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P r o d i n g s of the ZOO4 IEEEInternationalConknncs on Robotlu6 AutomationNow Orleans. IA *April 2004

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Nonsmooth analysis and sonar-based implementation ofdistributed coordination algorithmsCraig L. Robinson, Daniel Block, Sean Brennan, Fran cesco Bullo, a n d Jo rg e C o d s

General Engineer ing and Coordinated Science Labora tory, Universi ty of Illinois at Urbana -Champa ignDepartment of Mechanical Engineering, The Pennsylvania Sta te University

sbrennan@mne . su . dutc1robnsn.d-block,bullo,jcortes}@uiuc.edu

Ahsfmet-Thii nan er investigates the behav ior of a erou~ this collection of techniaues is well-suited to studv a numberr~ .~ ~~~ -~~~~~of autonomous robots evolving in a polygonal enviro&&taccording to a move away from the closest neighbor h euristic.We dem onstrate that this dis tribu ted coordination algorithm op- of coord ination problem s. The read er is referred to. [4] for thecomprehensive m athematicaltimizes an aggregate cost function that measure how unifotmiydistributed are the robots in their en\ironment Our technicalapproach based on non-smooth analysis and computationalgeometq unveils a sphere-packing problem. The algorithm isimplemented in a testbed of indo or mobile robots equipped withsonar. We develop novel approaches for improving single pointsonar scan performance: These algorithms are then shown tohave improved reliability, resolution and speed in distributedenvlronments as eompared to other scanning methods.

I. INTRODUCTIONOne fundamental capability of future mobile and Nnablenetwo rks of rob ots will be the ability to perform :spatially-distributed sensing tasks including coverage, surveillance, ex-ploration, target detection, and search. These future networks

of autonomous vehicles will he able to adapt to changingenvironm ents and dy namic situations. They will provide guar-anteed quality of se rvice in the presence of failures, and w illoperate via limited-bandwidth ad-hoc communication links.The algorithms required to achieve these desirable capabilitiesmust be amenable to impleme ntation in a coope rative setting,i.e., they are required to be distributed, asynchronous, ada p-fiv e, and verifiably corre ct. Samp le references on coordinationproblems for m ulti-vehicle networks include [l], [2], [3].

The premise of this paper is based on a class of coverageand d eploym ent problem s for networks of mobile robots. Oncethe optimal sensor coverage is formalized as an aggregateperformance metric via methods from geometric optimization,a class of cooperativecontrol algorithms can be designed bygeneralizing the classic Lloyd algorithm from quantizationtheory. The resulting control laws are interaction protocolsbetween the mobile sensors and include behaviors such as,move away from your closest neighbor, and, move towardthe geometric center of your sensing region.

Th e first objectiv e of this paper is the stud y of the %eveaway from your closest neighbor algorithm in groupof robotsand is developed in Section II.Tools fromnonsmooth analysis,. . stability and convergence analysis via a LaSalle Invariance,Principle and geometric optimization concepts such as sphere-packing and disk-coverin g functions are used. It turns out that

The second objective of this paper is to investigate thepractical issue s that arise in the implementation of these algo-rithms. A decentralized sensor based testbe d consisting of sev-eral mobile DSPplatforms (discussed in Section III) equippedwith rotating sonar sensors is used. Successful implemen tationof the algorithms is largely dependent on the properties andlimitations of the inexpensive wide-angle sonar sensors used.These problems are reviewed in Section rV which combinedwith the implementation limitations in Section IV-A providedthe impetus to develop improved methods for single pointsonar scan resolution and reliability. Hence, the final objec-tive of this paper is contained in Sections IV-C and IV-Ewhere two new algorithms called the Max Filter an d MaxResolver as well as an extension of the E E R U F methodby Borenstein and Korne [ 5 ] are described. These methodsprove to b e very effective at improving sonar reading reliabilityand resolution within the context of this investigation in anenvironm ent with large amounts of interference. Finally, Sec-tion V discusses the combined implem entation of the sensin gand coordination algorithms, as well as the modifications,adaptations and tradeoffs of the final implementation.

11. INTERACTION A L G O R I T H M SA. Preliminaries

Le t (1 . 1 denote the Euclidean distance function and let2) w denote the scalar product of the vectors u ,w E R N.Throughout the paper, versus(zI) will denote the unit vectorin the direction of 2) # 0, i.e., versus(u) = u/11u11. Recall thata set S c P N is said to be convex if Xz + (1- X)y E S, fo rall X E [0,1]nd all 5, E S. Given S c W N , he convexhull of S, co(S) , s defined as

co(S) = (2 E R N I 2 = xz+ (1 -X)yfor some X E [0,1]nd 2, / E S ) .

Obviously, if S is convex, then S = c o ( S ) . n general, S Cco(S) . I f S is a convex set in WN. et projs: R N + S denotethe ortbogona l projection onto S and let Ds RN -* W enotethe distance function to S. Also, let Ln(S) denote the least-norm vector in S (see Figure I ) .0-7803-8232-31041517.00 02004 IEEE 3000

mailto:tc1robnsn.d-block,bullo,jcortes%7D@uiuc.edumailto:tc1robnsn.d-block,bullo,jcortes%7D@uiuc.edu

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Fig. I .grey. The vector w is.the east-norm element in co(S).The convex hull of the set S = { V ~ , V ~ , Z ~ ~ , V ~ , I I ~ )s shown in

Le t Q be a convex polygon in W2. enote by in tQ i t sinterior set and by Ed(&) = { e , , . . e M } its set of edges.Given an edge e E Ed(Q), we let ne denote the unit normalto e pointing toward the int(Q). Let P = (PI.. .. pn ) E Q"denote the location of n generators (or robots) in the spaceQ. The Vomrtoipartition V ( P )= (Vl(P), ..., Vn(P)) f Qgenerated by the points @, ,. . p,) E Q" is defined byK ( P ) 5 q E Q I IIq-PilI 5 llq-pjll; V J ' # ~ } .

For simplicity, we refer to K ( P ) s V,. If V, an d I$ share anedge, i.e., V, nVj is neither empty nor a singleton, then p i iscalled a neighbor of p j (and vice-versa). Given a polytope Win R N , ts incenter set, denoted by IC(IV), is the set of thecenters of maximum-radius spheres contained in W.B. The geomerric optimization pmb lem

Consider the optimization problem consisting of maximiz-ing the following aggregate cost functionH ( P ) = ' m in {~llpi-pjll,De(pi)}i , j ( I , .., )i#j, e E d ( Q)

This geometric optimization problem corresponds to the situa-tion where we are interested in maximizing the coverage of thepolygon Q in such a way that the radius of the robots do notoverlap (in order not to interfere with each other) or leave theenvironment. Following [4], this problem can be restated asa sphere-packing pmblem: bow to maximize the coverage ofa region with non-overlapping disks (contained in the region)of minimum radius. The problem reads:-m={R I ,..., > Bb i ,R) C QBb i ,R )n B ( p j :R) = 0) (1)

{ q E RZ/ /q-plJ < R} and wherehere B(p;R) =B ( p ,R) s its closure.Remark 2.1: In the definition of 71 we employ a 112correction factor in comparing the pairwise robot distances,lJpi- j l l , with the distances to the walls, D,(pi). This factoris necessary to establish the equivalence between minimizing'H and solving the sphere-packing problem ( I ) .

C. The distributed coordination algorirhrmHere we design two coordination algorithms with the prop-erty that the corresponding closed-loop systems are guaranteedto monotonically increase the value of the performance mea-sure H. et i 11,.. n}. At a configuration P &", thefollowing set describes the distances of the generator p i to the

rest of generators and to the boundary of Q,~ ( p ) -pi11 I E 11:. . . - n }\{Q}

U 1 e E Ed(&)} .( 1Le t Ri(P) enote the minimum of Afi(P). ote that H ( P )=m i n i ~ ~ l , , , , . n ); ( P ) .Consider the set Si(P)defined by

versus(pi -p j ) E S i ( P )t) 1Ipj -pill = R , ( P ) ,'

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ne E &(P)t) e b ; ) =R i ( P ) .Note that if there is a single element (generator or edge ofQ) which is nearest to pi, he set Si(P) is formed by asingle vector. If several elements are equidistant to p i , ,thenS,(P) is composed of several vectors (see Figure 2). Forimplementation purposes, it is also worth noticing that in ordertocompute S , ( P ) ,wen eedt o compare $Jlpi-pjJJwith&@,)(see Remark 2.1).

Fig. 2.generator s the only one such that Ln(co(S~P ) ) )= 0.Illusnation of the vmor fi eld (2s). AI this confi guralion. the finl

Now, consider the following dynamical systems describingthe evolution of the generators i E { l ? . . in}.pi = Ln(co(Si(P))) ' (WPi E C ( K ( P ) ) .

The first system is nearesr-neiRhbor-di.~tri~uredn the sensethat L n ( c o ( S ; ( P ) ) )epends only by the position of p , andits nearest neighbors. The second system is Vamnoi-neighbor-distribured in the sense that IC(v(P)) depends only by theposition of p , and its Voronoi neighbors. Note also that thesolution to both systems must be understood in the Filippovsense [ 6 ] . Classical notions in differential equations suchas existence and uniqueness of solutions, invariant sets andstability analysis can also be exten ded to dynamical systems ofthis type. In particular, we need the notion of weakly invariantse t to state the next result: a set A f is said weakly invarianrif for each xo E A f , Af contains a maximal solution of thecorresponding dynamical system in (2).Theorem 2.2: For the dynamical systems (2a) and (2b), thegenerators' location P = (p,, . . ,p,) converges asymptoti-cally to the largest weakly invariant set contained in the closure

of A ( Q ) = { P E Q" 1 R i ( P ) = H i ( P ) * p i E IC(V,)}.

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The configurations P E A(Q)c QN have the property thatall the robots with the smallest value of R,(P)re located atthe incenter of their Voronoi regions.111. INDOOR MULTI-ROBOT TESTBED

A group of 5 mobile robots was developedin the UIUC Control Systems Laboratory.The robots are based ona 150MHz 32bit floatingpoint Texas Instruments DSP6711 mounted on a TI DS Pdevelopment board. A daughtercard was built in house toprovide an interface point for aBX24 Micro controller, PWMoutput, encod er input, AIDand D/A converters and serialcommunication. The BX24 Microcontroller co-ordinates a set ofthree Daventech SRFW ultrasonicsensors as well as three infrared distance sensors. The sensorsaie mounted on a RC servomotor that rotates to incrementallyscan the area around the robot. Pinman motors drive therobot wheels and HP optical encoders provide odometryinformation. A Matrix Orbital LCD screen displays onboarddebugging information and data is sent over a RS232 wirelessserial connection to a base station PC and to the other robots.Texas instruments Code composer and C compiler were usedto program the robots. Figure 3 contains a diagram illustratingthe hardware architecture of the multi-robot testhed.

Fig. 3. Hardware diagram of multi-robot testbed.,IV. DISTRIBUTEDO N A R S E N S IN G

A. Implementation limitationsIt is important to emphasize the constraints imposed bythe problem definition on the implementation. Firstly, thealgorithm s are to be completely decentralized. Hence, althoughthe robots might communicate among themselves or with abase station, their behavior must not dep end on this. Secondly,due to the dynamic nature and undetermined size of theenvironment, the robot cannot expect to build and keep an

accurate updated map. The occupancy grid methods describedby 171, [SI do account for a ch anging environment but severalreadings are required to remove an object from the map ifit has moved. This is not acceptable in this implementation.Decisions concerning the environment surrounding the robotmust be made from only one horizon of sonar readings at onelocation. This was the main stimulus to improve single pointresolution in sonar scans.B. Sonor advantages and disadvantages

The Daventech SRFW Ultrasonic range sensor has severaldistinct advantages over the Polaroid 6500 sensors which havecommonly been used, e.g., see 191, 171, [8]. These advantagesinclude a larger range of measurements (3cm to 6m), lower.power consumption (during both firing and quiescent periods),and smaller size. Any ultrasonic sensor comes with a set ofproblems [IO], 1111 such as wide beam angle, interference,multiple and specular reflections. Much of the previous workon these issues has focused on map building, e.g., 1121 andpath planning problems, where probabilistic approaches [SI,Kalman filters [12], and stereoscopic approaches [I31 havebeen shown to significantly improve the overall scanningperformance. Unfortunately, the distributed aspects of ou rsetting l i t s he applicabili ty of these approaches.C. Improving scanning reliability

One of the problems with ultrasonics sensors is direct andindirect interference from stray sonar pulses and reflectionscausing very erroneous readings. It is a serious factor in sonarusage and occurs when vagrant sonic pulses are mistakenfor an intended objects reflection. This problem is partiallyaddressed by the error eliminating rapid ultrasonic firing(EERUF) procedure described by Borenstein an d Kam e in 151.In the approach sonar sensors are fired in a sequence withdifferent timing. The timings differ between all sonars onthe same robot as well as between successive firings of thesame sonar. A predetermined timing schedule for a knownnumber of sensors is developed. The variable timing schemeensures that interference in successive readings is not constant.Scanning then continues until a pair of readings agree withinsome tolerance. Barenstein and Kame implement EERUFwith a total of 32 sonar firing on two separate robots. Theiralgorithm successfully rejects 97% of all erroneous readingsthat would affect naive firing methods. The work 151 containslittle indication of the measurement length at which theseresults are achieved; to the best of OUT nderstanding it appearsthat these successfu l results were obtain for distan ces typicallyless than 1 0 0 n . This is an important consideration for thecomparison of results in Section IV-D.

In a truly distributed system it is difficult to determinethe global number of sensors as well as to obtain globalsynchronization of the robots internal clocks and hence itis very difficult to devise a global timing schedule for sonarfiring. In addition, in this investigation it was found that in anenvironment with several (5) static mobile robots each with3 Sonar implementing the EERUF method there was sufficient

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Fig. 4.5 roboa in the environment.Error disvihution at a d ismce of 120cm with 15 Sooar mounted on

cross talk to continuously cause interference and often preventtwo successive sonar readings from being within tolerance.The result is very long delays (in the order of seconds) forobtaining a valid EERUF reading. In Other words the EERUFmethod appears to be not scalable in distributed systems.To overcome these problems, we propose, and later com-p e , a number of s ing le-p int sonar-=,an algorithms. We referto the following scheme as the Queue Checker algorithm.

Let e > 0 be a tolerance, and let n be the size of a queuewhere we record successive sonar readings in a FIFO fashion.If a new reading differs by less than e from a certain numberI; < n of readings in the queue, then the reading is accepted asa valid Queue Checke r measurement. In addition, the fi M gschedule is randomized and halts as soon as a valid measure-ment is obtained. This allows other sensors to fire alone, faster,and with less probability of interference. The randomizationreduces the possibility of consecutive interference occurring asa result of other sonar firing at the same rate. The performanceof the algorithm c an be tuned by appropriate selections of theparameters e , k , and n.

After considering a set of naive sonar data as in Figure 4itis evident that the m easurements are strongly left skew ed, i.e.,erroneous measurements are always smaller than the actualdistance. Hence, the greaiesi distance observed i s relative tothe closest object. Using this observation, thc Max Filterfilter collects a series of sonar distance readings and choosesthe maximum value as a valid measurement.

A further means to improving the sonar performance isto adjust the sensitivity and maximum range of the sensors.Limiting the maximum sensitivity ignores weaker reflectionsand interference. Scanning is also faster as the sonar does nothave to wait as long to detect reflections. The sensitivity can beset dynamically and was implemented throughout t h i s paper.D. El-perinrenral performmice of sensing dgoritlims

The performance of each sonar scan aIgorithm was testedin a reproducible environment that simulated the testbedenvironment. The experiments were conducted with a varietyof distances, objects and number of sonar. A brief summaryof the results are shown in Table I.

Considering Table I it i s significant to note the following:. nfiltered sonar measurements produce very poor mea-surements particularly with longer distance measure-ments.

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The EERUF method proves to be reliable but slow withrespect to the other methods. This is because a sequenceof readings differ from one another according to theoverall error distribution; see Figure 4. However, theEERUF algorithm was found to be more effective atshorter distances (i100cm); this is consistent with theresults reported in (51.The Queue Checke r approach improves the readingsreliability only at sh ort distances. At larger distances themeasurement error distribution is such that it is morel i e l y to match a series of incorrect readings than selectthe correct distance. In comparison, the Max F i l t e ralgorithm improves reading reliability at all distances.. he time required fo r a reading using the QueueC h e c k e r method increases exponentially with increas-ing number of values to match where as the MaxF i l t e r time increases linearly.

E. lmpmving scanning resolutionA persisting problem of using sonar is the wide beamangle as shown in Figure 5. n many approaches to improving

Fig. 5 . As a result of the wide beam angle (i ) distant objects are not observedand ( i i ) the size of an ohjed i s increased.

sonar resolution the robot moves tbrough an environment anduses readings from multiple, individual and subsequent sonarfirings, from different locations, to improve the resolution ofthe observed objects. A popular approach i s the occupancygrid method developed by Moravec and Elfes [S] n which theprobability of an object being located in discretized regionsof space is computed. Certainty i s added along an arc atthe distance recorded by the sonar (the apparent object sizein Figure 51, and subtracted in wedge shape of free spacebetween the robot and the object. A recent approa chb y Chbset,Nagatani and Lazar [9 ] represents each wide sonar reading asa single arc. Intersections of subsequent arcs then indicatethe location of an object. McKerrow 1141 proposes finingtine segments throngb successive sonar arcs that met certaincriteria. The result of al l of these methods is a representation ofthe environment that is significantly better than a naive sonarmodel. However, their application to ou r problem is severelylimited as there is only one se t of sonar readings from anunknown location available at any time (cf. Section IV-A).

As a result of these limitations, we developed a new ap-

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,... . ... . .. ..,-.mu_-

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Fig. 6.xesolver illgorithm.Il luslat ion of improvement scanning resolution using the hiax

proach called the Max Resolver n which a set of adjacentreadings in a single scan are compared and used to im prove theresolution of the complete horizon of distance measu rements..The effect of a wide beam angle is to record a distance asthe minimum distance to an y object within the beam angle as.shown in Figure 5 . This has the effect of enlarging an objectin the sonar view.,The Max Resolver algorithm uses themaximum distance in a set of readings that would lie withinthe beam angle as the actual distance. This approach attem ptsto inven the wide beam angle effect and has the effect ofsharpening the sonar readings. Figure 6 shows the original!and modified sonar distance measurements and illustrates theeffect of implementing the Max Resolver on the robot in !the actual environment.

v. E X PE R I M E N T A LESULTSThe steps in implem enting the algorithm are outlined below:(i) The micro-controller fires the 3 ultrasonic sensors foruse in the Max Filter or Queue Checker algo-rithms.(ii) A valid reading is returned to the DSP and stored. TheRC servo then rotates the sonar to obtain a full 360horizon of distance measurements. Ultrasonic scanningis balled.(iii) The Max Resolver algorithm modifies the distancehorizon and finds the centroid of the closest object.Neighboring measurements are incorporated into theobject if they are sufficiently close and the the new

object centroid is computed. (Object sue is limited to90 subtended at the sonar.)A correction to ensure that the robot does not perceivean adjacent wall segment as the next closest object isapplied and the previous step is repeated until 3 uniqueobjects have been identified.If the distance to all three objects are within tolerance therobo t stays in the cen tral region: Otherw ise it movesaway from the closest object (or bisector of the twoclosest) according to (2a).Visualization data is sent to the PC and the robot returnsto step I .

In some implementations only one robot would scan at anytime. This was done to reduce the possibility of sonar in-terference but was slow. The algorithms were successfullyimplemented with up to 5 robots as shown in Figure 7.Thereare several tradeoffs that need to be made for a successfulimplementation:Number of readings: 36 was sufficient. Fewer readingscaused the Max Resolver algorithm to eliminate someobjects while taking more increased scan time but did notsignificantly increase accuracy.Size of cen tral region: Should initially be large and thengradually reduced. This reduces close proximity colli-sions due to robots m oving simultaneously but slows theperformance.Dis tanc e tn Move: Reducing the distance moved each steptakes longer but reduces close quarter collisions andproduces behavior which closely represents the simulatedresults.Parameters of Max Filter: Reliable and fast results wereobtained for e = 5 m , k = 4, and n = 10.So nar Sensitivity a nd Range: Can be reduced for speed andaccuracy but needs to be dynamically adjusted.Rob ot Recognition: To implement the 112 distance rule forthe sphere packing (Remark 2.1) robots and wall objectsmust be identified. Leonard and Durrant-Whyte [ IO]use regions of constant depth (RCD). In this investigation

a RCD of up to 60 was found for walls and 20 to4 5 O fo r a robot. Hence, any object smaller than 45was considered to be a robot. This was successful butsometimes inconsistent.

Local minima play a larger role in the implementation(compared to simulation) because the introduction of a centralregion enlarges the space where a robot is in equilibrium.

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Fig. 7.computer simuliltion ran in Mathematica with identical initial condition.Th e upper left (right) tig u m illustrate the initial (final) locations of S robots in a typical 5 mho1 implemenvation. The lower ti pr es r rpresent a

A typical run (Figure 7) took about 3 minutes hut can REFERENCEShe reduced to less-than a minute by making the tradeoffsdiscussed above. Finally, there are often several possibleequilibrium configurations with different regions of attraction.While at equilibrium an erroneous sonar readings might cau sean unexpected movement. However, quite conveniently, thesystem was found to often settle into another equilibriumconfiguration with a larger region of attraction.

VI. CONCLUSIONS

We have formalized optimal senso r coverage as an aggregateperformance metric via methods fro m geometric optimization.We characterized the asymptotic behavior of the "move awayfrom your closest neighbor" algorithm using tools from nons-mooth analysis. We have presented experimental results froma fully distributed implementation on a sonar based testhed.These successful results were dependent on good sonar sensordata and two novel methods: the Max F i l t e r and theQueue Checker . We found the Max F i l t e r algorithmto be faster and more reliable at longer distances than theEERUF and Queue Checke r methods. We also developand implement the Max R e s o l v e r algorithm for improvingsingle point sonar scanning resolution. Our experimental setupprovides valuable insight into practical issues regarding therealization of distributed sensor based exploration algorithms.We refer the interested reader to [I51 for more details.

Acknon' ledperirsThis material is based upon work supported in part byAR O Grant DAAD 190110716, ONR YIP Award N00014-03-1-0512. an d NSF SENSOR Award IrS-0330Q08.

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