance be-tween objects being large compared to the range of therobot’s perception sensors (hypothesisH1). We fur-ther assume that the robot has no access to absoluteposition information and no a priori knowledge of itsenvironment (hypothesisH2). This set of assumptionsis representative of the operating conditions that canbe found for instance in underwater robotics where:

HHH111--- Distinctive perceptual features are sparse, whilethe range of perception sensors (sonar, vision) is small.

∗ Corresponding authorE-mail addresses:[email protected] (M.J. Rendas),[email protected] (S. Rolfes)

HHH222--- External positioning information such as GPS isnot available, and a priori information — when avail-able — is extremely unreliable.

Combined,H1 and H2 impose the need for impor-tant periods during which proprioceptive data is thesole means to localize the robot, i.e., the areas insidewhich perception-based navigation is possible are dis-connected, and represent a small fraction of the entireworkspace. The largest risk associated with platformsoperating in this kind of environments is toget lost,being unable to come back to their designated hom-ing region (where, for instance, they must be pickedup or re-charge their batteries). The control of theplatform is then best addressed as a multi-criteriaoptimization problem, which attempts tominimizerisks, while at the same timeminimizing consumed re-sources(energy, traveled distance, or execution time).In this paper we concentrate on risk minimizationonly.

0921-8890/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved.PII: S0921-8890(99)00082-2

44 M.J. Rendas, S. Rolfes / Robotics and Autonomous Systems 31 (2000) 43–52

Our approach to the problem is supported by thefollowing observations concerning the periods ofdead-reckoning navigation: (a) uncertainty about therobot’s state increases with the degree of maneuver-ing required by the trajectory, and is at a minimumwhen the reference values of the motor control loopsare held constant; (b) uncertainty growth is dictatedby the platform’s mechanical characteristics, sensornoise levels and sampling rate: the only way to min-imize it at the end of a motion is to minimize theuncertainty affecting its starting configuration. Ob-servation (a) restricts the kind of nominal motionsduring dead-reckoning periods, while (b) constrainsthe possible initial points of these motions to a dis-crete set of points in the configuration space of therobot, which we designate by “gateways”, and onwhich self-localization is optimal in a sense to bemade precise later. These two restrictions reduce the(infinite) search space of the optimization problem toa tractable set.

Under these constraints, we propose a navigationcontrol strategy thatguarantees the recovery of theplatform, and which is based on three fundamentaltools:1. the possibility ofestimatingthe currentloss risk;2. the ability tolimit this risk;3. the ability toactively searchfor supporting percep-

tual features.During dead-reckoning periods the robot continuouslysupervises its security, by determining theprobabilitythat it is able toreturn to its homing region. If thisprobability falls below a prescribed level, the motion isaborted, and the robot is directed to one of the knownobjects, from where it is sure (up to a given successprobability) that it will be able to reach the homingarea in a safe manner.

To travel between known objects, our controlstrategy leads the robot along the path that presentsthe smallest predicted risk, i.e., for which the prob-ability of reaching the designated target object islargest. We point out that this criterion automati-cally yields a compromise between dead-reckoningand perception-based navigation. If dead-reckoningerrors, or the uncertainty about the relative positionof the present and target regions, are large, a path isfound that relies heavily on the possibility of periodicrepositioning using perception. Otherwise, the lengthof dead-reckoning segments is maximized, and a path

close to the shortest path between the initial and finalpositions is found, as our experimental results show.

When accessing target points outside explored re-gions, the robot makes sure that it discovers usefulperceptual features that guarantee that it will not getlost. If necessary, it will execute an active explorationof the workspace, searching for new perceptual fea-tures that can safely support its progression towardsits goal. This active exploratory behavior leads to aneffective extension of the autonomy range of the robot(defined as the size of the region that it can accesswithout getting lost) and is necessary for safe opera-tion in sparse environments.

The paper is organized as follows. Below, we pointout the major innovative aspects of our work. The nextsection defines terminology, enumerates a set of basicfunctionalities that the platform must exhibit in orderto implement our control strategy, and explains its im-plementation on the basis of aplanning graph. Section3 formally defines the nodes and arcs of the planninggraph. Section 4 presents experimental results demon-strating the effectiveness of the approach. Finally, Sec-tion 5 summarizes the paper’s contributions.

Many authors have addressed the problem of us-ing perception for robot navigation under uncertainty.Four major aspects distinguish the work presented herefrom previously published studies:1. Open space contexts. A significant number of re-

ported studies address navigation in dense environ-ments, where perception can be continuously usedto guide the robot, for which the central issue oflosing the robot is not relevant. Research in openfield robotics [5] has mainly been developed for ter-restrial outdoor robots, where availability of GPSguarantees spatially homogeneous and boundedlocalization errors, limiting the risk of platformloss. For underwater (or space) robotics this riskis real, and must be mastered to enable large scaleuse of autonomous platforms in undersea applica-tions. However, the larger error conditions that canbe expected in this type of environments, and thatpreclude direct application of common ExtendedKalman techniques are often ignored, see forinstance [2].

2. The need for exploration. Many studies addresspath planning using an existing (eventually uncer-tain) environment representation, and under un-bounded configuration uncertainty: how to take the

M.J. Rendas, S. Rolfes / Robotics and Autonomous Systems 31 (2000) 43–52 45

robot from an initial uncertain configuration to abounded vicinity of a target configuration, using thegathered perceptual data [3,6] in the most efficientway. The majority of these studies assume a pas-sive attitude of the robot. Recent studies [4] haveaddressed the problem of controlling the platformor its sensors in order to minimize uncertainty (inworld knowledge or the robot’s sate). They differfrom our work in two major aspects: (i) they searchfor the action that yields thelargest informationgain, irrespective of the necessity of the resultinguncertainty decrease; we try to match the level ofuncertainty to the operational requirements of themission: uncertainty is tolerated if it does not affectthe robot’s safety; (ii) they ignore the problem ofcontrolling the platform evolution insideunknownregionsin order to guarantee that enough informa-tion is gathered to prevent platform loss, which isa central issue in our approach.

3. Collision free paths. The determination of colli-sion free paths has received significant attentionfrom the robotics community. In our opinion, ob-stacle avoidance is a robust local low-level built-inbehavior present in all modern robotic systems,and maintaining a safety distance to obstacles isno longer a central issue for mobile platforms. Onthe contrary, the possibility of observing an ob-ject when moving from one place to another canhelp correcting drifting errors, and thus keep therobot directed to its target region. In our approach,obstacles are considered as information sources,and a balance between the amount of informationthey convey and the cost of accessing it must befound.

4. Uncertainty models grounded on signal processingtools. Our navigation strategy is deeply groundedin the application of formal signal processing toolsto characterize all sources of uncertainty affect-ing the robot’s state: the accuracy of the controlloops, and of the a priori and acquired environ-ment representation, and the precision of robot’spositioning using perceptual data. Most previousworks are either based on heuristic bounded-errormodels [3], or simply ignore some of these uncer-tainty sources. Our approach is general, specify-ing formal tools/measures of uncertainty that canbe particularized to distinct platforms and sensingsystems. In particular, the anisotropic nature of the

positioning errors justifies our definition of “gate-way points”, which captures in a unified index themorphological properties of the workspace, the ac-curacy of the sensing systems (proprioceptive andexterioceptive) and the reliability of low-level con-trol loops.

2. Assumptions and control strategy

In this section we describe our control strategy, ad-dressing separately the problems of making the robotreturn to an object previously detected and of control-ling its progression in the regions of the workspace notyet visited. Before, we summarize some assumptionsand present some definitions.

Built-in behaviors. We assume that the robot canexecute the followingperception-based behaviors:1. detect and acquire an internal representation of per-

ceptual features;2. dock into a given configuration defined with respect

to a known object;3. recognize a previously seen object, and reposition

itself with respect to it.In the experiments presented in Section 4 we use a

small mini-robot (diameter of 5 cm) with two drivingwheels, equipped of myopic distance sensors (maxi-mum range of about 10 cm) and odometric counters.Using the distance sensors, the robot can detect ob-jects, and incrementally make a description of theirshape, by executing a trajectory parallel to the object,details can be found in [7]. It can subsequently usethis acquired representation to place itself at a config-uration defined relative to the object. An algorithm forshape recognition [10] allows the robot to recognizeobjects that have previously been mapped and reposi-tion itself with respect to them.

Mission. With generality, we consider that the robot’smission is defined by a sequence of target pointsTn, n = 1, . . . , N , defined relative to its initial posi-tion and orientation, and which it must reach sequen-tially. The homing region contains the initial positionof the robot, and an object is located there (servingas a “lighthouse” of limited range). This assumptionis not restrictive, since, if the robot can be placed ata given position, then it should also be possible to


place an object there. Information about this object iseither given to the robot, or autonomously acquiredas the first step of its mission.

Environment. As we stressed in the Introduction, ourwork addresses natural sparse work-spaces, where typ-ical objects’ separation is large. Mathematically, wemodel the environment as a stationary isotropic ran-dom closed set [11], also known asgerm-grain orcoverageprocesses:Ξ = ⋃∞

n=1 (Ξn + xn), where thegermsxn are random points ofRd(d = 2, 3), thegrainsΞn are random compact subsets ofRd , and thenotationA + x denotes the translation of subsetA byvectorx. We do not assume that either the intensityλ of the point process or the distribution of the grainsare known. In our context, we refer to the individualgrains ofΞ, Ξn, as objects.

Internal representation. Throughout its mission, therobot builds an internal representation of all detectedobjects ofΞ , that we designate byE :

E =M⋃

m=1

Tpm [Om],

whereTpm is the rigid motion defined by the configu-rationpm, andOm describes the morphological prop-erties (shape, and eventually, other attributes) of theobject.

Each object defines a local frame, inside of whichthe robot’s uncertainty is fundamentally dependent onits perceptual features. The description of the uncer-tainty affecting the objects’ morphological propertiesis sufficient to determine the positioning errors ex-pected when navigating inside its field of influence,i.e., during perception based navigation.

Besides each object’s position and attributes, wealso assume known a measure of confidence of the rel-ative positions and orientations of all pairs of objects,described by conditional distributionsP(pm|pn).When moving from one frame to another the robot’suncertainty is governed by these conditional distribu-tions.

Gateways. With each objectOm we associate adiscrete set of gateway points{gmj }Mj=1 in itsfield of influence, which verify the two followingconditions:

c1: the robot can position itself ingmj with boundeduncertainty. This bound is dependent only on itstrajectory inside a neighborhood ofgmj , and notof all its distant past trajectory;

c2: gmj is the configuration that leads to the largestlower bound on the probability of reachingOj

when moving in one of the nominal allowed mo-tions starting ingmj :

gmj = argmaxp Pr [reachingOj with constant

control references fromp].

Condition c1 implies, in particular, the existence ofa configurationp∗ where local perceptual data canunambiguously localize the robot (by this we meanthat the probability distribution of the robot’s con-figuration given the data acquired along a trajectorycontained in a neighborhood ofp∗ has a single lobe,and is independent of its configuration uncertaintywhen entering it). This condition is important, en-abling independent optimization of the transitionbetween distinct local frames. We relegate the presen-tation of the tools used to define the gateway points toSection 3.

2.1. Moving insideE : The planning graph

Let G = (N ,A) be a labeled graph, whose nodesare the complete set of gateway pointsN = {gij }Mi,j=1,where we arbitrarily makegii = pi , the configura-tion that defines the global position and orientationof Oi . We call G the planning graph. Two sets ofarcsA = Au ∪ Ad are defined between a subsetof elements ofN . Au is a set ofundirectedunitweight arcs{βij }i=1,... ,M;j 6=i , linking gii to all gate-ways of Oi . These arcs correspond to executing asequence of perceptual behaviors inside framei. Theother set ofdirected arcs, αij ∈ Ad, correspondsto the allowed set of dead-reckoning motions, andlink gatewaygij in object Oi to the nodegjj . Theweights of these arcs,a(i, j), express the probabil-ity of passing directly from one object to another.Their computation is addressed in the next section.Fig. 1 gives an example of a planning graph forM = 3.

The minimum risk trajectory that takes the robotfromOi toOj is the maximum length path inG linkingnodegii to nodegjj . Let tij = [i, t1

ij , . . . , tkij , j ] be


Fig. 1. Planning graph.

one of the paths inG linking nodesgii to nodegjj .According to the probability algebra, the confidenceassociated to this path is

C(tij ) = a(i, t1ij ) · a(t1

ij , t2ij ) · · · a(tkij , j).

OnceG is defined, any standard minimum length pathalgorithm can be used to find the optimal trajectorybetween any two pairs of objects.

2.2. Exploring new regions: the augmented graphGe

The planning graphG is not useful for handling sit-uations where an unknown region of the workspacemust be reached. We extendG by augmenting it witha “virtual node” Vn that represents the target pointTn. For each objectOi , we define a new set of con-figurationsein in the corresponding local frame, thatwe designate by “exploration points”. The arc linkingein to the “virtual node”Vn is labeled by anutilitymeasureu(i, n) that reflects its usefulness as a start-ing point for active exploration in the direction ofTn.Determination ofu(i, n) is discussed in the next sec-tion. We define the augmented graphGe = (N e,Ae)

where

N e =N ∪ {Vn, ein, i = 1, . . . , M},Ae =Au ∪Ae

d,

Aed =Ad ∪ {(ein, Vn)}Mi=1.

Exploration points are jointly defined by two condi-tions similar to those defining the gatewaysgmj :d1: uncertainty bound dependent on local trajectory

only;d2: maximal utility:ein = argmaxp u(p, n).

When trying to attain a virtual objectVn, the robotdetermines the exploration pointein which offers thelargest utilityu(i, n). It directs itself to that node (us-ing the safest route indicated byGe), and docks atein,in the direction of the object, initiating an explorationof the unknown region in the direction ofTn. We donot address here the problem of defining the actualexploration strategy that is used at each “explorationpoint”, assuming just that: (i) during the active explo-ration the robot never gets lost; (ii) if an object existsinside the reach ofein the robot will find it, i.e., thesearch procedure is complete.

Assumption (i) above is verified by continuouslysu-pervising the robot’s risk level, determining the proba-bility that it is able, from its current uncertain position,to return to one of the known objects. If this proba-bility falls below a prescribed level (close to one), thecurrent motion is aborted. The exploration pointein isupdated, along with its utility value, to reflect the factthat no object was found. The robot is directed againto the node with largest current utility, from whereit initiates a new exploration. To compute the risk ofbecoming lost, the robot must be able to evaluate theprobability of attaining any of the objects already inits map from its current position. As we report in Sec-tion 3, this probability is computed using analyticaltools that provide upper bounds for the evolution ofthe uncertainty during dead-reckoning navigation, seealso [8,9].

When executing an exploration in the direction of atarget pointTn, three end conditions may happen: (i)the robot can directly reachTn from the explorationnode; (ii) a new object is detected within the autonomyrange ofein; (iii) all u(i, n) become zero. In case (i) therobot reaches its goal. It builds a new augmented graphfor its next targetTn+1 and re-initiates the controlledexploration just described. In case (ii)G is augmentedby a set of nodes corresponding to gateway points toand from the new object and exploration points fromthe new object to the target, and a new exploratoryprocedure is repeated from the node with largest utilityvalue. In case (iii) the mission is aborted, declaringunreachability of the target point.

We stress that this control strategy necessarily leadsto environment representations for which the corre-sponding planning graph is connected, i.e., it guaran-tees that at least one safe path between any two objectsexists.


3. Uncertainty modeling and perception

In this section we address the definition of the gate-ways and exploration points,gij andein, and the prob-lem of computing the associated performance indexesused by our architecture,a(i, j) andu(i, n). Lack ofspace precludes the detailed derivation and presenta-tion of all analytical results involved in these defi-nitions. We give, instead, an outline of the approachused, referring the reader to published references forindividual results.

3.1. Performance prediction: The risksa(i, j)

Although the presentation below can be generalizedfor other nominal trajectories, such as arcs of circledescribed at constant velocity, we restrict it here, forsimplicity, to the case of linear trajectories.

A basic tool to be able to predict the risks associatedwith dead-reckoning motions is obviously a model ofhow uncertainty grows along the traveled path. Pre-vious studies [8] have established analytical approx-imations to the temporal evolution of position uncer-tainty over linear motions for both the component ofthe error process orthogonal to the direction of motionand along the direction of motion. For instance, forplatforms with a compass it has been shown that the or-thogonal component is well approximated by a Brow-nian process, and its error covarianceQ(t) verifies:

q(t) 6 q(t0) + σ(t − t0), t > t0. (1)

Expressions forσ can be found in [8].The probability of reaching a given object from

an uncertain initial configuration is formally relatedto a first passage time of the position estimate pro-cess. The determination of the statistics of first pas-sage times of multivariate processes is a difficult prob-lem, for which the majority of the analytical resultsconcern asymptotic behavior. Instead, we work with aone-dimensional projection of the error process, anduse available results on scalar Gauss–Markov pro-cesses [1] to determine a lower bound on the proba-bility of reaching a desired target object.

Let g = (x, y, θ) be the initial robot’s configurationin the vicinity of objectOi andQ(t0) its uncertainty.Let g′ = (x′, y′, 0) be the configuration in the coordi-nate system that has origin at(x, y) and with axisx′

aligned with the direction of motionθ . We defineTR

as the time required to reach the center of the target, ata distanceD, with constant nominal velocityv0, witha given probabilityγ ' 1:

TR = inft>t0

Pr(x′(t) > D) > γ, (2)

which is given by the distribution function of the firstpassage time of the error process along the directionof motion for a boundaryD − v0t , and can be deter-mined using the model for the evolution of the posi-tion uncertainty along the direction of motion.

Using the value ofTR defined by (2), the followingexpression yields a lower bound on the probability ofreachingOj in terms of the orthogonal error compo-nent:

Pr(reachingOj |p)

> Pr(∀t ∈ [0, TR], y′(t) ∈ [−`R, `R]), (3)

where 2̀ R is the target’s extension in the directionorthogonal to the direction of motion. This equationgives a lower bound on the desired probability in termsof the probability of an event defined in terms of aMarkov time of the error process, for which closedform expressions are known [1]. Note that (3) yieldsindeed a lower bound on the probability of attainingR, since other sample paths of the process that reachR may exist, crossing the boundary|x′(t)| = `R aneven number of times. Finally, use of the analytic up-per bound on the covariance ofy′ in (3) allows thenumerical determination of the desired arc links.

We point out that the value ofa(i, j) is dependent onthe distance between the starting and final objects (D),the morphology of the target object (`R), the initialconfiguration uncertaintyQ(t0) — which is related tothe quality of docking at the gateway — and of theaccumulation of control and sensor noises.

3.2. Gatewaysgij

Gateway points, as it has been stated before, arechosen as the configurations in the field of influenceof Oi that yield the lowest value ofa(i, j). As theprevious subsection shows, to compute these labels wemust be able to quantify the uncertainty affecting thedeparting configuration.

We assume with generality that the estimates of theposition and orientation of the robot with respect to


Oi are obtained by maximizing the likelihood of theobserved perceptual and proprioceptive dataz giventhe robot’s final posep and the known object’s char-acteristics,

p̂ = argmaxpf (z|p, Oi),

and approximate the covariance matrix of the dockingerror by the Cramer–Rao lower bound,

E[(p − p̂)(p − p̂)T] > J−1(p),

whereJ is the Fisher information matrix.The gateway points can then be found by search-

ing for the paths in the field of influence ofOi withfinal configurationgij at whichJ−1 leads to a mini-mal value ofa(i, j). This approach, although optimal,leads to a nonlinear multi-variate optimization prob-lem, which is far too complex to be solved in real time.

We use a suboptimal approach, and, considering arestricted set of docking maneuvers, we search for theset of target configurationsg that are local minima of

E[1y(g)2|pj ],

1y(g) = εy(g) + (D(g) − εx(g)) tan(εθ (g))

whereD(g) is the distance fromg to the center ofthe target objectOj — at configurationpj — andwe assume that the error vector [εx(p), εy(p), εθ (p)]has covariance defined by the Cramer–Rao bound aspresented above.1y(g) is the distance between thetarget point and the point actually attained.

It may happen that several local minima of the quan-tity above exist. The gateway is chosen as the configu-ration that yields the lowest value ofa(i, j) computedas described in the previous section.

3.3. Exploration pointsein and utility measure

The exploration points must provide large auton-omy range in the direction of the target, and a largeprobability that enough intermediate objects will befound at distances smaller or equal to the autonomyrange of the robot, effectively supporting the safe pro-gression of the robot. Autonomy range in the direc-tion of the target is determined using the positioningperformance measures and prediction of evolution ofdead-reckoning errors presented above.

At present, we use a simple definition of utility asthe predicted autonomy range normalized by distance

to Tn. For similar objects, this leads to choosing theexploration points as those where local positioning —at the departing object — is optimal and which arecloser to the target. Very small objects, or objects withno distinctive features are discarded in favor of largerobjects, even if these are at slightly larger distances.Using the random closed set model, we are currentlyworking towards the definition of probabilistic basedutility measures, which take into account the observedcharacteristics ofΞ to determine optimal searchingstrategies.

4. Experimental results

The next figures illustrate our control strategy,showing experiments realized with the mini-robotKhepera, a small circular robot (Φ = 5.2 cm) withtwo driving wheels and equipped with a belt of eightinfra-red sensors that were calibrated to be used asdistance sensors.

The robot starts from the neighborhood of the smalltrapezoidal objectO1 shown in the lower left cornerof the left plot in Fig. 2, its mission consisting in at-taining, by that order, pointsT1 andT2 (marked with∗ in the plot). Note that the size of the rectangular re-gion displayed is about 2m×2m, much larger than therobot’s size, and than the range of its sensors (widthof the rectangular regions in the left plot).

The left plot in Fig. 2 illustrates the active explo-ration of the workspace, that leads to learning a saferoute to the (unmapped) target points. The plot on theright concerns planning the motion between objectsthat have already been mapped.

The robot starts by observing and mapping the ob-ject close to its initial position, and defines the ex-ploratory gateways for the two targets. It positions it-self ine11, and moves towardsT1 (along the path indi-cated by an arrow). During this motion, it is stopped bythe risk surveillance procedure, and redirected againto O1, from where it starts — after repositioning withrespect toO1 — the exploration in new directions, al-ternating on each side of the direct route toT1. Theseexplorations are still aborted two more times, untilO2 is detected, observed and mapped. New gatewayspoints (betweenO1 andO2) and exploratory gateways(from O2 to the target points) are defined. The robotthen selects the most useful exploratory gatewaye21.


Fig. 2. Mission with autonomy control, active exploration, and minimal risk routing.

The same exploratory procedure is launched from thatnode, leading directly to the detection of the objectplaced atT1. Note that sinceO2 is larger thanO1, therange of autonomy out ofe21 is larger than out ofe11.

The robot includes this new object,O3, in its worlddescription, deletes all exploratory gateways leadingto T1, and defines gateway points to and from the newobjectO3. It then selects the exploratory gateway toT2 with largest utility, and starts the exploratory be-havior, reachingT2 in the first movement out ofO3,and incorporating the description of the object placedthere,O4, in G. The rectangular regions between ob-jects in Fig. 2 represent the areas sensed by the robotduring the exploration.

The robot must now return toO1. Fig. 3 illustratesthe selected route inG. The robot selects the path that

Fig. 3. Routing graph for returning toO1 from O4.

leads it fromO4 to O1 throughO2, which offers thesmallest risk. Note that this does not correspond toexecuting in the reverse direction the same path as inits progression toO4. The relatively small distancebetweenO4 and O2, and the large apparent size ofO2 seen fromO4, compared to the small attractionregion ofO3, make this route preferable, in spite of thelarger uncertainty in the relative position of these twoobjects (which the robot only knows “through”O3).The right-hand side of Fig. 2 displays the estimatedtrajectory during the return fromO4 to O1.

The robot docks atg42 and progresses in the direc-tion of O2. When it reaches it, it confirms that the ob-ject perceived is indeedO2 by comparing the observedshape — in this case two linear sides if the polyg-onal object — with its representation. Subsequently,it docks atg21, proceeds towardsO1, checks that itreached it and stops.

Fig. 4 illustrates the execution of the same missionwithout autonomy control. The two objects placed atT1 andT2 (O3 andO4) are detected, but represented atwrong positions and/or orientations, leading to robot’sloss when it tries to return directly toO1. This can beseen by the fact that the robot’s estimated path goesthroughO1, revealing an error at least as large as theobject’s size.

Fig. 5 illustrates the safety of the learned representa-tion for navigation between objects, showing motionsbetween the two most distant objects,O1 andO3. Theplots at the bottom show the planned path, and theupper diagrams the position of the robot after it rec-


Fig. 4. Executing the mission of Fig. 2 without autonomy control.

Fig. 5. Navigating between known objects:O1 → O3 (left) andO3 → O1 (right).


ognized the target object, and acquired its position inthe local frame attached to it.

5. Conclusions

We presented a strategy for the navigation of mo-bile robots in open sparse uncertain environments. Ourwork assumes that the robot has no access to externalinformations about its absolute position, and that theenvironment consists of a set of objects whose sepa-ration is large compared to the range of the percep-tion sensors. Our approach is based on: (i) maximizingthe probability of successful transition between anypairs of known environmental components, by requir-ing, if necessary, that the robot uses other objects asintermediate landmarks, optimally combining periodsof deliberative motions and perceptual behaviors, and(ii) guaranteeing, by controlling the extension of pur-posive robot’s motions, and resorting if necessary toactive search for perceptual information, that a reli-able path to the homing region always exits. The paperpresents experimental results that clearly demonstratethe increase of robot’s autonomy resulting from ournavigation strategy.

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Maria João Rendas received the Engen-heiro Electrotécnico degree in 1980, theM.Sc. degree in Electrical and ComputerEngineering in 1985, and the Ph.D. degreein Electrical and Computer Engineering in1991, all from Instituto Superior Técnico(IST), Lisbon, Portugal. She was a visit-ing researcher at Carnegie Mellon Univer-sity (CMU), Pittsburgh, USA, from 1987to 1989. In 1991, she joined the faculty

of IST as an Assistant Professor and was promoted to AssociateProfessor in 1994. She held a Fellowship from the European Hu-man Capital and Mobility program from 1993 to 1995 at theCNRS-UNSA Laboratoire d’Informatique, Signaux et Systèmes(I3S), Sophia Antipolis, France. Since 1995, she is a Chargée deRecherche at CNRS (I3S), being the leader of the SAM (Au-tonomous Mobile Systems) group. Her research interests are instatistical signal processing, fuzzy and probabilistic uncertaintycharacterization, and geometry of statistical inference, with ap-plications to the perception and navigation of autonomous robotsand inversion problems in underwater acoustics. Her research hasbeen supported by several European programs, including ESPRIT,MAST and ACTS. She is currently an Associate Editor for theIEEE Transactions on Signal Processing, and a member of theSensor Array and Multi-channel Technical Committee of the IEEESignal Processing Society. She was a member of the ProgramCommittee of the IEEE Conference OCEANS’98.

Stefan Rolfes was born in 1968 inCloppenburg, Germany. He studied Com-puter Science at the Technical Univer-sity of Braunschweig where he graduatedin 1995. He held an ERASMUS (Euro-pean Union) Fellowship at the Laboratoired’Informatique, Signaux et Systèmes (I3S)in 1995, where he worked on algorithmsfor shape representation. From 1996 to1997 he has been a Research Assistant at

the Instituto Superior Técnico (IST), Lisbon, Portugal, workingon shape recognition. Since 1998 he is a Ph.D. student at theCNRS-UNSA (I3S), France, working on the navigation of au-tonomous vehicles based on uncertain acquired environment rep-resentations.

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