Navigation Around an Unknown Obstacle for Autonomous Surface Vehicles Using a Forward-Facing Sonar

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1 Navigation Aroun an nknown Obstacle for Autonomous Surface Vehicles sing a Forwar-Facing Sonar Patrick A. Plonski, Joshua Vaner Hook, Cheng Peng, Narges Noori, Volkan Isler Abstract A robotic boat is moving between two points when it encounters an obstacle of unknown size. The boat must fin a short path aroun the obstacle to resume its original course. How shoul the boat move when it can only sense the proximity of the obstacle, an oes not have prior information about the obstacle s size? We stuy this problem for a robotic boat with a forwar-facing sonar. We stuy two versions of the problem. First, we solve a simplifie case when the obstacle is a rectangle of known orientation but unknown imensions. Secon, we stuy a more general case where an arbitrarily shape obstacle is containe between two known parallel lines. We stuy the performance of the algorithms analytically using competitive analysis an present results from fiel experiments. The experimental setup is relevant for harbor patrol or autonomous navigation in shallow water. I. INTRODCTION Imagine an autonomous robot equippe with a forwar facing sensor. The robot etects an obstacle in front of it. How can the robot go aroun the obstacle as quickly as possible? This is a classical robot navigation problem. Now imagine the robot is a boat, equippe with a forwar facing sonar (Figure 1). The narrow with of the sonar, the noisy an sometimes ambiguous sonar reaings, an the motion constraints of the boat make the problem challenging in this omain. In the literature, there are two primary approaches in solving this type of online navigation problems. BG Algorithms [1] provie easy to implement strategies which work well in simple environments with small obstacles. However, they o not have strong performance guarantees an their performance can be arbitrarily ba as the obstacles grow larger. In contrast, online navigation algorithms with provable performance guarantees have been propose in the literature [2], [3], [4]. However, these algorithms are rarely implemente on real robot systems since they o not aress motion an sensing constraints. In this paper, we present novel navigation strategies which are both provably efficient an suitable for fiel implementation. We first with the case of a rectangular obstacle with unknown size. The justification for this assumption is that many artificial objects (ships, ocks etc.) have rectangular shapes. Moreover, long shorelines or bounaries of vegetate areas can be approximate by a line. Obtaining the orienta- The authors are with the Department of Computer Science an Engineering, niversity of Minnesota, Minneapolis, SA. This work was supporte by a MnDRIVE RSAM grant an National Science Founation Awars # an # s: {plonski, jvaner, peng175, noori, Fig. 1. The autonomous boat (top), with sonar visible (bottom). tion of the line or the face of the rectangle amounts to fitting a line to initial reaings. We provie an aaptation of the oubling strategy (explaine in Section III) which accounts for motion an sensing constraints an analyze its performance. While testing the algorithm in fiel experiments, we realize that in some cases it becomes very har to fit a line to initial reaings. Moreover, the assumption of linearity is violate in most settings. Therefore, we also stuy the problem in a more general setting where we are given two parallel lines bouning an arbitrarily shape obstacle. Here the parallel lines represent a weak prior on the extent of the obstacle. Our analysis shows that this generality comes at the expense of slightly reuce theoretical performance. However, fiel experiments emonstrate the effectiveness of the strategy. II. RELATED WORK Several techniques have recently been evelope to enable obstacle avoiance in sonar equippe Autonomous Surface Vehicles (ASVs).

2 Heiarsson an Sukhatme [5] implemente the Vector Fiel Histogram (VFH) [6] metho for sonar-base obstacle avoiance. They also introuce an echo filtering process, which average the local maximas. Vector fiel methos are subject to local minima an the same vehicle with the same potential function in a ifferent environment can result in success or failure with no guarantee. Calao et al. [7] use Histogramic In-Motion Mapping (HIMM) [8] to buil gri maps of the sense obstacle. A convex polygon obstacle is built base on the line segments extracte by Hough transform. Given the constructe map, a potential fiel metho was use for navigation. Petillot et al. mappe the surrouning obstacles by segmenting an extracting obstacle features from the sonar image [9]. Then vehicle motion planning was converte into a nonlinear programming problem while the extracte obstacles were treate as inequality constraints [1]. Nonlinear programming approaches can be less succeptible to local minima than vector fiel methos. However, they require prior knowlege of the location an size of the obstacles. They also o not guarantee the length of the path an the time require to avoi an obstacle with arbitrary shape an size, whereas our approach can hanle arbitrarily large obstacles an still reach the target in finite time with a near optimum competitive ratio. III. PRELIMINARIES An online algorithm is an algorithm which oes not have access to its entire input in avance. Instea, the input is reveale uring the execution of the algorithm. For example, a memory management algorithm must choose which pages to retain in the cache without knowing future page requests. The obstacle avoiance problem stuie in this paper is an online problem since we must choose a motion strategy that reacts to the sonar measurements receive uring execution without knowing the exact geometry of the obstacle in avance. The performance of an online algorithm A is measure using its competitive ratio which is given by A(σ) c(a)=max σ OPT(σ) where σ varies across all inputs, A(σ) is the performance of A for input σ an OPT(σ) is the optimal offline performance i.e. the performance of an optimal algorithm which has access to the entire input σ in avance. The competitive ratio is a measure of worst-case eviation from the optimal offline behavior. Further information on online problems can be foun in [11]. The lost-cow problem is a classical online optimization problem which highlights important aspects of online algorithm esign. In this problem, a short sighte cow is lost an tries to fin the only gate on a straight fence. The problem is formulate as follows. The cow an the gate are on a line, the gate s location is unknown, an the cow s from x = in orer to fin the gate. The true location is chosen by an aversary. Note that the cow can not simply pick a irection (1) an move until fining the gate as this strategy woul have unboune competitive ratio (the aversary chooses the gate location in the opposite irection.) Instea, the cow can follow the so-calle oubling strategy to effectively fin the gate: Initially, at roun i= the cow is at f =. It moves in such a way that at the i th roun, the cow is at location f i where f i =( 2) i 1 for i 1. In other wors, in o rouns the cow is to the right of the origin while in even rouns the cow is to the left of the origin (Fig. 5). sing elementary computations, it can be shown that the oubling strategy has a competitive ratio of 9 [12]. Due to sensing an motion uncertainty an constraints, esigning a similar online algorithm for obstacle avoiance is non-trivial. In this paper, we present novel online strategies, analyze their competitive ratios an valiate them in fiel experiments. IV. SYSTEM DESCRIPTION Our test system is an Oceanscience Q-Boat 18D 1. The Q-Boat is 1.8m long, with a cruising spee of about 1 m/s an a turning raius of R t = 5m. During the experiments, we move at a slower spee of about.5 m/s. The Q- Boat is autonomously controlle with a laptop computer. Localization was achieve through the use of a GPS unit an a compass, filtere with an Extene Kalman Filter. A forwar-looking, single-beam sonar unit was mounte on the bow of the vehicle, as shown in Figure 1. The sonar unit is an Imagenex 852 Digital Echo Souner 2. It has a conical acoustic transucer with 1 egree beam with. The sonar soun frequency is 675kHz. As configure the sonar unit provies 5 measurements for each pulse, each measurement representing the intensity of the sonar return in an evenly space range bin etermine by the configure maximum range. For the maximum range of 5m, each range bin represents a.1m increment. The sonar pulses were sent once a secon. After each pulse, we smooth the ata returne by convolving the 5 bin vector with a Gaussian filter that has σ = 15. For each return we etect the istance that has the maximum return intensity. If this intensity is above our threshol of 3 out of 127, we place an obstacle point at the appropriate istance in the appropriate irection from the estimate position of the boat. Heiarsson et al [5] inicate that tilting the sonar by 1 egrees towars the lake surface or towars the lake bottom ha little effect on the etection of obstacles. However we foun that in shallow water, tilting the sonar coul cause the bottom to be etecte as an obstacle even if the water was clearly eep enough for our ASV to comfortably operate (1.5 meters in epth). In the en we ecie to operate with the sonar tilte ownwar slightly. This ensure that the ASV coul clearly etect the rising lakebe before the shore, but i not cause any false positives when the ASV was facing away from the shore.

3 Fig. 2. The setup for Problem 1. The robot, moving from the left, encounters the rectangular obstacle R. With no information other than the orientation of the leaing ege, the robot must fin the path past the obstacle (L) or one of comparable length. The robot is equippe with a range sensor which can sweep out an angle Θ, an etect obstacles up to range D. Fig. 3. Left: Problem setup for Problem 2. The optimal path is shown as an arrow, which lies tangent to one of the extreme points of the polygon. The optimal path, therefore, has length at least 2 + s 2. The robot will move up an own the line l, probing forwar for the extreme points of the polygon by travelling towar line l 2, up to istance s, or returning to l if it has encountere the obstacle. Right: An example execution of Avance- Retreat. The goal is to make sufficient progress s past the obstacle. The robot chooses points to probe accoring to the oubling strategy. Each point is checke until the robot is able to make sufficient progress. V. MOTIVATION AND PROBLEM FORMLATION We aress an online navigation problem, where a vehicle similar to the ASV escribe above etects an obstacle. First we escribe the moel use for the ASV. See also Figure 2. Motion Constraints: The ASV can position its ruer to irect thrust from a single motor. The maximum angle of the ruer, β, etermines a minimum turning raius r. The forwar velocity can be set inepenently of the ruer. In the remainer of the paper, we consier the forwar velocity fixe at a constant, labele v. In practice, the forwar velocity etermines the time to complete a turn, but has little effect on the turning raius. Thus, we assume a constant turning raius for the boat, given by R t 5m. Sensing Constraints: The ASV is equippe with a sonar (a range sensor), which etects the ranges to objects in a cone of angular with Θ, an up to a maximum range D. Both Θ an D are measure from the front of the vehicle. In our system, Θ is on the orer of 1 egrees. Next we efine the obstacle avoiance problem: We with an iealize case where the obstacle is a rectangle R. Suppose from the initial measurement, the ASV can infer the line containing a sie of R. How can the ASV go aroun the nearest corner of R as quickly as possible? We formalize this problem as follows Fig. 4. The Sweep maneuver. The robot with maximum sensing istance D, an beginning the maneuver from istance t+r t can search an a portion of the obstacle with with 2u 1 = 2 Rt 2 + D 2 t 2. Problem 1 (Rectangular Obstacle): Given a rectangle R, a linelcontaining the nearest ege of R, an a vehicle subject to motion an sensing constraints escribe above, compute a strategy for the vehicle to reach the nearest corner of R as quickly as possible using online sensor measurements (i.e. without knowing the exact shape of R in avance). The problem setup is illustrate in Figure 2. In the next section we present an online algorithm for Problem 1, show that it has constant competitive ratio an emonstrate its performance on the fiel. The strategy relies on two assumptions (1) the line l supporting the obstacle can be obtaine from initial reaings, an (2) the shape of the obstacle is a rectangle. Fiel experiments reveal that these assumptions can be too strong in some cases. This lea us to a secon, more general problem. Problem 2 (Arbitrary Obstacle Boune by Lines): Given two parallel lines, l an l 2, which are known to contain a single obstacle, an a vehicle subject to motion an sensing constraints as escribe, compute a strategy for the vehicle to reach the secon line l 2 as quickly as possible using online sensor measurements. This secon problem is illustrate in Figure 3. In Section VII, we present an extension to our algorithm for Problem 1 which can solve the secon problem. The generality comes at the expense of increase competitive ratio, but fiel experiments reveal the algorithm is effective in practice. VI. STRATEGY FOR PROBLEM 1 AND ANALYSIS In this section we present a solution to Problem 1. We are require to use motion primitives that respect the minimum turning raius of our ASV. The first, Sweep, is use to search the portion of the obstacle which is near the robot. If a corner of the rectangle is etecte, the robot can plan a clear path aroun the obstacle. The secon, GoTo, simply moves the robots between two points. As we will show later, robots o not always precisely execute their motion primitives. However, in practice, this only multiplies by a constant factor the true time require to execute a motion strategy. Sweep: The robot, moving on line l, begins a turn towar the obstacle along a circle with raius R t. When the turn is 3 4 complete, the robot re-aligns with the line l, an moves in the opposite irection. This maneuver is illustrate in Figure 4.

4 Fig. 5. The oubling strategy. At i th roun, the boat is at x i =( 2) i 1. At each point, a Sweep maneuver is performe to search a portion of the obstacle for the corner en 5 5 en 5 en Fig. 6. A Google Maps satellite image of the Lake Staring shoreline, an three trial executions of CircleSweep, using the shoreline of Lake Staring as the object to avoi. The initial point the object was etecte was assume to be at (4, 6). The crosses give the orientation of the line the robot moves along, an they are space apart by 25 meters, corresponing with our = 25m. The shoreline is in re, the etecte sonar objects are x markers, an the path followe by the ASV is labele. The ASV etect the corner of the object if it performs a sweep an observes no close sonar returns. On the left an in the mile, the corner of the object is mistakenly etecte at the thir Sweep operation (however for the test we allowe the ASV to continue sweeping). This is because the sweep was performe in eep water, out of sonar range of the shore. On the right, the corner of the object is mistakenly etecte at the fourth Sweep operation, for the same reason. If the ASV were allowe to continue on, it woul etect the shore again as a ifferent object, an all competitive guarantees woul be lost. This inicates that the algorithm must be extene if it is to perform well in this practical situation. GoTo: The robot, which is on line l an oriente parallel to l, moves straight ahea until a estination point on l is reache. Some simple calculations reveal the following properties of these operations. 1) Sweep traces out a path of length 2πR t. 2) After Sweep has been execute, the robot is facing the opposite irection to the irection it was facing initially. 3) Sweep searches a portion of the obstacle of with 2 R 2 t + D 2 t 2, when the maneuver is e at istance t+ R t, as shown in Figure 4 4) GoTo follows a path of length equal to the Eucliean istance between two points. We can now escribe our algorithm for fining a corner of a rectangular object. We call the algorithm CircleSweep. Let the robot etect the obstacle at position x, which we treat as the origin. Similar to the lost-cow algorithm escribe in Section III, the robot will GoTo waypoints which alternate left an right from x, such that each point x i is at position ( 2) i 1. At each of these points, incluing the first etection point, the robot will execute a Sweep, Algorithm 1 CircleSweep 1: θ orientation of the line l 2: i 3: x i (,), origin at the point of first etection 4: Sweep 5: while obstacle etecte at x i o 6: i i+1 7: x i (( 2) i 1 cosθ,( 2) i 1 sinθ) 8: GoTo x i an Sweep 9: en while which simultaneously scans the object for the ege an turns the robot towar the next waypoint. The algorithm terminates when the robot either etects the ege, or etects that it has passe beyon the ege. See Algorithm 1 for the etaile steps of this algorithm. The analysis of the cost of this algorithm procees in two parts. First, we boun the istance travelle uring all GoTo phases, then, we show that the cost of all the Sweep operations is boune with respect to the optimal cost. Combining these, we prove a competitive ratio in Theorem 1.

5 Lemma 1 (GoTo cost): The GoTo cost is upper boune by 12, where is the istance to the closest point from which the robot coul etect the corner. Proof: The last GoTo operation occurs immeiately before the robot performs the Sweep operation that etects the corner of the rectangle or etects that the robot has move beyon a corner of the rectangle. Let be the nearest point to the origin, from which the robot coul etect the corner while performing a Sweep. In the worst case, the robot travele to a position just short of etecting the corner, (i.e., ε). Note, the lost-cow algorithm will arrive at the point while traveling no farther than 9 [12]. However, we cannot stop at because we cannot continually sense the obstacle. Instea we must continue on to the next Sweep location. Because the robot travels to the point ε in step n 2, then to 2 in step n 1, the final sweep location is at 4. Thus, we have overshot the location by an aitional a travel istance of 3 in the worst case, which we a to the cost of the lost-cow algorithm. Lemma 2 (Sweep cost): Let be the istance to point from which the nearest corner of the rectangle can be etecte. Then the total cost of all Sweep operations is less than 4πR t log 4. Proof: Let be the nearest point to the origin, from which the robot coul etect the corner while performing a Sweep. We will just consier the positive steps, with x i > an ouble the result. Then each positive step reaches x i = 4 i. The algorithm terminates when the robot reaches a point beyon an completes a Sweep operation. If the n th step is the first to pass, then 4 n 1 4 n, which implies n= log steps are require on the positive sie, or at most n=2 log 4 steps are require overall. Since each sweep has a cost of 2πR t, the lemma statement follows. Theorem 1: CircleSweep has a competitive ratio of π R t Proof: By combining the previous two lemmas, we see that the total cost of using CircleSweep to fin the corner point is less than 12+4πR t log 4. Then, the robot must travel past the obstacle, which as a cost less than L. Diviing this cost by the optimal cost, L the resulting ratio is 12 L + 4πR t log 4 L + L L. Since < L, c(circlesweep) 13+ 4πR t log 4 log = 13+4πR 4 t = 13+4π R t log 4 Also note log b x x 1, proucing the theorem statement. Note, the last sweep operation will orient the boat so that it can travel on a straight path past the obstacle, thus Theorem 1 oes not inclue an aitional turning cost. A. Fiel Experiments For our first fiel experiments we execute Circle- Sweep at Lake Staring, Minnesota, SA. We use the north (2) (3) (4) east shoreline of the lake as a proxy for a large-scale obstacle to avoi. At first we attempte to fit a line to the sonar returns observe uring the initial circle performe after etecting the obstacle, but we foun that there was not nearly enough information to ecie the irection of the line. Our strategy requires a goo estimate on the line irection or it may simply retreat away from the object an eclare the problem solve. In the en, to execute CircleSweep we neee to manually efine the line to walk along. We use t = 1m an = 25m; this is reasonable as long as D 15.21m. D is nominally 5m, but in practice, given our tilt angle, 2 meters is a goo estimate of the true D at which we can reliably etect an obstacle. The paths followe by the robot along with the estimate positions of all etecte sonar objects are shown in Figure 6. The sonar returns inicate that the shallow water close to the shore is correctly etecte as an obstacle to avoi. Our strategy assumes the obstacle is a rectangle with known orientation but unknown imensions. In practice this is almost never the case, an the basic strategy performs poorly when its assumptions are violate: In each of our three trials it woul have incorrectly etecte the rectangle corner an terminate early. VII. STRATEGY FOR PROBLEM 2 AND ANALYSIS Algorithm 2 AvanceRetreat 1: θ orientation of line l 2: i 3: x i (,), origin at the point of first etection 4: probe forwar by travelling towar l 2 5: while obstacle etecte uring last probe towar l 2 o 6: complete probe by returning to x i 7: i i+1 8: x i (( 2) i 1 cosθ,( 2) i 1 sinθ) 9: GoTo x i 1: probe forwar by travelling towar l 2 11: en while Given the insights from our first fiel experiments, we now solve the more general case presente in Problem 2. In this section we propose an analyze an algorithm for moving past an arbitrarily-shape object. This object is paramaterize by two parallel lines l an l 2 which are inferre base on the assume with an orientation of the obstacle. The safety linelis assume to lie on the same sie of the obstacle as the robot, an the goal line l 2 is assume to lie on the opposite sie. The istance between the lines is s. In practice, both l anl 2 can be set upon initial etection of an object. The goal line shoul be set perpenicular to the esire irection of movement, an s shoul be selecte base on the maximum object with. First, we iscuss the optimal path for a given obstacle. Consier the polygonal obstacle shown in Figure 3. The optimal path, ing at position x, oes not enter the convex hull of the polygon. It passes through a corner point a

6 en en en 3 en Fig. 7. Four trial executions of AvanceRetreat, using the shoreline of Lake Staring as the object to avoi. The initial point the object was etecte was assume to be at (4, 6) for the first two, an (2,) for the secon two. The crosses give the orientation of the safety line l, an they are space apart at the unit istance = 25m. The portion of the shoreline we are avoiing is assume to lie between parallel north-south lines l an l 2, which is assume to be 5m east of l. The shoreline is in re, the etecte sonar objects are x markers, an the path followe by the ASV is labele. In each probe that is not a feasible route aroun the object, the shoreline is successfully etecte. In the last two trials, the boat successfully navigates aroun the shoreline to the other sie of l 2. obstacle shoul not have very narrow channels in which the robot can get stuck. In most settings, this assumption is not restrictive. We now analyze the cost of this strategy. First, note that the istance travele while searching for the closest location past the corner point is at most 12 as given in Lemma 1. What remains is to analyze the cost of each probe step. Fig. 8. Close up of the experiment area. Approximately 18 meters of shoreline was use as a proxy for a large obstacle. The shore curve east, proviing a convenient corner to serve as the ege of the obstacle. point which is a maxima or minima with respect to the line l, as shown. Thus, the optimal path has length at least 2 + s 2. Our algorithm, calle AvanceRetreat, procees as follows. From the ing location, which is efine as the origin, the robot will move forwar (perpenicular to l) up to istance s, an return if an obstacle is etecte. This is calle a probe step. pon returning, the robot will move to a location which is istance ( 2) i 1 along the line l an repeat the process (see Figure 3 an Algorithm 2). There are two important ifferences from CircleSweep. First, the robot cannot sweep out a portion of the obstacle, because uring a probe it may only etect a very small part of the obstacle. In this case, we terminate when we can etect no part of the obstacle, not just when we etect the corner. In practice, can be represente by the beam with, an a nonetection is simply no echo returns. Secon, we assume that the robot can always turn aroun to return to the line, thus the Lemma 3 (Number of Probes): During an execution of AvanceRetreat, the robot makes 2 log 4 probe steps, each with cost less than 2s+2πR t. Proof: This cost of each probe step is upper boune using s the max istance from l to the object plus the cost to make four quarter turns. Without loss of generality, let be the point on l from which the probe operation woul first succee in passing the obstacle. The algorithm terminates when the robot travels past an executes a probe step. sing the same analysis as Lemma 2, we obtain an upper boun of log 4 probe steps require on the positive sie, or 2 log 4 probes require in total. Theorem 2 (Cost of AvanceRetreat): AvanceRetreat has a competitive ratio that is Θ(log ). Proof: We know the robot travels no more than istance 12+(2s+2πR t ) log 4, by combining the travel (Lemma 1) an probe (Lemma 3) steps. Note that the optimal path is at least length 2 + s 2, as illustrate in Figure 3.

7 Thus, c(avanceretreat) = 12+(4s+4πR t) log s 2 = s + (4s+4πR t) log 4 2 ( 2 + s log πR t ), 2 + s 2 which proves the theorem statement. Note that the part of the competitive ratio that grows logarithmicly is the cost from the logarithmic number of probes. If s is small, the cost of each probe is also small an the competitive ratio of AvanceRetreat is linear. Next we present results from repeate fiel tests of the propose algorithm. A. Fiel Experiments For our secon fiel experiments we execute Avance- Retreat at Lake Staring. The obstacle to avoi was the same north east part of the shoreline as before. The safe line l was selecte manually but the trajectory of the ASV was etermine online from the sonar measurements. As before, we use = 25m an t = 1m. The paths followe by the robot, along with the estimate positions of all etecte sonar objects, are shown in Figure 7. The boat successfully navigate aroun the obstacle in both of the trials where it was run to completion, an at every probe the boat correctly etermine whether or not there was an object present. These results inicate that our strategy is in practice a feasible metho to fin a path aroun an object. We foun that AvanceRetreat performe much better in practice. This was because CircleSweep requires that the obstacle is a line with known orientation, an the bounary of Lake Staring is poorly approximate by a straight line. However, the northeast bounary of Lake Staring is well approximate as a polygon that stays within istance s of a straight line; this was the setup of Problem 2 for which we use the AvanceRetreat strategy. We expect many common obstacles are well approximate by the parallel bouning lines l an l 2. VIII. CONCLSION In this paper we presente strategies for an Autonomous Surface Vehicle equippe with a fixe angle sonar to navigate aroun an obstacle. We first stuie a simple case where the obstacle is a rectangle of unknown size. For this problem, we presente a strategy with a constant-factor competitive ratio. In other wors, the ratio of the istance travele to the istance travele by the optimal offline solution is boune by a constant. In the fiel, we observe that fitting a line to the sonar sensor reaings was ifficult. This mae it ifficult to etermine the orientation of a straight obstacle. Furthermore, a realistic non-straight obstacle was hanle poorly by our first strategy, even when the line was a reasonable fit for its bounary. Therefore, we also stuie a more general case in which the obstacle has arbitrary shape but it is containe between two boune lines. The competitive ratio of our algorithm for this case epens on the istance between the lines. In fiel experiments, we showe that the algorithm can be execute in a robust fashion to navigate aroun large, unknown obstacles incluing shallow waters aroun a shoreline. In this work we use the bouning lines l an l 2 as ing an ening conitions so that we know when a particular obstacle has been circumvente. In a practical eployment, however, an ASV will have a ing position (perhaps when the first object is etecte), a goal position, an perhaps some number of obstacles. These obstacles might or might not nee to be circumvente, as long as the goal position is reache. Thus, a valuable irection of future research is the extension of online competitive exploration to a full navigation solution. This coul be particularly interesting if the other obstacles are allowe to move, such as if they are boats. REFERENCES [1] J. Ng an T. Bräunl, Performance comparison of bug navigation algorithms, Journal of Intelligent an Robotic Systems, vol. 5, no. 1, pp , 27. [2] P. Berman, A. Blum, A. Fiat, H. Karloff, A. Rosén, an M. Saks, Ranomize robot navigation algorithms, in Proceeings of the seventh annual ACM-SIAM symposium on Discrete algorithms. 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